The ³H(α,γ)⁷Li reactions at low energies Citation Brune, Carl Richard (1994) The ³H(α,γ)⁷Li reactions at low energies. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/RWJ4-CP08. https://resolver.caltech.edu/CaltechTHESIS:11082010-092253508 Abstract The absolute cross section of the ³H(α,γ)⁷Li reaction has been measured for 50 ≤ E_(c.m.) ≤ 1200 keV. Specially prepared Ti-³H targets were bombarded with an α⁺ beam, and γ rays were detected using an 85% high-purity germanium detector. Total S-factors and the branching ratios for radiative capture to the two final bound states are reported for the entire energy range. Angular distributions of the capture γ rays were measured for nine energies in the range 115 ≤ E_(c.m.) ≤ 1200 keV. Legendre coefficients extracted from fits to the angular distributions are also reported. The preparation and characterization of the radioactive Ti-³H targets are discussed. This experiment is motivated by models of big-bang nucleosynthesis, which require the ³H(α,γ)⁷Li reaction rate for computing the primordial ⁷Li abundance. Our results have considerably smaller uncertainties and reach lower energies than previous experiments. For E ≤ 150 keV, we find the S-factor to be considerably smaller than indicated by previous experiments. The measured branching ratio is found to be approximately energy independent, with a value of 0.45. The energy dependence of the measured S-factors is in agreement with existing theoretical calculations. The new results are used to calculate the thermonuclear reaction rate for temperatures below 10 GK. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: physics, big bang nucleosynthesis, lithium-7, radiative capture, tritium Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Kavanagh, Ralph William Thesis Committee: Unknown, Unknown Defense Date: 17 May 1994 Funders: Funding Agency Grant Number National Science Foundation PHY88-17297 National Science Foundation PHY91-15574 Record Number: CaltechTHESIS:11082010-092253508 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11082010-092253508 DOI: 10.7907/RWJ4-CP08 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6174 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 08 Nov 2010 18:14 Last Modified: 21 Dec 2019 01:26 Thesis Files Preview PDF - Final Version See Usage Policy. 94MB Repository Staff Only: item control page

The 3H(a,7)/Li Reaction at Low Energies Thesis by Carl Richard Brune In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1994 (Submitted 17 May, 1994) ii Acknowledgments I would like to thank the faculty, staff, and students of Kellogg Radiation Labora- tory for creating a productive and enjoyable environment for carrying out research. Particularly, I would like to thank Ralph Kavanagh, my advisor, for his guidance, support and assistance. His knowledge and confidence were especially important for undertaking the tritium-target fabrication at Caltech. It was a pleasure to collaborate with Claus Rolfs in the early stages of this experiment. I have benefited from several enlightening discussions with Charles Barnes and Karlheinz Langanke. My work would not have been as productive or enjoyable, were it not for my colleagues in the sub-basement: Steve Kellogg, T. R. Wang, Michael Smith, Patricia Wrean, and Kevin Hahn. I would also like to acknowledge the technical contributions of several people: Bob Carr, for maintaining the accelerator, and assistance design- ing equipment; Pat Huber, for help with the computer systems; Al Massey, for his electronics expertise; Jim Pendlay, for fixing things that broke, and also for teaching me AutoCAD; Jack Richards, for his work in the machine shop; and Dick Azuma, for supplying the titanium-tritide targets used in the preliminary phase of this work. I would like to thank my parents, who throughout my life have been a constant source of encouragement. I am grateful to my wife Roxanne, for all of her love and support. Financial support for this work was provided in part by the National Science Foundation, Grant Numbers PH Y88-17297 and PHY91-15574. ill Abstract The absolute cross section of the 7H(a,7)‘Li reaction has been measured for 50 < Ezm. < 1200 keV. Specially prepared Ti-°H targets were bombarded with an at beam, and 7¥ rays were detected using an 85% high-purity germanium detector. Total S-factors and the branching ratios for radiative capture to the two final bound states are reported for the entire energy range. Angular distributions of the capture y rays were measured for nine energies in the range 115 < E.m. < 1200 keV. Legendre coefficients extracted from fits to the angular distributions are also reported. The preparation and characterization of the radioactive Ti-*H targets are discussed. This experiment is motivated by models of big-bang nucleosynthesis, which re- quire the 7H(a,7)’Li reaction rate for computing the primordial ’Li abundance. Our results have considerably smaller uncertainties and reach lower energies than previ- ous experiments. For E < 150 keV, we find the S-factor to be considerably smaller than indicated by previous experiments. The measured branching ratio is found to be approximately energy independent, with a value of 0.45. The energy dependence of the measured S-factors is in agreement with existing theoretical calculations. The new results are used to calculate the thermonuclear reaction rate for temperatures below 10 GK. iv Contents 1 Introduction 1.1 Big-Bang Cosmology ..... 1... 0. pe ee eee ee ee ee ee 1.2 Wuclear Physics oi ccc ewe 1.2.1 Previous Experiments ...........5.0 05022 eae AZO TORY cc ee 1.3 Experimental Overview... 2... 2... eee eee ee ee ee ee 2 Tritium Target Preparation 2.1 Description of Apparatus... .....-.-.-.0. 0020000222 eee 2.2 Target Production Procedure ..........-.-20-+2+5222000- 2.5. Target Characterization ...6 6 ses we reer eae eens eens 2.3.1 TiAreal Density .......0 00.2 cee eee ee eee 2.3.2 7H Areal Density .........--0--2 00+ ee ee eee BER "Hdl Delig wc ct 2.4 Results and Discussion ..... 2... 2... eee eee ee ee eee 3 Experimental Apparatus and Procedures Sol Derget Clmibers 2.222 ed ee i: ar 3.3 y-Ray Detection... 2... ee eee 3.3.1 Detector Geometries .......... 02.00.0022 eee 3.3.2 Photopeak-Efficiency Measurements............... on 15 21 23 25 27 30 32 32 34 37 3.3.3 Photofraction Measurements.........-.-.++2+02- 48 3.3.4 y-Ray Energy Calibration ..........-..-2-0.-004. 50 3.3.5 Detector Shielding .............0-2.-222-02200-. 52 Ba SOI Ck Ke 53 Data Analysis 57 Sl KGAA ccc ee 57 4.2 y-Ray Detection... 2... ee ee eee 59 4.2.1 Analysis of Spectra... . 2... . ee eee eee eee 59 4.2.2 Monte Carlo Simulations ...............-2-..2-4.- 64 4.2.3 Detector Efficiency ..........--2.2-222+0220208- 66 4.3 Target Deterioration .... 2.2... ...-.-02.020 2022+ +e eee 70 4.4 Extraction of Cross Sections ...........2.-.-2200-0-02-0- 71 4.4.1 Angular Distributions .............-2.2..-2006. 75 44.2 0° Data .... 2... ee ee ee eee 76 443 AllData........... 2.2.2.0 2.202 e eee ee eee 77 Results 78 Bel ‘Solel Geieeter oc ek wee eR 78 tial Eeamehiog Helios cc i 79 5.3 Angular Distributions. ............2.2.2.2.02 0020008] 80 5.4 Comparison with Previous Experiments................. 81 5.4.1 Total S-factor .. 2... ee ee eee 81 5.4.2 Branching Ratio .... 2... 2... 2.2.2.2. ee ee eee 81 5.4.3 Angular Distributions ............-..222.-0004- 81 5.5 Comparison with Theoretical Calculations ............... 82 SA.1: Tole Gfegiet pce cm 82 BES Gimiine Male ne tem eee aa 82 5.5.3 Angular Distributions .................20224, 83 vi Conclusions 92 6.1 Nuclear Physics... 2... 2... ee ee ee 92 6.2 Astrophysics... . 1... ee ee 94 Sensitivity of Standard Big Bang Calculation to the 7H(a,7)’Li Rate 98 Axl Nucleosynthesig «.csac ede ee Eee we ww 98 A.2 Temperature Sensitivity ...........2.-0.220 0228 e ee 99 A.3 Freeze-Out Calculation .. 66-6 ee ee es 101 Neutron Background 105 B.1 Target Normal to Beam, Ignoring Energy Loss ............. 106 B.2 Arbitrary Target Angle, Including Energy Loss. ............ 109 Summary of Experimental Parameters 114 Attenuation Factors 118 Photopeak Efficiency 125 Summary of Experimental Results 130 F.l Sand R From Each Run...........2.-.2.20 00002 ee eee 130 Bud sapere Weenie ce Ree 133 F.3 Final Results for Sand R .. 1... cee ee ee ees 134 F.4 Angular Distribution Plots... ............ 000022 eee 135 Bibliography 140 vii List of Tables 2.1 3.1 4.1 4.2 5.1 5.2 5.3 5.4 C.1 C.2 C.3 F.1 F.2 F.3 F.4 3H and Ti areal densities. ©... 0.0. ee eee ee 38 Summary of target deterioration. ............-2+.-.-2+0-.- 54 Yield correction factors... 2... 6.2 ee ee eee ee ee ee ee 73 Systematic errors .. 2... .. . ee 76 Comparison of targets and geometries... .....-...2-...004. 79 Fits to Legendre coefficients ...........2.5.2.025020000. 80 Comparison to previous angular distributions ............. 82 Theerctical fia to STB) cc cite eee eee 83 Summary of experimental parameters ..............-2--- 115 Summary of experimental parameters ..........--.-.-+--- 116 Summary of experimental parameters ...............062. 117 Summary of Sand Rforeachrun...............2206. 131 Summary of Sand Rforeachrun..............02e006. 132 Yo and 7, Legendre coefficients. .............-2-200-- 133 ani wesdiia Tee Swed Bin ec caw ewe RR 134 Vill List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 lot 21 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.9 Evolution of abundances with temperature ..............-. 3 Li abundance observations ........-..-.-.-2.-2.0+0+0202000- 5 Calculated “Li abundance .........-..---.-022.2 000 eee 6 "Li energy-level diagram ........-.-0 000 eee eee eee 7 Previous direct measurements .........--.-.+-+++++-e+0088. 10 Coulomb dissociation measurements. .........-.2+2-e208- 13 Theoretical calculations of S(F).. 2.2... 2. ee ee ee eee 18 Target production apparatus ............-2200220000. 26 Gas-handling system ... 1... 2.2.2... 2. ee eee ee eee ees 27 Ta boat for Ti evaporation... .........-2.0 00022 eee 28 Scattering chamber .... 2... 0.2... 2 ee eee ee ee ee ee ees 31 Rutherford-backscattering spectra... 2... 2. eee ee ee ee 33 7H(*He,a) spectrum .........-. 02.22 eee ee ee ees 34 SH(d,a) spectrum... 2... ee ee 35 3H(d, a) excitation functions for targets 1,3-5 ..........20-. 36 3H(d, a) excitation function extended to lower energies ........ 37 Details of geometry A .... 2... 2.2.2.2. 2 2 ee eee ee ee ee 40 South-30° beamline... 2... 2. ee ee ee ee 42 Geometry © 2... ee es 46 Photofraction results .. 2.2... ee ee ees 51 3.6 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 6.1 A.l A.2 B.1 B.2 B.3 B.4 ix Carbon profiling with C(a,n) .. 2.0... ee ee ee ee 56 Run 096 spectrum (high energy). .......----++22----e 60 Run 096 spectrum (low energy) .....------+ 222 eee tees 60 Run 095 spectrum (high energy)... ...-----2-+- +22 es 62 Run 095 spectrum (low energy) ......-----+++2222200- 62 Run 137 spectrum (high energy)... 1... ..----+22 eee eee 63 Run 137 spectrum (low energy) .......----0020+00000- 63 Example EGS4 calculation .. 2... 21. ee ee ee ee 65 Coincident summing corrections ...........20 2022 ee ee 68 Comparison of targets and geometries. ..........-.22+24:5 84 Wimel restlis for occas ci wah 85 Final results forS............ cee eee eee eae 85 Results for Z =o,/02....-.-... 0.2.20. ee eee ee ee ee eee 86 Results for R 2... ee ee 86 Yo angular-distribution coefficients. ..............-2224. 87 7, angular-distribution coefficients. ..................-. 88 ‘2 angular-distribution coefficients. ...........-.-.-2-+2048- 89 Comparisons with previous data..........-...2-02-0200- 90 Theamiicn! Pilate SUB} csc eee 91 Thermonuclear reaction rate... 1... 2... ee ee ee ee es 95 The perturbing function 2... cs te 100 The effect of changing (ov)... 2... . eee ee ee ee eee 101 Definition of variables for the normally incident case. ......... 107 Calculated neutron yield for target normal to beam .......... 109 Definition of variables for arbitrary angle of incidence ......... 110 Experimental and calculated neutron yields for target at angle with WG WO BER ew eRe A 112 B.5 D.1 D.2 D.3 DA D.5 D.6 E.1 E.2 E.3 E.4 E.5 E.6 E.7 F.1 F.2 F.3 F.4 F.5 F.6 F.7 F.8 F.9 Calculated neutron yield as a function of target thickness ....... 113 Photopeak attenuation factors, geometry A............2.-.. 119 Photopeak attenuation factors, geometry B............... 120 Photopeak attenuation factors, geometry C.........-..-2--. 121 Total attenuation factors, geometry A. ..........-..2-2--. 122 Total attenuation factors, gometry B...........-..2204. 123 Total attenuation factors, geometry C.............22-4. 124 Photopeak efficiency, geometry A ..............-22004. 126 Photopeak efficiency, geometry B .............-2.2224. 126 Photopeak efficiency, geometry C,0° ...............20.4. 127 Photopeak efficiency, geometry C, —45°..............-.. 127 Photopeak efficiency, geometry C, —-90°...............2.. 128 Photopeak efficiency, geometry C,+90°.............-..-. 128 Photopeak efficiency, geometry C, +135° .............2.. 129 E = 116 keV Angular Distribution .................0.. 135 E = 245 keV Angular Distribution ................20.4. 136 E = 331 keV Angular Distribution ................... 136 E = 417 keV Angular Distribution .................2.. 137 E = 560 keV Angular Distribution .................2.. 137 E = 676 keV Angular Distribution ................... 138 E = 848 keV Angular Distribution ................0... 138 E = 1021 keV Angular Distribution .................-.. 139 E = 1189 keV Angular Distribution ..................4. 139 Chapter 1 Introduction Observational evidence supporting the big-bang model of the universe comes primarily from four sources: (1) the Hubble expansion, (2) the age of the universe, (3) the cosmic microwave background radiation, and (4) the relative abundances of the light elements 'H, 7H, *He, *He, and “Li. The role of "Li in the big bang, and in particular its synthesis via the 7H(a,7)’Li reaction, is the primary motivation for this thesis. We describe laboratory measurements of the 7H(a,7)’Li cross section, at energies relevant for big-bang nucleosynthesis. 1.1 Big-Bang Cosmology An important cosmological quantity is the critical density, 3H? where Hp is the Hubble constant and G is the gravitational constant. The density parameter (2 is defined by n=, (1.2) where p is present density of the universe. The cases 2 > 1, 2 = 1, and 2 < 1 corre- spond to a closed universe (which will eventually recontract), a flat universe (which will expand forever), and an open universe (which will expand forever), respectively. Models of inflationary cosmology predict an (12 = 1 universe. Clearly, a firm observa- tional determination of 2. would be very interesting, but this has proven very difficult. Observations of luminous matter find Q;yy ~ 0.01 or less, but other determinations (for example the matter distribution inferred from the IRAS survey) indicate values up to 2 = 1 [Kol90]. The first big-bang nucleosynthesis calculations were made by Alpher, Bethe, and Gamow [Alp48], who assumed only neutrons for the initial state. Hayashi [Hay50] pointed out that neutrons and protons will be in statistical equilibrium for sufficiently high temperatures and densities. Further refinements were made by Alpher, Follin and Herman [Alp53] and Hoyle and Tayler [Hoy64]. The discovery of the cosmic microwave background at ~ 3 K in 1965 by Penzias and Wilson [Pen65] provided incentive for further calculations. Peebles [Pee66] and Wagoner, Fowler and Hoyle [Wag67] made detailed calculations of the resulting light-element abundances. Since this time the input data have undergone much improvement, but the essential physics has remained unchanged. Recent standard big-bang reviews have been published by Krauss and Romanelli [Kra90], Riley and Irvine [Ril91], Walker et al. [Wal91], and Smith, Kawano, and Malaney [Smi93]. Primordial-nucleosynthesis calculations provide some stringent tests of several of the assumptions underlying the standard big-bang model. A wide selection of variant models has been reviewed by Malaney and Mathews [Mal93]. Considerable recent at- tention has been focussed on the “inhomogeneous big-bang models,” which take into account the effects of baryon-density inhomogeneity resulting from a phase transition (or transitions) in the early universe (for example, the quark-hadron transition). In the remainder of this Section we outline several aspects of standard big-bang nucle- osynthesis. Standard calculations assume the correctness of general relativity, and the ho- mogeneity and isotropy of space. Important input data include the known particles (e.g., three neutrino species), nuclear reaction rates, and the neutron lifetime (used TTTT TT T T T porn T T potrrr 109 » jan} o Q *) q fo) 1 1 VA 1 1 | 1 1 L 1 | 1 i | | I 1075 T | T T T T | T T T T 6 3 H oO oO & oO E Shi 10715 1 : a SE ee | 1 1 1 bes ee i i 4 10! 109 1071 10-2 Tg Figure 1.1: Evolution of the light-element abundances with temperature (measured in GK) in the big bang, for 719 = 3.0. to determine the rates for weak n <> p processes). The present number density of background photons, n,, can be determined from the measured temperature of the background radiation (T, = 2.736 + 0.017 K [Gus90]) using kT,\3 Bag ct 167¢(3)(—*) ; (1.3) where ¢(3) = 1.20206. The ratio of baryons to photons, 7 = ng/n., is expected to have been constant since e+ —e~ annihilation [Kol90]. The calculated abundances are normally presented as a function of 7, the only free parameter in the standard big- bang model. The baryon density parameter (.g can be calculated from T, and 7 using Equations 1.1-1.3; however, the present factor of two uncertainty in Hp introduces a factor of four uncertainty in Og. A sample calculation for 719 = 3.0 is shown in Figure 1.1 (m0 = 10'°7). This calculation and the one shown in Figure 1.3 were made using Kawano’s nucleosyn- thesis program [Kaw92], with reaction rates from Caughlan and Fowler [Cau88], and a neutron lifetime of 888.5 sec. It is seen that the big bang produces mostly ‘1H, with some *He (about 25% by mass), and traces of other isotopes. Note that the radioactive isotopes 7H and 7Be decay eventually to *He and "Li. Walker et al. found that the calculated abundances of 7H, *He, and ‘Li (relative to 'H) agreed with observations for corresponding to 0.01 <Qg < 0.10. (1.5) The *He abundance is only weakly dependent upon 7. Including the uncertainties in 7 (Equation 1.4) and the neutron lifetime, Walker et al. found the calculated *He mass fraction, Y,, to be bounded by 0.236 < Y, < 0.243, (1.6) in good agreement with the observational value me = 0.23 + 0.01. Understandably, the observational determinations of the abundances of less common isotopes are more difficult, and have larger uncertainties. The reader is referred to the recent reviews [Kra90, Ril91, Wal91, Smi93] for detailed comparisons between observation and calcu- lation. The arguments supporting the observational determination of the primordial "Li abundance are outlined in the following paragraph. The “lithium plateau” in metal-poor halo stars was discovered in 1982 by Spite and Spite [Spi82]. Over a wide range of temperatures they found a constant Li abundance, [Li] ~ 2.0, where [Li] = 12 + log,.(Li/H). Their work has subsequently been confirmed by many observers; data from several sources have been compiled by Walker et al., and are displayed in Figure 1.2. The data set has been restricted to very metal-poor stars, [Fe/H] < —1.3, where [Fe/H] = log,)(Fe/H). — log, )(Fe/H)o. The stars with effective temperature T.g > 5500 K show a constant Li abundance. The spread in the data appears to be consistent with the observational errors, although there may be some evidence for underlying dispersion [Del93]. The observed Li is 8.0 yr rg er ee pt rer 2.56 4 o 2.05 oof . mai 1.5- 4 1.0F aA 4 [Li] —0.5F sa en ae - ee ee ee ee ee ee ee saad 4800 5000 5200 5400 5600 5800 6000 6200 6400 Tett (K) Figure 1.2: Observed Li abundance versus T.g, for the most metal-poor stars ({Fe/H] < —1.3). The diamonds are upper limits to [Li]. The horizontal lines corre- spond to Equation 1.7. (Adapted from Walker et al. [Wal91].) believed to be essentially “Li [Spi82, Reb88]; also note that the calculated big-bang ®Li yield is predicted to be negligible compared to “Li (Figure 1.1). The observed plateau supports the hypothesis that the abundance of Li in halo stars (Population II) is of primordial origin. Using stars with T.g > 5500 K, Walker et al. find a mean value of [Li] = 2.08 + 0.07, (1.7) where the error represents a 20 uncertainty. This 2c range is shown as horizontal lines in Figures 1.2 and 1.3. A larger range, 2.04 < [Li] < 2.36, has been derived by allowing for some depletion during stellar evolution [Del89]. The ‘Li abundance calculated as a function of 7 is shown in Figure 1.3. The separate contributions of “Li and “Be are shown as dashed lines. The calculated abundance agrees with the observations (horizontal lines) only for a narrow range of n, consistent with Equation 1.4. (Note that Equation 1.4 allows for some uncertainty in the reaction rates.) It is seen that "Li production dominates for mio < 3, while 7Be dominates for mo < 3. It is interesting to note that the observed 7Li abundance Li/H Figure 1.3: Calculated ‘Li abundance as a function of the baryon-to-photon ratio 7 (solid curve). The dashed curves on the left and right show the separate contributions of “Li and “Be, respectively. The horizontal lines correspond to limits on the ‘Li abundance from observations (Equation 1.7). coincides with the minimum in the calculated abundance at 419 ~ 3 (this minimum is absolute for at least 0.1 < m0 < 100; see Figure 12 of Walker e¢ al.). The two-body thermonuclear reaction rate N4(ov) used in astrophysical calcula- tions is calculated from the cross section o using [Fow67] 8 )"" Na ah — i Eo(E£) exp(—z) dE, (1.8) Na(ov) = ( (kT)3/ where Ny, is Avogadro’s number, p is the reduced mass in the entrance channel, k is Boltzmann’s constant, T is temperature, and F is the center-of-mass energy. The uncertainties in the abundance calculations arising from nuclear-data input have been studied in detail by Krauss and Romanelli [Kra90] and Smith, Kawano, and Malaney [Smi93]. Smith, Kawano, and Malaney have identified 12 reactions which significantly affect light-element production in the big-bang; all of these reactions were found to affect “Li production. Of the 12, 7H(a,7)’Li is by far the most uncertain; the authors estimated a 20 uncertainty of up to 55% in the cross section. Note also 2000 L- 4630 7/27 g Fl 1000 F J oae7-9 SH+a Y1 |Vo 477.61 1/27 y¥2 _ 0 3/2 "Li Figure 1.4: “Li energy level diagram (all energies are given in keV). that 9H(a,7)’Li is most important for m9 < 3, as ‘Be production dominates at higher 1. An investigation of the sensitivity of the abundance calculations to the 7H(a,7)‘Li rate is presented in Appendix A. We find that calculated ‘Li abundance is sensitive to Na{ov) for 0.1 S Ts S 0.6, where Ty is the temperature in GK. These temperatures provide guidance as to what energies in Equation 1.8 are important for determining the big-bang ‘Li yield, as discussed below. 1.2 Nuclear Physics The energy-level diagram for "Li is shown in Figure 1.4. The *H(a,7)‘Li reaction proceeds by y emission to the two bound states: the 3” ground state (yo) or the + first excited state (7). The first excited state decays by y emission (72) to the ground state. The (a,7) cross section is expected to be non-resonant for E < 2100 keV, since there is no energy level between the first excited state and the 5 state (Tm. = 9348 keV [Ajz88]) at 4630 keV. The next channel, °Li+n, opens at an excitation energy of 7250 keV. In this work, energies without a subscript label refer to the center-of-mass system. At low energies, the reaction cross section for charged particles is conveniently parameterized in terms of the S-factor S(E), defined by o(E) => amp 2). (1.9) where Eg is the Gamow energy, Eg = 20’ Z) 250° p, (1.10) and Z, and Z, are the charges of the reacting nuclei, a is the fine-structure constant, and p is reduced mass (in energy units) of the reacting nuclei (Eg = 6735.0 keV for 7H + a). This parameterization removes two energy-dependent factors: the de Broglie wavelength factor 1/E and the (approximate) Coulomb-barrier penetration factor exp(—1/Ec/E). The S-factor for a non-resonant reaction such as *H(a,7) is expected to be a slowly varying function of energy. The partial S-factors S; for the 7H(a,7) reaction are calculated from the cross sections o; for producing 7;, where 1 = 0, 1, or 2. Note that S, equals S2 since the first excited state always decays by emission. The total S-factor S = So + 5S; is the important quantity for astrophysics since it determines the fusion rate for 3H + 4He — ‘Li. The cross sections o; then determined by S and the branching ratio R=0;/00. For constant S(£), the integrand in Equation 1.8 peaks at kT \ 2/9 Ey = EL?(=) (1.11) and has an approximate 1/e width EokT \ 1/2 : ) ; (1.12) =4 aaa The temperature range 0.1 S To S 0.6 found above corresponds to 50 SHS 165 keV. For Ty = 0.1, A = 48 keV, and for JT, = 0.6, A = 213 keV. Thus knowledge of the 7H(a,7)’Li cross section over the range 25 S E S 270 keV is most important for the standard big bang. We will not confine our interest to this range, as alternative big-bang models may have different requirements. Measurements over a wide range of energies are also useful for testing theoretical calculations. While most of the energy range can be covered by laboratory measurements, energies as low as 25 keV will require some extrapolation of the experimental data, so a validation of the theoretical extrapolation is valuable. 1.2.1 Previous Experiments Direct Measurements The first measurements of the *H(a,7) reaction were reported in 1959 by Holmgren and Johnston [Hol59]. The experiment was performed using *H-filled gas cell which was bombarded though a Ni entrance foil by an a beam. The y rays were detected using an Nal scintillator. The experiment covered the range 200 < EF < 560 keV, and claimed to achieve 30% precision. The measured values of S varied between 5(200 keV) = 0.040 and S(560 keV) = 0.023 keV-b. The *H areal density was determined by measuring the absolute pressure in the gas cell. Improved measurements were made by Griffiths et al. [Gri61], who bombarded solid Zr-*H targets with an a beam, and again detected y rays with a Nal scintillator. The measurements covered 150 < E < 780 keV, with an estimated error in S of 20- 25%. Their results for S and R are reproduced in Figure 1.5. The results for S were a factor of 2 to 2.5 higher than found by Holmgren and Johnston; within the accuracy of the measurement, S was found to be constant with a mean value of 0.064 keV-b. From the yo and 7; yields, the branching ratio was found to be independent of angle (within 5%) and approximately independent of energy, with an average value of R ~ 0.40. The separation of 7; from yo required the use of a lineshape fit to the Nal spectra, 0.160 0.140 0.120 — 0.100 V—b & 0.080 S (k 0.060 0.040 0.020 0.770 0.60 0.50 ~ 0.40 0.30 0.20 0.10 fe) 0 10 # — Schréder et al. [Sch87] — A — Burzynski et al [Bur87] Y — Griffiths et al [Gri61] 4 4 | 1 ! 1 it 1 | 1 | 1 | 1 | 1 | 1 ! 100 200 300 400 500 600 700 800 900 1000 E (keV) Figure 1.5: The previous results [Sch87, Bur87, Gri61] for S and R as a function of the center-of-mass energy E. The errors on the S data points of Schroder e¢ al. and Burzynski et al. do not include respective systematic errors of 15% and 14%. Griffiths et al. only quoted uncertainties for a few R data points. The solid curve in the upper panel is Equation 1.16, normalized to 5(500 keV) = 0.060 keV-b. 11 due to the proximity of 7, to the yo first-escape peak in the Nal spectrum. Two measurements of the yo angular distribution were also reported: relative intensities at 0° and 90° (angles relative to the incident-beam direction) were determined for E = 241 keV, and relative intensities at 0°, 45°, 90°, and 135° were determined for E = 567 keV. The 7H areal density was determined by measuring the 7H(p,7) yield at E, = 800 keV, and using the (p,y) cross section found by Perry and Bame [Per55]. Two new measurements have been reported in the late 1980’s, undertaken as a result of the importance of *H(a,7) for big-bang nucleosynthesis. The new mea- surements were improved by using thinner Ti-°H targets and by detecting y rays with solid-state Ge detectors. The primary advantage of Ge detectors is the greatly improved -y-ray energy resolution, which allows the y-ray lines of interest to be unam- biguously separated from various background sources, with a much improved signal- to-noise ratio. The +, line is separated from the troublesome ‘yo first-escape peak by ~ 33 keV under ideal circumstances. The experimental resolution for these peaks is determined by three factors: (1) resolution in the center-of-mass energy, (2) energy spread from angle-dependent Doppler shift, and (3) detector resolution (+ 3 keV at E, = 3 MeV for Ge detectors). The use of thinner targets reduces the amount of energy lost by the beam in the target, and hence the energy resolution in the 7H + a center of mass. The improved center-of-mass resolution also reduces the error in the determination of the effective reaction energy. The determination of the effective energy at low energies is very important due to the strong energy dependence of the cross section. Using Equation 1.9 and neglecting the energy dependence of S(F) one finds 1 do 1 1 /Ecg cdE = 5(-1 + 9 =). (1.13) For example, a 1-keV error in E at E = 100 keV translates to a 3% error in a (or S). Measurements for 79 < E < 464 keV were published by Schréder et al. in 1987. The results for S and R are shown in Figure 1.5; the errors on the S data points do not include an additional 15% systematic uncertainty. The results for S show a significant increase with decreasing energy (in contrast with the data of Griffiths et 12 al.); the authors estimated $(0) = 0.14 + 0.02. Using yo and the weighted average of 7, and 72, the branching ratio was found to be independent of energy with an average value of 0.32 + 0.01, a significantly lower result than that found by Griffiths et al. Angular distribution measurements showed the yield to be isotropic (within 20% [Rol91]). The reported results were based upon 0° data, analyzed assuming isotropy. The effective reaction energy was determined using the measured Yo energy. The 7H areal density was determined in a manner similar to that of Griffiths et al., by measuring the *H(p,-7) yield at E, = 1000 keV and using the (p,-) cross section of Perry and Bame. Also reported in 1987 was the experiment of Burzynski et al. [Bur87], over the energy range 297 < E < 856 keV. The results for S and R are shown in Figure 1.5; the errors in S$ do not include an additional 14% systematic error. Using yo and +y;, the branching ratio was found to be energy independent, with an average value of 0.51+0.04. The measurements were performed at 0°; the E < 500-keV data were analyzed assuming isotropy, and the F > 500-keV data were analyzed using the angular distribution found by Griffiths et al. at E = 567 keV. The 7H areal density was determined by measuring the 90° *H(d,n) yield at Ey = 1000 keV and using the known (d,n) differential cross section. While this technique should be reliable, it could give misleading results if there is any *H distributed in the backing of the target, as has been encountered previously in our laboratory [Bru91]. The agreement between the measurements of S over the range of their mutual over- lap is reasonable if the systematic errors are taken into account (not including the early results of Holmgren and Johnston). However, only the experiment of Schréder et al. reaches the low energies needed to determine big-bang “Li production. The dis- agreement between the various branching-ratio measurements (results vary between 0.25 and 0.55) is rather puzzling. Another disturbing development is the new measurement by Feldman et al. [Fel90] of the °H(p,-) cross section used by Griffiths et al. and Schroder et al. to determine the 3H areal density of their targets. The new data cover 2 < E, < 15 MeV, and 0.50 T T T T T T T ] T T T T T T T T T 0.45 0.40 0.35 0.15 0.10 ke oO rw) on PETE EEETUpPETTCpeLeTprerepeereyprereyrrer perry oo O- y Bit o a oO te a ae, +o 7O posit itiretirrritrrir tire tir ir tipi pti pit titi ti 0.05 Pie aes | 1 j 4 l i i J. | 4. 4 { 1 i } 1 100 200 300 400 500 600 700 800 900 1000 E (keV) Oo fo) Figure 1.6: The Coulomb dissociation results of Utsunomiya et al. [Uts90]. The solid curve is Equation 1.16, normalized to $(500 keV) = 0.060 keV-b. Note that the vertical scale has been expanded compared to Figure 1.5. are ~ 30% lower than the cross sections found by Perry and Bame (who quoted an uncertainty of 7%). The ramifications of this measurement for the absolute normal- ization of the Griffiths et al. and Schréder et al. S-factors are not clear, as the new measurement has not been independently confirmed, and it only covers higher en- ergies than used in the (a,7) experiments. A possible systematic error in the Nal scintillator efficiency used by Perry and Bame is discussed by Griffiths et al. Coulomb Dissociation Measurements The Coulomb dissociation approach for determining radiative capture cross sections for reactions of the type a+b — c+v7 has recently received much attention. The basic idea is to study the inverse reaction, c-+ y > a+, by bombarding a target with a beam of c, and using the Coulomb field of the target nuclei as the source of (virtual) photons. The relative energy in the a + b center of mass must be reconstructed from 14 the detected momenta of a and 6. For example, the *H(a,7)’Li cross section could perhaps be determined by measuring the 7°Pb(‘Li, aH)?°8Pb breakup cross section. The practical advantage of the Coulomb dissociation approach appears to be that the event rate is significantly higher than for radiative-capture experiments, under realistic experimental conditions. The realization of this advantage largely remains to be seen, as only a few experiments have been performed to date, and experimental techniques have not been optimized. The approach can also provide access to reac- tions for which radiative-capture experiments are very difficult, for example due to the presence of a short-lived isotope in the entrance channel. The Coulomb dissoci- ation technique can necessarily only give information about radiative capture to the ground states of nuclei. Several theoretical issues complicate the use of the Coulomb dissociation method. Coulomb breakup dominates over nuclear breakup only for high incident energies and very forward scattering angles. Bertulani and Hussein [Ber90] suggest that bom- barding energies of + 100 MeV/nucleon will optimize Coulomb-breakup contribution. The number of virtual photons seen by the projectile is multipole dependent, so the extraction of radiative capture cross sections requires an understanding of the re- action mechanism. For the case of "Li breakup, Typel [Typ93] has shown that E2 breakup significantly effects the absolute yield and angular distribution of the breakup fragments (£2 transitions are expected to contribute < 1% of the radiative-capture cross section). “Postacceleration” effects, due to the differential acceleration of the breakup products in the Coulomb field, may complicate the reconstruction of the relative energy. Experimental challenges include the separation the breakup yield from the much more intense elastic-scattering yield, and the reconstruction of the relative energy with sufficient precision and resolution. In 1990 Utsunomiya et al. [Uts90] published results for the dissociation of 42- MeV “Li ions incident on 7’Al, *®Ni, and !?°Sn targets, and 63-MeV “Li ions incident on '*4Sm and ?°8Pb targets. Absolute cross sections for 7H(a,7) were not reported, 15 apparently because of interference from nuclear breakup. However, the reduced tran- sition probabilities extracted for different bombarding energies and targets were con- sistent with a nearly universal energy dependence. The authors presented simple theoretical arguments that the energy dependence of the nuclear breakup should be similar to the Coulomb breakup. On this basis, values of S were reported for 80 < E < 980 keV, assuming 5(500 keV) = 0.060 keV-b for the absolute normaliza- tion. The results, shown in Figure 1.6, indicate an increasing S-factor with decreasing energy, similar to that found by Schroder et al. Theoretical uncertainties and the large low-energy error bars preclude any definite conclusions about the *H(a,) cross section from being drawn. Experiments on the breakup of 54-MeV “Li ions incident on !*C and 1%’ Au targets were published in 1992 by Gazes et al. [Gaz92]. Large nuclear-breakup contributions were found in the case of '*C, and large distortions in the energy spectra (attributed to “postacceleration” effects) were observed in the case of '*’Au. No attempt was made to extract information on the *H(a,7) cross section. 1.2.2 Theory All theoretical calculations of the 7H(a,7)’Li cross section share several common features. The interaction Hamiltonian for y-ray emission is given by 1 ~ Hin, = —-J- A, (1.14) Cc where j is the nuclear charged-current density, and A is the vector potential of the y-ray field. The -emission interaction is sufficiently weak compared to nuclear and Coulomb forces that the capture cross section can be calculated using Fermi’s golden tule: o x |(U +| Hine!) |’, where |¥;) is the initial (elastic scattering) wavefunction, and |W;) is the final (bound state) wavefunction. The calculations proceed by de- composing H;,, into multipoles (#1, M1, E£2,...) and |W;) into partial waves. The 3H(a,7)"Li cross section primarily results from the E1 capture of s-waves, with a 16 substantial contribution at higher energies also coming from the E1 capture of d- waves. Due to its small binding energy, the “Li wavefunction has a relatively large tail (of primarily 7H + a structure) at distances beyond the range of the nuclear force. At low energies the capture matrix element primarily depends upon this part of the bound-state wavefunction. All calculations involve some form of approximation, as the calculation of |¥;) and |Wy) is in general a seven-body problem, which as yet has not been solved for realistic nucleon-nucleon forces. We note that *H — a elastic- scattering phase shifts, which determine |V,;) beyond the range of the nuclear force, have been measured [Spi67, Iva68] for higher energies, F > 1550 keV. The first calculations were reported in 1961 by Christy and Duck [Chr61] and by Tombrello and Phillips [Tom61], and included the F1 capture of s-waves. An im- proved calculation was published in 1963 by Tombrello and Parker; the £1 capture of s- and d-waves to final p-states was included, and several computational improve- ments were made. The internal structures of the triton and a particles were ignored, and only the extra-nuclear contribution to the capture matrix element was calculated. In this region, the radial dependence of the final bound states is given by Whittaker functions. The s and d scattering waves were calculated using the phase shifts from a hard-sphere potential with a 2.8-fm radius. This choice of radius gave a good descrip- tion of the mirror *He — a scattering reaction, and was shown later [Spi67, Iva68] to give a reasonable description of higher-energy 7H —a scattering. The two bound-state reduced widths, which determine the normalization of the Whittaker functions, were left as free parameters. The calculation was able to simultaneously describe the values of S and R measured by Griffiths et al. as well as the values of S and R found for the mirror *He(a,7)’Be reaction by Parker and Kavanagh [Par63], using the same values for the reduced widths. Unfortunately, the calculation for *7H(a,7) was presented as ao versus & plot, making it impossible to ascertain the predicted low-energy behav- ior. The resulting angular dependences of yo and +, were also calculated. Due to the assumption of s- and d-wave capture only, the predicted dependences were of the form 1 + a2P2(cos 6.m.), where P: is the £ = 2 Legendre polynomial. 17 Using the formalism of Tombrello and Parker for s- and d-wave F1 capture, we further investigated the origin of the a2 coefficient, assuming that both d-wave phase shifts are equal. The result is 1 sy — «/2y cos Ad —2 lity ; (1.15) a2 = where y = og/o, and Ad = a2 + 62 — ao — & is the phase difference (Coulomb + nuclear) between the s and d waves. The presence of ,/y, due to interference between s and d waves, means that there can be significant anisotropy even if the d-wave contribution is very small compared to the s wave. By taking y from Mohr et al. [Moh93] and using hard-sphere nuclear phase shifts (2.8-fm radius), we were able to reproduce the a2 coefficients calculated by Tombrello and Parker within 5%. Odd terms in the angular distribution are possible, due to interference with M1 or E2 capture from odd partial waves. Terms with £ > 4 are expected to be negligible, as higher multipolarities are predicted to be negligibly small in all calculations. The mirror reaction, *He(a,7)"Be, is also very important in another branch of astrophysics, as the number of ‘Be and °B neutrinos produced by our sun is dependent upon its cross section (at FE ~ 20 keV). Due to the persistence of the solar neutrino problem, as well as interest in the rates of 9H(a,7)’Li and *He(a,7)’Be in the big bang, several theoretical calculations of both reactions have been published in the last ten years. The first resonating-group method (RGM) calculation of radiative a capture to A = 7 nuclei was made by Liu, Kanada, and Tang [Liu81], who calculated S(E) for 3He(a,7)’Be. Subsequent RGM calculations of S(E) for 7H(a,7) have been reported by Kajino and Arima [Kaj84], Kajino [Kaj86], Mertelmeier and Hofmann [Mer86], and Altmeyer et al. [Alt88]. An RGM calculation of S(0) = 0.154 keV-b has been reported by Chopovsky [Cho89]. This technique allows the nucleonic substructure of the *H and a clusters to be approximately taken into account, and guarantees adherence to the Pauli principle between nucleons. The scattering- and bound-state 18 0.100 0.080 0.060 0.040 4 0.020}- =] ' | 1 | 4 | | { | \ | | | \ | n S (keV—b) | 0.100+ ——_———— [Alt88] MHN [Moh93] 0.080 0.060 = 0.040; = 0.020}- as 1 i 1 | 1 | 1 | 4 l L i 1 | 1 | 1 { 1 ° 100 200 300 400 500 600 700 800 900 1000 E (keV) Figure 1.7: Theoretical calculations of S(E) for 7H(a,7)’Li by Kajino, Toki, and Austin [Kaj87], Langanke [Lan86], Mertelmeier and Hofmann [Mer86], Altmeyer et al. [Alt88], and Mohr et al. [Moh93]. We have used the Modified Hasegawa-Nagata (MHN) interaction calculation of Altmeyer et al. The curves have been normalized to S(500 keV) = 0.060 keV-b. 19 wavefunctions are calculated in a unified manner, and the full capture matrix ele- ment can be calculated. The predictive power of the method is however limited by the choice of the effective interaction. The calculated S-factor can vary by nearly a factor of two, depending on the choice of effective interaction [Kaj86, Alt88]. However, the calculated energy dependence of the S-factor is nearly the same for all calcula- tions (see below). The choice of effective interaction can be constrained by requiring that the calculation reproduce the measured scattering phase shifts and ‘Li bound- state properties, perhaps reducing the uncertainty in the calculation. The various calculations differ in the model space and effective interaction(s) used. Several other recent studies also warrant mention. Langanke [Lan86] has cal- culated S(#) using a microscopic potential model. Kajino, Bertsch, and Kubo [Kaj88a] have estimated 5(0) = 0.097 + 0.038 keV-b, using the measured electric- dipole polarizability of “Li. Kajino et al. [Kaj88b] have found a constraint on S(0), 0.083 < S(0) < 0.15 keV-b, using the experimental matter radius of “Li and the correlation found in RGM calculations between S(0) and the matter radius. Kajino, Toki, and Austin [Kaj87] give an expression valid for EF < 1500 keV, S(E) = S(0)exp(—2.056F)(1 + 2.2875E? — 1.1798E° + 2.5279E*), (1.16) with EF in MeV, that reproduces within 3% the energy dependence found in RGM calculations [Kaj84, Kaj86]. Buck and Merchant [Buc88] calculated S(0) = 0.09 + 0.03 keV-b using a potential model. Another calculation of S(£) using a potential model has been reported by Mohr et al. [Moh93]. All of the calculations predict an increasing S(E) at low energies. Several of the calculations are shown in Figure 1.7. The curves have been normalized to 5(500 keV) = 0.060 keV-b. It is seen that the energy dependences of the calculations are consistent within ~ 10%, due to the validity of the extra-nuclear approximation used in the early calculations. The uncertainty in absolute magnitude primarily arises from the normalization of the asymptotic “Li wavefunction. None of the calculations 20 predicts as large an increase in the low-energy S-factor as found by Schroder et al. or Utsunomiya et al. It should be noted that calculations of the shape of S(£) for the *He(a,7)’Be reaction are in good agreement with experiment (see for example Kajino, Toki and Austin [Kaj87]). Through 1985, the recommended thermonuclear reaction rate [Cau85] for 3H(a,7)’Li was calculated assuming a constant S equal to 0.064 keV-b as found by Griffiths et al. The most recent compilation [Cau88] uses S(E£) found by Langanke [Lan86], which agrees with the data of Griffiths et al., and increases at low energies to S(0) = 0.105 keV-b. Several astrophysical calculations [Kaw88, Kra90, Smi93] have used different reaction rates based on the more recent experiments. Low-Energy Logarithmic Derivative Using assumptions similar to those of Tombrello and Parker [Tom63], Williams and Koonin [Wil81] calculated an expansion for the low-energy logarithmic derivative, 1 dS Using a hard-sphere radius of 2.8 fm, the authors found a = —2.034 MeV and b = —3.709 MeV~?. Schréder et al. fit their low-energy data with this energy de- pendence, finding 5(0) = 0.162 + 0.024 keV-b. However, Kajino [Kaj88c] has ar- gued that the Williams-Koonin calculation contains a computational error; he finds a = —1.15 MeV with what appears to be an identical calculation. The negative sign of b is inconsistent with the positive curvature found in all other calculations. Val- ues of a from other calculations (—2.056 MeV~! Equation 1.16 [Kaj87], —2.0 MeV~! [Buc88], —1.02 MeV~! [Moh93]) vary by a factor of two, even though the overall agreement of the energy dependences is within ~ 10% (Figure 1.7). 21 Branching Ratio Calculations of the branching ratio show much less model dependence than does the absolute cross section. RGM calculations [Kaj86, Alt88] find nearly energy- independent branching ratios in the range 0.41 — 0.44. Kajino, Mathews, and Ikeda [Kaj89] estimated R(0) = 0.47 + 0.07 using a kinematical direct capture model and the measured *He(a,7)’Be branching ratio. Mohr et al. [Moh93] also found R to be nearly independent of energy, with a value of 0.43. All of these calculations are significantly larger than the experimental value 0.32 + 0.01 found by Schréder et al., but are in reasonable agreement with the data of Griffiths et al. and Burzynski et al. 1.3. Experimental Overview The goal of the present experiment was to measure the absolute *H(a,)’Li cross section, with particular emphasis on the energies relevant to the big bang. The measurements were extended to as low an energy as practical, and covered energies ranging over a factor of 20. Our experimental approach, using Ti-*H targets, an a beam, and a Ge y-ray detector, is similar to that used by Schroder et al. and Burzynski et al. Our experiment does offer several technical improvements compared to the earlier work. The y rays were detected using a high-purity-Ge detector with 85% relative efficiency, giving a substantial increase in detection efficiency. The determination of the total cross section from the measured y-ray yield (which necessarily covers only a limited angular range) requires knowledge of the angular distribution. Angular dis- tributions were measured at nine energies for this reason as well as for comparison with theoretical calculations. The systematic uncertainty in the absolute normaliza- tion was reduced by using the well known *H(d, a) reaction to determine the 3H areal density. Several targets were bombarded to test the reproducibility of the absolute normalization. The data analysis is improved in many respects, particularly in the treatment of coincident y-ray detection effects. 22 Preliminary measurements were made in 1990 using a 35% Ge detector and a com- mercial Ti-*H target which had been extensively bombarded with “Li in an unrelated experiment. Due to problems from the implanted ‘Li and also target deterioration the results were not of high quality. Considerable difficulty was then experienced in obtaining suitable Ti-*H targets, which led us to develop our own production tech- nique as described in Chapter 2. The results reported here were taken in 1992 and 1993 using the 85% Ge detector and four targets made with our apparatus. 23 Chapter 2 Tritium Target Preparation The preparation of metal hydrides for use as hydrogen-isotope targets for ion- accelerator experiments has a long history [Gra49, Con52, Roc52, Con52, Mas57, Sco59, Coo60, Smi67, Kob89]. The metals Ti, Zr, and Er have been used; Ti has been the most common in recent times. The targets are most often used for neutron production through the *H(d,n), *H(p,n), or 7H(d,n) reactions. Typically, a (0.2-2)- mg/cm? metal layer is evaporated onto a solid substrate, which is then heated in the presence of the desired hydrogen-isotope gas. When the metal temperature reaches ~ 400 °C, hydride formation occurs rapidly. Stoichiometries approaching H:Ti = 2.0:1 have been reported in bulk samples of Ti [Smi48]. As discussed in the Introduction, we chose to measure the *H(a,)‘Li reaction using Ti-*H targets, an a beam, and a high-purity-Ge detector to measure the emitted + rays. Due to the small cross sections involved (~ 50 nb at E, = 150 keV), the *H areal density of the target should be high to maximize the y-ray yield. On the other hand, the energy lost by the a beam in the target is proportional to the areal density. Due to energy loss, *H(a,7)’Li reactions will occur at varying E,, depending upon the depth of the interaction in the target. The reaction kinematics (E, ~ 3E, + Q) imply that the a energy loss contributes to the width of the energy spectrum of the emitted capture y rays. The °H areal density is thus limited by the choice of an acceptable y-ray line width in the detection system. Furthermore, the cross section 24 far below the coulomb barrier varies rapidly with energy, necessitating an accurate determination of the effective reaction energy, which is subject to less systematic error for small energy losses. In principle the best choice would be a windowless *H-gas target, but this option was deemed impractical in our laboratory due to radiation- safety considerations. We determined that a Ti-*H target of ~ 20 ug/cm? would provide the best com- promise of energy loss and y-ray yield. Ti is preferable to the other materials because it has the smallest stopping power per 9H atom (assuming comparable stoichiometry for all materials could be obtained). The *H contributes only 25% to the energy loss for Ti*H, targets. Attempts to obtain targets with the above characteristics from commercial sources were unsuccessful. The majority of companies or laboratories that historically pro- duced *H targets had either temporarily or permanently stopped this service. The targets that we were able to purchase had poor stoichiometries (*H:Ti < 0.5) and/or unsuitable Ti thicknesses. The 20-ug/cm? target desired was much thinner than the targets commonly used for other purposes. A survey of the previous literature indi- cated that a common problem affecting target preparation is poisoning of the Ti layer with contaminant gases before hydriding. It is reasonable to suppose that a very thin target would be more susceptible to these problems, particularly if the contamina- tion is primarily near the surface. The technique most often described in the cited references is the evaporation of a Ti layer onto the substrate in one vacuum system followed by transfer of the target in air to another for hydriding. We believe that some of the difficulties commercial suppliers experienced in preparing a target to our specifications may be due to contamination of the Ti during the transfer step. The systems and procedures that we developed to fulfill our needs are described in the following sections. In our process the Ti evaporation and hydride formation take place in the same apparatus, without breaking vacuum. The system has been extensively tested, first making Ti-’H targets, and finally Ti-*H targets. Methods for accurately determining the 7H, 3H, and Ti areal densities are also discussed. An 25 accurate measurement of the 3H areal density is crucial for obtaining absolute *H(a, 7) cross sections. 2.1 Description of Apparatus The production of a Ti-H target consists of two operations: the evaporation of Ti onto a substrate, and the heating of the Ti and substrate in an atmosphere of hydrogen isotope to induce the formation of hydride. The boat used for Ti evaporation and the target substrate are each heated using a 3-turn induction coil centered either at the boat or target position. Radio-frequency power at 50-200 kHz is supplied to the coil by a 2.5-kW commercial supply (Lepel LSS-2.5). A drawing of the apparatus is shown in Figure 2.1. The closed-ended quartz tube (42.1-mm o.d., 38.1-mm i.d.) hanging below the gas-handling manifold is sealed with a Viton O-ring. All other seals are metal gaskets or ferules. The high vacuum in the system is supplied by a 20-L/s ion pump. A 1.3-cm long cylindrical quartz tube (36.3-mm o.d., 30.0-mm i.d.) supports the boat from the bottom of the closed tube. The target substrate is supported above the boat a distance of 5.1 cm by a length of identical cylindrical tube. Several holes were ground through the walls of the inner quartz tubes in order to increase the pumping speed. With all valves closed, the interior volume is ~ 300 mL. Additional features of the system are shown schematically in Figure 2.2. Pressures between 10-3 and 10° Torr are read by a Granville-Phillips Series 275 “Convectron” gauge, using supplied calibration curves for various gases. Non-radioactive gases are pumped away using a standard mechanical pump. The charcoal-filled trap, when chilled to LN2 temperature, is used to pump away excess °H gas for eventual disposal. A 150-mL cylinder of high-purity He gas is attached for use as a coolant (described in more detail below). The locations of the vent and the hydrogen-isotope gas supply are also shown. The boats used for Ti evaporation merit special comment. A top view of the 0.38- 26 to 20-—L/s ion pump an ate veadve manifold target substrate —zy)\ AAAARAAARARAN RAAAAAAEAREEEEEEEEEEN CRRRRRERURRUESS Ta boat SSSuSess Sra é Sy Ay AY N N N N= N | SSS SSS nS SSS SSS SS SSSA induction coil O00 6 Oe) ZZ bK—-5 em—+4 Figure 2.1: Side view of the target-production apparatus. The four ports connected to the gas-handling manifold are not shown (see Figure 2.2). 27 Convectron auge vent , / O CRI] manifold } RE mechanical charcoal pump trap helium hydrogen isotope Figure 2.2: Schematic diagram (top view) of the apparatus. The ion pump and quartz tube are not shown (see Figure 2.1). mm-thick Ta boats used is shown in Figure 2.3. The hemispherical dimple for the Ti source is positioned so that it is in the center of the quartz tube. The shape of the boat is designed to maximize the boat temperature at the source position for a given power output from the induction-heating power supply. New boats were thoroughly outgassed for several minutes in vacuum at temperatures in excess of those attained during Ti evaporation. 2.2 Target Production Procedure The Ti was evaporated onto substrates of 31.7-mm-diameter Cu (0.81-mm thickness) or Ta (0.76-mm thickness). All of the targets used for the 7H(a, ) reaction were made using Cu substrates. The Cu substrates were cleaned with detergent, etched for 30 s in 3M HNOs, and finally rinsed with distilled HO. Ta substrates were treated in the 28 —— 3.18 cm <a dimple for source Figure 2.3: Top view of the Ta boats. The Ti evaporation source is placed in the hemispherical dimple at the center. same manner, except the HNO3 etch was replaced by a 15-s etch in 5M HNO3 : 48% HF = 3:1 (by volume), followed by a 15-s etch in 8M HNO3. The Ti evaporation sources (typically ~ 3 mg Ti) were cut from wire stock that had been previously outgassed in high vacuum by passing current through it, heating it to just below the melting point. The Ti source and substrate were then installed in the quartz tube assembly, and the system was pumped to high vacuum (base pressure ~ 4x 10-® Torr, as determined by jones current). At this point the substrate was outgassed by heating it for five minutes (orange heat for Cu, yellow for Ta) with the induction coil. Next, the Ta boat and Ti source were outgassed by heating the boat for 20 minutes at a temperature somewhat below that required to melt Ti. The system was then allowed to pump and cool for several hours in order to achieve the best vacuum possible. The Ti source was evaporated by slowly raising the power applied to the boat, until the source was observed to melt. The power was then left constant for two minutes in order to evaporate the Ti. This method proved effective in evaporating ~ 70% 29 of the Ti (the remaining fraction appeared to amalgamate with the Ta). During evaporation, the pressure at the ion pump remained below 5 x 107’ Torr. After roughly 10 such evaporations, the boats typically developed holes in the bottom and required replacement. Following evaporation, the gate valve was closed and ~ 1.0 Torr of high-purity He gas was admitted to the quartz tube. The presence of the gas increased the thermal conductivity inside the tube, greatly reducing the time required for the objects inside the tube to return to room temperature. After 20 minutes the He was pumped away, and the system was returned to high vacuum (gate valve open). Next, the gate valve was closed, and 1.5 Torr of 7H or *H gas was admitted (corresponding to ~ 1.5 Ci for 9H). The substrate was then heated for three minutes. The temperature reached by the substrate is not known, but was reproducible in terms of the induction-heater control settings. It was found that if the Cu substrate was heated to a visible red temperature the Ti and Cu would amalgamate, which invariably led to a poor H:Ti ratio. We used a power level that, in vacuum, would heat a Cu substrate from room temperature to red heat in three minutes. The progress of hydride formation was monitored by recording the gas pressure versus time. Upon admission of the hydrogen-isotope gas, some spontaneous absorption of the gas by the Ti occurred (i.e., without heating the substrate). The amount of sponta- neous absorption that takes place is apparently very sensitive to the vacuum condi- tions during Ti evaporation. Absorption at room temperature was not anticipated, as the previous references [Gra49, Lil51, Roc52, Con52, Mas57, Coo60] stated that heating to ~ 400 °C was necessary for hydride formation. This effect is somewhat undesirable since it allows absorbtion by Ti which was evaporated onto surfaces other than the substrate, and because its extent is not reproducible. Attempts to control the effect by precooling the substrate and inner quartz pieces prior to the admission of hydrogen-isotope gas (by filling the quartz tube with 1.0-Torr He, immersing the tube in L.N2 for ~ 30 minutes, and then pumping out the He) were unsuccessful. On one occasion we tested the 7H areal density of a target which was not heated. A ?H:Ti 30 ratio varying between 0.6 and 1.6 over the surface of the target was found. All of the other targets were heated in the hydrogen-isotope gas, improving the uniformity and reproducibility of the process. Following the heating of the target, the system was allowed to cool until a constant pressure was reached (~ 15 minutes). The remaining gas was pumped away, using the LN -cooled charcoal trap in the case of 7H gas, or the mechanical pump for 7H gas. Then the residual pressure was pumped using the ion pump. The gate valve was then closed and the system was vented with air, at which point the target was transferred to the accelerator facility for testing. Following each use of 7H gas, the system was reassembled with another substrate, but without a new Ti source, and pumped to high vacuum. The boat was then heated for five minutes at yellow heat with the gate valve closed and the valve to the L.No- cooled charcoal trap open in order to outgas as much of the residual *H radioactivity as possible and collect it in the trap. The final step was to prepare the system for its next use by removing the Ti deposits from the quartz pieces with acid and distilled-H2O rinses. Five targets (numbered 1-5) were produced on Cu substrates. Targets 1 and 3-5 were used for the *H(a, 7) measurements (insufficient Ti was evaporated on target 2). An additional target on Ta backing was produced for some unrelated experiments. All procedures involving *H gas are done in a fume hood in our isotope laboratory, with disposal of radioactive waste according to regulations in our license. The air in the personnel operating area is continuously tested with a tritium monitor during 3H-handling procedures. 2.3 Target Characterization The use of accelerator techniques to determine the hydrogen-isotope content of metals has been extensively reviewed [Kha89]. The 7H, 3H, and Ti areal densities were determined using the 7H(*He, a), *H(d,a), and Ti(a, a) reactions, respectively. The 31 }~——— 10 em ———>| Si detector —400 V 3.c—-mm dia. collimator 1.6—mm dia. collimator Figure 2.4: Scale drawing of the pertinent parts of the 0°-beamline scattering cham- ber. proton, deuteron, Dif, *Het, and *Het beams were supplied by the Caltech 3-MV Pelletron Tandem Accelerator. The beam is described in more detail in Section 3.2. The experiments were performed using the scattering chamber on the Pelletron accelerator’s 0° beamline. A scale drawing of the pertinent components is presented in Figure 2.4. The charged particles were detected at 4.4 = 165° with a silicon surface- barrier detector collimated to have a solid angle of 1.194 + 0.018 msr. The beam, collimated to a 1.6-mm diameter, was incident normal to the target. The target was biased at +300 V in order to ensure accurate beam-current integration by preventing secondary electron emission, and the collimator was followed by a —400 V suppression ring. A pulser peak was inserted above the spectrum to monitor dead-time corrections (typically less than 5%). The beam-current integration and detector solid angle were checked by measuring the Cu(p,p) yield from a thick Cu substrate for 2000 < E, < 2600 keV, where the cross section is known to follow the Rutherford formula [Gol67], and where the proton-stopping power [Zie77] is known to 1%. The expected height of the plateau at its edge can be calculated from the differential cross section and stopping power as 32 described by Sargood [Sar82]. The measured values were in excellent agreement with calculated values over the entire range; for example, the measured and calculated values at E, = 2200 keV were (9.22 + 0.17) x 10* and (9.03 + 0.09) x 10% protons- keV~'pC~'sr~!, respectively. 2.3.1 Ti Areal Density The Ti areal density was determined by elastic a scattering for 1500 < Ey, < 2250 keV, where the cross section is given by the Rutherford formula. A spectrum obtained at EF, = 2000 keV from a Ti-*H layer on a Cu substrate (target 1) is shown in Figure 2.5; also shown for comparison is the spectrum from a blank Cu substrate. An areal density of 2.96 x 10'’ Ti atoms/cm? at the center of the target was found using the number of counts in the Ti(a,a) peak (the mean energy of the a particles was used in evaluating the Rutherford formula). Figure 2.5 shows a shift in the edge of the Cu(a,a) scattering plateau as a result of the Ti-*H layer. Using 5.52 x 107” 3H atoms/cm? (determined as described below), the above Ti areal density, and as- suming stopping-power additivity [Zie77], a shift of 46.2 keV was calculated, in good agreement with the observed 47.5 keV. The uniformity of the Ti layers was excellent; the areal density was found to decrease slowly with distance from the center of the target (6% lower at 12 mm from center), consistent with the substrate — Ti source evaporation geometry. The Ti(a, a) spectra were also useful for profiling the Ti and Cu composition. In some early tests the Ti and Cu were amalgamated, an effect clearly visible in the spectra. 2.3.2 7H Areal Density For Ti-*H targets, the 7H areal density was determined by detecting a particles from the 7H(*He, a) reaction with E[*He] = 650 keV. This energy corresponds to the peak of a broad maximum (Ij ~ 600 keV), where the cross section is known to ~ 3% [Mol80, Dav80]. We have assumed 94(6j., = 165°) = 34.3 mb/sr in the following 33 T T T T T T T T T T T é 3 T T T T T T ij T T T T | T T T T 700 4 i Cu substrate p 600}- + 500};- 4 400}- “| 300;- = 200}- 4 100} 4 oO I counts / uC ~~) fs) ro) | Ti-°H on Cu substrate J 600 500 400 300 200 100 4 1 1 i | 4 350 400 4 1 4 l 4 100 150 1 | rt 1 | 4 250 300 channel 1 | 1 200 Figure 2.5: Spectra of scattered 2000-keV a particles observed at 6,,, = 165°, from a blank Cu substrate (top panel) and target 1, a Ti-*H layer on Cu substrate (bottom panel). The peak in the bottom panel at channel ~ 295 is from the Ti layer. The leading edge of the Cu-scattering plateau in the lower panel is shifted by 10.0 channels due to the presence of the TiH layer. 34 T TTT i on me meee! a TTT 1 rout 104 TT La | Lot faatinie 103 counts a Se Teeny ee tut 10? 2! ul 10! ToT TTT Lid Litiul 4 | 700 800 L 10° 1 a 1 4 TE POP | 0 100 200 300 400 500 600 channel Figure 2.6: Si-detector spectrum resulting from the 650-keV-*He bombardment of Ti-*H on Cu substrate. The peak at channel ~ 415 is due to a particles from the 7H(*He, a) reaction. The counts in channels > 550 are due to high-energy protons from 7H(*He, p) that deposited a fraction of their energy in the detector. analysis. The 7H(*He, a) spectrum obtained from Ti-?H on a Cu substrate is shown in Figure 2.6; the 7H and Ti areal densities were determined to be 5.05 x 10'” and 3.24 x 10!” atoms/cm”, respectively, yielding a 7H:Ti ratio of 1.56:1. 2.3.3 °H Areal Density The 7H areal densities of the Ti-°H targets were determined using the ?H(d, a) reac- tion. The total cross section and center-of-mass Legendre coefficients for this reaction were taken from the evaluation of Drosg and Schwerer [Dro87]; the uncertainty is es- timated to be 1.5% for Eq < 400 keV, increasing to 4% for higher energies. The majority of the Ti-*H targets were tested over the range 250 < Ey < 700 keV, with a deuteron beam produced using the accelerator’s tandem mode (thus avoiding ?H gas in the terminal ion source, which could lead to 7H} contamination of the at 35 6 1 (0) T T T T T T T lj T T T T T T T TT TtTt 10° T TTT : oe tiul 104 TT TTT Ae featin 103 counts T TTT UL | oe Liou 10? TTT TITTY | eo piu 10! 10° i | | | L | 1 | n he | n ! | 0 100 200 300 400 500 600 700 800 channel 1 bo pl TT LL | Figure 2.7: Si-detector spectrum resulting from bombarding Ti-*H on Cu substrate (target 1) with a 400-keV deuteron beam. The peak at channel ~ 505 is due to a particles from *H(d, a). beam; see Appendix B). A sample spectrum obtained at Ey = 400 keV for target 1 is shown in Figure 2.7. The observed spectra were consistent with the 3H being uniformly distributed in the Ti, with no 7H in the target backing. In all cases the energy dependence of the a yield (including small corrections for energy loss) was consistent with the evaluated cross sections [Dro87]. From the average of runs at 7 energies between 250 and 700 keV, the 7H areal density of target 1 was determined to be 5.52 x 10!” atoms/cm’, yielding a *H:Ti ratio of 1.87:1. The *H(d, a) excitation functions from targets 1, 3-5 are shown in Figure 2.8, along with the fits to the energy dependence of Drosg and Schwerer [Dro87]. The 3H:Ti ratio was found to be constant within 3% over the surface of the targets. In order to test the evaluated cross section [Dro87], the *H(d,a) measurements were extended down to Eq = 70 keV to include the peak at Eq ~ 110 keV (where the cross section is best known). The required deuteron energies were obtained by using 36 10° T | qT | qT ] + | T | qT T T d N q — target 1 | i A — target 3 > — target 4 y — target 5 Q 2 ~~ N 2 = g lr 3 a z ° 3) . j A 10! r | 1 | r | 1 | 1 | 1 | l 100 200 300 400 500 600 700 - 800 Figure 2.8: *H(d,a) excitation functions obtained from targets 1, 3-5. The solid curves are the recommended cross sections [Dro87], normalized to the data. 37 7 T T T T T T T | T | T T T | T + g 3H(d,a) 4 Q = 108+ 4 9 L 3 5 a r 4 © SB r 4 ws, 5 4 Be L J si | ° oO 10®- 4 a 0 100 200 300 400 500 600 700 800 Eq (keV) Figure 2.9: *H(d,a) excitation function obtained using deuteron (circles) and Df (squares) beams. The solid curve is the recommended cross section [Dro87], normal- ized to the data. the molecular ion Dj from the accelerator’s terminal ion source. These measurements were performed after the 7H(a,7) experiments, since the D2 gas was not installed in the ion source during the (a,7) work due to concern about deuteron contamination of the a beam. In this case, a target prepared on a Ta substrate was used. A 560-yg/cm? Ni foil was placed in front of the detector to reduce the count rate due to elastic scattering. The resulting excitation function (corrected for energy loss) is shown in Figure 2.9; the data agree with the shape given by [Dro87] (solid line) within 4%. 2.4 Results and Discussion The technique described is capable of consistently making targets with hydrogen isotope to Ti ratios of > 1.5:1. The results for the four Cu-backed 7H targets used for the *H(a, 7) measurements are shown in Table 2.1. During the (a, 7) measurements, 38 Target Ti 3H N x? (10'’/cm?) (10'’/cm?) 1 2.96 5.52(4) 13 13.1 3 4.67 1.32(2) 10 17.4 4 2.86 4.33(4) 14 17.5 5 1.90 3.23(3) 13 13.2 Table 2.1: The 3H and Ti areal densities determined for the Cu-backed targets used for (a,7) measurements. Also given are the number of data points N and the x? (using only statistical errors) for the fits to the 7H(d, a) cross section. The numbers given in parentheses with the 3H areal densities are the statistical errors in the least significant digit, scaled by ./ 5. the targets were directly water cooled, and the a beam (5-30 A) was rastered over ~ 1 cm?. The targets proved very stable under bombardment; the observed deterio- rations are given Chapter 3 (Table 3.1) and the corrections applied are described in Chapter 4. The deterioration is attributed to sputtering; blister formation at the Ti — Cu interface was not observed (a problem encountered with commercial targets). One improvement in the apparatus would be the addition of an infrared temper- ature sensor to monitor the heating of the substrate. The technique currently used to reproducibly heat the substrate requires that the applied heat and gas-pressure history be reproduced, since the temperature reached by the substrate depends on pressure-dependent convection as well as the delivered radio-frequency power. An- other improvement would be the addition of a mechanism to transfer the substrate to a separate chamber after Ti evaporation (still without breaking vacuum). This procedure would conserve the *H gas by preventing it from being absorbed by the Ti that was evaporated onto surfaces other than the substrate. The nuclear-reaction-analysis techniques used were capable of determining abso- lute 7H, 9H, and Ti areal densities to ~ 4%. 39 Chapter 3 Experimental Apparatus and Procedures The preparation of the Ti-*H targets proved to be the most difficult and time- consuming part of the experiment. The remaining aspects of the experiment, bom- barding the targets with at beam and detecting 7 rays, were comparatively straight- forward. However, a careful consideration of all experimental aspects is important for achieving the most accurate results. The following sections describe the target chambers, beam, detection, and gen- eral procedures followed. Particular attention is paid to calibration of the high-purity Ge y-ray detector. The availability of large-volume detectors has led to higher ef- ficiencies than were previously possible, but has also increased the importance of coincident-summing corrections. A detailed description is given of the photofraction measurements, which are crucial for making coincident-summing corrections. 3.1 Target Chambers Several factors were considered in the design of the target chambers. A knife- edge vacuum seal to the target was used in order to achieve the best possible (and hydrocarbon-free) vacuum. The backs of the Cu targets were cooled with flowing 40 -———_10 em———-+ Middl water Ge cooling 4 A out i target WW cr cooling NX in Figure 3.1: The details of geometry A, showing the perpendicular target chamber and the position of HPGe y-ray detector. chilled water to dissipate the beam power. The amount of material between the Ti- 3H layer and the y-ray detector was made as small as practical in order to minimize y-ray absorption. The target flange (with knife edge) and the water-cooling jacket were machined from stainless steel. The target flange was welded onto stainless-steel 3.81-cm o.d. vacuum tubing. Two target chambers were used during the course of the experiments. The first chamber fixed the target perpendicular to the incident beam direction and was used for the experiments using targets 1, 3 and 5 which measured the y-rays at zero degrees relative to the incident beam. This chamber is depicted in Figure 3.1. For the angular-distribution measurements using target 4, a chamber which fixed the target at an angle of 44.41° with respect to the beam direction was used. Hereafter, this chamber will be referred to as the “45-degree” target chamber. This chamber was oriented so that target substrate was rotated about the vertical diameter. The angular-distribution measurements were performed with the y-ray detector in the horizontal plane. 41 For the measurements using the perpendicular target chamber with target 1, the beam was collimated by a 12.7-mm diameter Ta collimator 75 cm upstream from the target. For the measurements using this target chamber with targets 3 and 5 the same collimator was used at a distance of 62 cm upstream. For the measurements using the “45-degree” target chamber, a rectangular Ta collimator (4.0-mm horizontal x 15.9- mm vertical) was located 53 cm upstream from the target. For all measurements, a suppression ring held at —400 V was placed ~ 12 cm downstream from the collimator. This voltage helped insure accurate beam-current integration by preventing secondary electrons from leaving the target chamber, and by preventing electrons from entering the chamber from upstream. The target chambers were installed at the end of the South-30° beamline of the Pelletron accelerator, as shown in Figure 3.2. The beamline’s cryopump maintained a base pressure of ~ 4 x 10°, measured 5-cm upstream from the collimator. However, the pressure rose to © 3 x 10~” torr when running a beams due to the presence of *He gas from the accelerator’s ion source which the cryopump did not efficiently pump. The probability for charge-changing processes in the residual gas between the target- room optics and the target (which could lead to an error in beam-current integration) was calculated to be < 0.002, using tabulated [AII58] charge-exchange cross sections for Het ions in He gas. An inline 2.36-cm-i.d. hollow Cu tube held at LN2 temperature was installed ~ 20 cm upstream from the target in an attempt to reduce residual hydrocarbon vapors in the vicinity of the target. The presence of hydrocarbons in the residual gas will lead to an undesirable beam-induced carbon buildup on the target. The cold tube was used only with the experiments done with target 1 — with disappointing results: ~~ 5 x 10'” atoms/cm? of carbon were observed on the surface of target 1 after the 3H(a,7) measurements. The carbon buildup observed on the targets is discussed in more detail below. After the experiments with target 1, the cold tube was removed, and the target chamber and collimator were thoroughly re-cleaned. No carbon buildup (< 1.5 x 107° atoms/cm?) was observed during the remaining experiments. ae | 42 1m ~— magnetic quadrupoles magnetic steerer (X) magnetic steerer (Y) cryopump beam-—profile monitor Faraday cup collimator target chamber Figure 3.2: A scale diagram of the Pelletron accelerator’s South-30° beamline, show- ing the location of the optics, vacuum system, beam-handling equipment, and target chamber. 43 3.2 Beam The proton, deuteron, Df, 7He*t, and *He* beams were supplied by the Caltech 3- MV Pelletron Tandem Accelerator. The beam energy was defined by a 90° analyzer magnet and NMR magnetometer. The energy calibration (+0.1%) was established using the 483.91 + 0.10-keV resonance [Ajz87] in '°F(p, ay), the 991.86 + 0.03-keV [End90] in ?’Al(p,7), the 606.0 + 0.5-keV resonance [Wan91] in '!B(a,n), and the 1530.03 + 0.15-keV resonance [Maa78] in *4Mg(a,7). This energy calibration was used for the experiments described in Chapter 2; the energy scale for the *H(a,7) measurements was deduced from energy of the capture y-rays. The number of inci- dent particles was determined by beam-current integration. The calibration of the integrator was tested with current sources over the range 20 nA — 30 wA, and found to be accurate within 0.5%. The effect of leakage current through the H,O-cooling lines used during the *H(a,7) experiments was measured to be < 0.5%. The target chambers used for the *H(a, 7) experiments were aligned by installing a quartz window in the target position (in place of the usual target and water-cooling jacket), and bombarding the quartz with a ~ 20-nA a beam. The beam-induced fluorescence in the quartz indicated where the beam was striking the target. The target chamber and collimator were positioned so the beam, when positioned at the center of the collimator, would strike the center of the target. It was also verified that the beam was confined to the center of the target; it was found that the area of the target accessible to the beam was ~ 50% larger than the area of the collimator. The alignment was tested using a beams with energies between 200 and 1500 keV; no energy-dependent effect was noted. The layout of the beamline components is shown in Figure 3.2. In order to bom- bard the target uniformly and reproducibly, the beam was rastered over the area of the collimator by the magnetic steerers. The beam was sharply focused at the col- limator position (with the aid of a beam-profile monitor ~ 1 m upstream from the collimator), and then amplitudes of the horizontal and vertical rasters were increased 44 until ~ 10% of the beam struck the collimator. In addition, to help insure a repro- ducible trajectory, the beam was tuned so that it passed through the center of the target-room quadrupole magnets. This positioning was accomplished by tuning the beam so that the quadrupoles had no steering effect on the beam. These procedures were tested during the quartz alignment procedure, and found to yield good results. After removing the used targets, the area of the targets struck by the beam was ob- served to have a different appearance from the unstruck portions. For all targets, the beam spot was observed to be centered within 2 mm; for targets 1, 3, and 5 the beam spot consisted of a roughly square area of 10mm x 10mm; for target 4 (used with the “45-degree” chamber), the rectangular beam spot (on the target) was roughly 6 mm (horizontal) x 17 mm (vertical). The energy of the incident a beam was varied between 140 and 2790 keV during the *H(a,7) experiments. In order to achieve this range of energy, the pressure of the insulating SF, gas in the accelerator’s pressure vessel was varied between 10 and 60 psi. In addition, a variable length of the accelerating tube could be shorted in order to increase the voltage gradient for a fixed terminal potential. This technique was essential for the obtaining the needed beam intensity and stability at low terminal potentials. For example, at E, = 140 keV and with 8/9 of the accelerating tube shorted, we averaged 19 wA on target for 11.5 hours. The beam intensities varied between 5 and 30 wA over the course of the experiment. 3.3. y-Ray Detection Gamma rays were detected using an 85% relative-efficiency high-purity Ge detector (ORTEC serial no. 31-TP40242A). The signals were processed by an ORTEC Model 972 spectroscopy amplifier and stored in a 4096-channel analyzer (Tracor-Northern TN-7200). The spectra were transferred to a computer for off-line analysis. The energy resolution of the detector was typically 2.1 keV at E, = 1332 keV. 45 3.3.1 Detector Geometries The detector was used in three different setups. Using targets 1, 3, and 5 and the perpendicular target chamber, extensive (a,y) measurements were performed with the detector at 0° in close geometry for maximum efficiency (source to crystal front face 2.1 cm). A 1.02-cm lucite spacer-ring mounted flush between target’s water- cooling jacket and the detector housing’s front face insured precise reproducibility of the detector — target distance. This setup (hereafter referred to as geometry A) is depicted in Figure 3.1. In anticipation of large coincident-summing corrections and/or angular- distribution-attenuation effects, measurements were also performed at 0° on targets 1, 3, and 5 using the perpendicular target chamber with the detector much farther away, where these effects would be minimal (source to crystal front face 11.6 cm; hereafter referred to as geometry B). This change only required the moving of the detector. For the angular-distribution measurements using target 4, the “45-degree” target chamber was used. The y-ray detector was placed on a rotating table whose rotation axis was placed directly below the target position. A fairly large source — detector distance was used in order minimize attenuation of any anisotropy of the angular distribution (source to crystal front face 10.1 cm). The detector could be placed at five different angles: 0°, —45°, —90°, +90°, and +135°. This setup (geometry C) is depicted in Figure 3.3. It is estimated that the uncertainty in the angular positions is +2°. In this case the geometry was much easier to reproduce, as target or radioactive-source changes could be made without moving the detector. 3.3.2 Photopeak-Efficiency Measurements The photopeak efficiency, €, for a particular geometry is defined as the probability of a 7 ray emitted by an isotropic source depositing its full energy in the Ge crystal. The photopeak efficiency was determined for the needed range of y-ray energies in each 46 Figure 3.3: A scale diagram of geometry C, showing only the target and sensitive Ge. geometry (including each angle for the angular-distribution setup) using the following sources: Eu, °6Co, and 74Na. The commercially prepared '°?Eu source covers the range 122 < E, < 1408 keV. The source was taped onto a blank Cu substrate which was sealed in the target position for the efficiency measurements. The °°Co source was produced via the *°Fe(p,n)°°Co reaction by bombarding a 0.51-mm-thick Fe disk with 1 mC of 5950-keV protons using a different beamline. The source was installed in the target position with a 0.25-mm-thick Cu disk behind it (the y-ray absorption of this combination is very close to that for 0.81-mm-thick Cu). The primary advantage of this source is that it extends the calibration to higher energies: it covers 847 < E, < 3548 keV. The strength of this source was determined to be 2350 decays/sec by comparison with other sources. The isotope *4Na has a relatively short half-life (15 hours), and emits two y-rays (1369 and 2754 keV) per decay in one-to-one coincidence (other decays take place ~~ 0.06% of the time). Using the coincident-sum peak and the two photopeaks, the absolute strength of the source can be determined as described in Chapter 4. The ?4Na was produced using the 7*Na(d, p)*4Na reaction. For the perpendicular target chamber, the 74Na was produced in the “45-degree” 47 chamber, using a different beamline, and then transferred to the perpendicular cham- ber for counting. A target consisting of unknown amount of Na2WO, evaporated on a 0.81-mm-thick Cu disk was bombarded by 2000-keV deuterons, producing an initial activity of ~ 2x 10° decays/sec. In the case of the “45-degree” target chamber used for the angular-distribution measurements, the 74Na was produced in the target chamber. The y rays from a source produced in this way should have the same spatial distribution as the y-rays from the *H(a,7) reaction, because the same collimator and beam-tuning procedures were used. The target was prepared by evaporating 1 mg/cm? Na,WO, onto a 0.81- mm_-thick Cu disk. Using the technique of Skelton and Kavanagh [Ske84], a 1-mg/cm? Au layer was then evaporated over the Na,WO, in order to insure that all of the 4Na produced would remain in the target. The source was produced by bombarding the target with 1.1 mC of 2200-keV deuterons, producing an initial activity of 1.3 x 10° decays/sec. One concern about using this technique was neutron damage to the Germanium crystal caused by neutrons from (d,n) reactions. According to the detector instruction manual, a cumulative neutron flux of > 10° n/cm? on the detector will cause degradation of resolution. During production, the y-ray detector was rotated 180° in its dewar (to increase the distance from the target to detector), and the target and detector were heavily shielded with borated paraffin. Using a portable neutron monitor, it was estimated that the detector received ~ 2 x 104 n/cm? during the 74Na production. It was determined that < 0.04% of the “Na was lost from the target by measuring the background after removing the the target. The 74Na-efficiency measurements were performed both before and after the 7H(a, 7) angular-distribution measurements. Care was taken during the y-ray calibrations to use the same Pb-shielding ar- rangement as during the *H(a,7) measurements, so that the total efficiency, which is needed for making coincident summing corrections, was the same. The analysis of the photopeak-efficiency calibration data is described in Chapter 4, and the results are plotted in Appendix E. 48 3.3.3 Photofraction Measurements The photofraction, ¢, is defined as the ratio of photopeak efficiency to total efficiency, n. The total efficiency for a particular geometry is defined as the probability of a y- tay emitted by an isotropic source depositing any energy in the Ge crystal. The total efficiency includes events where y-rays interact in the Ge crystal, but do not deposit their full energy, as well as events where radiation is scattered into the detector by material in the vicinity of the target and detector (possible “scattering” mechanisms include the Compton effect, pair production, 511-keV radiation from the annihilation of positrons, bremsstrahlung, and X-ray fluorescence). It is clear that the presence of the target chamber and Pb shielding increases the total efficiency relative to a bare detector. In order to make (7n-dependent) coincident-summing corrections to the *H(a,7) and e(E,) measurements, the photofraction as a function of E., was measured using various sources. An ideal source would emit isotropic, monoenergetic rays, with no other radiation. In practice, such sources are rare, so use has been made of several “non-ideal” sources, with appropriate corrections. Due to the presence of noise in the detector electronics, a low-energy threshold (13-100 keV, depending on gain settings) was required in the detector spectrum. The extrapolation of the spectra to zero pulse height involved a correction of 3-9%. The spectra were corrected for background from the room as well as source impurities and non-coincident 7 rays from the source. Most of the measurements were performed in geometry A, where coincident-summing effects were most important. A °’Co source (EF, = 122 keV) was produced via *°Fe(d,n)®’Co by bombarding a 0.51-mm-thick Fe disk with 30 mC of 4000 keV deuterons. The source was installed in the perpendicular target chamber using the 0.25-mm-thick Cu disk as described in Subsection 3.3.2 for the *°Co source. A source of ®'Cr (£, = 320 keV) was prepared using the *!V(p,n)*!Cr reaction. A target consisting of 0.4 mg/cm? V evaporated onto a 0.81-mm-thick Cu disk was bombarded by 61 mC of 2180 keV protons. 49 A 1% Au source (E, = 412 keV) was produced by thermal-neutron activation of a 25-ym_-thick, 1.1-cm diameter Au foil using a 7°?Cf neutron source and H2O moderator at an undergraduate physics laboratory on campus. The ‘Li(p, p;) reaction at E, = 1200 keV was used as a source of 478-keV y¥ rays. The target was prepared by evaporating ~ 5 yg/cm? Li on a 0.81-mm-thick Cu disk. Lithium-hydride was used as the evaporation source material; the evaporated film was a metallic grey color assumed to be metallic Li. Upon exposure to air the film became translucent, presumably reacting to form LiOH. The '?C(p,7)'3N reaction at E, = 461 keV was used to provide 2370 keV 7-rays. The y rays were produced by a steady 1-~A bombardment of a target consisting of ee 2 wg/cm? C on a 0.81-mm-thick Cu disk. The spectra required correction for the unavoidable background arising from '*N(G*) decay. By acquiring timed spectra with the beam off and on target, and using the known half-life and '*N production rate, the required correction was readily made. The ¢(£,) measurements were extended to E, = 6130 keV using the '°F(p, a7) reaction at E, = 340 keV. A 1.5-yg/cm? LiF target on a 0.81-mm-thick Cu disk was used as the target; no effects from the ‘Li(p,) reaction were observed. The 19F(p,a,) reaction, which populates the pair-emitting first excited state of 1°O, is not resonant at this energy, and is expected to make a negligible contribution to the detected spectrum. Additional measurements were made using the following commercially prepared sources: '3’Cs (E, = 662 keV), °4Mn (E., = 835 keV), and ®°Co (E, = 1172 and 1332 keV). The 1172- and 1332-keV ¥ rays from ®°Co are emitted in one-to-one coincidence; because the energies are fairly close together, the data were analyzed using $(1172 keV) / $(1332 keV) fixed at at a value determined from other sources. Coincident- summing corrections were also made to the data as described in Chapter 4. The results from the **Na source (Subsection 3.3.2) were used to obtain $(2754 keV) by using ¢(£.,) from other sources to estimate $(1369 keV) and making coincident-summing corrections. 50 It should be noted that the commercial sources were prepared on 0.3- to 1.0-mm stainless-steel or plastic backings. The presence of the extra material could cause increased scattering into the detector, systematically lowering the measured ¢. Monte Carlo simulations (Chapter 4) indicate that the effect of the source backing is < 5%. No corrections were made for this effect. The results for geometry A are shown in Figure 3.4; the error bars reflect estimated systematic uncertainties. The solid curve is an empirical fit used in subsequent analysis. Also shown are the experimental results from geometry B; the solid curve was used in the analysis of data using geometries B and C. 3.3.4 y-Ray Energy Calibration At low energies, the *H(a,7) cross section is strongly energy-dependent due to the coulomb barrier. An accurate determination of the mean reaction energy is thus very important. The mean reaction energy was determined from the centroid of the yo peak and reaction kinematics (the details of the analysis are described in Chapter 4). An accurate determination of the detector’s energy calibration is a crucial ingredient in this determination, particularly in the range 2500 < E., < 2700 keV, corresponding to the lowest-energy *H(a,7) measurements. The energy calibration was primarily based upon the accurately known 74Na lines at E., = 1368.633(6) and 2754.030(14) keV [Gre79]. The calibrations were performed daily during the *H(a,7) experiments by placing a *4Na source as near the target position as practical. The calibrations were also performed before and after most low-energy (Eem. < 150 keV) measurements. Gain shifts were observed to be less than 0.2 keV in almost all cases (a change in calibration of + 1 keV at 2754 keV was observed during run 096). The calibration was taken to be a linear fit to the centroids of the two 74Na lines. The duration of the low-energy runs was long enough to acquire sufficient statis- tics in the radio-thorium background line at 2614.533(13) keV [Gre79] to test the calibration. The measured energy was within 0.2 keV of the correct value in all cases. 10° T T T T T T T T T T T T T T T T Jae : geometry A : 107! od a g ° z © 10°* $$$ +1 t+ a : i | 5 4 = b geometries Band C jj a L 4 1071L 2 107° 4 1 4 1 4 Kill cilall 4 1 1 1 1 ee Oe 10° 108 104 E, (keV) Figure 3.4: The photofraction for geometry A is shown in the upper panel. The squares are the experimental points, and solid line is the curve used in subsequent analysis. The squares on the lower panel are the experimental results for geometry B; the solid curve on the lower panel is used in the subsequent analysis for geometries B and C. 52 The energy calibration for run 096 (during which the gain shifted slightly) was fit using this line. Tests using *°Co and '°?Eu sources indicated that the straight-line fit from **Na described the calibration within 0.5 keV over the range 400 < EF, < 3500 keV. The energy calibration in the range 2500 < E, < 2700 is assigned a 0.3-keV systematic uncertainty. 3.3.5 Detector Shielding The natural y-ray environment gives rise to background extending to E, = 2614 keV. At higher energies, the detector background is primarily due to cosmic-ray muons. The most important regions of the *H(a,7) spectrum are E., = 478 keV (72 tran- sition) and 2000 < E, < 2700 keV (capture transitions for low incident energies). For E, < 400 keV, it was anticipated that beam-induced background would be neg- ligible due to the large coulomb barrier for a-induced reactions on contaminants in the targets. The detector was initially set up in geometry A with ~ 10 cm Pb shielding in every direction (except the direction of the incident beam). In addition, a 65-cm x 58-cm x 2.5-cm-thick plastic scintillator paddle was centered over the detector, and used to reject cosmic-ray events. A background spectrum obtained with this setup is shown in Figure 3.5; also shown are the results for the detector with and without the Pb shielding. With 10-cm Pb in place, the cosmic-ray veto reduced the background by an additional factor 2 at E, = 2550 keV, but only 20% at E, = 478 keV. Also note that the Pb shielding has little effect on the background for E, > 2614 keV. When the *H(a,7)‘Li experiments were performed with target 1, it was found that there was significant beam-induced background in the low-energy runs. For example, at E, = 200 keV the background at E, = 2700 keV was a factor of 13 higher with beam compared to without. It was determined that the background was due to neutrons from the 7H(*H, 2n)a reaction, where the *H projectiles resulted from 3H(a, 7H) elastic scattering. This effect is discussed in more detail in Appendix B. 53 a oO ES Te TTT T TTT Let 1) _ © sv) me TTT pol — oO nun TT TTT _ oO _ pop il ae La counts / keV / hour — o o TT VPTTHy peti! Postal sent pnt la aye B toe pen. 1077 pot |e pa ta | en Eee | { : ES SR oe 1 1 1 1 L i 4 1 Linh i a ee ae | 1 1 } 0 500 1000 1500 2000 2500 3000 3500 4000 E, (keV) T CPE 4 rosin Figure 3.5: The y-ray detector (beam-off) background. The highest curve (A) is the result for no detector shielding. The middle curve (B) is the result for ~ 10-cm Pb shielding. The lowest curve (C) is for Pb shielding identical to curve B and cosmic-ray veto. The spectra have been re-binned to reduce statistical fluctuations. This additional background made the effect of the cosmic-ray veto negligible, so it was discontinued for the remaining experiments. For the angular-distribution measurements, only 5 cm of Pb shielding was used, except for the E, = 290-keV runs, where it was increased to 10 cm. 3.4 Procedure After each target was made, its 9H and Ti areal densities were determined as described in Chapter 2. The target was then installed in the target chamber for (a,y) mea- surements. The incident beam energies were chosen so that the 2614-keV background line would not interfere with the capture 7 rays. For each run, the incident charge, time, NMR magnetometer reading, and the y-ray spectrum were saved; a summary of all runs is given in Appendix C. The most extensive measurements were performed 54 Target Q °H (before) “°H (after) (C) (10'7/cm?) (10'”/cm?) 1.573. 5.52(4)-+5.36(6) 1.036 1.32(2) —_—‘1.32(4) 5.905 4.33(4) —-2.55(11) 0.690 3.23(3) 2.97(3) ao Fk WO eH Table 3.1: Summary of measured target deterioration. The column labeled Q gives the total at charge incident on each target. The “after” results have been corrected for 7H(@~) losses. The numbers in parentheses are the estimated relative uncertainties in the least significant figures. using target 1. Targets 3 and 5 were primarily used to test the reproducibility of the absolute cross section and to make additional measurements at low energies. Many repeated measurements at E, = 1000 keV were performed to test for possible target deterioration over time. Measurements using targets 1, 3, and 5 were made in ge- ometries A and B for each. The measurements using target 4 were made in geometry C. The 7H areal-density determinations using *H(d, a) were repeated after the (a,7) measurements for targets 1, 3, and 5. The areal density of target 4 was observed to deteriorate by ~ 40% over the course of the experiment, as evidenced by the decrease in yield at E, = 1000 keV. The before-and-after results are shown in Table 3.1. The corrections for target deterioration are described in Chapter 4. During the experiments with target 1, the capture 7 rays for the repeated E, = 1000-keV runs were at a slightly lower energy in the later runs compared to the earlier runs. It was suspected that this energy shift was due to a buildup of carbon on the target. This speculation was confirmed by measuring rays from the !*C(p, 7) resonance at E, = 461 keV. The carbon layer was studied with an a beam by measuring neutrons from the 13C(a,n) reaction. The measurements were performed using the “45 degree” target chamber and a polyethylene-moderated neutron detector [Wre94] at the 0° beamline. The beam collimation was carefully adjusted to reproduce the beamspot of the (a,7) 55 measurements. The results from the vicinity of the E, = 1053-keV resonance are shown in Figure 3.6; also shown are the results from a pure 'C foil [Bru92] taken under nearly identical conditions. The resonance profile indicates that either the carbon layer is not uniform, or that the carbon is mixed with the target material. We believe that the former explanation is most likely. The (p,y) and (a,n) measurements consistently indicated the existence of a carbon layer of 5 x 10'” atoms/cm”, averaged over the area of the beamspot. The carbon buildup problem was corrected as described in Section 3.1. The 12C(p,7) measurements indicated that < 1.5 x 10'© atoms/cm? of carbon was de- posited on the remainder of the targets. 56 14.0 T ] T T T T T T T T T T T T T _ 12.0- a target 10.0 ° ~ 8.0 a 4 6.0 a "ag o 4 4.0- a 4 2.0F = counts / uC ° 1200+ pure 8C foil 1000+ 5 3, 4 800} ° ~ 600+ J 400 _ T L oo 0G 4 200+ - 4 o 2 (@) [ 1 1 1 1 nl 1 | i 1 ! | | | l 1000 1020 1040 1060 1080 1100 1120 1140 1160 E, (keV) Figure 3.6: The '°C(a,n) yields are shown in the vicinity of the 1053-keV resonance. The upper panel is the result for target 1; the lower for a pure /°C foil. Note that natural carbon is ~ 1.1% 13°C. 57 Chapter 4 Data Analysis The data analysis consists of finding the absolute y-ray yields, and then normalizing by the 3H areal density and number of incident particles to obtain cross sections. The determination of the effective reaction energy from the observed energy of the Yo transition is also important. The reaction kinematics are discussed in Section 4.1. The y-ray analysis is then described in Section 4.2. In Section 4.3 the small corrections for target deterioration due to beam damage and 7H decay are discussed. Finally, the pieces are put together, and the cross sections are extracted in Section 4.4. 4.1 Kinematics The energy of the capture 7 rays as a function of the a-particle lab energy FE, and the lab angle 0, is given by Q(Mr + M, + M7) +2Mr Ea 2[Mr + Ma + Eq — cos 814/Eo( Ea + 2Ma)) E.,,( Ea, cos 6,) = (4.1) where My, M., and M;, are the rest masses (in energy units) of the 9H, a, and ‘Li final state. The “Li nucleus may be left either in its ground or first excited state (£, = 477.61 keV [Ajz88]). The masses and Q-value, Q = My + M, — Mz, are taken 58 from a recent evaluation [Wap93]. The ground-state Q-value used (2467.9 + 0.7 keV) is significant because it is used in the extraction of the effective reaction energy from the observed yo energy. The error in Q leads to a +0.7-keV systematic uncertainty in the center-of-mass energy. This Q-value and its uncertainty are derived directly from the “Li mass excess (A = 14906.9 + 0.7 keV), since the uncertainties in the °H and *He mass excesses are negligible in comparison. The differential cross section in the center-of-mass system is given by do. t Ae. 0; ‘ where og; is the total cross section for producing y;. The normalized angular distribu- tion, W?,, , is expanded in Legendre polynomials Weim. = 1+ ¥° a,P;(c0s6..m.)- (4.3) f=1 The decay of the first excited state is isotropic and uncorrelated with the feeding y-ray; hence the aj should be identically zero. The lab and center-of-mass angles are related by cos 6; — £ 6. mn. = >) ’ co8te- 1 — B cos 6, (44) and dQ, (1—fcos6,)?’ (4.5) where the relativistic parameter ( is given by E.( Ea “f 2M.) B= v. . (4.6) Mr + Ma. + Ee The differential cross section in the lab system is do; Oo; ; Fae = Gwin), (4.7) 59 where the normalized angular distribution in the lab system is calculated from Equa- tion 4.3 using Equations 4.4 and 4.5: Wi = 1+ 2GP,(cos6z,) + S> ai{ Pe(cos Or) + alte + 1)(£+ 2)Pr41(cos 67) =] —£(l — 1) P-1(cos 6:,)] \, (4.8) to first order in G. Note that @ < 0.022 in this experiment, so this is an excellent approximation. The lifetime of the first excited state of “Li (7 = 105+3 fs [Ajz88]) is such that the ion velocity is significantly reduced (on average) before decaying. We use G.g = 0.758 for this transition, where 0.75 is the Doppler-shift attenuation factor found [Pau66] for this state with similar-velocity “Li ions stopping in Ni. The center-of-mass kinetic energy E is related to E, via _ 2(My + M,)E + E? 7 2Mr ; E, (4.9) For use in subsequent error analysis, we note the following two approximate relations: dE, —) .10 pat AE +Q) (4.10) and dE, B cos 6, Q ae ™ 1 5 (3 + E (4.11) 4.2 y-Ray Detection 4.2.1 Analysis of Spectra Examples of the y-ray spectra obtained are shown in Figures 4.1 and 4.2 (at the lowest energy measured); Figures 4.3 and 4.4 (at an intermediate energy); and Figures 4.5 and 4.6 (at the highest energy). The y-ray peaks were analyzed using a technique 60 250 i T T T T T T T T T J 5 4 200 T B— 4 150+ Yo J Be bE al S| F 4 3 L 4 9 L 4 100+ | _ 50L _ 1 1 1 4. i | \ 2200 2400 2600 2800 3000 3200 channel Figure 4.1: High-energy portion of run 096 spectrum obtained with target 5 and E. = 140 keV, showing the 7 and 7; transitions. The peak labeled B is a background line at 2614 keV from 7TI decay (7°?Th decay product). Peak C is a background line at 2204 keV from ?'*Bi decay (??°Ra decay product). Peak D is the first-escape peak associated with peak B. 1000 T T T T T T T T T T T T T T 800+ 4 L 511 keV 4 Y2 “~~ ann. rad. 600 counts 400 200 i 1 1 i i 1 1 1 1 I 1 1 1 i 200 550 600 650 channel Figure 4.2: Low-energy portion of run 096 spectrum, showing the +2 transition and the 511-keV positron annihilation peak. 61 similar to that described in Appendix B of J. L. Osborne’s thesis [Osb83]. The background under each peak was estimated by making a linear fit to user-defined background regions on both sides of the peak. The least-squares fitting was performed iteratively. The error AN in the number of counts in a channel N was AN = 1//N for the first pass, and for subsequent passes the errors were generated from the fit, AN = 1/ J Nyie- This was done to avoid improper weighting when there are only zero or a few counts in each channel. Then the number of counts in the peak and the peak centroid (with statistical errors) were extracted using a user-defined peak region. The centroid was converted to a mean y-ray energy using the calibration described in Subsection 3.3.4. In almost all cases, the 7H(a,) peaks were free from background lines, and the background was only a slowly varying continuum, which was fit as described above. Ina — cases, the 7, peak included a contribution from a room-background line. These cases were easily handled by subtracting from the peak area the contribution measured during beam-off background measurements. In the angular distribution measurements with FE, > 1500 keV and detector angles —45°, —90°, +90°, or +135°, the capture y-ray peaks were Doppler-broadened such that the first-escape peak from ‘Yo interfered with the 7, peak (see, for example, Figure 4.5). These cases were treated by extracting the combined number of counts in the Yo first-escape and 7 peaks, and then subtracting the 7 first-escape contribution calculated from the number of counts in the yo peak and the ratio of first-escape to photopeak. This ratio was determined for geometry C using the *°Co source and the 0° 3H(a,7y) data (where the Doppler- broadening was much smaller). The only beam-induced background lines identified in the spectra were traced to the }°B(a,py) and Ti(a,a’) reactions, both of which were observed for E, > 1500 keV. There was no case in which these lines interfered with *H(a,7) peaks. 62 140;- 4 120; Yo 4 100}- = 71 80 | counts T lL 60}- 7 rn L 1 3400 3600 _ 1 4 | | 3000 3200 channel I 2600 2800 Figure 4.3: High-energy portion of run 095 spectrum obtained with target 5 and E. = 1000 keV, showing the yo and 7 transitions. The peak labeled A is the yo first-escape peak; peak B is a background line at 2614 keV from ?°°T] decay (7°?Th decay product). The energy calibration is the same as for run 096 (Figures 4.1 and 4.2). 400 T T T T T T T T T T T T —T 350 V2 511 keV ann. rad. 300 250 200 counts 150 100 a per tirrri tori tinier tirir tir si teri ta 4 | 1 B00 550 600 650 channel Figure 4.4: Low-energy portion of run 095 spectrum, showing the 2 transition and the 511-keV positron annihilation peak. 63 60} 7 counts 20 7 OK ! Nie | 1 1 j 1 1 1 | 1 | 2700 3200 3300 3400 | , Nl 2800 2900 3000 3100 channel Figure 4.5: High-energy portion of run 137 spectrum obtained with target 4 and E. = 2790 keV, with the detector in the —90° position. The yo transition is cleanly resolved, but the 7; transition and the first-escape peak from the 7 transition overlap (peak A). The energy calibration is different from runs 095 and 096 (Figures 4.14.4). 1400 = 011 keV 4 1200- Ye ann. rad. sa 1000 800 counts 600 400 200 | nl | n | 1 | 440 460 480 500 channel ! | 400 420 Figure 4.6: Low-energy portion of run 137 spectrum, showing the 72 transition and the 511-keV positron annihilation peak. 64 4.2.2 Monte Carlo Simulations Monte Carlo simulations of the y-ray detector response were used to supplement the information found using radioactive sources. In particular, one can determine the distribution of y-ray emission angles for detected 7 rays using the simulation. This information is important for making finite detector-size corrections to the angular distributions, and also for extracting the effective reaction energy from the observed energy of the yo transition. This information is not readily attainable experimentally from radioactive sources. The simulations were performed using the Electron Gamma Shower (EGS4) code developed at Stanford [Nel85]. The details of the target chamber, detector, and Pb shielding (as shown for ex- ample in Figure 3.1) were included in the simulation. All simulations were performed using an isotropic, mono-energetic source, uniformly distributed over the experimen- tal beamspot (Section 3.2). The simulations reproduced the measured photopeak efficiency € and photofraction ¢@ within 15% over the range 120 < E, < 3500 keV for each geometry. Over the range 2000 < E, < 3500 keV, the agreement was within about 5%. For each y ray interacting with the detector, the energy deposited in the detector and the y-ray emission angle 6, were saved. The 0° direction is defined by a line from the center of the target, through the center of the detector. The results for 2750-keV 7 rays and geometry A is shown in Figure 4.7. Note that a photopeak event’s first and subsequent interactions must all take place within the active Ge, and hence are confined to cos 6, > 0.5 by the beamspot — detector geometry. From the distribution of all detected events it is clear that scattering “in” to the detector is significant. Using the simulated distributions of y-ray emission angles, the angular- distribution attenuation factors [Ros53] are easily calculated: 1 Gh = WV > Pe(cos 6i), (4.12) 1=1 where WN is the number of detected events, cos 6,; is cos 6. of the i** detected event, 65 500 T T T T T | : T — T ] T T T T T T T Tv T T photopeak events 400} 4 300 }- 4 200+ =I 100+ aay L all events J number events 1500; 4 1000}- = 500 }- = I 1 4 4 4 | 1 4 it n l 4 _t 4 n | 4 1 4 n —1.0 —0.5 0 0.5 1.0 cos 0, Figure 4.7: The distribution of y-ray emission angles for detected y-rays, calculated with EGS4 for a 2750-keV source and geometry A. The upper panel shows the re- sults for full-energy (i.e., photopeak) events, and the lower the results for all events depositing energy in the detector. 66 and P; is the Legendre polynomial of order £. We will distinguish between photopeak attenuation factors, Q§, which are calculated from events which deposit the full y- ray energy, and total attenuation factors, Q?, which are calculated from all detected events. The factors Q§ and Q? for 1 < £< 4, 100 < E, < 6000 keV, and geometries A, B, and C are plotted in Appendix D. The calculations for geometry C were performed using an idealized geometry where the target was normal to the beampipe, as the axially-symmetric geometry was much easier to input to EGS4. The factors calculated for the 0° detector position were assumed to hold for the other positions of geometry C as well. 4.2.3 Detector Efficiency 24Nia Source The 74Na source emits two 7 rays (1368 and 2754 keV) per decay in one-to-one coincidence. The cascade consists of a 4+ —+ 2+ — 0+ sequence, where both decays are £2 transitions. The y-ray angular correlation for this sequence is readily found [Fer65] to be 5 W(@)=1+ ag F2{cos 6) + FazPaleos 6). (4.13) The photopeak efficiency at the two y-ray energies can be found using the number of counts in each photopeak and in the coincident-sum peak. A constraint on the total efficiency can also be found using the total counting rate in the spectrum. The results were analyzed using the relations Ny = Se{1—mll + ZORIQH2) + ZOO} (4.14) N, = Sani GUO) + ere (4.15) sai HEP) (4.16) Nit = S{m +72 —- m [1 + (1) 312) + AVI), (4.17) = No = Sererll + 2O5(1)Q5(2) + 67 where N, and Np» are the number of counts in 1368- and 2754-keV photopeaks, Nj. is the number counts in the coincident-sum peak, and N;,,; is the total number of counts in the spectrum (corrected for the experimental effects of dead time, pileup, background, and extrapolation to zero pulse height). The quantity S is the number of source decays in the counting interval, c; and 7; are the photopeak and total efficiencies for energy i, and Q§(z) and Q(z) are the attenuation coefficients for energy 7, from Subsection 4.2.2. These equations include all coincident summing effects. Equations 4.14-4.17 were iteratively solved for €), €2, S, and 72, using 7; = 0 for the first pass. Then 7 was calculated from ¢, and ¢; (¢; is assumed known from other sources); 72 was found using Equation 4.17. This process was continued until it converged to the solution; the convergence is quite rapid, because 7; is small com- pared to unity: 7; ~ 0.15 for geometry A, and is an order of magnitude smaller for geometries B and C. For geometry C, the photofraction was assumed to be known at both energies, so €,, €2 and S were calculated without using Equation 4.17. The values of S obtained from different measurements of the same source were checked for consistency with the known [End90] 14.958 + 0.004-h half-life. All of the comparisons were consistent within 1%. The measurements included comparisons of geometries A and B or different angles of geometry C. The before-and-after measurements in ge- ometry C agreed within ~ 1%. The combined error due to statistics and inaccuracies in the photofractions and attenuation coefficients is estimated to be 2%. The 7 rays from the ©Co source used for the photofraction measurements have the identical spin sequence and multipolarities as 74Na. The ®°Co data (taken in geometry A only) can thus be analyzed using Equations 4.14-4.17. Using other sources, it was estimated that $(1173 keV) / $(1332 keV) = 1.09. The photofractions for °°Co were extracted using this constraint. *6Co Source The absolute strength of the *°Co source was determined by comparison with the 182Fiu source described below, in geometries A and B. The relative y-ray intensities 68 107! T T T T T T T T T T T T T T T T T L J L 4 — 152Ru(EC,B*) 4 | a A — 182pu(g7) ] | q — Migs | v _ -4Na L 4 oe raw data o E io : aT : ‘3 107! SEE CLE + pot v L 4 a, : J ro ~ L 4 ° G A, L 4 corrected data 107? “1 1 ss 1 1 1 1 4 it 1 Ei 1 1 i i L 1 a 10° 104 E, (keV) Figure 4.8: The photopeak efficiency for geometry A. The upper panel shows the results without correction for coincident summing, while the lower shows the results including summing corrections. The solid curve on both panels is an empirical fit to the corrected data with 300 < E, < 4000 keV, used in subsequent analysis. 69 were taken from Yoshizawa et al. [Yos80], for FE, < 2750 keV. The relative intensities for higher energies were taken from Appendix 4 of Table of Isotopes {Led78]. The relative intensities of the stronger lines are known to better than 1%. The remaining details of the decay scheme (§-decay feeding factors, internal-conversion coefficients) needed for making coincidence-summing corrections were taken from Nuclear Data Sheets [Jun87]. 152Au Source The absolute strength of the "Eu source was previously determined to 1.5% by comparison with NBS source #4275. The relative y-ray intensities (known to better than 1% for stronger lines) were taken from Trzaska [Trz90] where available; the remaining intensities were taken from Iwata et al. [Iwa84]. This source decays by (8*, EC) to ?Sm and also @~ to ’Gd. The §-decay feeding factors and internal- conversion coefficients were taken from Nuclear Data Sheets [Pek89]. °6Co and °?Eu Coincident-Summing Corrections The results from these sources were corrected using the technique of T. M. Semkow et al. [Sem90]. Using decay-scheme information, the full coincidence correction from all possible y-ray combinations is calculated. The effects of “summing out” (loss of photopeak counts due to coincidently detected radiation) and “summing in” (increase in photopeak counts of a crossover 7 ray due to coincident photopeak detection of the cascade 7 rays) are included. The effects of annihilation radiation from coincident *-particles are also included. The angular correlation between ¥ rays is neglected in this calculation. This assumption is justified on the grounds that in a close geometry, when the coincidence correction is large, the angular correlation is much reduced due to the attenuation coefficients. In a distant geometry, the coincidence correction is small, so an error in the correction due to angular correlation is less significant. The effects of coincident X-rays (from internal conversion or electron capture) and §-particles are not included, and are estimated to be negligible. 70 The decay-scheme input and the calculations are done in matrix form. The /- decay feeding factor, y-ray branching factors, and internal-conversion coefficients were supplied for each level. When necessary, the factors were adjusted slightly to repro- duce the aforementioned relative y-ray intensities (which are more accurately known). The Eu (EC, G+) branch was analyzed separately from the 8~ branch, since y rays from one branch are never in true coincidence with y-rays from the other. The ex- perimental input to the calculation consisted of the number of source decays in the counting interval, the number of counts in a selected set of photopeaks, the ¢(E,) function, and an estimate for e(£,). The photopeaks selected were those with suffi- cient statistics and also free from interfering lines. For example, the 1360.2-keV line from *°Co was not used because of interference from the 846.8 + 511-keV coincident- sum peak. The estimate of e(£,) was used in calculating the effects of lines which were not supplied; the estimate was iteratively improved as data from more sources became available. The calculation of the photopeak efficiency for each member of the set is then iterated self-consistently, using ¢(E.,) to estimate 7 at each step. The results for geometry A are show in Figure 4.8. The uncorrected efficiency is simply the number of counts in the photopeak, divided by the number of 7 rays of that energy emitted. The uncorrected efficiencies show considerable scatter, and are usually lower than the true efficiency. The solid curve is an empirical fit to the corrected data that is used in subsequent data analysis. The corrections for geometries B and C were much smaller (+ 3%), due to greater source distances. The corrected efficiencies for each geometry are presented in Appendix E. The solid curves are empirical fits for 300 < E, < 4000 keV. The detection efficiency is assigned a 3% systematic error. 4.3 Target Deterioration The °H areal density was observed to deteriorate under a bombardment (see Table 3.1 for before-and-after results). Corrections were made assuming that the 7H areal 71 density after cumulative charge Q (measured in pC) is (nt)rF,(Q), where (nt)+r is the initial areal density and F,(Q)=1- [ ae (4.18) 0 [E(Q')|*™ (nt)r- The quantity F, is a constant, E. is the incident a energy in keV, and (nt)z; is the initial Ti areal density in 10'’/cm?. The sputtering rate is expected [Win76] to be proportional to the nuclear stopping power. The nuclear stopping power for a’s in Ti [Zie77] is approximately « E>°’4 (over our range of energy), so this dependence is included in Equation 4.18. The constant F, is 4.0 x 10~° for targets 1, 3, and 5 (perpendicular target chamber) and 1.8 x 10~° for target 4 (“45-degree” chamber). These values were found to reproduce the deteriorations observed after the (a,7) measurements. The error in the correction was assumed to be 40% of the correction, ie., AFr(Q) = 0.4[1 — F,(Q)]. From the positions of the centroids and the widths of the yo peaks obtained from target 4, it was determined that the Ti was lost from the target in the same proportion as the 7H. The 3H areal density was also corrected for radioactive decay using D log 2 1/2 Fa(D) = exp(— ); (4.19) where D is the number of days since the (d, a) areal-density determination, and t/2 is the half-life in days. This correction was never more than 1%. 4.4 Extraction of Cross Sections The effective reaction energy E is found from the observed yo centroid Eo by in- verting Equation 4.1 to find E,, and then calculating E from E, using Equation 4.9. In Equation 4.1, we use E, = Ey and cos 6; = Q§(E,0) cos 9p, where Q§ is the photopeak attenuation factor (Subsection 4.2.2) and @p is the angle of the center of the detector with respect to the incident beam. In principle, finite detector-size ef- 72 fects should be calculated by expanding Equation 4.1 in terms « P;(cos 6,,), and then replacing P;(cos9,) by Q§(E0)P:(cos 9p). However, our approximation is sufficient since the £ > 1 terms are very small. This analysis also assumes that the radiation is isotropic; the effects of including anisotropy in the E extraction were investigated, and found to be negligible for our case. At EF = 50 keV, the estimated 2% uncertainty in Q§ contributes -+0.2-keV uncertainty to E. The effect of finite detector size on the observed yo energy was tested using the E,, = 1600 keV runs on target 5 (numbers 098, 099, and 100), where the detector was in geometry A, geometry B, and geometry A, respectively. The runs were performed in succession, using the the same beam tune for each run. Thus the energy distribution of the 7 rays should be identical for the three runs. The centroid was observed to shift by E.,0(B) — E,0(A) = 5.3(4) keV, where the error includes statistics and an estimate of the systematic uncertainty. For E., = 3200 keV, the calculated attenuation factors are Q{(A) = 0.880 and Q§(B) = 0.987. Using E, = 1600 keV and cos 6; = Q§ in Equation 4.1, we calculate E,9(B) — Eo(A) = 5.8 keV, in reasonable agreement with the measured value. The observed angular distributions Wi(6 ), which include the effects of finite detec- tor size, are given by Equation 4.8, where P;,(cos 9) is replaced by Q3(E,;)P:(cos 9p), and 6 = € or 7 determines whether photopeak or total attenuation factors are to be used. We define €; to be the photopeak efficiency evaluated at the observed centroid of 7; (the energy for 7 is calculated from yo and reaction kinematics). The total efficiencies y; are calculated from ¢; and ¢(E,;). The effective efficiencies €;, which include coincident-summing corrections, are defined by - W7(e)Wz(e) W7(e) & = e,[1— mW7(n)l, (4.21) and & = €o[1—mW;(n)]. (4.22) 73 RUN £E, f (keV) 063 400 0.998 070 200 80.979 074 500 = =0.998 075 300 §=—.0.992 O77 150 380.944 079 250 0.984 089 500 =0.998 092 280 0.991 096 140 380.992 145 290 0.993 146 290 0.994 147 290 0.995 148 290 0.996 149 290 §=0.996 Table 4.1: The correction factors f. All of the runs with |1 — f| > 0.001 are shown. The run numbers, incident energy, and correction factors are given. The term added to €o includes the effect of coincident photopeak detection of 7, and 2; the factor 0.44 is branching ratio, 0, /o0 (Chapter 5). The terms subtracted from €, and € correct for the lost counts due to the coincident detection of the other cascade member. The number of counts in y-ray peak 7 per incident a, Y;, is related to the experi- mental cross section at E, o;(F), by Yi = f (nt)r Fx(Q) Fa oi() &: W1(€), (4.23) where f is a deconvolution factor defined below, and Q is the cumulative charge up to the half-way point in the run (as determined by charge). If the cross section is known and the target consists of a homogeneous Ti-*H layer, 74 the expected yo yield YF can be calculated: if, AEa o0( EZ) éol Erol Ba | [the (E!, )) dE’ fee_an, [Be( BL) 4Ey, = (nt)r Fr(Q) Fa W,(€) , (4.24) where a is the effective stopping power for a’s in Ti-*H and AE, is the energy lost in the target. The quantity Eo(£/,) is calculated using Equation 4.1 with cos 6, replaced by Q{ cos 4p. The energy of the yo centroid can also be calculated: Sie a6_ Exo Ex) 0( Ea) €o(Eyo(Ea)) [e(Ba))* 4B ES, = ” SE2-an, %0(E,) €o[Eo(E%,)] [82(E4)]-! dEy, (4.25) The calculated y-ray centroid ES, can be converted to an effective reaction energy E° as described in the beginning of this Section. In order for the calculated yield to be consistent with Equation 4.23, the correction factor f must be given by: Sica oo( Ei) €o[Eo( Ha)] [e(Ba)l* AB o(E*) &o( E50) Sean, (Se(Es))* dE, f= (4.26) The energy dependence of op in Equations 4.25 and 4.26 is assumed to be given by Equation 1.9, with the energy dependence of S(E) from Equation 1.16. The attenuation factors used implicitly in Equations 4.25 and 4.26 are evaluated at the experimental y centroid Eo in view of the negligible energy dependence of Q§(E,). The calculation of f was further simplified by using €9 in place of €) in Equations 4.25 and 4.26. This analysis of the correction factor ignores straggling of the beam and the energy dependences of the angular distribution and branching ratio. The factor f arises because the use of FE (determined from Eo) does not com- pletely deconvolute the effects of the distribution of reaction energies. It is primarily the strong energy dependence of the cross section that makes the correction neces- sary. The magnitude of f can be estimated in a simple model: (1) ignore the energy dependences of € and ae in Equations 4.25 and 4.26, (2) expand o(E,) to first or- der, i.e., “hee) = o(F£o) + | 2, (Eo — Eo), and (3) simplify Equation 4.1 to be E,( Ea) = 1 «irs wea Ha + Q. The quantity Ep is defined to be EF, — AE, /2. The factor 75 f is then analytically calculable using Equations 4.25 and 4.26: f= ! (4.27) 7 AE2 1 de )* 1+ 12 (io zele It is clear that the factor f will be significantly different from unity when the target is sufficiently thick and/or oa is sufficiently large. For targets 3, 4, and 5, f was calculated using Equations 4.25 and 4.26, with E, calculated from the NMR-magnetometer calibration. For target 1, the calculation was modified to take into account the energy loss and energy spread due to the nonuniform carbon layer that was deposited. The carbon-layer thickness was adjusted for each run; the values used were found by interpolating between values inferred from the Eo shifts in the repeated E, = 1000-keV runs. The correction factor is only appreciable for the lowest-energy runs, as it arises primarily from the strong energy dependence of the cross section. All of the runs with |1 — f| > 0.001 are shown in Table 4.1. The uncertainty in f is taken to be 40%, i-e., Af = 0.4|1 — f|. 4.4.1 Angular Distributions The factors W;,(5) used for summing corrections in Equations 4.20, 4.21, and 4.22 were calculated by taking a} and a} from Tombrello and Parker [Tom63], setting a? = 0.0, a = 0.05, a} = 0.08, a} = —0.05, and setting a},, to zero. These choices will be justified by the results presented in Chapter 5. For each y-ray line, o(E)W,(e) was extracted using Equation 4.23. This quantity was then converted to S;(E Wile) using Equation 1.9. For each nominal energy where angular distributions were taken, the effective reaction energies determined at each angle were then averaged together to give EF, the nominal center-of-mass energy. For each energy the quantities S;(E)W;,(e) were fit using Equation 4.8, with P,(cos 6;,) replaced by Q§P;(cos6p). The parameters S;, ai, a}, and a} were varied; aj,,, were set to zero. The fit was fully determined, since data were available at four independent angles. For each fit the needed Q¢ were calculated once at the average 76 Source of Error Error (%) 3H areal density 3H(d, a) cross section 3 detector solid angle 1.5 current integration 2 statistics 0.8 uniformity 1 “41 41 y-ray detection efficiency 3 current integration 1 angular-distribution correction 3 total 6 Table 4.2: Systematic errors in the absolute cross section. y-ray energy. The neglect of £ > 3 terms in the angular distribution is justified on the grounds that the terms are very small due higher multipolarity and/or initial angular momentum required. The yo and 7; angular distributions are plotted in Appendix F (Figures F.1-F.9). The errors in the a} are dominated by statistics; the systematic error due to uncertainties in the detection efficiencies, detector angles, and attenuation coefficients is estimated to be 0.03. The repeated E, = 1000-keV measurements (runs 139 and 150) were used for monitoring target deterioration, but were not analyzed further. The majority of the deterioration of target 4 took place during the E, = 290-keV measurements (runs 145- 149). Due to the large uncertainty from target deterioration, these runs were analyzed by normalizing the yo and 7; yields to the y2 yield. The 2 angular distribution was assumed to be isotropic in center of mass. Absolute cross sections were not extracted for these runs. 4.4.2 0° Data As above, the factors W;(5) used in Equations 4.20-4.23 were calculated by taking a} and a} from Tombrello and Parker [Tom63], setting a? = 0.0, af = 0.05, at = 0.08, 17 ai = —0.05, and setting aj,, to zero. These choices will be justified by the results presented in Chapter 5. The cross sections o;(E) were then extracted using Equation 4.24, and converted to 5;(E£) using Equation 1.9. 4.4.3 All Data Since S; must equal S2, the weighted average of S; and S> (using statistical errors) was taken to form Sj. Then Sp and Sj. were added together to give S, the total S- factor. The error explicitly included with S results from errors due to statistics, AF, Af, and uncertainty in the center-of-mass energy (from the Q-value, y-ray energy calibration, statistical error in centroid, and attenuation coefficients), combined in quadrature. Additional systematic uncertainties are listed in Table 4.2. The branching ratio R is calculated by dividing the weighted average of 7, and 72 by yo: R = Si2/5So. The error explicitly included with R is statistical; the systematic error in R is estimated to be 4%. 78 Chapter 5 Results 5.1 Total S-factor The total S-factors obtained with the different targets and in different geometries are shown in Figure 5.1 and tabulated in Tables F.1 and F.2 of Appendix F. The data taken under different experimental conditions are seen to be in excellent agreement. It is also seen that the energy dependence of the data is well described by Equation 1.16. We adopt this energy dependence as a means of comparing and combining the different data sets. The results for fits to Equation 1.16 are shown in Table 5.1. The data sets were combined by first slightly renormalizing each set so that the fitted S(0) = 0.1067 keV-b, and then taking the weighted average of S and E for data points whose energies were within < 10 keV. The weighted averages for o,/o2 and R (presented below) were also formed using statistical errors and the same energy binning used for S. The systematic uncertainty in S due to the Q- value was not included in making the weighted average, but was included after doing so. The final results for o and S are shown in Figures 5.2 and 5.3; the numerical values for S are listed in Table F.4 of Appendix F. The error bars do not include an additional systematic error of 6% (Table 4.2); of the total systematic error, the energy- dependent contributions due to y-ray efficiency and angular-distribution corrections are estimated to be < 4%. 79 Data Set N 5(0) x? (keV-b) target 1, geo A 19 0.1060(6) 23.0 target 1, geo B 2 0.1067(18) 0.008 target 3,geo A 8 0.1074(9) 7.96 target 3, geoB 2 0.1049(21) 2.44 target 4, geo C 9 0.1053(8) 5.23 target 5,geoA 8 0.1109(12) 4.17 target 5, geoB 1 0.1116(33) 0.0 all data 49 0.1067(4) 64.3 Table 5.1: The resulting 5(0) values found from fits of Equation 1.16 to data taken under different experimental conditions. Also given are the number of data points N and the x” for each fit. The numbers in parentheses are the statistical errors in the least significant digits. 5.2 Branching Ratio As a test of internal consistency, the ratio Z = 01/02 is shown in Figure 5.4; Z = 1 is required since the cascade through the first excited state is one-to-one. The results are seen to be consistent with Z = 1, although the dispersion is greater than the statistical errors: x” = )_[(Z; — 1)/AZ,]? = 31 for 17 data points. An additional error of 4% (in quadrature) $e required to make x? = 17; this error is within the estimated systematic error in detection efficiency and angular-distribution corrections. It should be noted that the statistical error in 7 is about twice that of y2, so the weighted average used in calculating R strongly favors y2 compared to 7. The results for the branching ratio R are shown in Figure 5.5, and are consistent with a constant; a fit of the 17 data points to a constant using statistical errors yields R = 0.453(2), with x? = 40.1. A fit to a linear dependence yields an improved fit: R(0) = 0.437(5), dR/dE = 3.4(9) x 10-° keV~', with x? = 27.1, but this energy dependence is within the estimated 4% systematic error. The results for R are tabulated in Table F.4 of Appendix F. 80 Coefficient Function Result x? Adopted a? constant 0.007(10) 8.9 0.0 ay [(Tom63] ~ 9.3 [Tom63] ay constant 0.05(2) 8.2 0.05 ay constant 0.08(2) 13.2 0.08 as [Tom63] - 13.6 [Tom63] a} constant -0.05(5) 12.9 -0.05 Table 5.2: Fits of the yo and 7, Legendre coefficients (9 data points for each coefficient) to either a constant or the Tombrello-Parker [Tom63] calculation. In the case of the Tombrello-Parker calculation, there are no variable parameters. The parameterization used in the analysis of the zero-degree data is listed in the column under Adopted. The numbers in parentheses are the statistical errors in the least significant digits. 5.3 Angular Distributions The measured Legendre coefficients for the yo and 7; transitions are shown in Figures 5.6 and 5.7, and are tabulated in Table F.3 of Appendix F. The results for +, are similar to yo, except that the errors are significantly larger due to poorer statistics. The results for a, are seen to be in reasonable agreement with the calculation of Tombrello and Parker [Tom63]. Comparisons of the a}, coefficients to the Tombrello- Parker calculation as well as fits to a constant for a} and a}, are described in Table 5.2. The Tombrello-Parker calculation and the values obtained from fits to a constant for the odd terms have been adopted for the analysis of the zero-degree data. The constant fits were chosen for simplicity; the Legendre coefficients are expected to be slowly varying functions of energy, but the limited statistics mask any energy dependence. The values adopted have little impact on the final results: setting ai = a’, = 0 changes the extracted values of S and R by ~ 1 — 2%. The measured coefficients for 2 are shown in Figure 5.8, and are consistent with the required isotropic distribution. The systematic error in the a} coefficients is estimated to be +0.03. The fits to the y and 7 angular distributions and the Legendre coefficients extracted are given in Appendix F. 81 5.4 Comparison with Previous Experiments 5.4.1 Total S-factor The present results for S are compared to the previous direct measurements in Figure 5.9. Our results are seen to be consistent with all previous data for E 2 200 keV, if systematic errors are taken into consideration. However, for E < 150 keV, the present data are ~ 40% lower than the data of Schroder et al. [Sch87] (the only previous experiment in this range). For E 2 250 keV, our results for S are consistent with the Coulomb-breakup results of Utsunomiya et al. [Uts90] (Figure 1.6), but are considerably lower at lower energies. The agreement with the Coulomb-breakup data at higher energies is not surprising since their results are normalized to $(500 keV) = 0.060 keV-b. The 6% systematic error in the present experiment is considerably smaller than the 14 — 25% systematic error in the previous experiments. 5.4.2 Branching Ratio The present results for R are compared to previous data in Figure 5.9. The results are consistent with the data of Griffiths et al. [Gri61] and Burzynski et al. [Bur87], but are about 40% higher than those of Schroder et al. The statistical and systematic errors are considerably reduced compared to the previous experiments. 5.4.3. Angular Distributions The only previous data on the angular distributions come from the limited measure- ments of Griffiths et al. They found that the angular distributions of y) and 7 were identical within 5%; the present data (Figures 5.6 and 5.7) and the calculation of a’, by Tombrello and Parker are consistent with this result. The present data are in agreement with the measured ‘yo angular distributions of Griffiths et al., as shown in Table 5.3. 82 Intensity Ratio Griffiths et al. [Gri61] Present Experiment Calculation E (keV) Result E (keV) Result 0°/90° 241 1.05(6) 245 1.17(5) 1.18 0°/90° 567 1.28(4) 560 1.33(5) 1.36 45° /90° 567 1.06(6) 560 1.10(4) 1.14 135° /90° 567 1.02(6) 560 1.12(4) 1.16 Table 5.3: The previous yo angular-distribution measurements of Griffiths et al. are compared to present measurements taken at nearly identical energies. The ratios calculated with the angular distributions used for the analysis of the 0° data are also given (a; = 0, except a} is taken from the Tombrello-Parker calculation [Tom63], and a3 = 0.05). All of the results are in the center-of-mass system, except for the 241-keV result of Griffiths et al., where it is not clear if corrections for finite detector size and center-of-mass motion have been made. The numbers in parentheses for both experiments are the statistical errors in the least significant digit. 5.5 Comparison with Theoretical Calculations 5.5.1 Total S-factor The present data are compared to several theoretical energy dependences in Figure 5.10 and Table 5.4. The S-factor rises at low energies, consistent with the theoretical calculations. The data are consistent with all of the energy dependences, except perhaps that of Mertelmeier and Hofmann [Mer86]. The absolute magnitude of the S-factor is consistent with the rather large range allowed by theoretical calculations described in Subsection 1.2.2. 5.5.2. Branching Ratio The branching ratios found in the present experiment are consistent with all calcula- tions [Kaj86, Alt88, Kaj89, Moh93], which predict R in the range 0.41 — 0.47. The strong theoretical objections [Alt88, Kaj89] to the branching ratio found by Schréder et al. have proven to be well founded. 83 Data Range Reference S(0) S(500 keV) N x? (keV-b) (keV-b) all data [Kaj87)0.1067(4) 0.0604(2) 17 18.2 100 < E < 1000 keV [Lan86] “= 0.0613(2) 12 12.4 E > 100 keV [Mer86] - 0.0603(2) 14 82.9 50 < E < 600 keV [Alt88] MHN - 0.0609(3) 13 20.3 E < 1000 keV [Moh93] 0.9744(4) 0.0608(2) 15 47.4 Table 5.4: Fits of our data to various theoretical calculations of S(E). We have used the Modified Hasegawa-Nagata (MHN) interaction calculation from Altmeyer et al. [Alt88]. Some of the calculations do not cover the full range of the data, so the range of data and number of points N are given. The normalization at E = 500 keV, x’, and when possible $(0) are also given; the numbers in parentheses are the statistical uncertainties in the least significant digits. The fits are plotted in Figure 5.10. 5.5.3 Angular Distributions As discussed in Section 5.3, the measured a}, coefficients are consistent with the calculations of Tombrello and Parker. There are no theoretical calculations of the a‘ and ai, coefficients, but the results are consistent with the expectation that ai, and a’, would be less significant than a}, due to relative weakness of the interfering M1 or £2 amplitudes. In the case of the mirror *He(a,7)’Be reaction, Tombrello and Parker calculated |a‘| and |a3| to be less than 0.1. 84 T T T T ] T T T T T | alan target 1 0.050 qT T T T | qT T qT T ] T 0.100 0.050 T T qT qT | T LU T T | T “~~ T 8 oH —_++—_}_+___+—__}—_+—_+—_+—_ | — wn target 4 0.100 0.050 T T T T ] T T T UJ | q 0.100 0.050 T T qT T | qT T T T ] T | 1 ! 1000 1200 | 1 | 200 400 600 E (keV) 1 1 | 800 oO Figure 5.1: Total S-factors obtained from targets 1, 3-5. The circles, crosses and triangles correspond to data taken in geometries A, B, and C, respectively. The solid curves are Equation 1.16, normalized to S(0) = 0.1067 keV-b. 85 F 5 ° a L ° 4 | o O° ° J 0° o L 3 4 ° 10° ° 4 FE ° 1 oe 6 | & [ o ] S 4 r (o) 2 al 10 ere) q Pw 4 10! 1 | 1 | 1 { 4. ! 1 | 1 { 0 200 400 600 800 1000 1200 E (keV) Figure 5.2: The final result for o, found by combining data sets shown in Figure 5.1. 0.100};- = im | 0.080 T ~oO- | S (keV-b) fo} 8) fo) cs T | 0.040 0.020 7 4 it i 1 4 4. { ° 200 ! i | i 600 800 1000 1200 E (keV) L 400 Figure 5.3: The final result for S, found by combining data sets shown in Figure 5.1. The error bars do not include the additional systematic error of 6%. 86 1.40 = 1.30}- wal 1.20; = sia Td | 0.901 fit Z \ J 0.80} + 0.70- = 0.60- - 4 i 1 i 1 | 1 i 1 I 4 | 0.505 200 400 600 800 1000 1200 E (keV) Figure 5.4: The ratio Z = o;/0; the error bars include only statistical uncertainties. Note the suppressed zero. 0.50; 7 0.40} me 0.30; 4 0.20; 4 0.10}; 5 1 n 1 1 1 ! 1 ! 800 1000 1200 0 200 600 E (keV) ! 400 Figure 5.5: The branching ratio R. The error bars are statistical; the systematic error is estimated to be 4%. 87 0.40}- - 0.20}; i ~i.20 4 —0.40-- i oles -_ ™ 0.40;- = 0.20 =0.20 - —0.40- ae 0.40}; 4 er ! 1 —0.20- “7 —0.40- 4 1 { n 1 | n | n ! 1 | 1 | . 0) 200 400 600 800 1000 1200 1400 E (keV) Figure 5.6: yo angular-distribution coefficients. The errors are statistical; the system- atic uncertainty in a? is estimated to be +0.03. The solid curve is the calculation of Tombrello and Parker [Tom63] for a9. 88 0.40- “1 0.20} —0.20-- “4 —0.40- “4 0.40 0.20 —0.20;- = —0.40} a 0.40; 4 0.20 4 a I) Mi ca enn se ects Bf Sy Se tee nesta beeen om RIES SS etl: SE ae oe en —0.20;- 4 —0.40- | “q L | \ | \ | ! | i | 1 | 0 200 400 600 800 1000 1200 E (keV) 1 | 1400 Figure 5.7: +, angular-distribution coefficients. The errors are statistical; the system- atic uncertainty in a) is estimated to be 0.03. The solid curve is the calculation of Tombrello and Parker [Tom63] for a}. 89 T £ | | : | r T ' | ‘ | 0.40 4 0.20 - ; 9 4 & | Ns nit TL LLG LLL LLL LLL a a aaa —0.20- =f —0.40- 4 1 | n | \ | | | n | 1 | l | ' ] ‘ ] ‘ | ‘ | ‘ ] : | : i 0.40- 4 0.20 4 fav) ee en ee —0.20- 4 —0.40+ i 1 | 1 | 1 i 1 | 1 | 1 | 1 | T | T ] qT | 1 ] T ] qT ] T i] 0.40 > 0.20 _ wo (of eepep ptetenpertetne —0.20F > —0.40F 4 n | t | t l t | Ll | n | 1 I @) 200 400 600 800 1000 1200 1400 E (keV) Figure 5.8: 2 angular-distribution coefficients. The errors are statistical; the system- atic uncertainty in a? is estimated to be +0.03. 90 © — Present Experiment 0.140, L i — Schréder et al. [Sch87] 0.120- } A — Burzynski e¢ al. [Bur87] | v-— T -—oO— =e a Bos > 0.100 Y — Griffiths et al [Gri61] ~ 2 0.080 ; 0.060 4 S (k a —<—o £ 0.040 };- 0.020; - 0.70b “4 0.60 Z Vv 0.50 4 % 0.40- o.30L. 7 b 0.20 J 0.10- = | 1 1 ! 600 800 1000 E (keV) | | 400 1200 Figure 5.9: The present results for S and R, compared to previous direct measure- ments. 91 0.100 0.080 0.060 0.040} 4 0.020}- 7 0.120 +——__+—_|—_+—_+_+-—} S (keV-b) ——— [altss] MHN 0.100 FJ ins [Moh93] 0.080 — 0.060 ! | pP l 0.040 = I 0.020 T | , ] 600 800 1000 E (keV) \ ! ° 200 ! | 400 1200 Figure 5.10: Fits to theoretical calculations of S(E). The details of the fits are shown in Table 5.4. 92 Chapter 6 Conclusions 6.1 Nuclear Physics We have measured the *H(a,7)’Li cross section for 50 < E < 1200 keV, with a sys- tematic uncertainty estimated to be 6%. Nine angular distributions were measured between 115 and 1200 keV. The present results disagree at low energies (E < 200 keV) with two previous experiments [Sch87, Uts90], which show a larger increase in the low-energy S-factor than predicted by theoretical calculations of the energy depen- dence. Our measurements of the branching ratio (we find R ~ 0.45) are in reasonable agreement with Griffiths et al. [Gri61] and Burzynski et al. [Bur87], but not with Schroder et al. [Sch87] (who found R ~ 0.32). The systematic error in the present experiment is reduced by a factor of 2-4 compared to the previous measurements, leading to a much better determination of the absolute normalization of the cross section. As discussed in Section 5.5, the energy dependence of the cross section, the magnitude of the branching ratio, and the angular distributions are in reasonable agreement with the available calculations. The serious disagreement between our results and those of Schroder et al. for the low-energy behavior of the S-factor and also the absolute magnitude of the branching ratio prompted an investigation into possible explanations for the discrepancies. The discrepancy in energy dependence at low energies is probably due in part to 93 the neglect of the effects of finite detector size on the calculation of the effective re- action energy from the observed mean ‘yo energy. Schroder et al. apparently assumed cos 6; = 1 in their analysis of their 0° data (see Section 4.4 for the analysis used in the present experiment). For their geometry (104-cm* detector, front face ~ 1.0 cm from target), we estimate Q{ = 0.83. Using this estimate for cos #; in Equation 4.1, the extracted value of E is increased by 2.2 keV, for E = 79 keV (the lowest energy measured by Schroder et al.). This change in energy will reduce the extracted value of S by about 10%. For E = 200 keV, the extracted values of E would be increased by 3.9 keV, but this reduces the value of S by only about 4%, due to the decreased slope of the cross section (see Equation 1.13). The inclusion of this effect would clearly improve the agreement of the Schroder et al. data with the present experiment. This example shows the importance of understanding the energy scale at low energies, where the cross section is highly energy-dependent. The discrepancy in the branching ratio probably results in part from the neglect of coincident-summing corrections by Schréder et al. For their geometry, these effects would decrease the apparent 7; and 7 fluxes by ~ 15%, and increase the yo flux by =~ 3% (see Section 4.4 for the analysis used in the present experiment). These corrections would increase branching ratio by ~ 18%, to R = 0.38, still lower than our result, but in the right direction. The neglect of coincident-summing effects will also affect the determination of the detection efficiency using calibration sources which emit 7 rays in coincidence. There exists some indication [Rol91] that there is an error in the relative y-ray efficiency used by Schroder et al. Major improvements on our experiment seem unlikely, at least within the confines our experimental approach and available detector technology. Continued increases in the efficiency of Ge y-ray detectors is to be expected. Improvements in the low-energy data using the present techniques would be especially difficult, as the uncertainties from target deterioration and energy-scale determination are at least as important as statistical considerations. A windowless *H-gas target could be helpful here, but radiation safety would be a serious challenge. A better determination of the “Li 94 mass excess would eliminate part of the systematic error in determining the effective reaction energy, and would somewhat reduce the errors in our low-energy 5(£) data. The angular distributions could be measured more accurately using a fixed array of Ge detectors, thereby greatly increasing the rate at which data is obtainable, and eliminating systematic errors in the normalization of the *H areal density and number of incident particles for each angle. While the data are in reasonable agreement with the theoretical calculations of the energy dependence [Kaj86, Lan86, Mer86, Alt88, Moh93], it would be useful if the existing ~ 10% discrepancies (see Figure 5.10) between the calculations could be resolved. However, it may be that this uncertainty represents the best that can be achieved with current theoretical techniques. An improved understanding of the energy dependence would reduce the uncertainty in the cross section at low energies, where the data have larger uncertainties, or are unavailable. However, it is important to note that the present experiment covers nearly all of the energy range needed for standard big-bang nucleosynthesis, so astrophysical conclusions do not depend very much on the extrapolation of data. It would also be interesting to perform another "Li Coulomb-breakup experiment, under improved kinematical conditions, in order to test the accuracy of radiative-capture cross sections determined in this approach. 6.2 Astrophysics For the calculation of the thermonuclear reaction rate, we assumed S(£) to be given for E < 1200 keV by Equation 1.16 normalized to our data, and for higher energies by a linear extrapolation: Equation 1.16, with S(0) = 0.1067 keV-b E < 1200 keV S(E) = ; (6.1) 0.0337 + 2.85 x 10°-°E (keV-b) E > 1200 keV where FE is in keV. Using this parameterization, the reaction rate N4(ov) was then calculated by numerically integrating Equation 1.8. The results are shown in Figure 95 104 Ef T Tt prey T T i a T T T T TTT ntl eal oe oOo Oo 0 | | ° nw a UU UL La | TUNA TUT VeTTay TU atta TUTTnY TOTTI TTT TTT ry a5 ~- (ov) (em3s~!mole~!) 31075 10-6 107” 10-8 107° tt ptt aaaee eee ee oe ee 1.40; ss 1.20, 4 0.20 (@) i i 1 1 4L i ee I 1 4 1 1 1 Lei | L i 1 1 | on oon 107 107 10° 10! Tg Figure 6.1: The thermonuclear reaction rate N,4(ov) calculated from our data is shown as a function of temperature in GK in the upper panel. The ratio of the present reaction rate to that given by Caughlan and Fowler [Cau88] is shown in the lower panel. 96 6.1. Energies between 50 and 1200 keV (i.e., the range covered by this experiment) contribute at least 50% of the integral in Equation 1.8 for 0.09 < Ty < 6. Also shown in Figure 6.1 is the ratio of the present reaction rate to that given by Caughlan and Fowler [Cau88]. It is seen that the present reaction rate is excellent agreement with the Caughlan-Fowler rate for Ty < 1, but is considerably lower for higher tempera- tures. The excellent agreement at low temperatures is due to Caughlan and Fowler’s fortuitous use of Langanke’s calculation of S(£) [Lan86], which is in excellent agree- ment with the present experiment. Our numerically integrated reaction rate is given within ~ 1% for Ty < 10 by the following expression: _ 8.080 Na{ov) = 8.79 x 10°T5 Pexp(— Tips) 9 x(1 + 0.05167,/° — 0.7117;’° + 0.3677» — 0.005817). (6.2) The S-factor and reaction rate are now determined much more accurately than previously. For E < 1200 keV, the S-factor is well described by Equation 1.16, with 5(0) = 0.1067 keV-b. A reasonable estimate for the fractional 1-o uncertainty in the S-factor is AS “7 = 0.06 + 0.06 exp(—0.0052), (6.3) where F£ is in keV, the constant term is the estimated systematic error in the present experiment, and the increased error at low energies reflects the increased uncertainty in the low-energy data and also the uncertainty in theoretical extrapolation. The resulting temperature-dependent fractional uncertainty in the reaction rate is ap- proximately given by A(ov) _ _ 2/3 (ox) = 0.06 + 0.06 exp(—1.1673/%). (6.4) This uncertainty is much smaller than previous estimates; see for example Smith, Kawano, and Malaney [Smi93], who estimated the l-o uncertainty to be 29% for Ty, — 0. The uncertainties in Equations 6.3 and 6.4 are only valid for E < 1200 keV 97 and Ty X< 6, respectively, due to the lack of knowledge of the cross section at higher energies. However, as discussed in the Introduction and Appendix A, big-bang ‘Li production is only sensitive to the reaction rate for 0.1 S Ty < 0.6. Our new reaction rate does not differ significantly from the Caughlan-Fowler compilation in the tem- perature range needed for standard big-bang nucleosynthesis, so the calculated ‘Li abundances described in the Introduction will not change significantly. The error in the *H(a,7)’Li reaction rate is now comparable to other nuclear-physics uncertain- ties, reducing a major source of uncertainty in the comparison of big-bang calculations with the observed light-element abundances. 98 Appendix A Sensitivity of Standard Big Bang Calculation to the ?H(a,7)‘Li Rate An analysis of the sensitivity of the standard big bang calculation to the 7H(a,7)"Li tate is presented in this Appendix. We are particularly interested in finding the temperature range of N4(ov) to which the final calculated “Li yield is sensitive. The existing literature is not clear on this point: Schroder et al. [Sch87] and Utsunomiya et al. [Uts90] state that an effective energy of Ey ~ 10 keV (corresponding to Ty ~ 0.009) is most important for 7H(a,)’Li in the big bang, while Walker et al. [Wal91] use Ty = 0.9 as “a temperature characteristic of nucleosynthesis” for comparison of reaction rates from different sources. It is clearly important to have some idea of what temperatures are important, in order to focus nuclear-physics research on the important energy range. A.1 Nucleosynthesis In this section we review some of the fundamentals used in nucleosynthesis calcula- tions. The rate of change of species k, due to the reaction i+ 7 — k +1, is given by Ny = NN; (OV) i;+kl; (A.1) 99 where n; is the number density of species i, the dot represents differentiation with respect to time, and (ov),;.4, is the temperature-dependent reaction rate. It is con- venient to factor out the effects of expansion from n, by defining the number fractions Y; = n,;/ng, where ng is the baryon number density; then Equation A.1 becomes Ye = Vi¥; [ig — bl], (A.2) where the rate factor [17 — kl] is defined by [17 — kl] = ng(ov)ij;u- (A.3) In general the time evolution of a particular species is governed by a sum of terms of the type A.2, including all reactions which produce or destroy the species. These equations, along with the evolution of other quantities such as temperature and ng, form a complicated network which is normally solved by numerically integrating the equations forward in time (see for example Kawano [Kaw92}). A.2 Temperature Sensitivity We investigated the temperature sensitivity of big-bang calculations to the 7H(a,7)"Li rate numerically, i.e., by systematically varying the rate used in the Kawano code. Specifically, the existing reaction rate [Cau88] was multiplied by a temperature- dependent factor 1 T<%-4 f(T, To) = L+e[}-+ $25m — (Play? Ty-4<T<T+4, (A.4) l+e T>%M+4 where the parameter Tp is the “perturbing temperature”, A = 0.17), and e = 0.1. This function is essentially 1 + «O(T — Jo), where O(z) is the step function, except that the step is smoothed out over a width A, as shown for Ty = 0.5 GK in Figure 0.20; 4 1 ! 1 i i | 1 | i i | i ! 4 I i | all. 0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Tg 0 Figure A.1: The perturbing function f (Equation A.4) as a function of temperature in GK, for To = 0.5 GK. A.1. The time step used in the nucleosynthesis calculation was adjusted so that the change in temperature for each step was small compared to A. The ‘Li fraction Y; (not including ’Be) was calculated as a function of Ty. The resulting fractional change 6(T9) was then calculated using aH) = a cr (A.5) Note that the Ty) — oo limit corresponds to making no change in the reaction rate, while Ty — 0 corresponds to multiplying (ov) by a constant 1+. The results for Mo = 1.0, 3.0, and 10.0 are shown in Figure A.2. For 1 < mo < 10, it is seen that the calculated abundance is sensitive to the position of the step for 0.1 S To S 0.6 GK. Changes in (cv) outside of this range do not change the final “Li abundance. The range of sensitive temperature is also seen to shift to lower temperatures with increasing 7. We find that the opposite change in the reaction rate, « = —0.1, has 101 | mio = 1.0 ] 0.100}- a a” i = 3.0 b | 0.080} 7710 "a : — 1 L a J o po 0-060 — ry - = PF L A q | Nig = 10.0 A 5 0.040; comm, A ° 4 A oa . aA 4 A 0.020 a - s 4 4 OF 4 | ‘ L 1, ' ! ; ! | 0 0.20 0.40 0.60 0.80 1.00 Figure A.2: The fractional change in the calculated ‘Li abundance as a function of the perturbing temperature Ty in GK (see text for details). the effect of changing the sign of 6. To summarize, we find that the calculated “Li abundance is sensitive to the 3H(a,7)’Li reaction rate for 0.1 < T, < 0.6. It is not expected that alternative functional forms for variations of the rate would lead to significantly different con- clusions. These conclusions could change if any of the assumed reaction rates are seriously in error and are only valid for the standard big bang. A.3 Freeze-Out Calculation An alternative approach to calculating big-bang abundances has been given by Es- mailzadeh, Starkman, and Dimopoulos [Esm91]. The technique is approximate, but can be done analytically, so the relevant physics is made clear. The authors have suc- ceeded in calculating the big-bang abundances within a factor of three of the exact numerical solution for a wide range of baryon densities. 102 We will outline their idea, using ‘Li as an example, assuming that ‘Li is only produced by *H(a, y)’Li, and destroyed by “Li(p, a). The evolution of the “Li fraction Y7 is then given by Y7 = YrY,[Ta > 77] — ¥,Y;|Tp > aal, (A.6) where Yr, Y,, and Y, are the 9H, “He, and 'H fractions.’ Provided that the destruction rate Y,[7p — aa] (i.e., the "Li reciprocal lifetime) is much greater than the expansion rate of the universe H(t), the “Li fraction will be given by _ YrY,[Ta — Ty] Y; = (A.7) Y,[7p > aa] ° By comparison with Equation A.6 it is clear that this solution corresponds to the case Y7 = 0, where the production and destruction nearly cancel. The regime where this solution is valid has been dubbed quasi-static equilibrium (QSE). As the universe expands and cools, the reaction rates slow down, and eventually Equation A.7 will no longer be valid. Esmailzadeh, Starkman, and Dimopoulos claim that a reasonable approximation to the final abundance can be found by evaluating Equation A.7 at the “freeze-out” time (or temperature), defined as the time when the expansion rate and the destruction rate are equal, i.e., when H(t) = Y,[7p > aa]. (A.8) Assuming three massless neutrinos, the expansion rate after et — e~ annihilation is given in terms of temperature by H(T,) = 0.002817; sec”. (A.9) ly, here is different from the Y, used for the primordial *He mass fraction in Section 1.1. 103 Using Equation 1.3, the baryon number density is np = 2.029 x 10°T3qi9 cm. (A.10) Assuming Y, = 0.75, the freeze-out temperature Tos for “Li is found by solving 0.0028173, = 0.75 x 2.029 x 10'°T3sm10(00) 7pa03 (A.11) where (ov) is in cm*sec™’. For mio = 1.0, 3.0, and 10.0, we find To to be 0.45, 0.34, and 0.27, respectively. The 7 dependence comes from the fact that reaction rates are faster (and freeze out at lower temperature) for higher density. The resulting ‘Li fraction is estimated using ~ Y7TYq (ov) Ta377 (A.12) “ah Yp (Ov) 7p—00” where Yr, Y,, Y,, and the reaction rates are evaluated at To;. For Tos = 0.34 (m0 = 3.0) and assuming Yr = 4 x 10-7 and Y, = 0.063, we find Y7 ~ 4 x 107", in good agreement with the exact value Y,**! = 4.8 x 10-'!. In order to do this calculation from scratch one would of course have to first estimate Y,, Yr, and Y, (the values we have assumed are taken from the exact calculation at Ty = 0.34). This analysis indicates that the final abundances are dependent on the reaction rates in the vicinity of the freeze-out temperature. The freeze-out temperatures found for “Li coincide very well with range of sensitive temperatures found in the previous Section (including the dependence of Ts on 7). The fact that the final abundance does not depend on the reaction rate at higher temperatures can be understood by considering the QSE solution. As long as “Li is in QSE, Y; only depends on the 3H(a,7)‘Li rate at that instant, and not early times (higher temperatures). Thus the final abundance only depends upon the *H(a,7)‘Li rate at times when the QSE condition fails. The lower limit on the sensitive temperature occurs because nuclear reactions cease completely once density and temperature are sufficiently low. 104 The exact solution of Equation A.6 for the final "Li fraction is given by [Dim88] ¥;(00) = / ” Yo(t)¥,(t)[Ta > 7y\(€) exp — / * Y,(#’)[7p > aal(t’) at'| dt. (A.13) 0 t While this equation is not practical from a computational standpoint, it does provide some insight into the dependence on the reaction rate. Since Y7 < Yr, Y,, and Y, at all times, the evolution of Y7 is decoupled from Yr, Y,, and Y, (i.e., changes in [Ta — 77] or [7p — aa] affect Y7, but have negligible impact on Yr, Y,, and Y,). Thus from Equation A.13 we would expect that multiplying the 7H(a,+)‘Li rate by a constant factor would result in the final “Li abundance being scaled by the same factor. However, in the exact calculation we do not always see a linear relationship: observe the J) — 0 limit in Figure A.2 (6 = 0.1 is expected for a linear relationship). While the dependence is linear for 719 = 1.0, a less than linear dependence is seen at higher 7. This result occurs because we have ignored the ’Be(n, p)’Li contribution in Equations A.6 and A.13. 105 Appendix B Neutron Background During the *H(a,7) measurements at low energies (EF. < 400 keV), an unanticipated neutron background was encountered (see Section 3.1). The *H(a,n) channel does not open until Ey > 11.13 MeV. Furthermore, for FE. < 400 keV, it is expected that that a-induced reactions on contaminants in the target would be negligible due to the large coulomb barrier. Initially, we suspected that the neutrons were from *H(d,n), where the incident deuterons arose from either D{ or DH} contaminants in the a beam (these molecular ions have the same magnetic rigidity as at ions). Also note that the cross section for this reaction is very large at low energies: Omar © 5 b at Ea = 110 keV, so only a = 10~’ contamination is required to explain the observed neutron yield. We attempted to minimize this problem by using high-purity He gas in the ion source, and by removing the hydrogen (natural abundance) source bottle from the ion source. In spite of these efforts, the background remained unchanged. A neutron back- ground was alias wiles] tes the 7H(a, 7) experiment of Griffiths et al. [Gri61]. The authors attributed the neutrons to the 7H(#H,2n) reaction, where the incident *H particles arose from *H(a,*H) knock-on reactions. Similar effects have been reported for the proton bombardment of heavy-ice [Jen50, $in59] and beryllium [Jen50] targets. After the (a,7) measurements using target 1, the neutron yield from the target was measured for 150 < E, < 1250 keV, using a 4m neutron detector [Wre94] and 106 the “45-degree” target holder. The purpose of the experiment was to quantify the neutron background, and also to profile the buildup of carbon on the target with the 13C(a,n) reaction. The efficiency of the detector was assumed to be the same as for a *°2Cf source. In the calculations that follow, we show that the low-energy neutron yield is explained by the two-step process suggested by Griffiths et al. First, we calculated the neutron yield for the azimuthally-symmetric case of the target perpendicular to the a beam. In this case it is a reasonable approximation to ignore energy loss, and an analytic result is obtained. We then calculate the yield for the situation when the target is at an arbitrary angle with respect to the a beam. This case is more complicated (energy losses cannot be ignored), but is the case for which we have experimental data. A somewhat similar calculation has been done by Carraro et al. [Car90] as part of an investigation of “cluster-impact fusion.” B.1 Target Normal to Beam, Ignoring Energy Loss The ingredients for the calculation are the: target-beam geometry, the a — 7H elastic scattering cross section and kinematics, and or, the *H(?H,2n) cross section. The geometry is displayed in Figure B.1. For sufficiently thin 7H layers, the neglect of energy losses of the 9H and a particles should be reasonable. The probability of neutron production, P,,, for a triton of energy Ero incident on a 7H layer of thickness £, is then P,( Ero, £) = 2nfor( Ero), (B.1) where n is the 7H number density and the factor of 2 accounts for the fact that two neutrons are emitted per reaction. Experimental data on the *H(*H, 27) cross section 107 ee Oe 8 Y Figure B.1: Definition of variables for the normally incident case. The target is viewed perpendicular to normal. are summarized by Typel et al. [Typ91]. We take the cross section to be given by o7(E) = “exp (-|£2) (B.2) where EF = 0.5Er7 is center-of-mass energy, 5 is the S-factor, and Eg is the Gamow energy (1476.6 keV for 7H + 7H). By choosing So = 165 keV-b, we are able to reproduce the experimental cross sections within 20% for Ey < 800 keV. Assuming the Rutherford formula, the differential cross section for knock-on *H scattering in the laboratory system is doko Z22Z2e'(Mr + Ma)’ dQ = 4M2.E2 cos? 6’ (B.3) where Z; and M; are respectively the charge and mass of i = T or a, e€ is the electronic charge, E, is the alpha particle energy, and 6 is the scattering angle with respect to 108 the incident beam. The initial energy of the scattered triton is given by E7o = Em cos” 6, (B.4) where 4MrM. En = sO En B.5 is the maximum energy that can be transferred to the triton. Tritons scattered at angle 9 and depth z in a *H layer of thickness t, will traverse a thickness —2 A(x, 0) = (B.6) cos 9 before leaving the *H layer. The probability of neutron production, Y,,, for an a particle of energy E, normally incident on a 7H layer of thickness t and number density n can now be evaluated: =2n [ ndz f dy Wiel Zard) p Pp[Ero( Eas ¥)s€(2; 9), (B.7) where y = cos9. Integrating over dz and changing the integration variable from y to Ey, we obtain _ 2x(nt)? Zz ze e1 M3/? i o7(Er) dEr ~ (Mr + Ma)Mi MR 1/2 ES? (B.8) With the parameterization (B.2), the integral is elementary and we find 8a (nt)?Z2.Z264M2/? So ( Ne 4! (ey (B.9) (My + M.)Mil? Bi??(2Rg)8/2 XV Em EX RIN Em) " The calculated neutron yield per pC of incident beam as a function of FE, is plotted in Figure B.2, for a target with nt = 5.52 x 10!” 9H/cm?. Also shown is a calculation including energy losses that is described below. It is interesting to note that the yield is proportional to the square of the 7H areal density, (nt)?, as opposed to the linear dependence associated with a cross section. This proportionality will of course not be valid for targets sufficiently thick that energy losses cannot be neglected. 1.20 T T T T T T T T T T T T T T T T T T T T neutrons / uC 0.40 0.20 L 1 i L i 1 i 1 | iL | J. 1000 1500 2000 Eq (keV) 4 L 1 1 | °9 500 Figure B.2: The calculated neutron yield is shown as a function of the incident Ey. The solid curve corresponds to the calculation ignoring energy loss (Equation B.9), while the squares show the effect of including energy loss (Equation B.15). B.2 Arbitrary Target Angle, Including Energy Loss The neglect of 7H energy losses is expected to have the largest effect on the neutron production in the longer trajectories. With the target normal to the incident beam, the tritons scattered ~ parallel to the target plane (which have very long trajectories) correspond to 6 = 90°, and have very low energies and hence contribute little to neutron production due to the vanishing *H(*H, 2n) cross section (making the neglect of *H-energy losses a reasonable approximation). We now consider the case when the target is tilted at an angle y with respect to the normal direction. The geometry is displayed in Figure B.3. It is now not generally true that tritons scattered parallel to the target plane are unimportant. Let 110 me—& —-" Figure B.3: Definition of variables for arbitrary angle of incidence. The target is viewed perpendicular to normal. Note that the distance @ is not necessarily in the plane of the paper. us assume that the target consists of 9H and Ti with r = n(*H)/n(Ti). A more realistic calculation of P, for a triton with incident energy E7o incident on a tritium layer of thickness £ (compare to Equation B.1) is L P,(Ero,£) = 2 | ndzor[Er(z)], (B.10) 0 where L is the lesser of £ and the triton range. The function E7(z) is found by integrating dE: 1 = meh(Er/3) + ~ey(Er/3)), (B-11) subject to E7(0) = Ero. The quantities ef, and ej}, are the electronic stopping pow- ers [Zie77| (in units of energy loss per number of target atoms per unit area) for equivalent-energy protons in hydrogen and Ti, respectively. The energy of the a 111 beam at depth z in the target is similarly calculated by integrating dE. dz = nféi(Ba) + —h(Ba)] (B.12) subject to E,(0) = Eao (the incident energy), and the quantities «fj and ef; are the electronic stopping powers [Zie77| for a’s in hydrogen and Ti, respectively. The target thickness traversed by a scattered triton now depends on the azimuthal angle ¢, as well as the scattering angle 0, and depth z of scattering. The angle of the scattered triton with respect to the target normal is given by: cos a = cos 6 cos x + sin 6 cos dsin x. (B.13) The target thickness traversed is then (t—xcosx)/cosa cosa >Q0 t(2,0,¢) = (B.14) —z cos x/ cos a cosa < 0 where cosa > 0 corresponds to the case when the triton exits the target layer into the substrate, and cosa < 0 to the case when the triton exits from the target surface. The neutron yield per incident a is now given by y, = [Onde / * ad i ty WrolBal®)4 p.(epo(Ba(2),9),A(2,>)) (B.15) The integral is evaluated using Monte Carlo techniques. The results for nt = 5.52 x 101” 3H/cm?, r = 1.89, and the target normal to the incident beam (x = 0) are shown in Figure B.2 (these 7H and Ti thickness are the values for target 1). The inclusion of energy loss is seen to decrease the calculated yield by 15 — 20%. The results for nt = 5.52 x 10'” 3H/cm?, r = 1.89, and x = 45° are shown as the solid curve in Figure B.4 (target 1 in “45-degree” target chamber). The experimental (a,n) data are shown as squares. The calculation is seen to reproduce the data very well for E, < 800 keV. The excess yield at higher energies is clearly due to 112 [ &, 1 r a J } o Tas L - a oa 9 10) & 3 C i a n 3 G 5 fe) & ® — 3 5 o f=) 0 10" F /, ~ SO 7 Ef ee 1071 1 i] i | n ! 1 I 1 I i (0) 200 400 600 800 1000 1200 E, (keV) Figure B.4: The neutron yield is shown as a function of the incident E,, with the target at 45° with respect to the incident beam. The squares are the experimental points. The solid curve is the calculation including energy loss (Equation B.15). The meaning of the dashed curve is described in the text. the 3C(a,n) reaction, confirmed by the resonant yield seen at Ey = 1053 keV. The dashed curve in Figure B.4 is the energy dependence expected if the neutron yield was due to *H(d,n) reactions from E,-independent D} or DH{ contaminants in the at beam. The calculated neutron yield, for E, = 200 keV and r = 1.89, is shown as a function of target thickness in Figure B.5. The yield becomes constant for large target thicknesses due to the stoppage of all of the particles in the Ti*H. We have succeeded in quantitatively verifying that the neutron production for low incident a energies is due to the *H(#H, 27) reaction initiated by energetic tritons created by knock-on elastic a scattering. The calculation could be easily improved by including atomic and nuclear effects in the knock-on scattering cross section, and by using a more accurate representation of the *H(*H,2n) cross section. The inclusion of multiple scattering on the particle trajectories may also be important. Experimen- 10° E T T T T T T T T T T T T T VS i fo} ga000090| L googaee | 10! E 5 - oS) 3 ; 2 L q ™ . 4 n ; ° J q 0 10° b = uw c em 5 r J o : °° 4 q : 5° J L ° 4 1071 J z = 1072 1 1 n 1 11 1 1 1 1 1 1 — a 107 10° 10! nt (10!8/cm?) Figure B.5: The neutron yield is shown as a function of the target thickness, (nt), for E, = 200 keV. The squares and circles are the results using Equation B.15 for x = 0° and 45°, respectively. The solid curve is the result using Equation B.9 (x = 0°). tally, the effect could be investigated by bombarding 7H or *H targets with other ion beams. The effect is fundamental to any experiment using *H as a target, although it can be minimized by using a thin target [due to the proportionality to (nt)?]. We note that the targets used by Griffiths et al. [Gri61] were 3 — 10 times thicker than the targets used in the present experiment, so the neutron background would have been more apparent. 114 Appendix C Summary of Experimental Parameters A chronological summary of the 7H(a,7) measurements is given in this Appendix (Ta- bles C.1, C.2, and C.3). The column labeled RUN gives the “run number” associated with each measurement. The target number and geometry (including angular posi- tion for geometry C) are given under the TAR and GEO headings, respectively. The incident a-particle energy (determined by NMR-magnetometer calibration), incident charge, and run duration are given under the F,, Q, and At headings, respectively. The measured energy of the centroid of the yo transition is given under Eo, where the number in parentheses is the statistical uncertainty in the least significant digit. The net number of counts in the yo, 71, and 72 peaks are given under No, Nj, and N2, respectively, where the statistical uncertainties are given in parentheses. 115 RUN TAR GEO E, Q At Exo 7 Yields (keV) (mC) (sec) (keV) No N, N2 060. 1 A 1000 9.75 1868 2919.9(2) 1054(33) 416(26) 1197(37) 061 1 A 800 12.51 1408 2827.2(2) 1145(35) 464(29) 1289(38) 062 1 A 600 15.48 2218 2735.4(2) 1018(33) 394(24) 1088(35) 063 1 A 400 33.20 3473 2645.1(2) 1047(33) 379(24) 1015(35) 064 1 A 1000 9.78 1623 2916.6(2) 1079(33) 441(27) 1120(36) 066. 1 A 1600 7.50 961 3189.8(2) 1155(37) 477(29) 1268(44) 067 #1 A 2000 5.15 625 3370.8(3) 873(37) 386(34) 928(59) 068 1 A 1200 7.56 1844 3007.5(2) 1000(33) 407(25) 998(35) 069 1 A 1000 9.77 1765 2916.0(2) 1085(34) 488(27) 1086(35) 070 1 A 200 292.32 20292 2554.1(2) 1019(37) 390(31) 1079(42) 071. 1 A 1000 9.76 1518 2915.6(2) 1012(32) 452(26) 1109(36) 072 1 A 900 12.12 991 2869.5(2) 1151(35) 533(28) 1260(38) 073 1 <A 700 14.12 1768 2778.6(2) 1065(33) 461(26) 1180(36) 074 1 A 500 24.85 2218 2687.3(2) 1081(35) 459(28) 1258(38) 075 1 <A 300 81.05 5672 2597.2(2) 1115(35) 443(28) 1270(40) 076. #1 +A 1000 9.75 775 2914.8(2) 1124(34) 462(27) 1147(36) 077 1 A 150 617.95 33664 2530.9(3) 529(31) 227(31) 577(41) 078 1 <A 1000 9.75 856 2913.6(2) 1018(33) 409(26) 1210(37) 079 «#61 A = 250 153.07 9540 2574.8(2) 1091(36) 511(31) 1199(42) 080 1 B 1000 159.39 10527 2916.6(1) 2553(53) 1218(44) 2742(63) 083. 1 B 1000 77.84 5969 2916.6(2) 1233(37) 665(32) 1298(44) 08 3 B 1000 257.76 11299 2923.1(2) 1009(33) 541(28) 996(40) 086 3 B 2000 162.07 8836 3378.6(2) 1035(39) 542(39) 966(92) 087 3 A 1000 41.03 2512 2919.2(2) 1105(34) 489(27) 1176(37) 088 3 A 700 56.77 2739 2782.4(2) 1049(33) 485(27) 1115(36) 089 83 A 500 88.43 5103 2691.4(2) 1018(33) 425(26) 1088(37) 0903 A 900 49.14 2411 2873.4(2) 1265(36) 549(28) 1270(38) 091 3 A 1000 41.02 2031 2918.6(2) 1076(33) 471(26) 1148(37) 092 3 A = 280 - 268.29 18739 2591.9(2) 766(33) 349(26) 884(39) 093 3 A 1000 38.83 2484 2918.3(2) 1024(33) 457(26) 1121(37) 094. 3 A 1200 32.90 2266 3010.1(2) 1033(34) 441(28) 1118(39) 09 5 A 1000 15.65 3200 2926.5(2) 1080(33) 450(26) 1096(37) 096 5 <A 140 782.08 41235 2534.4(1) 554(29) 248(28) 535(45) 097 5 A 1000 16.20 1126 2925.9(2) 1107(33) 419(25) 1046(34) 098 5 <A 1600 12.45 961 3197.4(2) 1074(33) 423(25) 1140(36) Table C.1: Summary of experimental parameters (see text for explanation). This Table is continued in Tables C.2 and C.3. 116 RUN TAR GEO E, Q At Eo 4 Gels (keV) (mC) (sec) (keV) No N, N2 099 ) B 1600 76.93 5873 3202.5(1) 1038(33) 527(25) 1073(40) 100 5 A 1600 13.10 1042 3197.2(2) 1152(35) 507(27) 1269(38) 101 5 A 2000 11.04 2190 3378.5(3) 1020(35) 407(26) 1238(44) 1022 5 A 1200 14.51 1756 3017.5(2) 1126(34) 442(25) 1172(36) 103 5 A 1000 17.64 2036 2926.4(1) 1166(34) 513(27) 1251(38) 104 4 C(O) 1000 62.85 4251 2922.2(2) 1052(34) 559(29) 1170(48) 105 4 C(—45) 1000 65.76 4367 2909.5(2) 1043(34) 531(30) 1311(47) 106 4 C(—90) 1000 77.66 5064 2882.9(3) 1040(35) 473(33) 1363(53) 107 4 C(+90) 1000 79.94 4939 2883.4(3) 1157(37) 508(61) 1678(57) 108 4 C(+135) 1000 74.98 4375 2858.1(2) 1055(34) 545(36) 1474(51) 109 4 O(+135) 2000 50.65 3458 3272.7(3) 1086(34) 506(30) 1661(58) 110 4 (0) 2000 47.38 3220 3377.2(2) 1343(39) 578(30) 1363(55) 111 4 C(—45) 2000 51.06 3347 3357.4(3) 1111(37) 577(62) 1630(54) 112 4 C(—90) 2000 58.90 3952 3314.0(3) 1028(34) 452(57) 1649(59) 113 4 C(+90) 2000 54.67 3239 3314.9(3) 1023(35) 462(58) 1980(59) 114.4 C(490) 1600 58.95 3819 3142.5(3) 1098(36) 454(30) 1781(62) 115 4 C(+4135) 1600 56.89 3773 3106.9(3) 1050(36) 473(28) 1540(57) 116 4 (0) 1600 45.20 2722 3195.5(2) 1132(35) 568(58) 1152(47) 117 4 C(—45) 1600 51.42 3675 3179.0(3) 1042(35) 537(55) 1390(51) 118 4 C(—90) 1600 63.08 4398 3142.2(3) 1052(36) 464(29) 1469(55) 119 4 G(—90) 600 138.63 7026 2711.3(2) 1141(37) 555(36) 1264(53) 120 4 C(+90) 600 120.96 5144 2712.7(2) 1081(35) 516(34) 1352(53) 121 4 C(4135) 600 124.93 6745 2694.4(2) 1042(34) 424(32) 1218(58) 122 4 (C(0) 600 131.99 6155 2740.0(2) 1367(40) 582(40) 1371(56) 123 4 C(—45) 600 117.30 4758 2731.3(2) 1069(35) 607(32) 1292(49) 124 4 C(—45) 1330 61.14 3171 3057.5(3) 1127(35) 616(52) 1468(51) 125 4 C(—90) 1330 67.07 3674 3025.0(3) 1075(34) 477(32) 1368(49) 126 4 C(+90) 1330 67.11 3313 3026.7(3) 1119(36) 533(58) 1784(54) 127 4 C(+135) 1330 62.35 4109 2996.0(2) 1085(35) 492(29) 1406(53) 128 4 (0) 1330 54.03 2667 3072.8(2) 1218(37) 591(57) 1331(46) 129. 4 (0) 2400 37.11 3345 3561.0(2) 950(40) 470(33) 1222(59) 130 4 C(—45) 2400 42.24 3123 3536.2(3) 1011(36) 519(62) 1420(61) 131 4 C(—90) 2400 55.40 3518 3485.2(4) 1054(39) 587(69) 1639(63) 132 4 C(+90) 2400 51.88 3349 3486.9(4) 1080(38) 568(71) 1886(76) 133 4 C(+135) 2400 50.26 3364 3440.2(3) 1103(37) 609(37) 1636(67) Table C.2: Continuation of Table C.1; continued in Table C.3. 117 RUN TAR GEO £E, Q At Eo 7 Yields (keV) (mC) (sec) (keV) No N, N2 134. 4 C(+4135) 2790 45.09 3854 3601.5(3) 1080(35) 426(32) 1636(69) 135 4 (0) 2790 41.39 2998 3737.9(2) 1199(39) 566(32) 1473(70) 136 4 O(—45) 2790 42.81 3175 3710.8(4) 1022(36) 486(68) 1453(66) 1387 4 O(-90) 2790 54.38 3797 3652.8(4) 1107(36) 599(73) 1800(69) 1388 4 C(+90) 2790 48.81 3509 3655.8(4) 1039(37) 528(71) 1768(94) 139 4 C(+90) 1000 80.00 3736 2884.7(3) 1075(35) 558(57) 1660(54) 140 4 C(+90) 800 86.96 4406 2799.0(3) 1063(34) 542(35) 1451(56) 141 4 C(+135) 800 98.24 3786 2776.4(2) 1109(36) 516(40) 1393(51) 142 4 C(0) 800 86.09 3038 2831.2(2) 1214(37) 635(57) 1216(49) 148 4 C(—45) 800 89.66 3228 2821.2(2) 1093(36) 570(31) 1439(49) 144 4 C(—90) 800 106.09 3768 2797.6(3) 1101(37) 570(35) 1410(49) 145 4 C(-90) 290 523.67 27348 2581.8(2) 858(40) 484(38) 999(46) 146 4 C(0) 290 514.69 22983 2601.1(2) 961(36) 490(33) 1054(46) 147 4 C(—45) 290 521.90 23692 2596.1(2) 880(41) 398(30) 1076(48) 148 4 O(+4135) 290 708.95 30758 2571.8(2) 1072(38) 492(41) 1307(55) 149 4 O©(+90) 290 639.70 27168 2584.0(2) 917(39) 467(41) 1044(45) 150 4 C(+490) 1000 134.93 6516 2886.6(3) 1177(37) 504(54) 1665(45) Table C.3: Continuation of Tables C.1 and C.2. 118 Appendix D Attenuation Factors The calculated angular-distribution attenuation factors are presented in this Ap- pendix. The photopeak attenuation factors, Q§, and total attenuation factors, Q?, are shown in Figures D.1-D.3 and D.4—D.6, respectively, for geometries A, B, and C. The data points are the results from the EGS4 simulations; the error bars reflect the estimated statistical error. The solid curves are empirical fits used in subsequent data analysis. 119 1.00 T T i| T | T | 0.80 a ; f=2 0.60 + £=3 0.40 e=4 0.20 \ | 1 | 4000 6000 8000 E, (keV) l 0 2000 Figure D.1: Photopeak attenuation factors, geometry A. 120 1.05 T T | qT ] T T 1.00+ 5 e—e—_9—_9_8-_8—8 © C #=1 a r £=2 ose eres £=3 @ 0.90 ad £=4 0.85 0.80 0.751 1 | 1 | in | l | 0 2000 4000 6000 8000 E, (keV) Figure D.2: Photopeak attenuation factors, geometry B. 121 1.05 T T T T T T T T 1.00 Fg oe 6 ee 0 2=1 F ; £=2 0.95 - £=3 @ 0.90 £=4 0.85 0.80 6) 2000 4000 6000 8000 Ey (keV) Figure D.3: Photopeak attenuation factors, geometry C. Q? 122 0.80b 0.60F £=1 0.40 £=2 0.20- “oo aa BOO 2 Q eS £ = 4 1 | | | 4000 6000 8000 E, (keV) | O 2000 Figure D.4: Total attenuation factors, geometry A. 123 1.00 T T | T | T T 0.90 0.80 kK Z=1 0.70 @ 0.60}¥ £=2 0.50 £=3 0.40 fama 0.30 0.20 4 | 1 I 1 1 | 0 2000 4000 6000 8000 E, (keV) Figure D.5: Total attenuation factors, geometry B. 124 0.90 0.80 0.70 £= 1 & 0.60 0.50 £=e 0.40 £=83 £=4 0.380 0.20 1 | 1 | nl | n | 0) 2000 4000 6000 8000 E, (keV) Figure D.6: Total attenuation factors, geometry C. 125 Appendix E Photopeak Efficiency The measured photopeak efficiencies for each geometry are given in this Appendix. The results for geometry A are shown in Figure E.1, for geometry B in Figure E.2, and for geometry C (at each angle) in Figures E.3-E.7. The points are the results (corrected for coincident summing effects) measured with the indicated radioactive sources. The solid curves are empirical fits used in subsequent data analysis. 126 1071! T T T T T T T T | T T T T T T T T b og o — '?Ru(EC,6*) ; i 4 — '?Eu(67) L ¢ — °°Co | y = *4Ng | w L oat geometry A 1072 1 n 1 1 1 pi it 1 1 af 4 1 a | E, (keV) Figure E.1: Photopeak efficiency, geometry A. T T T T T T TT T T T T T T T T T io ¢ — 152hu(EC,pB*) 4 4 A — 182u(¢-) | L § — 6c, A v _ 24Naq w® 5 “ geometry B 2x1073} : 4 4 1 1 4 4 [i | it i i 4 4 4 A. 4 4 10 103 104 E, (keV) Figure E.2: Photopeak efficiency, geometry B. 127 T T T T T T T T T T T T 7 T T F 10-2 ¢ — '®Eu(EC,6*) 4 | ° ° 4 — *2Eu(67) : L 5 — 6c, v _ 24nNVa @) b a geometry C L on J 2x1073 4 adi i 4 4 A 4 : ae | 1 i i i 4 4 1 i 10 103 104 E, (keV) Figure E.3: Photopeak efficiency, geometry C, 0°. T T T T T T T T T T T T T T T T T 10-2} ¢ — °?Eu(EC,6*) . rE a 4 L A — 82Fu(p7) d r o 56Co 7 vy — *4Na @® L = geometry C —46° 2x1073} : AL E 3 4 i 4 4. J. 1 | 1 i 4 i 1 1 i i 107 108 104 E, (keV) Figure E.4: Photopeak efficiency, geometry A, —45°. 128 T T T T T T T T | T T T T T T 7 T 107? ¢ — 1°Eu(EC,@*) 4 [ A — 182mu(g7) | _ 4 _ 5609 4 ; vy — 4Na r = @ r 4 geometry C | —90° 2x1073} 4 i BS 1 i 1 i 4 1 i 4 ail 1 4 4 4 p 1 10° 103 E, (keV) Figure E.5: Photopeak efficiency, geometry C, —90°. T T T T T T T T T T T T T T T T T 1072 0 ¢ — 1°?Eu(EC,8*) * i A — 182y(g7) | L 5 — 58¢o J vy — *4Na | w L 4 geometry C | +90° 2x1073} . wail 4 1 4 1 i 4 J i 4 i i 1 i 4 (a 3 10 108 E, (keV) Figure E.6: Photopeak efficiency, geometry C, +90°. 129 T T T T T T T T T 7 T T T T T T T 10-7} 9 — 1?Eu(EC,6*) 4 a oa 4 L A — 152hu(g-) | L § — 8c, c Y a 24Na wW 5 4 geometry C +135° 2x1073- 4 i 41 4: 1 41 1. 4. 4. i i 4 1 Li L i i 1 E, (keV) Figure E.7: Photopeak efficiency, geometry C, +135°. 130 Appendix F Summary of Experimental Results F.1 S and R From Each Run The measured values S and R for each run are given in Tables F.1 and F.2. The run number (or range of run numbers for angular-distribution measurements) is given under the RUN heading. The “explicitly included” errors described in Subsection 4.4.3 are given in parentheses for the least significant digits of S and R. The additional systematic errors S and R are estimated to be 6% and 4%, respectively (see Table 4.2). These S$ data points are plotted in Figure 5.1. 131 RUN E S R (keV) (keV-b) 060 419.3 0.0614(14) 0.461(19) 061 331.2 0.0677(15) 0.461(19) 062 244.2 0.0741(18) 0.435(19) 063 159.1 0.0795(21) 0.395(17) 064 416.1 0.0621(14) 0.434(18) 066 676.3 0.0610(15) 0.450(20) 067 848.9 0.0613(20) 0.443(29) 068 502.6 0.0622(15) 0.420(19) 069 415.6 0.0624(15) 0.431(18) 070 74.3 0.0881(42) 0.424(21) 071 415.2 0.0601(14) 0.461(20) 072 371.3 0.0623(14) 0.465(19) 073 285.1 0.0691(16) 0.461(19) 074 198.8 0.0737(19) 0.474(20) 075 114.3 0.0857(26) 0.454(19) 076 414.4 0.0658(15) 0.428(18) 077 53.2 0.0944(79) 0.444(38) 078 413.3 0.0632(17) 0.480(20) 079 93.5 0.0849(33) 0.458(21) 080 412.3 0.0631(14) 0.450(13) 083 412.4 0.0629(17) 0.460(19) 085 418.5 0.0604(15) 0.440(20) 086 850.2 0.0619(20) 0.439(30) 087 418.6 0.0636(15) 0.450(19) 088 288.7 0.0683(16) 0.453(19) 089 202.6 0.0759(19) 0.441(19) 090 375.0 0.0673(15) 0.428(16) 091 418.0 0.0621(14) 0.450(19) 092 109.4 0.0848(32) 0.473(27) 093 417.7 0.0632(15) 0.461(20) 094 505.1 0.0632(16) 0.451(20) 095 425.5 0.0646(15) 0.428(18) 096 56.4 0.0951(70) 0.413(35) 097 425.0 0.0653(19) 0.396(17) 098 683.6 0.0600(17) 0.434(18) Table F.1: Summary of S and R for each run (see text for details). This Table is continued in Table F.2. 132 RUN E S R (keV) (keV-b) 099 683.2 0.0614(18) 0.447(19) 100 683.3 0.0623(18) 0.458(18) 101 856.2 0.0607(19) 0.473(22) 102 512.1 0.0647(19) 0.430(17) 103 425.4 0.0659(19) 0.452(18) 104-108 416.6 0.0613(8) 0.471(11) 109-113 848.3 0.0595(9) — 0.503(12) 114-118 676.0 0.0581(10) 0.472(12) 119-123 245.3 0.0708(17) 0.440(11) 124-128 559.5 0.0616(18) 0.460(11) 129-133 1021.2 0.0647(21) 0.477(13) 134-138 1189.4 0.0669(23) 0.464(13) 140-144 331.4 0.0674(27) 0.469(11) Table F.2: Continuation of Table F.1. 133 F.2 Legendre Coefficients 0 0 0 1 1 1 E ay a> a; ay a; a3 (keV) 116.1 0.01(5) -0.05(9) 0.09(12) 0.06(8) -0.28(13) 0.35(17) 245.3 0.04(3) 0.05(4) 0.06(6) 0.21(6) 0.05(9) —_-0.28(11) 331.4 0.03(3) 0.06(4) 0.08(6) 0.127) 0.00(9) —_0.10(13) 416.6 0.02(3) 0.10(4) -0.04(6) 0.00(6) 0.26(9) -0.10(11) 559.5 0.01(3) 0.13(4) 0.10(6) 0.10(6) + 0.24(10) -0.11(18) 676.0 0.04(3) 0.13(4) 0.10(6) 0.12(7) 0.29(9)__-0.04(21) 848.3 -0.02(3) 0.25(4) 0.10(6) 0.00(7) _0.36(13) -0.18(19) 1021.2 -0.02(3) 0.19(4) -0.08(7) -0.07(7) 0.17(13) -0.09(19) 1189.4 -0.05(3) 0.20(4) 0.05(6) —0.11(8) -0.06(16) — 0.18(24) Table F.3: Measured yo and 7; Legendre coefficients. The statistical errors in the least significant digits are given in parentheses. The results are displayed in Figures 5.6 and 5.7. 134 F.3 Final Results for S and R E (keV) S (keV-b) R 55.1 74.3 93.5 112.4 159.1 200.7 244.8 286.9 331.2 373.1 417.4 506.0 559.5 679.2 849.8 1021.2 1189.4 0.0929(62) 0.0887(42) 0.0855(33) 0.0855(23) 0.0800(21) 0.0748(14) 0.0731(12) 0.0687(11) 0.0682(13) 0.0647(10) 0.0625(4) 0.0626(9) 0.0624(18) 0.0592(6) 0.0605(7) 0.0655(21) 0.0678(23) 0.427(26) 0.424(21) 0.458(21) 0.458(12) 0.395(17) 0.457(14) 0.439(9) 0.457(13) 0.467(10) 0.444(12) 0.448(4) 0.433(11) 0.460(11) 0.457(7) 0.484(9) 0.477(13) 0.463(13) Table F.4: The final results for S and R found by combining the data in Tables F.1 and F.2. The “explicitly included” errors described in Subsection 4.4.3 are given in parentheses for the least significant digits of S and R. The additional systematic errors S and R are estimated to be 6% and 4%, respectively (see Table 4.2). These results are plotted in Figures 5.3 and 5.5. 135 F.4 Angular Distribution Plots The measured yo (circles) and 7; (squares) yields with statistical errors are plotted as a function of detector angle in Figures F.1-F.9. The results are not corrected for center-of-mass motion and finite detector size. The solid curves are the fits described in Subsection 4.4.1, used to extract the Legendre coefficients. The absolute scale of the 116-keV data (Figure F.1) is only approximate, as the data at each angle were renormalized to make the center-of-mass 2 yield isotropic. 0.08 T T 1 T T 0.07 - E=116 keV 7 0.06 + ; : T | 0.05 + 4 > « o = 9.04 + ef o 0.03 + : ” 0.02 F | 0.01 + ‘ 0.00 | | | | | l l | it | 0 20 40 60 80 100 120 140 160 180 6,, (degrees) Figure F.1: EF = 116 keV Angular Distribution. S Wie) (keV—b) SW e) (keV—b) i( 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 136 ! I I I T T I T E=245 keV | | | | | | | | | 0 20 40 60 80 100 120 140 160 180 6,, (degrees) Figure F.2: EF = 245 keV Angular Distribution. T I T I T I I T T ! E=331 keV | | | at | | | | 7 | 0 20 40 60 80 100 120 140 160 180 6, (degrees) Figure F.3: E = 331 keV Angular Distribution. (c) (keV-b) L i SW é) (keV-b) i SW 137 0.08 | T T T 0.07 - E=417 keV 0.06 + 0.05 - 0.04 - i a 0.03 F 0.02 + a al 0.01 - 0.00 l l l l l l L l 0 20 40 60 80 100 120 140 160 180 6,, (degrees) Figure F.4: EF = 417 keV Angular Distribution. 0.08 T T T T T T ea | E=560 keV 0.06 + 0.05 + 0.04 + 0.03 + 0.02 + ee 0.01 - 0.00 1 | | | | | | I | | 0 20 40 60 80 100 120 140 160 180 6, (degrees) Figure F.5: E = 560 keV Angular Distribution. S Wile) (keV—-b) S.Wi(e) (keV—-b) 1 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 138 E=676 keV Pe | | | | | | i t 1 1 0 20 40 60 80 100 120 140 160 180 6, (degrees) Figure F.6: E = 676 keV Angular Distribution. I I I I I I ! 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