The Elastic Scattering of Deuterons by Beryllium Citation Renken, James Howard (1963) The Elastic Scattering of Deuterons by Beryllium. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Q4A4-ZP21. https://resolver.caltech.edu/CaltechTHESIS:05032012-120004946 Abstract The differential cross section for the elastic scattering of deuterons by beryllium-9 has been measured for bombarding energies from 0.4 Mev to 1.8 Mev at the center-of-mass angles of 90°, 125°16', and 163°30'. The cross sections were found to be slowly-increasing functions of the bombarding energy and are compatible with the assumption that the scattering nuclei may be represented by nearly-impenetrable charged spheres. No resonance structure was observed. These results are in disagreement with earlier observations of Be^9 (d,d)Be^9 scattering from which the existence of two levels in B^(11), having excitation energies of 16.766 Mev and 16.912 Mev, respectively, was inferred. The results of this experiment indicate that the previously-observed elastic scattering anomalies were due to some shortcoming in the procedure of the earlier experiment. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Fowler, William A. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1963 Record Number: CaltechTHESIS:05032012-120004946 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05032012-120004946 DOI: 10.7907/Q4A4-ZP21 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 7001 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 03 May 2012 20:10 Last Modified: 09 Jan 2024 00:23 Thesis Files Preview PDF - Final Version See Usage Policy. 14MB Repository Staff Only: item control page

THE ELASTIC SCATI'ERING OF DEUI'ERON"q BY BERYLLIUlvl Thesis oy James H. Renken In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1963 ACKNOvTLEDGMENTS The author wishes to thank the staff and personnel of the Kellogg Radiation Laboratory for the advice and assistance which he received during the course of this inve~tieation . He especially acknmlledges his debt to Professor vl . A. Fouler, who suegested this experiment, and Professor C. A. Bn.rnes, \vhose suee;estions and cri ticisi:lS were instrumental in its successful completio~•· In addition, discussions with Dr. T. A. Tombrello have been of great benefit. TLc assistance of t-!rs . Ea=bara Zimmerman in performing the computer ca ~.culatior..s also is gratefully acknowledged. 7ne preparation of this thesis has been facilitated by the editing assis~~ce of the author's wire ~nda and his sister June Picci'-"CO. Finally, the author wishes to acknowledge the receipt of fellowships from the Hughes Aircraft Company and the National Science Foundation which provided financial support during the course of his graduate training. ABSTRACT The differential cross section for the elastic scattering of deuterons by beryllium- 9 has been measured for bombarding energies 0 from 0.4 Mev to 1 . 8 ~~vat the center-or-mass angles of 90, 125 0 1 6 •, and 163°30' . ~nctions The cross sectior.c!J were f'ound to be slmrly- increasing of the bombnrdi ~ ~~~rgy ~nd arc compatible with the assumption that the scc.-../..,cri~ nuclei rr:ay be represented by nearlyimpenetrable charged spheres. No resor.::mce structure l?as observed . .- These results mo.:. in disagreen:..a::J.t ·Hith earlier observations of' Be9 (d,d)Be9 scattering from which tr..e existence of two levels in 11 B , havi!'..J: excitation energies of 16 .766 Mev and 16 . 912 ?-1ev, respectively, was inferred . The results of this experiment ir..dicate that the previously- observed elastic scattering anomalies were due to same shortcoming in the procedure of the earlier experiment. TABLE OF CONTENTS ITTlli PAAT PAGE I. INTRODUCITON l II. EXffiRJ1.1ENTAL EQUIPMENT 4 l . 2. 4 6 6 8 3. 4. III . General Description The Electrostatic AP~yzer Energy Calibration The :M agnetic Spectrometer Energy Calibration The Semiconductor Detector EXPERIMENTAL PRO~URE 6. The Determination of Cross Sections ~rom Experimental Yields The l·~'T:l.etic Spectromei;er Acceptance Solid Angle The Scattering Target Data- T--c..king ProcedU':'e Corrections to the Experir.:.:=nta.l Yields Some Effects Causing Deviation from Rutherford 7. 8. 9. The Atomic Stopping Cress Section Results of the Cross Section Measurements Analysis of the Experimental Error 1. 2. 3. 4. 5• ScatteriP~ IV. COMEA.RISON OF THE OBSERVED CROSS SECTIO!>.TS WITH TRE SCA'1·~RIJ."'IJG DUE TO A CHARGED SPEERE l. 2. 3. V. Scattering by an Lrn:penetrable Charged Sphere Scattering by a Charged Sphere with Arbitrary Boundary Conditions at the Surface Dete!"'lll.ining the Parameters Required to Fit the Experimental Data DISCUSSIO!'I OF RESULTS 9 9 16 17 19 21 25 26 Z7 27 32 32 34 43 46 53 APPENDIX I: Sample Calculation of the Differential Scattering Cross Section From the Experimental Yield 55 58 60 - 1 I. INTROD~TION The major task in low-energy e~erimental nuclear-structure ... physics is the determination o~ characteristic properties o~ the energy levels ~ various nuclei. These properties include the excitation energy, spin, isotopic spin, parity, li~etime , and branching ratios ~or various dec~ modes. One o~ the most ~ruit~ul techniques ~or studying these level parameters is the observation o~ various rediationa ~rom the nuclear system which ~ollow charged- particle bombardment .~ o~ an appropriate target. In particular, d~termining the number and angular distribution of elastically scattered projectiles o~ten yields unambiguous values ~or many o~ the level parameters o~ inter est. Many canp<nmd-nucleus states resulting ~rom the bombardment o~ ' light nuclei with ·protons or alpha particles have been studied by observing the elastically-scattered particles. Much less elastic scattering work has been done with deuterons as projectiles. are perhaps three reasons: There First, in contrast to the proton and alpha particle, the deuteron is a relatively weakly bound system with large spatial extent. o~ What ~ be pictured classically as the noncoincidence the deuteron's center-0~-mass and center-0~-charge produces a di~- • :f'use distribution o~ the deuteron's charge . Such a distribution, as well as any spatial polarization or it by the electric rield of the scatterer, conceivably can have an obs~rvable e~~ect on the scattering cross section. Second, because o~ the small binding energy o~ the deuteron, the compound s~te:ms ~ormed by the target and projectile nuclei have relatively high excitation energies. Since the obvious course or action vas to iuvestigate low-lying states rirst, very little - 2 - incentive to use deuterons as scattering projectiles existed. Now as 1. theoretical techniques attain more sophisticati on, it becomes desirable. to investigate level parameters for states with high excitation energies. Third, the fact that the deuteron has unit spin somewhat complicates the a.nal.ysis of experimental data. difficulty and not a conceptual one. This is a practical However, this diff'iculty places stringent limits on the permissible error in the experimental data and often reqUires that information about the associated r eactions be available. A recently-completed measurement by Ford(l) of the Li7 (d,d)Li7 scattering cross section showed several broad anamolies. Unfortunately, the variation of the cross section with energy was sufficiently complicated to prevent the unambiguous determination of any level parameters. Both direct-type reactions and the formation of :sl-1 as a compound nucleus result from the bombardment of Be9 by deuterons. (See Fig . l for the Bll energy level diagram.) For deuterons having energies between 0.400 Mev and l.8o0 Mev, the excitation of the B11 nucleus will be between l6.l5 Mev and 17.29 Mev. In this region, two levels, lying at l6.77 Mev ~ l6.93 Mev respectively, have been inferred primarily from anomalies in the deuteron el~tic-scattering cross section.( 2 ) 10 In addition, the results of an investigation of the reaction Be9 (d,n)B indicated the presence of a level near l6.77 Mev.(3) A later investi- gation of this same reaction, however, indicated a smooth variation with energy of the neutron-production cross section, and, therefore, no resonant compound-nucleus states. ( 4 ) Same information about other reactions induced by deuteron - 3 bombardment of beryllium also exists. Unfortunatel y, t he quantity and quality of these data make it impossible to say wi t h c ertainty whether or not there are well-defined excited states at 16.77 Mev and 16. 93 11 Mev in B • 11 An examination of the B level diagram will r eveal that reactions of the types Be9(d ,p )BelO . Be9(d,t)Be8 Be9 (d,a.)Li7 Be9(d,n)~0 are all energetically possible. Since the residual nuclei are l ef't in various states of excitation, there are several groups of' emerging particles for each type of reaction. There are, i n f'act , f'ive positive- Q channels which yield neutrons and eight positive- Q c hannels which result in the em!ssion of a charged particle other than a de ute ron . The task of separating these various particle groups in order to measure the desired cross section is by no means trivial. As wi ll b e discussed later, the combination of a magnetic spectrometer and high-resolution solid-state detector provides a suitable means f'or distinguishing various reaction products. Clearly, the experimental information concerning the two previouslymentioned ex<..ited states of Bll was unsatisf'actory. the investigation described in t~s thesis. This fact l ed to - 4 II. 1. EXPERIMENTAL EQUIH-iENT General Description The layout of the experimental e quipnent is shown schematically in Fig. 2 . The beam of accelerated deuterons is provided by a 1. 8 million-volt Van de Graaff generator . (5), ( 6 ) The beam of particles whi ch emerges from the generator is a mixt ure of the ionic type.c H+ , D+., HD+, and n:n+. v The magneti c f i e ld creat e d b y a crossfield magnet separates these beam components so that the unwanted ones may be stopped by a slit system. The des i red beam component then enters a one-meter radius, ~degree, elect ros t at i c analyzer, the energy resolution of which is determined by the width of the entrance and exit slits. For this experiment, the sli ts were set so that the spread in energy of the transmitted particle s, BE, gave an energy resolution, E/5E, a£ about 1200. Horizontal and vertical s lits placed between the electrostatic analyzer and the target chamber provide additional collima.tion. Difference signals from these slits s upply c orrective informa- tion to systems which control the accelerating voltage and the current for the crossfield magnet. After the incoming particles are scattered b y t he target, they are ana.J..yzed by a double-focusing magnetic spectrometer . (7) The arrangement of the target chamber and magnetic s pectrometer is such that the particle beams entering and leaving t he t arget chamber are inclined, respectively, 10 degrees above and below the horizontal plane. (See Fig. 3) This permits continuous rotation of the spectro- meter about a vertical axis through nearly 18o degrees, thus providing laboratory scattering angles from 0 to 160 degrees . The magnetic - 5 spectrometer ~alyzes particles according to their momenta 7 and the spread in momentum, oP, which the particles emerging from the spectrometer possess is determined by the width of the exit slit. For this experiment, an exit slit 0.125 inches wide produced a momentum resolution, P/oP, of 302. A silicon ~n junction soli d-state detector was used to count the analyzed particles.(B) The output of this detector was then pro- cessed by a charge-collecting preamplifier, amplifier 7 and appropriate counting circuits. The amount of charge delivered to the target by the beam, and hence, the number of incident particles, is determined by a current integrator. This instrument collects a known amount of charge by using the beam current to discharge a capacitor which previously was charged to a known voltage. When this capacitor is completely dis- charged, a series of relays is activated which gates off the counting circuits, turns off a timer, and energizes a small magnet that deflects the beam off the target to minimize unnecessary deterioration of the target. In order to insure that the current which discharges the capa.citor is a true measure of the number of incident particles, tv1o precautions are taken: First 7 the target is raised to a potential of 300 volts with respect to the walls of the target chamber. This minimizes the effect of secondary electrons emitted when the beam strikes the target. Second, a screen with a negative potential of 300 volts is placed at the entrance to the target chamber. produced at the sl.i ts from reaching the target. This prevents electrons - 6 2. The Electrostatic Analyzer Energy Calibration The energy o~ the particles which pass through the electrostatic analyzer is determined by the potential di~ference between its plates . If a particle h~ving rest energy Me 2 and charge Ze passes through the analyzer, then the kinetic energy of the particle is given to sufficient accuracy by (l) where ke is a constant and Ve is a certain ~raction of the voltage across the analyzer plates. In practice V is measured with a potentiometer} e and the particle energy is calculated using an appropriate value of k e in Eq. (l). The coastant k e is determined by observing the gamma rays from 16 . Assuming a resonance the 873-Kev resonance in the reaction r9(p,cq)o energy of 872.7 ± 0.4 Kev for the incident proton, Brown( 9 ) measured k e to be k 3. e = 1.0047 ± 0.0006 Mev/decivolt The Magnetic Spectrometer Energy Calibration The momentum of a particle following a circular orbit in the presence of a magnetic field is proportional to the product of the magnetic induction and the orbital radius. The gecrnetry o~ the spectro- meter de~ines the radius; there~ore the momentum o~ a particle passing through the instrument is proportional to the spectrometer magnetic field. The measurement of this magnetic field co~s ists in balancing the torque that it produces on a small, current- carrying coil against the torsion produced in a quartz fiber to which the coil is attached. When - 7 the coil is in the equilibrium position under the influence of opposing torques, an optical lever system provides error signals which regulate the field-producing current in the magnet. S~ose the current passing thr-ough the small coil is determined by measuring the voltage drop V which it produces across a precision m resistor. Then, the kinetic energy of a particle which has rest energy 2 Me , charge Ze, and passes through the spectrometer is given to sufficient accuracy by (2) where ~ is the proton rest mass and km is a constant of' the spectrometer . The value of km is determined as follows : Protons are scattered f'rom a thick copper target prepared by evaporating the metal in a vacuum and allowing it to condense on a glass microscope slide. a typical target "profile." Fig . 4 shows This illustrates how the observed number of scattered particles varies as a fUnction of' V for a f'ixed incidentm proton energy. The energy of the scattered particles can be calculated from the kinematics of' the reaction. Then, by assuming that V at the m midpoint of the profile rise corresponds to the energy of particles scattered from the surface layer of' the target, Eq. (2) may be solved for k • m The value of k determined from many measurements is m 2 k = ,381200 ± 922 Mev - mv m In order to relate the observed number of' scattered particles and the differential scattering cross section, it i s necessary to know the acceptance solid angle of' the spectrometer. The discussion of - 8 this solid angle will be postponed until the general procedure for determining the differential scattering cross section from the observed yield is considered. 4. The Semiconductor Detector An important component of the apparatus used in this experiment was the junction silicon-diode detector, which was purcha.c;ed :from the Hughes Aircraft Company. Fig. 5 shows its essential features . The junction is formed by diffusing phosphorus into one side o:f a high ' resistivity p-type silicon wafer . By applying a reverse bias to the junction, a depletion, or space-charge, region is formed. nesses The thick- and X of the depletion r egions in the two types of silicon n depend on properties of the materials and the bias voltage; a typical value for ~/Xn is 1000. The maximum electric field pr esent in the 4 depletion region is of the order of 10 volts/em. A particle to be detected strikes the n-type layer and penetrates into the depletion region. Here it creates hole-electron pairs that are promptly swept apart by the strong electric field which is present . This produces a current across the junction and results in a negative voltage pulse whose height is proportional to the energy of the incident particle. Optimum results in terms of linearity and reso- lution are obtained when the incident particle loses all of its energy in the depletion ~ayer and the depth o:f this layer nearly equals the thickness of the silicon wafer. A mount was fashioned which held the detector in place and acted as a spectran:eter exit slit. Fig. 6 shows its general features. - 9 III. l. EXPERIMENTAL PROCEDURE The Determination of Cross Sections from Experimental Yields The cross section for a nuclear reaction is defined to be y 0=-- (3) nd t where Y is the reaction yield per incident particle~ nd is the number of dis integrable target nuclei per 1.mi t volume, and t is the target thickness. Similarly, the differential cross section per l.Ulit solid angle in the laboratory system is defined to be (4) where dY is the portion of the r eaction yield per incident particle which is emitted into an element of solid angle dn at the laboratory angle e. In the present experiment~ deuterons were incident on a thick target and the magnetic spectrometer selected the particles scattered by a particular lamina within the target. The only symbol in Eq. ( 4) which cannot be easily identified with some experimental quantity is the effective target thickness t. In order to express this thickness in terms of suitable quantities, consider the nuclear reaction sh~Nn in fig. · 7. I'he energy of the incoming particle as it leaves the elec- trc .... ~tic analyzer is E • e Beca:l.Se of the electron-suppression voltage on the target, the energy of this particle at the face of the target is ~ = Ee - z1 eVT. The existence of a contaminant layer on the surface can further degrade the particle energy, so the energy of' the - 10 particle at the surf'ace o:f the reacting material is E . 10 At a certain normal depth s in the target~ the incident particle with energy E reacts. 1 The emitted particle has energy E and reaches 2 the surface o:f the target •.,i th energy E ; the contaminant layer ·further 20 degrades the energy to E2R. Finally~ the energy of the particle which traverses the magnetic spectrometer is Em = E2R + z2 evT . In practice~ the electron-suppression voltage and thickness of the contaminant layer are kept small enough so that their effects on the particle energies are usually negligible. exist, Ee = E10 = El.B and Em = E2R = E20 • vlhen these conditions This is assumed in what :follows. The directions of the incoming and outgoing particles relative to the target normal are specified by e and 8 , respectively; these 2 1 angles are related to the laboratory angle o:f observation by 7{-8=8 1 +8 2 (5) B,y c c~ idering ' the kinematics o:f a reaction with particle 1 incident on particle 0 and producing particles 2 and 3 and energy Q, it can be shown that (6) where (7) We ascume that the energy degradation which the particles suf:fer in moving through the target may be found :from the expression - ll - dE = - n €(E) s dx (8) where n s is the density of stopping atoms and €(E) is the atomic stopping cross section. For the case of a target having uniform com- position, the variables in Eq. {8) may be separated and the result integrated to give rl dE ns €l(E) = s cos e (9a) 1 El.B s E20ns dE €iE) = s cos e (9b) 2 E2 where € 1 and € apply to t~e incident and emitted particles respectively. 2 It can be seen from Fig. 7 that the thickness of the target layer along the direction of the incident beam is given by t = 8s/cos el . We want to relate t to the spread in energy of the particles transmitted -, by the magnetic spectrometer; therefo~ in first approximation we have This gives (10) for the relation between the target-lamina thickness and the acceptanceenergy spread of the spectrometer. We now differentiate Eqs. (9) with respect to E20 while holding e constant. 'lhis gives - 12 - - cos e 1 (11) and (12) Because of Eq. (6) we can write (13) By combining Eqs. (11), (12)., and (13) we obtain (14) Substituting this into Eq. (11) gives (16) where the sign in Eq . (15) is unimportant and has been dropped. We can now relate all the quantities in Eq. ( 4) to experimental observables. Let the difference between the number of detected parti- cles at a point on the top of a thick-target profile and the number at the foot of the profile be called the resultant number of reaction products NR. The total number of incident particles which produces -13~ is ( 17) where C is the capacitance of the current -inteerator condenser .that is charged to voltage V and then discharged by the incident beam, and Ze is the charge on each incident particle . By definition Ze NR NR dY = Ni = (18) CV The energy spread of the particles passed by the magnetic spectrpmeter is related to its resolution by (19) According to theoretical predictions of the spectrometer performance, the resolution is qetermined by the width of the exit slit. The relation between these quantities is r R = 2(1 + m) where m is the magnification, r width of the exit slit. 0 0~ (20) the equilibrium radius, and or is the Finally, if we identify the element of solid angle appearing in Eq. (4) with the acceptance solid angle of the spectrometer nL' Eqs. (16), (18), and (19) may be combined to yield * (21) To use Eq. (21), it is necessary to know the energies E and E • 1 2 If suitable stopping cross section data are available, in principle, *This expression has been obtained by a procedure similar to that used by Bardin in reference 30 and agrees with what can be derived from his results. - 14these energies mey be :round by using Eqs. (6) and (9). For work in which cross sections are determined :rram yields measured near the ta.rget-pro:t>ile rise (i.e., the reactions producing the yield occur in the lamina near the sur:race of the target), a useful approximation :ror E may be derived from Eq. (14). 1 For small s we assume € (E ) = €1 (ElB), € (E ) = € (E20 ), and (d.E /d.E ) remains constant. 1 1 2 1 2 2 2 Then Eq. (14) may_ be integrated to give €2(E20) cos e1 1) 8 ElB + E20 cos + ~ e ElB- E2B €l{ElB) 2 El = €2{E20) cos e1 oE + €l(ElB) cos e2 1 e (OE (dE2) (22) where E2B is the energy o:r a particle produced by a reaction at the sur:race of the target. by Brown et.al. (lO) -. This expression is identical with that obtained Equation (7) shows that :ror elastic scattering reactions (23) This simplifies the use o:r Eqs. (21) and (22). found with Eq. Knowing E , E may be 1 2 (6). In order to facilitate the examination of the scattering data~ the well-understood variations of the cross section with energy and angle characteristic o:r Ruther:rord scattering were eliminated by dividing the measured cross sections by the appropriate Ruther:rord cross section. This cross section in the center-of-mass system is easily shown to be - 15 - millibarns steradian (24) where E expressed in Mev, is the laboratory energy of the incoming 1 particle, z 1 and M are the charge number and mass of the incoming 1 particle, and z and M are the same quantities for the target . 0 0 Before comparing Eqs. {21) and {24), however, both cross sections must refer to the same coordinate system. To convert a labora- tory cross section to the equivalent center-of- mass value , the former must be multiplied by 1 + x cos ec + 2X cos ec + 1 ) 3/2 (25a) (25b) where 8L and 8C are the laboratory and center-of-mass angles, respectively, and (26) The angles eL and ec are related by (27a) or (27b) - 16 2. The Magnetic Spectrome ~er Acceptance Solid .~gle B,y using Eq. (21) it is possible to determine absolute cross sections providing that all quantities on the right-hand side can be measured. It would be possible to calculate a value of' R using Eq. (20) .., and C could be measured; however the determination of' nL from the geometry of' the spectrometer would be difficult. Instead or attempting to determ~ne nL in this way, the following method was used. Protons were scattered from a copper target, and the scattering cross section was assumed to be Rutherford. Then the yield at the top or the resulting thick-target profile was used in Eq. (21) to calculate R/CnL. Using the nominal value of' C and a value of' R = 302 calculated with Eq. (20), the solid angle was found to be 1.47 X 10-3 steradians. This value was obtained from measurements performed at several energies and angles; Table I summarizes these measurements. The uncer+ainties listed are standard deviations calculated from repeated observations at each combination of' energy and angle. This uncertainty does not include the effect o~ the uncertainty in the atomic stopping cross section or copper. It may be argued that the cross sections calculated with nL determined in the a.f'orementioned fashion are not absolute. Strictly speaking this is true because the calculated cross sections are really expressed as ~ fraction of the Cu(p,p)Cu cross section. Obviously any difference between the true absolute cross sections for Be9(d,d)Be9 and the values quoted will depend on how much the Cu(p,p)Cu scattering deviates from the Rutherford law. The results in Table I lend confidence to the assumption of' Rutherford scattering by copper. - 17 3. The Scattering Target The target used in this experiment was a .small slab of beryllium whose dimensions were 1 1/2" X 1/4" X 1/16 ~·. target is illustrated in Fig. 8. The holder used f'or this The target is positioned in the target chamber by attaching the holder to the end of' a rcxl which is coaxial with the center axis of the target chamber. By using 0-ring seals, this rod is extended through the top of the target chamber so the vertical position and the angul.ar orientation of the target may be varied at any time. In addition to provision for holding the beryllium target, the target holder was built to hold a copper-on-glass target as well. By a simpl.e vertical movement of the target rcxl, either the beryllium or copper target could be exposed to the beam. Mozer(ll) bas shown that fine scratches on the surface of' a scattering target can reduce the observed yield. ' Although his work indicates that fine scratches should not be important at the laboratory scattering angles used in this experiment, sane effort was expended in trying to obtain a target whose surface was as mirror-like as possible. Preliminary attempts to polish the target surf'ace by rubbing it with fine grades of polishing ];81>er and then iron- oxide rouge impre.g nated in a kerosene-soaked rag were tmJ:;atisfactory. This method not only removed scratches very slowly, but new scratches were created during the polishing process by single pieces of extraneous grit which became embedded in the rag'. It vas finally decided to have the target polished by a commercial t1ra vhi.ch specilaJ.ized in polishing aetal.l.ic surf'aces • This - 18firm's method involved using a high-speed buffing wheel; the results were surprisingly good. A microscopic examination of the target sur- face did reveal a series of tiny pa.rall.el scratches • However, as sug- gested by Mozer's work, the effect of these scratches was minimized by holding the target during polishing so that the scratches would be approximate~ in the scattering plane'-when the target was positioned in the target chamber. Once the target surface vas prepared, the deposits of carbon contaminant which built up on the surface of the target during bombardment were periodically removed by lightly rubbing it with a rag containing iron-oxide rouge. After each such polishing, the target was succes- sively washed with kerosene, acetone, and distilled water. A target profile was taken to check tEe cleanliness of the target surface and to see if any impurities were present within the target. 'Ibis is shown in Fig. 9. By kinematics the two peaks in front of the beryllium elastic-scattering yield can be attributed to elastic scattering by ~era of carbon ataus and oxygen atoms, respectively, on the surface of the target. '!he small, symmetrical peak at V m = 580 is due to a very thin surface layer of some scatterer having a mass number of about 100. V m '!be constant background plateau which begins at = 592 is probably due to the presence of sane impurity distributed within the target. Judging fran the value of Vm where this background beglus, it seems likely that this impurity is either iron, cobalt or • nickel. The background yield indicates that the ratio of the number of berylllua at<DS to the number of impurity atoms is 2200:1. Such an illlpurity cOlleentration results in an effective stoppiDg cross section - 19 for the target which differs from that of pure berylli um by less than one-third ot one percent. 4. Data-Taking Procedure (a) Separation of the Reaction Products The number of possible breakup channels available to the Be9 + d system is large . Consequently the detecti on of elastically- scattered deuterons vas complicated by the presence of protons , tritons, and singly- and doubly- charged a lpha parti cl es which wer e produced by the canpeting reacti ons • The magneti c s pectro~ter alone could not di scriminate these different parti cl e groups because the reaction products from the many layers i n a t hic k target had a continuum of mcmenta . Although the singly-charged parti cles transmitted by the magnetic · spectrometer all had the same manentum, it is easy to show t hat their energies were not equal. I t the energy of the protons transmitted by the spectrometer is taken as the unit, then the energi es of the other particles transmitted at t.be same spectrometer setting are given by Ed = 1/2 Ep Et = 1/3 Ep Ea.+ = l/4 . Ep E a.++ = Ep where the subscript meanings are obvious . Thus the use of a detector possessing good energy-resolving properties provided t he necessary means tor distinguishing the various particles which passed through the spectrometer. The p-n junction detector used in this experiment has an energy - 20 - resolution of about 10 percent in the energy range where it was used. As can be seen from Fig. 10, this detector afforded the necessary separation between different particle groups. During the experiment the number of elastically-scattered deuterons was determined by setting the window of a single-channel pulse-height analyzer so that the deuteron pulses would be passed whilb the others would not. As the energy of the incident deuterons was lowered, the height of the detector pulses produced by the scattered particles also decreased. Finally the output- pulse height became comparable to the level of the electronics noise. When this condition existed, the entire detector- output spectrum was recorded by using a multichannel analyzer. TyJ>ical spectra obtained with this instrument are shown in Figs . 11 and 12. With records such as these, any necessary n6ise sUbtractions could be made and the yield of scattered deuterons obtained by summing the number of counts in the appropriate analyzer channels • It was necessary to be certain that the particle group being observed really was the elastically-scattered deuterons. verified in two ways : This was First, the observed positions of the scattering- profile rises agreed with the positions calculated from kinematics. Second, the observed pulse height for deuterons fran Cu(d,d)Cu (there being no question here that the observed particles were deuterons because of the intensity of the scattered particles) was compared with the pulse height of the particles thought to be deuterons from '7. Be9(d,d)Be9. The ratio of these observed pulse heights agreed with the calculated ratio of deuteron energies for the two scattering processes. 'lhus there can be 11ttle d·o tibt that the observed particles - 2l. - were reall.y deuterons • (b) ~ Determination !!.! !fa Values of~ corresponding to various incident ener gies and scattering angles 11J1J::f be obtained from the appropri ate scatter ing yield profiles. Figs. 13 to 20 show some typical profiles obtained under the noted condi tions. It is unnecessary to obtain a complete profile at each point where tbe cross section is to be ~asured. Si nce NR is determined from the difference in number of counts before and after the profile rise, it is sufficient to ~asure the number of counts at these two posi tions without being concerned wi th the detailed shape of the profile. ihe data-taking procedure cons i sted of the :following: The number of scattered particles was determined for two values of V on the profile m rise . ihis verified the location of the rise . Then V was increased to m a value corresponding to the peak of the profi le and the number of scattered particles was noted. The scattering yield was again recorded after making one or two additional. sma1.1 increases i n V ; the scattering m energy E was determined from the average of the values of Vm for which 1 yields at the profile peak were measured. This permitted v erification that the yield was being measured at the profi le peak and provided several. yield measurements which could be averaged . Finally, V was set m at a value below the profile rise and the background yield was measured. 5. Corrections to ~ Experimental Yields (a) Cba.rge Neutralization When a scattered particle emerges from a solid target , its effective cba.rge, which depends u;pon the number of electrons attached - 22 - to it, ma;y, in general, be one o-r several values . The magnetic spectr e- meter can pass only the particles o-r interest with a parti cular effective charge. To correct the yield -ror the particles o-r i nter est which have other e-r-rective charges, the -rraction of the emerging partic les in each charge state must be known. These -rractions, or charge equilibrium ratios, have been measured by Phillips ( 12 ) and -round to depend upon the energy o-r the particle and the type of atoms in the l ast few atomic layers at the surface of the solid. The results quot ed by Ihillips are· directly applicable to cases where the emerging particles are protons . By making the assumption that the probability of electron attachment to a moving ion is only a function of the ion's velocity and the number of other electrons already attached to i t , it is easy to show that an emergent beam o-r deuterons with energy E will have the same charge equilibrium ratios as a beam o-r protons with ener gy 1/ 2 E . Phillips ' experimental apparatus was such that he was able to maintain the cleanl.iness of the surface :from which t he i ons emerged. This enabled blm to detect small dif-rerences i n the charge equilibrium ratios fot" particles emerging :from di -r-rerent surfaces . The charge- equilibrium-ratio measurements which seem most consistent with the conditions in this scattering experiment are those f or a surface which Phillips called "dirt." For this measurement Ihillips did nothing to overcome the buildup of foreign matter on the sur-race from which the i on beam emerged. Although it is uncertain that Phillips' "dirt" and the sur-race contaminant present in this experiment bad the same c omposition, such an assumption seems more reasonable than t hat the charge equilibrium ratios were determined by a clean beryllium sur-race . 1he :traction or the total. number o-r deuterons which emerges - 23 f'rom a "dirt" sur:face as singly-charged ions is shown in Fig. 21. Several points at the high-energy end of' this curve are f'rom Mozer's work. (ll) He obtained these data by assuming the measured cross sec- tion f'or Cu(p,p)Cu deviated f'rom the Rutherford value at low energies because of' a reduction in the observed yield due to the f'ormation of' neutral and negatively-charged ions at the target surf'ace. For compari- son, Fig. 21 also shows the fraction of' singly-charged deuterons emerging f'rom a clean berylli'Wll surf'ace. Since the magnetic spectrometer was set to pass singly-charged deuterons, Fig. 21 provides correction f'actors f'or eliminating the ef'f'ect of' electron attachment. (b) ----- Scaler Dead Time ~~~----- Because of' the f'inite amount of' time required f'or the detection equipment to process the signal generated by each detected particle, the equipnent is "dead" during a certain f'raction of' the counting period. This gives an observed yield which is less than the true yield. A correction to the observed yield f'or the equipment dead time may be derived in the f'ollowing way: The f'raction of' the counting period f'or which the counting equipment is dead is given by NT/t where N is the number of' counts observed during a period t and T is the dead time per count; the number of' potential counts available during this dead period is NTM/t where M is the true number of' counts appropriate f'or the period t. Thus M must be given by M = N + (NT/t)M or (28) A convenient f'orm f'or expressing the me.gnitude of' this correction is - 24- lM M = M- N ~ M NT t (29) Using a value of T = 10-5 seconds as characteristic of the scalers emplo.yed to count the number of detected particles, the I Be9(d,d)Be9 counting rates were such that t:!M/M was never greater than 0. 3 percent. For these data the dead time was neglected. When it was necessary to record the scattering data with a multichannel analyzer, the percentage of dead time was calculated to be of the order of: one percent. The most important dead-time corrections were made to the Cu{p,p)Cu yield data taken for the solid-angle calibration. Corrections in this case were typically between one and three percent. (c) Scattering Ez the Detector Dead Iayer It is conceivable that the number of particles observed by the ~n junction detector could be less than the number incident on its face because of backscattering by the dead layer. Communication with the manufacturer of the detector revealed that the dead layer could have a thickness as large as 0.5 microns. A layer of this thickness is consistent with the observed difference between the pulse heights produced by equally-energetic protons and alpha particles as shown in Fig. 10. To check the effect of this scattering, a calculation was made of the number of ~Kev deuterons which would be lost by Rutherford scattering from an 0.5 micron layer of silicon. For these conditions it was found that the number of detected particles is lowered only 0. 07 perce:at, by dead-layer scattering. This loss is smaller at higher energies and tbus is completely negligible. - 25 6. Some Effects Causing Deviation f'rom Rutherford Scattering As mentioned earlier 1 the interpretation af the observed scattering cross sections was facilitated by f'irst removing the variation attributable to the ordinary Coulomb interaction. ' The possi- bility did remain that other well-understood, although more subtle, ef'f'ects might be influencing the experimental scattering. The following describes two such effects. (a) Screening Ez Atomic Electrons A calculation by Wenzel (l3) shows that the screening of' the target nucleus by its electron cloud will lower the scattering cross section f'ran the Rutherford value according to do doR = 1 - (30) where E is the center-of'-mass energy of' the incident particle and 6 is the absolute value of' the potential at the nucleus due to the electron Foldy(l4 ). has shown tbS.t 6 is well-represented by cloud. 6 where Z 0 = 34(Z0 )7 / 5 ev is the atomic number of' the nucleus • This screening ef'f'e.ct was negligible f'or the Be9(d,d)Be9 scattering. However, the cross sections measured f'or Cu(p,p)Cu to be used in determining the solid angle were altered by a small amount. The maximum correction to these data f'or screening was about 0. 7 percent. - 26(b) E:f:fects Resulting :from the Structure o:f the Deuteron The e:f:fect on the elastic scattering caused by the :finite separation between the deuteron's instantaneous center-o:f-mass and center-o:f-charge has been investigated theoretica~ by French and Goldberger(l5 ). A computer program to evaluate their expression :for the change in the cross section was written in this laboratory. A calculation using this program indicated that the scattering due to the Coulomb :force should slowly :fall below Ruther:ford as the incident energy is increased. However 1 at the highest energy used in this experiment, the predicted deviation :from Rutherford was slightly less than two percent. A second e:f:fect which can in:fluence the Coulomb-interaction scattering is the induction o:f an electric dipole moment in the deuteron by the electric :field o:f the scatterer. A Born-approximation estimate o:f the magnitude of thi~ e:f:fect has been made by Morinigo(l6 ). I This work indicate~ that the :fractional deviation o:f the cross section :from the Ruther:ford value should be r0ughly proportional to the polarizability and the momentum transferred :from the incident particle to the scattering potential during the collision. Using a reasonable estimate o:f the deuteron polarizability, the predicted cross section :for backward scattering o:f one-Mev deuterons is about two percent less than the Ruther:ford value. 7. The Atomic Stopping Cross Section The variation o:f the atomic stopping cross section :for protons passing through beryllium ,is given by Mozer(ll). In general the stopping cross section :for a light particle in a given material is - Z7 related to that of a proto~1 in the same material by (31) Thus the necessary valu€~ of the stopping cross section for deuterons in beryllium were obtained by using €d(E) = €p(l/2 E). This relation is considered to be accurate to within two percent. (l7) 8. Results of the Cross Section Measurements The results for this measurement of the Be9(d,d)Be9 elastic- scattering cross section are shown in Fig. 22. These data are expressed as the ratio of the observed cross section to the corresponding pointcharge Rutherford cross section. For measurements made at the center- of-mass angles of 90° and 125°16', the maximum particle energy which ' the magnetic spectrometer could analyze fixed the upper limits for the bombarding energies. 9. .Analysis of the Experimental Error (a) Relative Uncertainty in ~dcrR The possible error in the quoted values of dcr/dcrR for Be9(d,d)Be9 scattering can be determined from the uncertainties in the experimental quantities from which this ratio is calculated. To determine the rela- tive fractional standard deviation for dcr/dcrR the quantities whose l.Ulcertainties must be considered are where the symbols have their previously-defined meanings. - 28 - The reasons for the uncertainties in these terms are the following: NR: The error in this term results from the background subtrac- tion and the inevitable statistical counting error. e(E): The stopping cross sections are assumed to have a relative fractional standard deviation of two percent. Cos e /cos e + ~ [e(E )/e(E )] : The errors in~ and 1 2 1 2 cos e /cos e due to random errors in the various angle settings are 1 2 very small compared with the error in the stopping cross section ratio. Thus the main source of error in this term is the uncertainty in the stopping cross sections. E and E : The errors in these terms are due to the uncertain20 1 ties in the appropriate calibration constants. V: The uncertainty in this tenn results from fluctuations in the charging voltage for the condenser which is discharged by the beam current. The relative fractional standard deviations for these sources of error are shown in Table II . Assuming these errors to be independent and normally distributed, their combined effect gives a relative fractional standard deviation for do/daR of about 2.9 percent. At the lowest values of E , an additional source of uncertainty 20 in NR is the noise subtraction procedure. ,.. It is assumed that the greatest uncertainty due to this procedure is eight percent. It is further assumed that the char~e equilibrium ratio makes no contribution to the relative uncertainty in do/daR. - 29 (b) ~clute Uncertainty in £E/daR The values of do/daR measured in this experiment were normalized on the absolute scale by using the observed yield from Cu(p,p)Cu scattering. Thus the absolute uncertainty in do/daR is due to the uncertainties in the quantities c Nb do oc _!! Vc E20 d~ Nc vb Eb 20 R --- b € (E20) €c(E20) e:{E1 ~ eb [:coo 1 + a.b cos eb € (E2) 2 tos e~ + a.c ec(El ~ cos e~ 4 (Eb) 2 -1-2 (Ec) 1 €c(E2) where the superscripts b and c signify that the particular quantity is determined for Be9(d,d)Be9 or CU(p,p)Cu scattering respectively. The error analysis based on the quantities shown above is complicated by the possibilit~ of a systematic error in the angular settings of the magnetic spectrometer. As discussed by Ford(l), the protractor used to orient the spectrometer seems to be misaligned by about two-thirds of a degree. Although the angular settings used in this experiment were corrected by this amount, the possibility that a systematic error in the angle still exists cannot be overlooked . A systematic error of one-half of a degree would dominate any random error in the angle settings. For such a situation the errors in the square-bracket and sin ~C terms are not independent, and the usual technique for combining independent, normally-distributed errors in order to determine the probable error in an experimental result is not rigorously valid. The effect on the experimental uncertainty due to a possible - 30 systematic error in angle was investigated by finding the greatest possible error in a = cos a /cos e2 which could result from an error ~ 1 in the laboratory scattering angle. is to set e 1 as nearly equal to 9 2 When the experimental procedure as possible, it can be shown that a fractional error Ba/a = ~ cot(~L) can result. Using this fact, the error arising from the two square brackets was calculated in two different ways: First, it was assumed that Ba/a could be treated as the fractional standard deviation aa /a and the error in the ratio of the square brackets was calculated as due to a term of the form ~ = {a+f)/(b+g) where a, b, f, and g are independent quantities. Second, as an attempt to assess the importance of the interdependence . cos 9b/cos ab and cos 9 c/cos 9c , the error in the of the errors 1n 1 2 2 1 ratio of the square brackets was calculated as due to a term of the form 'If = (a+f) / ( a+g) whe:re a, f., and g are independent quanti ties • These two assumptions yielded approximately the same result for the fractional error resulting from the ratio of the square brackets. This suggested that the uncertainty in this ratio is only slightly dependent on the errors in the angle settings. The third angle-dependent term which contributes to the uncertainty is X = (sin ~~ /sin ~~) 4 . . If both center-of-mass angles have a systematic error v, then the fractional error due to this term is 'OX/X= 2v (cot . ~~- cot~~). Although there is no justification for combining t.':lis systematic error with the random errors in the other quantities, its relatively small size suggests that 'OX/X can be treated as random without invalidating the estimate of the fractional standard deviation for dcr/dcrR. - 31 The uncertainties in the various terms which contribute to the absolute fractional standard deviation in do/daR are shown in Table II. The uncertainties in the angle-dependent terms were assumed to result from a one-half degree systematic error in the angular settings. The combined effect of these uncertainties gives an absolute fractional standard deviation for do/daR of about 6.7 percent. At the lowest values of E uncertainty are present. 20 , two additional sources of These are the uncertainty in the charge b equilibrium ratio and the uncertainty in NR resulting from the noise subtraction procedure. - 32 IV. cc:.i?.i\RISO:; 0':! TI3 OI..i:SL.I. ::D CROSS S:t:CTICES \-liTH THE SCA'l:l.'ERii\G. Di.E '1.0 A CHARGED SPHERE Tne most striking f'e2..tu.r~ oi' the excitation functions nhmm in FiG. 22 is the lack o:f a:n:y p!·ur.:ounc:~ variation o:f the ratio- toRl.!therford ui th energy. rczpor~ible This s 1.:ggc::; ts that the mechanism primarily for the scatteri~ is only slightly dependent upon tte dcto.iled internal structure of the interacting nuclei . The additionn.l fc.ct tr...o.t the observed values .of do/doR are approximately unity indicc.tes that the Coulo~b int~raction is the dominant scattering z::ech::c..nism. so~e Since· the values of c;lo/doR do deviate from unity 7 however 7 interaction in addition to the pure Coulomb :field is present; a lostcal choice for tr.c s~urce o:f this edditional interaction is the finite size of the nuclei. 1. Scattering .£:y: an Inrrenetrable C'.r1a.r.;!:ed Sphere The first attempt to calculate scattering cross sections which agreed with the experimental values "i·Io.s m3de after assuming the scattering nucleus was an impenetrable charged sphere . In this case, results o:f Blatt and Biedenharn(lB) can be used to find. that the ratioto-Rutherford for the predicted cross section is given by sin4 ( e) + 2 2 k z 2 £+L L (2£+1)(2£'+1) [(.t.e•ool.t.t'L0)] 2 x .t '= 1£-Ll (32) ..X. It is assumed here and in what follovrn that the amplitudes of the various scattered partial waves depend only on the·.e-values of the respective lre.ves. - 33 where 8 is the center-of-mass scattering angle and (tt•ooitt'LO) is a Clebsch-Gordan coefficient. other symbols in tr.-i.s expression are defined by the following: (33) where v is the relative velocity of the two particles; (34) where M i s the reduced mass of the system; 1 (2 ME ) 2 em k = ----,.fl,..;-- where E em (35) is the kinetic energy of the two particles in the center-of- mass system; (36) where F£ and G£ are the regular and irregular Coulomb wave functions, respectively, and R is the radius of the charged sphere representing the nucleus; and where the Ruther:t'ord scattering phase shifts_ a£ are given by exp (2icr£ f£ +ill~ f£- l +iT}~ • • • fl +iT}~ ) =t - iT} t - l - ill • • • l - ill exp (2-;cr) .... o (37) so that tan-l .1 s S=l and (38) - 34 Although the radius of the charged sphere is not precisely determined by the theory, the idea inherent in the derivation is that the charged sphere represents a region of space from which the projectile is excluded by virtue of the finite sizes of the particles. Therefore the radius of the sphere should be equal to the sum of the radii of the two interacting particles. Call thi s radius the interaction radius. Equation ( 32) was evaluated for various deuteron energies and scattering angles by using a Burroughs 220 computer. The hard-sphere phase shifts given by Eq. (36) were obtained by using a computer program written by Dr. T. Tombrello. results is shown:in Fig. 23. A portion of the calculated Despite +.he fact that the radius values used in this calculation were about one-half' the estimated interaction radius for the Be9 + d system, the calculation yielded larger cross sections than the experiment. A value of zero for the total reaction cross section is likewise an unsatisfactory result of the hard-sphere model. This is particularly unrealistic in the case of Be9 + d because studies of the possible reactions indicate that a substantial total reaction cross section exists. 2. Scattering ~!: Charged Sphere with Arbitrary ~oundary Conditions at the Surface Since the results calculated with the hard-sphere model dis- agreed with experiment, the effect of changing the boundary condition at the nuclear surface was investigated. In order to get an expression similar to Eq. (32) for arbitrary conditions at the spherical surface, we follow the derivation o: Blatt and Weisskopf(l9 ). In doing this, we first derive expressions for the reaction and differential - 35 - scattering cross sections as functions of the complex amplitudes of the various scattered partial waves. Then we relate these amplitudes to the logarithmic derivative of the wave function at the spherical surface. This allows us to write the cross sections as functions of the logarithmic derivative of the wave function at this surface. The wave function representing the beam of incident particles is a plane wave exp(i~·~) where ~ is a vector from the center of the target to the incident particle and ~ is the wave vector whose magnitude is given by Eq. (35) and whose direction is that of the incoming beam. For simplicity we assume the z-axis of the coordinate system is parallel to ~· It is possible to expand the incident wave in terms of spherical harmonics; thus in the limit of large kr the plane wave is given by 1 11'2 exp (i k z) = kr I co (2£ + l)t il+l {exp [-i(kr - ~11')] l=O (39) Because of the presence of the scatterer, the actual wave function is not a plane wave. However, the distortion caused by the scatterer only affects the outgoing spherical waves which go asymptotically as exp(ikr). Thus in the asymptotic region we write the actual wave function as co I (2£ + l)t il+l {exp [-i(kr - ~11')] l=O - 1.e exp [i(kr- !11')]} Y.e,o (e) (4o) - 36 where r£ is a cpmplex number that determines the amplitude and phase o~ the outgoing £-th partial wave. ~unction The di~f'erence between the true wave and a plane wave is the scattered wave 'IT given by sc exp(ikz) - h 1(~ ~ (2£ + 1)l 1 £~ (1 - r£) exp[i(kr - ~)lY.e L 0 (e) ' - kr (41) £::0 The reaction cross section is given by Na/N where Na is the number ~ o~ particles removed f'rom the beam per second and N is the number o~ particles per unit area per second in the incoming beam. The value of' N is given by the net ~lux into a large sphere o~ radius r centered a o about the scatterer. This ~lux is f'ound by integrating the probability current density, as determined f'rom the complete wave f'liDction, over the surface of' the large sphere. In this way Na is ~ound to be (42) The ~lux o~ particles in a plane wave exp(ikz) is equal to the particle velocity v. This ~act combined with the result of' Eq. (42) yields (43) f'or the contribution to the reaction cross section f'rom the partial wave having angular-momentum quantum number £ • Since a IJ must not be r,x- negative, this result can be used to show that I rtl ~ l. The contri- butions f'rom the waves having di~f'erent £ values are incoherent, so the total reaction cross section is given by - 57 - (44) (] r,t The dirrerential scattering cross section is determined by rinding the number or particles Nsc scattered per second into an element or solid angle dn about e and then dividing this by the incident rlux v. Using 'ljr in the expression ror the probability current density, we can sc equate the result ror the rlux through an element or surrace on the large sphere to the number or scattered particles to get N sc * ( -:rsc 'llr* - ~ (},Jr~c 'ljr ) r 2 dn 2iM ur sc vr sc o 0 n (e) = -.- (45) From this we rind the dirrerential elastic scattering cross section to be L 00 2 (2t + l)! (l - rt> Yt,o (e) (46) We wish to relate the scattering and reaction cross section values to the boundary condition on the wave runction at the nuclear surface. Since only the asymptotic behavior or the complete wave fUnction 'l!r(~) is given by Eq. (40), we n~J determine an expression ror 'l!r(~) which is valid everywhere outside the nucleus. We know that the " general solution or the Schrodinger equation ~ be written in the form L 00 'l!r(~) = 'l!r(r,e) = u!{r) ,.t=O We' must now determine the proper radial wave functions ut(r). ' (47) - 38I£t us de:fine linear combinations o:f F,.e(r) and G,e(r) which correspond to outgoing and ingoing spherical waves. By examining the behavior of' F,e(r) and G,.e(r) :for larger, we see that the combination (48) has the desired outgoing spherical-wave character exp [i(kr - ~ .en)] in the asymptotic region. Similarly the combination ( 49) behaves asymptotically as an ingoing spherical wave. Using the :functions u f, ( +) and u f, (- ) we can now write the radial wave :function as the linear combination (50) To :find A and B we compare the large-distance behavior o:f Eq. (50) with Eq. ( 40). This shows that and B = - 'f,.eA (51) The logarithmic derivative of' u,e(r) at the nuclear surf'ace is de:fined by (52) In addition we de:fine quantities 6..e and s..e by (53) - 39 Using Eq. (48) we can express b..e and s.e in terms or F.e, G.e, and their derivatives by _ [G_e(dG.e/dr ) + F,e(dF_e/drj 6 .e - R 2 2 G.e + F.e . (54) r=R (55) Finally, we define a phase factor s .e by the expression (56) It should be noted that all of the quantities defined by Eqs. (53) to (56) are specified completely by the conditions outside the nucleus. We can now express the cross sections in terms of f.e by relating f.e to Y.e· ~llien Eqs. (50) and (51) are substituted into Eq. (52) the desired result is Yn = k r .e - b..e + i s .e f.e b..e - i s.e exp(2isn) k (57) In order to examine the effect on the scattering due to interactions within the nucleus and to the Coulomb field and finite size, it is helpful to define amplitudes for scattering processes which occur inside and outside the nucleus. We introduce (58) - 40 - (59) where, to within a phase, A~ ~n and At are the amplitudes for internal o and external scattering processes, respectively. We note that when t the wave function vanishes at the nuclear surface, i.e. , r £ = co, Ain Using Eqs. (58) and (59) the quantity vanishes as would be expected. .R, .R, (l- yb) ;., may be written as exp(2isn) ;., (Ao +A. ). ~n Thus the differential scattering cross section becomes 2 (60) Equation (60) is not suitable for computational purposes because the long range or the Coulomb interaction causes the sum to converge very slowly. In order to overcome this difficulty, we can replace the partial-'t:ave form or the amplitude for Rutherford scattering by an Manipulation or Eq . (60) yields equivalent expression. co I dCJ d!l= (2£ + l)~ [exp(2iot) - l]Yt ,o £=0 1 '1(2 + i k co I (2£ + l)~ exp(2iot) X £=0 { 1 - exp( 215£ - 2i a£ ) [ l - ' 1 Now Y.R, ' A~n l} Yt, 0 2 (6oa) (e) = [(2£ + l)/41C]2 Pt (cos e), so the slowly-converging term 0 in the sum becomes - 41 CIO l \ (2£ + 1) [exp(2icr.,) - 1] P.,, (cos 8) , 2ikL £=0 This may be replaced by the equivalent expression ~or the Ruther~ord scattering amplitude Arter ~actoring out the constant ~actor exp(2icr0 ), the expression ~or the cross section becomes* .- ~= I z csc2 (~~) exp [-2i~ .en sin (~)] 1 CIO 1 1(2 \ iit L (2£ + 1)2 exp (2icr.e - 2icr0 ) X (61) Be~ore squaring Eq. (61) we note that cr.e - cr0 = *.e and the quantities z, k, ~' and e are again de~ined by the expressions ~ollowing Eq. (32). In addition, Eq. (56) shows that G.e (R) - iF.e (R ) exp(2i<l>_e) = G.e (R) + iF_e(R) * The equivalent expression on p. 336 o~ B~att and Weisskop~ contains .e an error in the sign o~ Ares - 42 Thus In Eq. (61) the expression to be squared has the form IT1 2 = IT1 + T2 1 ; this ~an be expanded to yield ITI 2 2 2Re(T1 T* 2 ) where Re means the real part of the bracket. = IT 1 1 2 + IT2 1 2 + Using this and the expansion for writing the product of two spherical harmonics as a linear combination of spherical harmon:lcs, the final result, after dividing by the Rutherford cross section, becomes co 2 da 1- Re [i daR= sinkz (~) exp[2iT) tn sin(~)] L\ (2.£ + 1) exp(2iWn) X k '-=0 t+L LL L co co (2'-+1)(2.£'+1) exp(2i'irt,-2i'ljr.£) x L---0 '-=0 t '=I'--Ll [(tt•ooltt•L0)] 2 {l-exp(2i~t)[l-A~nJ}*{l-exp(2i~t•)[l-A~~J}PL (cos e) (62) Equation {62) has been derived vrithout considering the spins of the interacting particles . It can be shown that when no resonances in the scattering are present, the effect of the particle spins does not alter the expression for the scattering cross section which this model predicts. Although more sophisticated models can be used to describe the scattering interaction, the simplicity of the charged-sphere model is a virtue which cannot be overlooked . In view of the satis;factory results obtained with this simple model , little additional physical insight can be gained by analyzing these data with a more complicated -43model. 3. Determining :!?!ll;. Parameters Required ~ m_ ~Experimental~ The expression which has been derived for the scattering cross section depends upon the parameters A~ and the interaction radius. Aside from the value of zero for the A~ when the wave f'unction vanishes at the nuclear surface, it is difficult to decide what constitutes "reasonable" values of At under more general conditions. in The scattered-wave amplitudes 7 t, however, do have a more apparent physical interpretation. A program to evaluate Eq. (62) was written for a Burroughs 220 computer. By using several reasonable values of 1 t, the values that gave the best agreement between theory and experiment were found by trial and error. The relatively low incident energies used in this experilllent suggest that only modi:fication of the s- and p-waves by the internal region of the nucleus should be considered. ihus the 7£ for the d-waves and higher were assumed to be the hard charged sphere result 1£ .. exp(2i~.t). We see from Eq. (43) that the reaction cross section is determined by the magnitudes of the 7/,. In order to find these magnitudes, the variation of the total reaction cross section with energy was estimated from data available in the literature and is shown in Fig. 24. The data used in constructing this figure were taken ~om references 20 through 23. Because of the large number of possible reactions, these data are incomplete. Therefore, this canposite total reaction cross section curve is probably only a lower estimate. - 44 - The first attempt to fit the experimental results was made for the incident energy of 0.500 Mev. Here it was assumed t hat only the s-wave would contribute to the reaction cross section; this determined I)'ol uniquely . The variation in the cross section due to changes in the phase of )'0 and the interaction radius was examined and the best :fit to the experimental results was determined by minimizing E = Le [a (e) - a (e)J e a (e) 2 c (63) c where the subscripts refer to experimental and calculated values. The interaction radius which gave the best fit was 3.7 :fermis, although the minimum in Eq. (63) was rather broad. For comparison, the expression gives a rough estimate of 4.0 fermis :for the sum o:f the radii o:f the two particles • Using an interaction radius o:f 3·7 :fermis, the values of J'o and )'l required to fit the experimental numbers were determined :for higher incident energies. At 0.700 Mev a slight amount o:f p-wave contribution to the reaction cross section was introduced. This required that the .relative magnitudes o:f )'0 and )' also be determined by trial and error. 1 Figures 25 through 28 show the effect on the calculated cross sections due to the variations in the various quantities. The values :for the coefficients J'o and )'l which produce the best :fit to the experimental data are listed in Table III and plotted in Fig. 29 . It must be remembered that :for bombarding energies o:f 0. 700 Mev - 45 and higher 7 the availability of' f'our parameters 7 the magnitude and argument of' both r 0 and r :~ should permit an exact f'it of' four cross 1 sections having almost any values . Thus the significant fact learned f'rom the charged sphere calculations is no·t that a f'i t is possible 7 but rather that the parameters which give a f'it vary in a reasonable way as the bombarding energy is changed . Undoubtedly the unevenness in the variation of the parruneters shmm in Table III can be attributed to uncertainties in the scattering and reaction cross sections and the relative crudeness of the trial- and-error f'i tting procedure . Although it may be possible to interpret the elastic scattering of' deuterons in terms of' a more sophisticated meuel than that used here 7 it appears that the cross sections measured in this experiment are compatible with the assumption of' scattering by a slightly-absorbing charged sphere. - 46V. DISCUSSION OF RESULTS A comparison of the present results with those given by Juric 2 and Cirilov( ) reveals that the ~ro I!!easurements are in disagreement concerning the existence of two states in B11 formed at incidentdeuteron energies of 1.16 Mev and 1.34 Mev and having widths of 70 Kev and 120 Kev, respectively. To aid in evaluating the relative trust- worthiness of these two experiments, a few comments concerning the earlier experiment seem pertinent . Juric and Cirilov used photographic plates to detect the scattered deuterons. The tracks due to these deuterons were distin- guished from those due to the reaction products by examining the lengths of the tracks. However, at certain angles of observation, one or more of the reaction products had the same range in the emulsion as the scattered deuterons. This difficulty was supposedly overcome by determining the number of each type of particle in an angular region where no range overlap occurred and then extrapolating to the region where the different types of particles had the same range. An error of four percent which they attributed to this extrapolation procedure seems rather optimistic. / / Another factor which tends to cast doubt upon Juric and Cirilov's results is the target they claim to ...1.a.ve used. They quote a value of 0.9 mg/cm2 for the approximate thickness of their beryllium target. is possible that this thickness was misprinted. However, if the stated thickness is correct, such a target would have a thickness of over 300 Kev f'or a one-Mev deuteron. It The reaction products created at - 47 various depths in. the target would have a wide range of energies; this would complicate their identification according to the length of the emulsion tracks which they produced. In addition., a target of the stated thickness would make it very difficult to determine accurate values for the energies of the deuterons when they actually underwent scattering. In view of such difficulties., it is hard to understand hmr these authors could have obtained yield curves which vary with energy as rapidly as their results indicate. A possible explanation for the 1.34-Mev anomaly which Juric , and Cirilov observed is the elastic scattering of deuterons by a carbon contaminant l~er on the face of their target. possibility was made by the authors. No mention of this Recent data for c 12(d,d)c12 scattering( 24 ) indicate that a peak in the cross section occurs at about Ed = 1.30 Mev. Although t he deuterons scattered by carbon would have more energy when detected than those scattered by beryllium, the energies of these two groups of deuterons could differ., depending upon the scattering angle, at most by 20 percent . Whether the resulting differences in track length for the yields from Be9(d,d)Be9 and c 12(d,d)C12 could be reliably resolved is open to question. 12 Judging from the variation of the c (d,d)c12 cross section with energy, it seems unlikely that the anomaly at 1.16 Mev can also be attributed to scattering by a carbon contaminant layer. However, the presence of some other contaminant, such as o.xygen, is very possible. 16 unfortunately, the o16(d,d)o cross section at low energies has not been measured, so no comparison can be made with the results of JuriC' and Cirilov. -48Since these earlier d;;.;.ta vere available at the time of: the present experiment, the procedures used in making the present measure ~cnt~ >rcrc cxo.m:I.ncd c~osely t:or any possib~e sources of error. was f:ou.Tld. None In vieu ot: the results obtained in this e>.."'Periment, it must be concluded that the a~omalies in the Be9 (d,d)Be 9 elastic scattering cross section reported by Juric and Cirilov are spurious. An unusual t:eature of the excitation functions shown in Fig. 22 is the occurrence of: cross sections which t:all below the Ruthert:ord -v-alues at lo¥T bombarding energies. This sub-Rutherford effect is interesting because·s~ilar results were observed t:or a recent reeasurement of: the Li 7 (d,d)Li 7 scattering cross section . {l) e)~ct One might that as the bonbarding energy is decreased the projectile approaches less and less closely to the nucleus and the point-charge assumption of: the Rutherford theory is increasingly well satist:ied . Thus the fact that the measured cross sections dip below Ruthert:ord is not trivial. There are two possible 'vays to account for the observed cross section being less than the Ruthert:ord vafue . It is possible that the scattering cross section really is less than Rutherford at low energies because of absorption or some obscure et:fect peculiar to the structure of: the deuteron . In t:act the previously-described charged sphere calculations shown in Fig. 25 indicate that a sub -Rutherford cross section can result when the phase ot: the outgoing partial wave is shifted somewhat from the hard- sphere value . A second possible explanation t:or the observed sub -Rutherford cross sections is that the cross section really has the Rutherford - 49 value:~ but the measured values are 101:• because o:f some instrumental or other experimental difficulty :for vhich an adequate correction has not been made. For example;, it is possible that the charge equilibrium ratios are not yet accurat.ely knovm. In view of the great care taken by Phillips in his experiment:~ this is not considered likely. Another possibility is that some contaminant;, say oxygen:~ in the surface layers of the target increased the effective atomic stopping cross section from the pure-beryllium value. T'.aen when the cross sections are calcu- lated with Eq . (21) :~ the use of beryllium stopping cross sections gives scattering cross sections \-lhich are l01:-1er than the true values. This effect is possibly important at low energies because the thickness of tee target layer from which the magnetic spectrometer selects scattered particles is smaller at low energies than at high energies. This means that the thickness of the camtaminated region constitutes an appreciable fraction of the entire thickness o:f the target layer. Although this reasoning seems to account for the sub-Rutherford behavior of the Li7 (d;,d)L17 scattering;, the amount of contaminant present in this experimert is not sufficient to explain the observed results. Ford(l) has measured the scattering cross section for Cu(d,d)Cu with incident energies such that the energies of the scattered deuterons were comparable to the lowest scattered-deuteron energies in this experiment. T'nis measurement was performed with the same equipment used in this experiment, and the cross section was found to be within three percent of the Rutherford value. This appears to rule out gross instrumental effects and ::momalous effects due to the deuteron's structure as possible causes of the sub-RuthP.rford cross section. - 50, After examining possible reasons ~or the observed below-Rutherford values of the cross section at l~r ener g ies, the most reasonable explanation seems to be that the true cross section is below Rutherford. How- ever, small contributions to the observed lrnvering from the other causes which were mentioned cannot be ruled out completely. An important point to be considered is the possible failure to observe a bona fide resonant-level anomaly in the cross section. The data sho;.m in .Fig. 22 were measured at 20-Kev intervals, so it seems reasonable to assume that any appreciable anomaly having a spread greater than 30 Kev would certainly have been observed . In view of the high excitation energy 0~ the Be9 + d-> ~l * compound nucleus, the occurrence of an anomaly in this region with a spread less than 30 Kev, although :possible, is not likely. 'fllus there is little reason to sus- :pect that in this experiment an isolated resonance may have been overlooked . It should be noted, hcr...tever, that the gradual rise in the scattering cross sections with increasing bombarding energy may be due to the :presence of many broad, overlapping levels. o~ A rough estimate the expected level density can be made by assuming the nucleus is a degenerate Fermi gas of eleven particles co~ined within the nuclear Under these conditions, it can be shown that ~or an excitation 11 energy o~ 17 Mev the level density in B should be about 25 levels per volume . Mev. 11 The existence of one or two states in B having about 17 Mev excitation is suggested by some o~ the Be9 + d reaction data. It is of interest to consider whether the absence of structure in the Be9 (d,d)Be9 cross section can be explained in terms o~ the relative magnitudes o~ the :probabilities for decay o~ the compound nucleus by various channels . - 51 - From barrier penetration considerations the decay of a B11 compoundnucleus state by emission of e particle other than a deuteron is favored because or the large Q values for many of these reaction ch~nnels . The decay by neutron enission is ~urther enhanced by the absence of any Coulomb barrier . 11 For exa~le, if a state in B with spin and parity 3/2+ is formed in the excitation region under consideration, then the conservation laliS require p-wave deuterons for its formation. ~~t this state can decay by one of more than a dozen reaction channels for which £-values less than or equal to two are allowed. An estimate of the probability of deuteron re-emission can be made by assuming that the probability of decay by a channel c is proportional to k c r c P~(k k c , r c , and P~k (kc r c ) are the wave number, interk c r c ) where . action radius, and penetration factor, respectively, which are appropriate to channel c. In this way it is found that the probability of deuteron re-emission by a 3/2+ state formed at a bombarding energy of one Mev is about 0.004. The consequence of a small probability for the emission of a deuteron during the decay of the compound nucleus is a small value for the ratio of the deuteron-emission width to the total decay width, r d/r. Since the amplitude for scattering by compound nucleus formation near a resonance is roughly proportional to rd/r, it is necessary that rd/r be large enough to produce a noticeable interference with the Coulomb scattering if an anomaly is to appear in the scattering cross section. The effect on the scattering cross section caused by a state r.3.ving a very small value of' r d/r can be estimated. Suppose that a resonant state in B11 is formed when the incident deuteron energy is - 52one Mev, that its total decay width is 500 Kev, and that rd/r = 0.01. Then a simplified calculation, neglecting spins and :finite nuclear size effects, shows that under these conditions the scattering cross section varies only about two percent from the Rutherford value . Since other effects, such as the :finite nuclear sizes, also cause deviation :from pure Rutherford ~cattering, an ancmal.y of this order of magnitude would be difficult to identifY. It is interesting to note that the excitation :function measured , at 163° 30' indicates a slight dip in the cross section near a bombarding energy of one Mev. This may be due to the :formation of a compound- nucleus state with a very small value of rd/r. A rough estimate indi- cates that the cross section deviates fran the smooth trend by about two percent. Such a deviation is consistent with the previous estimate for the value of rd/r which would be expected :fror this reaction. How- ever, the small size of this anaDal.y me~ that ita presence can hard.1y 11 be considered proof' of the existence of' a state in B • - 53 - ' VI. SUMMARY The results obtained in this experiment offer no support for 11 the existence of levels in B with excitations near 17 Mev. It has been argued that the absence of anomalies in the Be9(d,d)Be9 cross section does not rule out the presence of such levels. The lack or structure in the scattering cross section does require that rd/r for a:ny levels in the excitation region of interest be very smalL That this is a reasonable possibility follows from the properties of the ' Be9 + d system. The present ·extent of the information concerning the B11 excitation region of interest is as follows: Same or the :ae9 + d reactions suggest the presence of levels in this region. The data. from which these inferences are drawn are by no means sufficiently clear-cut to yield a:ny information about spins and parities. Recent B10 + n data ( 25)' ( 26 ) show two small ma.x:ima which are about the size of the experimental uncertainty at incident neutron energies which .,_ correspond to a r excitation just below 17 Mev. However, the B10 (n,t 2~) cross section is in serious disagreement with earlier measurements ( 27)' ( 28 ) which showed a single, pronounced maximum in the cross section. Again the sizes of the anomalies preclu:ie a:ny analysis to determine spins and :parities. Perhaps the most conclusive evidence for a level in this region comes fran a study(29) of the 0 reaction Li7 (~,n)lf • In this case the neutron yield at zero degrees shows a pronounced maximum ,at a pooitioo. where levels have been suggested previously. ' . 0 The fact that the Li7 c~,n)r experiment indicates an appreciable - 54probability for compound nucleus fonnation by the Li1 + a. channel immediately suggests that the scattering e:x;peri.ment Li7 (a.,a.)Li1 might help to clarify the level spectrwn in ~ 1 • Although a laboratory ll energy of 14 Mev for the alpha pa.rticles is required to reach the B excitation considered here, this is not unreasonable for a doubly-charged alpha-particle beam from a tandem accelerator. Furthermore, if ana.lyzable results are obtained, the zero spin of the alpha particle will sim;plify considerably the interpretation of the experimental data. ,- - 55 - APPENDIX I Sample Calculation of the Differential Scattering Cross Section ~ the E;Perimental Yield The dif'ferential scattering cross section is calculated from the experimental yield by using (21) As an example of this calculation, consider the thick-target profile shown in Fig. 17. The pertinent experimental quantities are E 113 =1.005 Mev aL = 113°26• C = 1.00 X 10-6 farads V = 9.452 volts 91 = 92 Assume that the yield is measured at Vm = 609 mv. used to find Em = E = 0.5139 Mev. 20 Then Eq. (2) is From the data given by Mozer, ( the stopping cross sections € € are obtained. 1 ( E ) = 5.23 X 10-15 113 2 (E ) = 7.64 X 10- 1 5 20 For elastic scattering with eL = 113°26' and 2 = 2.015 amu ~ = M ,Mo =~ = 9.015 amu it follows that 11 ) - 56 - By substituting into Eq. (22), it is f'ound that E = 0.9960 Mev and 1 E 2 = aE1 = 0.5269 Mev. The stopping cross sections appropriate to these energies are £ £ 1 (E ) = 5.25 X l0- 15 ev- cm2 2 (E2 ) = 7·53 X l0- 15 ev- cm 1 2 Thus (Al) . Using the previO\Wly-discussed values of' R = 302 and nL = 1.47 X 10-3 steradians, it f'ollows that ~ = 2.05 X lOll f'arads-l- steradian-l (A2) L From Fig. 17 the net yield at Vm = 609 mv is f'oUJld to be NR = 1350 - 52 = l298 For the particular bomb1.rding energy aild scattering angle being considered here, the charge-neutralization and dead-time corrections are negligible. Thus farads Mev For a pure beryllium target ns/nd = 1. Finally, by combining (Al), (A3) - 57 {A2), and. {A3) the laboratory differential scattering cross section is found to be mb sterad. The center-of-mass scattering angle is :found :from Eq. {27b) to be ec = l25°l6', and the factor for converting the value of the cross section :from the laboratory to the center-of-mass system is found f'rom Eq. {25b) to be d.nL d.n = 1.236 c Thus mb sterad. The appropriate Rutherford cross section is calculated :from Eq. { 24) and found to be Thus the ratio o:f the observed cross section to the Rutherford value is da d~ = l.l3 - 58 References J. L. C. Ford, Jr., Ph.D. Thesis, california Institute of Technology, (1962) / / Juric and Cirilov, Bull. Inst. N. Sciences ·~ris Kidrich" §: 45 (1956) "t 3· Shpetnyi, Zh. Eksp. Teor. Fiz. ("Soviet Physics") 2= 357 (1957) 4. Neilson, Dawson and Sample, Bull. Am.er. Pbys. Soc. _.2: 323 (1958) 5· Lauritsen, Lauritsen and Fowler, Pbys. Rev. _22: 241 (1941) 6. E. A. Milne, Ph.D. Thesis, California Institute of Technology (1953) 1· Snyder, Rubin, Fowler and Lauritsen, R. S. I. 21: 852 (1950) 8. Friedland, Mayer and Wiggins, Nucleonics 18: 54 ( 1960) 9· R. E. Brown, Ph.D. Thesis, california Institute of Technology (1962) . 10. Brown, S~·der, Fowler and Lauritsen, Phys. Rev. 82: 159 (1950) 11. F. S. Mozer, Ph.D. Thesis, California Institute of Technology (1955) 12. Phillips, Phys. Rev. 21= 404 (1955) 13. W. A. Wenzel, Ph.D. Thesis, California Institute of Technology (1952) 14. Foldy, Phys. Rev. §2: 397 (1951) 15 • French and Goldberger, Phys • Rev. §1: 16. F. B. Morinigo, Private Camnunication 17. W. Whaling, Class Lecture 18. Blatt and Biedenharn, Revs. Mod. Phys. 24: 258 ( 1952) 19. Blatt and Weisskopf', Theoretical Nuclear Physics, John Wiley, New York, 1952, p. 311 20. De Jong, Endt and simons, Physica 18: 676 (1952) 899 ( 1952) - 59 References Bisgcr~t~f~ Ph .D. Thooio , Univoroity of Kentucky (~96l) 21. J . A. 22. Ne ilson~ Dawson~ 23. Bard.es and Ovren, Pbys . Rev . 120: 1369 . ( 196o) 24. Kashy ~ Perry and Risser~ Pbys . Rev . 117: 1289 ( 1960) 25 . Fossan ~ Walter~ Wilson and Barschall ~ Phys • Rev. 123: 209 ( 1961) 26. Davi s~ Gabbard.~ Bonner a!!d Bass~ Nuc. Phys. g]_: 27· Frye and Gmrmel~ Pbys. Rev . 103 : 328 (1956) 28 . Wyman , Bull. .Amer . Phys . Soc . 29 . Mehta~ 30 . R. K. Bardin ~ Ph .D. thesis~ Calif'or 'n ia Institute of' Technology (1961). Johnson and Sample~ Suf'f'ield Technical Paper No. 176, Suf'f'ield Experi..mcnt Station~ Ralston~ Alberta 448 (1961) 2: 187 ( 1958) H'tmt ~ Plend1 end D!l.vis , Bull. .Amer . Phys . Soc. 6 : 226 (1961); and Tar4em Accelerator Laboratory Annual Progress Report f'or 1961~ ~orida State Univ e rsity p. 12 - 6o- TABlE I Values of the Magnetic Spectrometer Acceptance Solid Angle Measured at Various Proton Energies and Laboratory Scattering Angles (in millisterad.ians) El in Mev 158° 53' 0.6028 1.469 ± 0.011. 1.466 ± 0.01.2 1.458 ± 0.014 1..486 ± 0.01.2 0.7033 0.9042 1.206 1.472 ± 0.006 " 1..466 ± 0.009 - 61TABIE n A. The Relative Uncertainty in do/dcrR Contribution to the Relative Fractional Standard Deviation in dcr/dcrR eL Uncertainty Due to: 128022' 2.0 °/o 112°26' 2.0 °/o 17.024' 0.92°/o 0.50 0.30 0.50 0.30 1. NR 2. v 3· E20 0.50 0.30 4. €(E20) 2.0 2.0 2.0 5· cos e /cos 92 + « [€(El)/€(E2)] 1 0.24 0.40 0.73 6. E2 1 0.40 0.40 0.40 2.9 °/o 2.9 °/o 2.4 °/o Noise SUbtraction When Determining NR 8 .0 8.0 0 At the Lowest Values of' E20 , Resulting Fractional Standard Deviation f'or dcr/d~ 8 . 3 °/o 8.3 °/o 2.4 °/o Resulting Relative Fractional Standard Deviation f'or dcr/dcrR At the Lowest Values of' E20 , Additional Uncertaintl is Due to: - 62~LE II B. (Continued) The Absolute Uncertainty in da/daR Contribution to the Absolute Fractional Standard Deviation in da/daR eL Uncertaint;y: Due to: 128022' 112°26' 11024' 2.0 °/o 2.0 °/o 0.92°/o 0-35 0.35 0.35 0.50 0.50 0. 50 0.50 0.50 0.50 E20 c E20 0.30 0.30 0.30 0.30 0.30 0.30 1· eb(E20) 4.40 4.40 4.40 8. ec(E20) 3·70 3·70 3·70 9· [:cos c ·c<ElTl 2.39 2.47 2.74 0.40 0.40 0.40 0.40 0.40 0.40 0.07 0.58 1.42 8.0 8.0 0 1. 2. 3· 4. 56. Nb R Nc R vb vc b e! eb(E [:cos b a. cos 82 10. 11. 12. 1 )] e~ e (E2 ) cos s~ b +a. (Eb)2 1 (Eb)2 2 (s1.n. ie'ds1.n. iec)4 C 2 2 ec(E2) Resulting Absolute Fractional Standard Deviation for da/d~ b At the Lowest Values of E20 , Additional Uncertainty is Due to: 1. Noise Subtraction When Determining ~ - 63 ~LE II {Continued) Contribution to the Absolute Fractional Standard Deviation in do/daR eL 113°26' 2. Charge Equilibrium Ratio b At the Lowest Values of E207 Resulting Absolute Fractional ~ Standard Deviation for do/daR ,... ,.J 2~1 + 20 2s 1 2s 0 + 1.1° 2s 0 + o.5° 0 2s 0 - o.7 0.9700 0.9471 0.9323 0.9062 0.8874 0.9064 0.9100 o.BBoo 0.9000 0.8700 215.0 258.4 289.2 328.5 354·7 0.900 1.000 1.100 1.200 2s 1 2s 0 + 1.00 2~1 2s 1 2s1 + 3° 2s 0 + 1.5° ' 2s 0 + 2.5° 0.800 0.9900 0.9049 166.4 2s 1 0.700 2s 0 + 4.3° 1.0000 0.9163 2s1 arg r 1 129·3 2s 0 + 6.5° arg r 0 1.0000 r1 0.600 0.9613 ro 74·3 in mb r (} 0.500 E1 in Mev Charged-Sphere Parameters Which Produce the Best Fit to the Experimental Cross Sections TABLE III \ X 10- 6 6 105 X 10- 61 X 10- 6 6 105 X 1082 X 10- 6 28 X 10- 6 25 2.8 X 10-3 3.8 X 10-3 E -I:" 0\ -64aTt-.BLE IV ec = 90° E 1 {da/dn)c in kev in mb-ster 357.4 378.1 397 .4 417.4 436 . 9 457 . 8 476 .2 496 .9 516.9 537.7 556.6 577.0 595.4 615.8 636.2 657 .4 676.7 694.1 714.9 735 .4 755 .3 770.1 9ll 821 755 688 623 574 529 492 454 414 380 358 34o 3ll 290 275 259 248 243 228 217 215 ( da/daR) -1 0.937 0 . 91~7 0.960 0.965 0.957 0.96 8 0.966 0.976 0.977 0.963 0.948 0.959 0.972 0.947 0.945 0.956 0.954 0.961 0.999 0.992 0,.992 1.03 E1 (do/dn) in kev in mb-ster -1 794.1 815 .5 834.4 854 .4 874 .7 894 . 8 91l.4 931.6 951 .5 970.5 992 .8 1013 1032 1052 1071 1091 lll1 1129 ll50 ll70 ll90 203 196 189 184 179 166 159 156 151 148 141 139 137 132 129 127 125 121 1l9 ll7 ll2 c ' (do/daR) l.03 l.05 1.06 l.08 1.10 l.07 l.06 l.09 1.~0 1.13 1.12 1.15 1.17 l.18 l.19 l.22 1.24 1.24 l.27 l.29 1.28 -64b- TA13LE IV (continued) 0 E l ( da/dn) c in kev 399 . 8 420.1 4!~o .o 458.9 479.8 499.4 520.1 539.4 560.3 579.8 59u.8 618.6 639.6 659.9 680.0 699 .0 719.3 739.2 759.8 779.9 799.3 818.9 838.7 859.1 879.0 899.5 918.3 937.3 956.6 975.4 997.5 in mb-ster 269 235 225 208 196 176 173 159 153 141 137 128 121 117 108 106 100 96.2 93.3 88.6 81 . 4 80.5 77.3 73.0 70 . 2 65.7 65.2 61.7 59.9 58.7 56 .9 c = 125° 16' (do/daR) -1 El in kev 0.862 0.831 0.871 0.880 0.904 0.878 o.9ln 0.926 0.959 0.953 0.980 0.983 0.991 1.02 0.999 1.04 1.04 1.05 1.08 1.08 1.0~- 1.08 1.09 1.08 1.09 1.07 1.10 1.09 1.10 1.12 1.14 1017 1036 1057 1077 1099 1117 1136 1156 1177 1196 12l5 1236 1256 1276 1296 1315 1335 1355 1376 1396 1416 1434 1454 1473 1494 1515 l534 l552 1572 1592 1612 (do/dn)c ( da/daR) in mb-ster -1 54 . 8 53.3 51.4 49 .9 47 . 2 47.2 45 . 6 44 . 2 42 .7 42 .2 4o.9 39 .8 39 . 4 36 .9 36 .4 35 .0 34 .0 33.8 33 . 2 32 .3 32 .8 31.6 31.1 30 .6 30 .4 29.0 28.6 26.1 27.1 27.8 28.2 1.14 1.l5 1.l5 1.16 1.14 1.18 1.18 1.18 1.19 1.21 1.21 1.22 1.25 1.20 1.22 1.22 1.22 1.24 1.26 1.26 1.32 1.30 1.32 1.33 1.36 1.34 1.35 1.26 1.34 1.41 1.47 . .:.64cTABLE I': ( continued) ec = 163° 30' El ( dcr/dn ) c in kev 4so .1 499 . 7 52l.O 540 . 8 5G0 .1 580 . 8 6ol.5 619 o3 639 . 8 659 . 8 680 . 9 70l.6 718 .8 740 .9 760 .0 781 . 3 800 . 1 820 . 8 839 .2 860.2 880 .6 900.1 903.7 921.9 941.8 960 .8 in mb -ster 9~2.. .0 1002 1022 1042 1061 1081 ll02 ll22 l20 109 101 98.1-~ 91 .6 88 .0 84 .5 . 81.4 77.4 74 .6 73.4 67.7 64.5 6l.3 58.5 58.4 52.6 5l.5 48 .9 47.1 42.9 41.8 41.7 39 .7 37.7 35.5 34.1 33.4 32.2 30 . 7 30.0 28.8 27 . 3 26.8 ( do/ daR ) El ( dcr/dn) e ( dcr/daR) 0 . 857 o . 8t:-o 0 . 851 0 . 891 0 . 889 0 .916 0 .945 0 .963 0.976 1.00 1.05 . 1.03 1.03 1.o4 l.04 l.10 1.o4 l.07 l.06 l.08 1.03 l.05 l.05 1.o4 l.03 1.01 l.01 1.o4 l.04 l.03 l.04 1.04 l.02 1.04 in kev ll42 1162 ll82 1.201 1.220 1.240 1.260 1.281 1.300 1.320 1.341 1.360 1.381 1.4oo 1.420 L44o 1.46o · 1 .479 1.498 1.519 1.539 l.56o 1.579 1.600 1.620 1.639 1.658 l.679 1.698 1 .718 l.738 1.758 1.778 1.798 in mb-ster -1 25.2 25 . 2 25 .1 23 . 8 22 .7 22.4 22 . 2 22 . 0 21 .3 20 .9 20 .7 20 . 4 20 .2 20 .1 19.9 19.4 18 . 9 18.5 18.6 17.8 18.0 17.5 17.5 16.9 17.0 16.7 17 .0 16.9 16 .1 l5.9 •15 .6 15.2 15.1 15.1 1.02 1.05 l.08 1.06 t.o4 1.06 l.09 1.12 1.ll l.12 1.15 1.16 1.19 1.22 1.23 1.24 1.24 1.25 1.29 1.27 1.31 1.31 1.35 1.34 1.38 1.39 L44 1.47 L44 L45 1.46 1.45 1.49 1.51 -l -65 - 21~ 20 lf t) 7 d" -- 16. .f6 Be"+d 1: c: ...0>:. 11.508 11.229 Bea+I:J Be" ~ 10 8 +d-j) 8.666 13- 1r 6.8 ~ c• T 1::2.121 5.198 8'0 + t-d "+d-a 1!20 ~85 'r Ey•20 2911; f +t -a En•4.5 61" 2 .14 ,,.2m.,c' 14.1 1 to.I 57 T Ect-4 2 I 8 11 +x -x' N14 + n -a 21.6 T '( -8.02.1__ Be0 +a-d -65a- Figure l Energy Levels of Bll 11 Toe bo~bardment or Be9 by deuterons rorms B as a compound nucleus. page. The energy levels or this nucleus are shown on the opposite Tne excitation region above 15.819 ~ev '~as investigated in this experiment . Note that the deute ron binding energy in Bll is consider- ably higher than the binding energies for other light particles . -66- BEAM FROM E. S. GENERATOR 80° E.S. ANALYZER SEE FIG.3 lOt INCH VARIABLE ANGLE SPECTROMETER (160° POSITION) - 66a Figure 2 layout of the Experimental Equipment The layout of the equipment used i n this experiment is shown in this figure . Descriptions of the various components may be :found in Part II o:f the text . -67- t:::" B ~;AM -- -- TARGET I I 160° POSITION 0° POSITION Figure 3 Target Chambe1 Detail 'I1li s :figure shows the geometry o:f the target chamber and the paths o:f the incoming and scattered particles relative to the scattering target . See p. 4 o:f the text. -68- I I I - 321- - - 28 ~ - 24 - - 20 ~ - - 16 - - Cll 12 - - 8~ - C /) -o c 0 C/) ::s 0 - .c. c ·- C/) c :J 0 {.) 0 '- ..0 E ::J z - o~~~~-~~~--~'-----~~----~ 815 820 825 Spectrometer Setting ,Vm (Millivolts) - 68aFigure 4 Cu(p,p)Cu Target Profile This target profile is typical of the profiles obtained from thick copper targets. It was taken with ElB = 0.6028 Mev and eL = 158° 53. Since the momentum of the analyzed particles varies inverse~ with V , m the abscissa energy scale increases to the left. The use of profiles such as this to determine the solid angle and calibration constant for the magnetic spectrometer is discussed on p. 1 and p. 16 of the text. Incident Particles ) \ \Dead Loyer n- type Silicon 1-+-Xn Depletion Region Junction Diffused Phosphorous Xp p- type Silicon K _ _.. - ~ Voltag e +__ a·lOS > Signal I m U) I - 69a - Essential Features of the p-n Junction Detector The physical processes which occur within the detector when incident charged particles are being observed are described on p. 8 of the text. -70- - ~-s0 c -Q) 0~ -.c. E Q) 0 en ..0 (/) a.- (/) ·- 0 <( ~c '- Q) "0 0 ::r: '0 0 Q) Q) 0 '0 0 Q) Q) 0 - 70aFigure 6 Detector Holder The holder to which the detector was attached and the flange which was placed in the output port or the magnetic spectrometer are shown here. The detector was attached first to the holder, and the holder was then joined to the flange as shown in the assembly. The connector for the detector-output lead is on the back side or the flange. - 7 I- Possible Contaminant Loyer -71a- Figure 7 ~ick-~set Reaction Geometry The various energies and angles necessary ~or a detai~ed description of the reaction process are shown in this ~igure. o~ For given va~ues El.B and aL' the magnetic spectrometer determines the depth o~ the lamina from which the observed reaction products originate. For the sake o~ clarity, the width o~ the reaction lamina relative to the width o~ the target is exaggerated. See pp. 9 and ~0 o~ the text ~or an expla- nation o~ the quantities shown in this i~ustration. 'I -72- Clamping~ Screw Copper Target ' I(] lJ Beryllium Target Target Holder - 72aFigure 8 Target Holder The target holder which permitted simultaneous mounting o:f two targets is pictured here . The targets are held in place by pressure applied to their back sides by the clamping screws. The target holder is positioned in the target chamber by attaching the upper end o:f the holder to the orientation rod which extends through the top o:f the target c)lamber. See p. l7 o:f the text. - en (.) 0 ::J c 610 Ql 400 1 • IJ 630 'r • 1 0 • A~IOO ft..:t= 650 I I 690 ( 10 12 C (d ,d )C 12 730 Spectrometer Setting, Vm (Millivolts) 670 o16 ( d ,d )0 16 I 1 750 9 9 Be (d,d)Be 790 - 73aFigure 9 Pro~ile o~ the Scattering Target This ~igure shows the scattering pro~ile produced by the beryllium target and the associated contaminants. ElB = 0. 6028 ~.ev and ElL = 113° 26' . clearly apparent . For this particular pro~ile, Scattering by carbon and oxygen is The constant background beginning at V m = 592 may be due to an impurity distributed within the target which has a mass number o~ about 57. See p . 18 of the text . -74- I I I I 9~ T I I - Elastically-Scattered Deuterons - 81- - -! J) 6- - 5~ - - - 3- - c: :::> ~ '0 '- . .0 '- <! - !J) c 4 :J 0 <..) - ~ -:~rticles ~~ons_ I~ 0 I 10 I L ~ I 50 Pulse Height 30 I l 70 AA 90 - 74aFigure 10 Pulse-Height Spectrum of the Detector Output Shown in this figure is the detector output spectrum produced by the particles resulting from Be9 + d which pass through the magnetic spectrometer. This spectrum was recorded when ElB = 1. 206 Mev, 8L = 77° 24', and E 20 = 0.8104 Mev for the deuterons. Although the pro- tons and doubly-charged alpha particles which pass through the spectrameter both have the same energy, the alpha pa.rt_cles lose more energy in passing through the detector dead layer than do the protons • This means that the alpha particles produce slightly smaller pulses than do the protons. In order to decrease the amount of multichannel analyzer dead time, the low-energy noise pulses were biased out by suitably adjusting the low- energy detection level of the analyzer. the text. See pp. 19 and 20 of u ::::> 0 z ..... (/) 0 , . ..,_r ' :=be 20 40 PULSE HEIGHT I 60 I 0 Qt 1 200 01 -t 100 I u 0 ~ 3001 (/) 400 100 200t- 300 400 500 600 Noise I Noise I I I .. 20 40 PULSE HEIGHT I I l \ ElasticallyScattered Deuterons I 60 I I I I (JI .., - 75aFigure 11 Output Spectrum Produced by Low- Energy Deuterons This £igure shows a detector-output spectrum typical o£ the kind used to determine the scattering yields at low bombarding energies. this case ElB = 0.3617 Mev, eL = 77° 24', and E20 = 0.2456 Mev. In The spectrum taken with V set before the profile rise shows the noise m spectrum and the spectrum of background deuterons scattered by impurities within the target . Under the conditions existing here, the noise sub- traction is easily accomplished. See p. 20 o£ the text. -76- BEFORE I PROFILE RISE I I I AFTER PROFILE RISE I 160f-Noise - 140- - 160 Elastically Scattered Deuterons - - en 80 - - 0 u ::J 0 (.) en c: ::J c: - 60 - - - 20~ ..,. " \·•. o~--~L-~·--·~~~·~~-~~~~--~~~ 0 20 40 Pulse Height Noise - 76aFigure 12 Output Spectrum Produced by Low-Energy Deuterons This is a detector-output spectrum obtained at low bombarding energy. The relevant experimental quantities were ElB = 0. 4019 Mev, eL = 113° 26', and E20 = 0.2090 Mev. In this case the noise subtrac- tion is complicated by the substantial overlap or signal and noise. A ma:x:i.mum tmcertainty in the yield or eight percent was attributed to the si~al-noise tmrolding procedure. S~e pp. 20 and 28 or the text. -g u c 1/) I I I ;JA" • 0 I I • 0 I • 0 I 0 ~ • ~ Spectrometer Setting, Vm (Millivolts) 0 ~----~----~8~0~5------8~1~0------~8~15~-----8J2_0_______8~2~5------8~30 _______8~3~5------~ 1000 2000 3000 7 ooor I -..J -..J - 7(a- Figure 13 'This prof'ile was obtained with ElB = 0. 4220 Mev and eL = 77° 24' . The counts observed at each setting of' V have not been corrected f'or m dead-time or charge-neutralization ef'f'ects. See p. 21 of' the text. - u 0 ::J c C/) Spectrometer Setting , Vm (Millivolts) 0~~-------L------~--------~------~------~------~------~------~--~ 905 910 915 920 925 930 935 900 940 200 400 600 800 1000 1200 1400 Figure 14 This profile was obtained with E1 B = 0 . 4421 Mev, and 8L = 113° 26' . The counts observed at each setting of V have not been corrected for m dead-time or charge-neutralization effects . See p . 21 of the text. - u 0 ::l en c 0 • • • Spectrometer Setting, Vm (Millivolts) 7 L-~------~~----~~------~~----~8~6 0------~8~6~5------~87~0~----~8=7~5------~8~80~ 840 • I <D ....,I - 79a Figure 15 This profile was obtained with E1 B = 0.5024 Mev, and eL = 113°26'. The counts observed at each setting of V have not been corrected for m dead-time or charge-neutralization effects. Seep. 21 of the text. 800 u :3 0 c (/) - ·- c ~ 0 c Q> -- 0\ L. ·-00 900 905 I 910 915 • 920 925 • 930 ~ Spectrometer Setting> Vm (Millivolts) 0~--------._------~--------~------~------~--------~------~----~ 895 ----~ · - 200 400 - soor- c (/) 1000 1200·- I I 0 (X) - 8oa. Figure l6 -, This profile was obtained with ElB = 0.5626 Mev and E>L = l58° 53'. The counts observed at each setting of Vm have not been corrected for dead-time or charge-neutralization effects. See p. 2l of the text. -81- IJ) c ::J 0 L) Spectrometer Setting, Vm (Millivolts) - 81aFigtn"e 17 Be9(d,d)Be9 Target Profile This profile was obtained with E1 B = 1 . 005 Mev and SL = 113° 26' . The deuteron yield obtained from this profile is used in the sample calculation sh01m in Appendix I . In this case the dead- time and charge- neutralization corrections are negligible . Appendix I. See p . 2l of the text and -82- • • • • • • (/) c - .Q 0 L... C'l <11 80 c (1, (1, ~ L:. 1-- c (/) c ::J 0 u 620 625 630 635 Spectrometer Setting, Vm (Millivoltc;) 640 - 82aFigure 18 Be9 (d 7 d)Be9 Target Pro~ile This pro~ile was obtained with E B = 1.186 Mev and eL = 158° 53'· 1 Dead-time and charge-neutralization corrections are negligible. o~ the text. See p. 21 -83- • • • !/) c :::J 0 u 0~~~------------~~------------~----------------~----------------~------------~~ 535 540 545 550 530 555 Spectrometer Setting, Vm (Millivolts) - 83aFigure 19 Be9(d,d)Be9 Target Profile This profile was obtained with ElB = 1. 256 Mev and eL = 113° 26' . Dead-time and charge-neutralization corrections are negligible . p. 21 or the text. ' See -84- • • • • • • • • (/') c -e 0 0'1 (U c Q) (U '- .s:=. 1c (/') c :::l 0 u 0~--~--------~--------~--------~--------~--------~~ 575 580 585 590 Spectrometer Setting, Vm (Millivolts) 595 600 - 84aFigure 20 Be9(d,d)Be9 Target Profile This prof'ile was obtained with El.B = l . 366 Mev and eL = 158° 53' . Dead-time and charge- neutralization corrections are negligible . p. 2l of' the text . See -as- = t: ~ ·0 'QJ = CD .f"'o (/) .... (/) '-a.c.. ·Q) · N := := -- - O..c..c ~D... D... >< • > - QJ .Jtl! <I 0 0 ~ c 0 '- Q) :J ~ 0 at c at '- QJ E w 0 ~ at '- Q) c w co 0 -86 - - .~iri. .. "'"' N- <bvt:z) ~CJ:t (D~ " -N «)<X) <()«) "" "" N . 0~ > mr-- " i II CDU~ w .. § 0 - ~ 0 Figure 22 The Ratio of the Experimental Be9 (d,d)Be 9 Differential Cross Section to the Rutherford Cross Section The result of dividing the experimentally-observed cross sections by the Rutherford Values is shot.n here. The relative and absolute frac- tional standard deviations in da/daR are about 2.9 and 6 .7 percent, respectively, over most of the range of bombarding energies. 64a, 64b , and 64c of the text . See pp. 27, -ar- - 8 I{) I {) C\J II 0 +0 0 0 +- c 4J E .... Q) Q. >< IJJ • ... - 0 ~ ~ - r() 0 r() w <.0 - II CJ::)u • I{) 0 ~.opj.Dp • I{) d - B7aFigure 23 Comparison of the Observe d Cross Section with the Scattering Resulting from an Impenetrable Charged Sphere This :figure shows the scattering cross section calculated by assuming the target nuclei to be impenetrable charged spheres. '!he co..lcula.ted result obvi ously is increasing with energy faster than the experimental. cros s section. See pp. 32 to 34 o£ the text. -88- 4oo.-------r-------r-------r-------.-------,-------~------~ • • - V> c • ~ 0 .0 - ~ c 0 • 2 u Q> CJ) • V> V> e u • c 0 u 0 <1J 0:: 0 ~ 0 ~------~------~------~------~-------L-------L-------J 500 600 700 800 Bombarding Energy 900 (kev) 1000 1100 1200 - 88e. Figure 24 Estimated Total Reaction Cross Section for Be9 + d These data were constructed from information found in the literature . The cross sect ions for several of the reaction channels have not been measured, so no contri butions were included for these channels. Hence the composite r esult shown here must be considered a lower limit for the total rea ction cross section. guessed ,~ but The uncertainty can only be +50 percent and -20 percent seem to be reasonable limits. See p . 43 of t he text. 0.80 2!0-5° 2 4 3 Arg Y0 ,3 2!JI0° 2(0 -5 4 -· . I -·w I b "tJ 8c =163°30 1 (X) ba: B.c = 125° 16' ~ 8c =90° E1 = 0 .500 MeV - 89aFigure 25 Calculated Values of da/daR E 1 = 0.500 Mev The effect on da/d~ produced by varying the amplitude and phase o:f 7 0 is shown here . The interaction radius and reaction cross section used in calculating each curve are specified by the numerical label.s • The values to which these labels correspond are 1: R = 4.90 f, a = 73.4mb 2:R=4.90f, a = 99.6 mb r 3: R = 3.70 f, a = 73.6mb r r 4: R = 3.70 f, a = 99.6mb r The three h oriz ontal lines show the value of the experimental cross section and limits which lie three percent above and below it, respectively . See pp. 43 to 45 of the text. ~ ~ b a: 1.20 1.4 Be = 90o E 1 = 0. 700 MeV 0 Arg Y Be= 125° 16' err = 166.4 mb Be =163° 30' I I 0 I.D - 90a. Figure 26 calculated Values or dcr/d~ E1 = 0.700 Mev, ar = 166.4 mb The errect on do/daR produced by varying the phase or r the interaction radius is shown here. 1 and The phase and interaction radius used in calculating each curve are specif'ied by the numerical labels. The values to which these labels correspond are 1: R = 4.10 r, 2: R = 4.10 :f., 3: R = 4.10 r, 4: R = 3.70 r, 1 = 2s1 arg r = 2s - 30 1 l arg r = 2s + 3° 1 1 = 2s arg rl 1 + 3° arg r The three horizontal lines show the value or the experimental cross section and limits which lie three percent above and below it, pp. 43 to 45 or the text. .......- respective~. See "'0 b "0 ........ bQ: 0.90 1.20 1.30 1.40·- 4 I 2 3 Arg Y0 E 1 = 0. 900 MeV I 4 2 3 4 I ~ - 9laFigure zr Calculated Value s of dcr/dcrR 1 = 0. 900 Mev, E arg"1 1 = 2s1 + 2° The ef'f'ect on dcr/d crR produced b . varying the magnitu:les of 1 1 1 is shOYm here . 0 and The magnitudes used in calculating each curve are specif'ied by the numer i cal labels . The values to whi ch these labels correspond are 1: 2: 3: 4: j1o 1 = o.8343, 1111 1 1 = o. 1856, 1 1 11 1 0 11ol= 0.9oo8, 11 11 l1o l= o.8562, 1111 = 0.97, crr = 258.8 mb = 0.97, crr "" 300.7 mb = 0.95, crr = 258.8 mb = 0.95, crr = 300.7. mb The three horizontal l i nes show the value of the experimental cross section and limits which lie three percent above and below it, respectively. See pp . 43 t o 45 of' the text. ba:. 'U b ........ 'U 0.101 1.20 1.30 I 2Co I 2!o -5o Be= 90° I 2!o +5o I I 2Co-so Yo 2Co I I I I 2Co + 5o 2lo -5o 8c = 125° 16' 'Yo'= Ed = 1.100 MeV 0.870 2!o I I 2!o +5o 8c = 163°30' I I I N IJ) - 92aFigure 28 Galculated Values of da/daR E1 = 1.100 Mev, j70 j = 0.870 _ The effect on da/daR prod~ced by varying the ampli tuie and phase of 7 1 is shown here. 'nle values of 7 1 are s pecified by the numerical labels. used in calculating each curve The values to v hich these labels correspond are 1, 3, 5: 2, 4, 6: = 0.9121, 17 = 0.8826, 7 1 11 11 1, 2: arg 71 = 2sl 3, 4· ' 5, 6· arg 71 = 2sl + lo arg 71 = 2sl + 20 ' a = 328.5 mb r a = 398.3 mb r The three horizontal lines show the .. value of the experimental cross section and limits which lie three percent above and below it, respectively. See pp. 43 to 45 of the text. -93- J?0 & 0 0 I I; A.o I 0 0.2 \ 0 y 0 • Y.I 0.4 0 .6 - 93aFigu.::-e 29 Values of -, 0 and 7 1 whi ch Produce the Best Fit to the Experimental Cross Sections This figure shows how the b e?>t-fit values of -, the incident- deuter on energy i s changed. 0 and -, vary as 1 The numbers placed adjacent to the arrows denote the d euteron energy to which the indicated values of r p, correspond. See Table III and p. 44 of the · text.