A Momentum Spectrum for High Energy Mesotrons Citation Safonov, George Michael (1949) A Momentum Spectrum for High Energy Mesotrons. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7rfj-5032. https://resolver.caltech.edu/CaltechTHESIS:06192025-013607600 Abstract By means of a two cloud chamber technique, a differential momentum spectrum was obtained for mesotrons with momentum up to ~ 40 billion electron volts. The existence of individual particles with energies of 16 billion electron volts is found by direct measurements. We find an apparent 20% positive excess of mesotrons moving in a nearly vertical direction. A technical description of the instrument is given along with the theory of its operation. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics and Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Minor Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Anderson, Carl D. (advisor) Smythe, William Ralph (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1949 Record Number: CaltechTHESIS:06192025-013607600 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06192025-013607600 DOI: 10.7907/7rfj-5032 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17474 Collection: CaltechTHESIS Deposited By: Ben Maggio Deposited On: 27 Jun 2025 21:23 Last Modified: 27 Jun 2025 21:26 Thesis Files PDF - Final Version See Usage Policy. 13MB Repository Staff Only: item control page

A FOR MOMENTUM HIGH SPECTRUM ENERGY MESOTRONS Thesis by George Michael Safonov In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1949 The author wishes to express his deepest appreciation for the guidance and encouragement given him by Professor Carl D. Anderson. During the recent war and for the last several years, the author has had the privilege of working under the supervision of both Professors Anderson and W.R. Smythe. This association has served to inspire him to continue his education and has sharpened his sincere interest in physics. It is with pleasure that the writer acknowledges the early work in creating the instrument used in these investigations. Con- tributors to this task have been Dr. Bernard Hamermesh and Mr. Donald Glaser. ABSTRACT By means of a two cloud chamber technique, a differential momentwn spectrum was obtained for mesotrons with momentum up to rv 40 billion electron volts. The existence of individual particles with energies of 16 billion electron volts is found by direct measurements. We find an apparent 20% positive excess of mesotrons moving in a nearly vertical direction. A technical description of the instrument is given along with the theory of its operation. TABLE OF CONTENTS -PART TITLE -PAGE I INTRODUCTION II THEORY OF THE EXPERIMENTAL PROCEDURE 3 III DESCRIPTION OF THE EQUIPMENT 1 IV DISCUSSION OF ERRORS 13 V DISCUSSION OF RESULTS 17 VI REFERENCES 21 VII APPENDIX A - THE APERTURE OF THE INSTRUMENT 22 VIII APPENDIX B - DERIVATION OF THE MOMENTUM FROM THE DEFLECTION AND DEI'ERMINATION OF THE DEFLECTION CONSTANT 26 IX APPENDIX C - MEASUREMENT OF THE DEFLECTION FROM PHGrOGRAPHS 30 l - 1 INTRODUCTION Since the work of Anderson and Neddermeyer (1) in 1934, there have been continued efforts to extend our knowledge of cosmic ray momentum spectra to higher and higher energies. Le Prince-Ringuet (2), Blackett (3), Jones (4), Hughes (5), and Wilson (6) have made major contributions toward the achievement of this end. The limit of momentum resolution achieved up to the present time is about 20 Ber, that is, the instrwnents used thus far have been unable to distinguish between a particle with 20 Bev momentum and one with infinite momentum. The accurate production and measurement of a truly represent- ative particle trajectory in a cloud chamber has always been a limiting factor. This and the small curvatures caused by the magnetic fields avail- able have restricted the range of investigation. The techniques of previous workers have a point in common. All have used a single cloud chamber immersed in a magnetic field for the study of singly occurring particles. The following text describes a technique employing two cloud chambers. These are placed above and below a magnetic field. Observation of the de- flection of a particle passing through both chambers allows the computat:bn of its momentum. The fact that the chambers are out of the field provides certain advantages. It is possible to concentrate a strong field in a small but efficiently used gap without an excessively large magnet . Also, *The momentum, P , of a particle multiplied by c , the velocity of light yields a quantity, Pc , which has the dimensions of energy. This quantity is numerically proportional to the momentum and is therefore a measure of the momentum. In this text we call out momenta in terms of Pc which we give in units of BeV, billions of electron volts. From this convention, one would obtain P in wiits of c.g.s. by multiplying by .533 :x:10-13 • - 2 - the magnetic field of such an arrangement cuts out a large amount of the soft component. It is a simple matter to enclose each chamber in a thermally isolated box, thus shielding it from the distortion-producing heat of the magnet coils. Each chamber may be photographed through a lens utilizing a small angle of view, thus making possible an accurate recording of the tracks. Photography in a magnetic field is usually ac- complished with wide angle lenses set close to the chamber. This is to avoid a large gap with a consequent loss in field strength. A minor ad- vantage of the two-chamber technique is the simplicity of computation for the aperture of the instrument to particles of various momenta (see Appendix A). An apparent disadvantage is the possibility of deflecting or scattering a particle in the material separating the two chambers. This possi- bility is shown to be remote in the section on 11 Discussion of Errors." It has been possible to double the past range of investigation with such a technique; hence, particles with 40 BeV mo~entum may be distinguished from those with infinite momentum. With a higher resolution available, we have made a preliminary investigation of the high energy component of sea level radiation. - 3 THEORY OF EXPERI:MENTAL PROCEDURE The distribution of momenta among cosmic ray particles is usually described by means of a function ICP) . I(P) (1) is defined by the equation d N = I (P) • -t: • c:I A • d .Q • d P In equation (l), dN is the number of particles that pierce the area d ,q in time t . The moment um of each member of the group, d N , lies in the interval d P about P . in the solid angle d The various directions of entry are included n. about the normal to cl A . To experimentally determine I(P) , it is necessary to account for the number, N , and momenta, P , of the particles that pass through an instrument. These data and a proper interpretation of the "aperture" of the instrument permit the computation of I(P) . N and P are determined for singly occurring charged particles by means of the instrument sketched in Figure 1. G, , G2 are identical Geiger tubes held with their cylindrical G, is a distance D directly above G2 . axes horizontal and parallel. Each tube has a useful length W and a diameter T . It is noted from Figure 1 that W and T are small compared to D . The tubes, working in coincidence, can enumerate the ionizing particles that pass through the system. for each of these particles, (2 ) ~ I(P) = If it were possible to determine P L(P) could be found from equation (1) as , tn, 2 dN t: lmi dP G, , G2 is a magnetic field. The direc- tion of the field is normal to the plane defined by the axes of G, , Gz . In a region midway between The magnitude of the field is essentially uniform in the region of length L - 4(see Figure 1). C, , C2, Elsewhere the field is weak and negligible. are cloud chambers situated in the field free spaces just below G, and above G2. . C, , C2, are viewed by cameras with their lens axes parallel to the magnetic field. The trajectory of a particle detected by G, , G 2 the figure. In C, it is rectilinear. is indicated in It is continually deflected as it passes through the region of the magnetic field. Finally it passes through C2. along a linear path. The operation of the instrument is initiated by a coincidence signal from G, , G2 . This causes the expansion of photography of the tracks produced. is observed in the photographs. C, , C2 and the eventual The net deflection of the trajectory This observation together with a knowledge of the field permit a computation for the momentum of the particle. Since both N and P may be determined by the device, it is possible to determine I(P) . However the magnetic field between G, , Ga. the instrument certain selective properties. gives That is, it may now partially or completely exclude particles that would pass through it in the absence of the field. (3) Above .12. For this reason, equation (2) must be modified to l(P) 2 = x_ (_!2_) i!f t: wr dP ¥ is a factor depending on the momentum, P , and the charge of the particles considered. the instrument used. Y is also a function of the geometry of r is evidently a correction or weighting factor that compensates for the selectivity introduced by the magnetic field. The number of particles, d N , with momenta between P and P ~ d P that pass • through the instrument is proportional to ; (~9 '2. . This quantity may be appropriately termed the aperture of the instrument. The aperture for no - 5 - wr) 2 ; thus V1 would be the relative aperture. magnetic field is ( O A general equation for t as a function of the momentum and of various geometrical parameters is derived in Appendix A. The relative aperture of the instrument under the conditions used is shown in Figure 2. The magnetic cut-off value of .3 BeV has been quite well confirmed; the lowest momentum measured is .351 BeV. The relative aperture for weaker particles is seen to be less than ·fo . We note that beyond 1.3 BeV the relative aperture is sensibly constant. Thus the correction to be applied is small for momenta greater than 1.3 BeV. relative aperture varies widely. Between .3 and 1.3 BeV the Unfortunately this is in the region where it is most difficult to determine ~P ply the correction with caution. accurately and one must ap- This does not hinder the progress of the present work which is aimed primarily at the investigation of high momenta. Putting i = 1 in equation (3), does not appreciably alter the determination of I (P) • fl be the net angular deflection of a particle with charge If caused by the field, then its momentum (4) .Q. P is approximately p This equation is derived in Appendix B. f( is the deflection con- stant of the magnetic field; it has the dimensions of gauss centimeters when ..12. is in e.m.u. and P is in c.g.s. A simple and unique method of determining this constant is given in Appendix B. the experiment, P may be fo'Wld in BeV from (5) where f p-:: . 12 6 yF is expressed in radians. Under the conditions of - 6 '/J is measured from the photographs of the two chambers. The essence of this measurement involves the reprojection of a picture of the track and a fine vertical reference wire. The angle between the track and wire is determined for each chamber; the differences between these two angles is If' . A rigorous equation treating the determination of fl' stereoscopically is given in Appendix C. Figure 3 shows photographs of a typical track passing through the appa~atus. Having determined P for several tracks, we enumerated the particles according to increasing momenta. A plot of the number assigned t o each particle versus its momentum is called an integral spectrum. the integral spectrum t~ The slope of gives the desired ~ifferential spectrum. - 8 - It might be noticed from Figure 11 that at 150 amperes the nominal field strength is 12.5 x 103 gauss. Thus at four times the power it is only possible to obtain a field increase of about 25%. At this high current, the heat from the magnet coils may cause objectional distortions in the cloud chambers. The resolution gained by higher fieldsis more than can- celed by the inability to produce straight tracks in the chambers. While in operation the magnet coils were protected by a circuit breaker which could be activated by various safety devices. These in- cluded two water-flOVf switches, a water temperature thermostat, and a circuit which detected a radical imbalance in voltage across the four coils. A series of selenium rectifiers absorbed the large magnetic field energy when the circuit was disrupted. Parts of the magnet are visible in Figures 15 and 16. THE CLOUD CHAMBERS The cloud chambers C, , C2 (see Figure 1) are about 30 cm in diameter and 10 cm deep. A typical chamber is shown in Figure 18. The expansion is due to the movement of a piston which is connected to the chamber proper by means of a 1/16 inch rubber diaphragm. The chamber mixture is argon with saturated vapor of absolute alcohol. The total pressure of the mix- ture is about 8 pounds per square inch when the chamber is in the unexpanded position. The piston is held "cocked" in this position by means of a solenoid-operated mechanical latch. The discharge of some 100 micro- farads at an initial voltage of ,...., 500 volts activates the solenoid and causes the expansion. A sweep field voltage of l'V 100 volts is continually applied between the body of the chamber and an aquadag ring painted on the cylindrical glass piece (see Figure 18). This clearing potential is dis- - 9 - rupted only upon expansion. The chambers are recompressed and cocked by means of air pressure (greater than the internal pressure) being applied behind the pistons. This pressure is discontinued when the pistons are in the compressed position and the mechanical holding latch is again engaged. Expansion ratios of about 1/10 are used. Illumination during photography is due to a rapid condenser discharge through gas-filled tubes. 1/2 inch in diameter. Each gas tube is about 14 inches long and The light is concentrated into a beam by a bank of three 4 inch diameter condensing lenses. 90 degrees to its axis. The beam enters the chamber at Black velvet mounted on a grid inside the chamber forms a background for sharp contrast during the photography. Great precautions were taken to insulate the chambers from possible heat emanating from the magnet coils. To minimize this possibility and to protect the chambers from local drafts a box enclosed each chamber and its two cameras. The box was built of 3/8 inch plywood with 1/2 to 1 inch of glass wool insulation as an inner lining. ~ CAMERAS Two 35 mm recording cameras are used to photograph each cloud chamber. Thus, a total of four cameras view the two chambers. cameras is shown in Figure 17. This battery of Each camera holds 100 feet of film, 3/4 inch being required for each exposure. The film is moved by means of an air piston after each exposure; thereby being ready for the next. lenses used are 4 inch f/4.5 Ektar made by Kodak. were left open continuously at f/4.,. The In operation, the lenses The shutter action during photo- graphy is due to the rapid firing of the gas-filled flash lamps. The half - 10 angle of view subtended by the chamber is about 6 degrees. At this small view angle, lens distortions are entirely negligible. The front nodal point of each camera is about 140 cm distant from the face of its cloud chamber. The axes of the two cameras viewing a single chamber are at the horizontal level of the chamber and spaced 25 cm apart. One camera views the chamber directly down the chamber a.xis. The second camera obtains an oblique view of the chamber at a stereoscopic angle of 10 degrees. The pair of pictures thus taken give data leading to the computation of the desired trajectory angle. The measurements from which this angle may be computed are discussed in Appendix c. Beside photographing the chambers, each camera recorded a number on a mechanical counter. This number enumerates the pictures and allows the correct pairing of pictures. The film used is the fast Eastman Linograph 0rthochromatic. It is developed for about 6 minutes in Eastman D-19 developer at normal room temperatures. THE CYCLING PROCESS The operation of the instrument is completely automatic. Once the chambers are properly set and in temperature equilibrium, the following cycle is initiated by a coincidence signal from the Geiger tubes: a. The solenoids trip the mechanical latches holding the chambers cocked; thus the internal pressure displaces the chamber piston and an expansion results; b. The displacement of the mechanical holding latch opens a switch which· cuts off the sweep voltage; c. The switch mentioned in b. starts a two-minute timer. Two - 11 minutes is the "dead time" of the chamber. During this interval, no coincidence detected by the Geiger tubes may set off a cycle. d. After the /\/ 1/40 of a second taken for the mechanical latch to experience its full displacement, a second switch is engaged which causes the flash lamps to illuminate the scene of the track; thus the photography is complete. e. In the following 15 seconds, the chambers are reset and the clearing potential is again applied. f. At this same time (as event e.) the films in the cameras are rolled into a position awaiting the next exposure. Also, the mechanical counter indicating the picture number is changed. g. Finally, approximately 2 minutes after the expansions, the dead time mechanism again permits coincident counts to initiate a cycle. THE PROJECTION AND MEASUREMENT Figure 19 shows two projectors which throw a full size image of each chamber onto a table. The black rectangles represent the Geiger tubes and the region occupied by the magnetic field. These are in their full scale positions relative to the projected pictures. rapid preliminary survey of the pictures. This arrangement allows a Accidental tracks which obvi- ously do not belong to each other in the paired pictures are discarded. Having made a preliminary survey, the numbers of the legitimately paired pictures are recorded. Finally, each picture is projected onto a flat which may be rotated about a vertical axis. One meter from this axis is a scale marked every mm - 12 along a one meter arc. On this flat is a family of parallel lines. The flat is first rotated until the image of the vertical reference wire is made parallel to the family. The scale reading is recorded. The family is then made parallel to the track and the scale reading is again recorded. The difference of the scale readings is the inclination of the track with respect to the vertical. The difference between the inclinations found in the two chambers is the approximate angular deflection due to the magnetic field. - 1.3 - DISCUSSION OF ERRORS The measurement of the momentum. of each particle is subject to certain errors. These may be classified under two main headings. One con- siders all the factors that limit the precise measurement of a trajectory that was undisturbed by other than the magnetic field. The second con- siders the possibility of an alteration in the trajectory by other means. An estimate of the nominal error in measuring the angular deflection of a trajectory shown in photographs of the two chambers is radians. ~ <.p ~ .00.3 This estimate includes the net effect of many factors such as chamber distortions, the overall process of photography, development, and projection for measurement, and the actual measurement itself. The main lack of resolution appeared to be due to one of the two chambers. This chamber yielded distortions of the same character independent of its position. Were both chambers of a quality equal to the best, it is felt that the nominal error in the deflection could have been halved. The error in determining the deflection constant of the magnetic field is estimated at about 1%. The total error in momentum may then be computed from equation (4) which involves only the angular deflection and the deflection constant. Let P be the momentum calculated by use of (4) from the measured deflection If . Due to an error 4 ~ in $" , the true momentum could actu- ally lie in an interval bounded by (6) and (7) P, ::: P [ I - 4.tp ...Q. K p ] -1 - 14 If we express P in units of energy (BeV), it is necessary to write .126 for .Q.i( in (6) and (7); this corresponds to the deflection constant K =4.2 x 105 gauss • cm which was used. A plot of P, and in Figure 4. P2, is given Since the deflection constant is known relatively accurately, the uncertainty in P due to measurements on each particle trajectory is quite well represented in Figure 4. For example, it is shown in that figure that a particle measured as 15 BeV momentum might actually have a momentum somewhere between P = 23.3 BeV and P =11.05 BeV. If we regard the useful limit of the instrument as being that at which the error in the measurement of (f) is equal to (,') , this limit corresponds to 42 Bev. The second type of error mentioned is due to an alteration of a particle's actual trajectory by causes other than magnetic deflection. Such an alteration would be due to the scattering of the particle as it passed through the material between the two chambers. We treat this problem by answering the question, "What are the chances that a particle will scatter so as to yield a deflection corresponding to a momentum than its true momentum, P ?" r times greater That is, we compute the probability that a net deflection of amount ( 'f - l) IP/; will result due to scattering alone. It is first necessary to decide whether single, multiple, or intermediate scattering must be considered. Janossy (7) gives a criteria by which one may determine the type of scattering involved. computation of This requires the N~r'. Here N is the number of scattering nuclei per square centimeter presented to the incident particle. t"' is the impact parameter corresponding to a Rutherford scattering resulting in a deflection of amount ( ~ - l) ~/~ . If this product is less than or equal to one, we have single scattering; if it is greater than one, we must consider intermediate or multiple scattering. The relationship between ( ~ - 1) 'f/f - 15 and ,... is (8) In (8) ~ is the atomic number of the scattering nucleus; .a. is the charge of an electron. (8) is written for particles moving with veloci- ties close to c. Substitution of (4) and (8) into the product N.,,- r'I. gives Above "10 and r-0 are the mass and classical radius of the electron respectively. For the apparatus used, N = 1.08 • 1023 cm- 2 and r""' 10; these values in (9) give 2 (10) N.,,..,-'L = I. a (er~,) · /O -4 For <; = 1.1, the above product is 2.2 x 10- 2 . larger values of f Hence, for this and , the problem is one of single scattering. (10) is the fraction of a square centimeter presented to an incident particle which will scatter this particle by an amount ( f - 1) 'P/'> or more. It is, therefore, the chance that this particle might be mistaken for one of momentum equal to or greater than ) P . For f = 1.1, this chance is about one out of fifty. Hence, we con- clude that one out of fifty particles could scatter so as to give a momentum incorrect to 10%. Only six out of a thousand will scatter so as to give a momentum in error to 20%. the momentum considered. These conclusions apply independent of - 16 Reference to Figure 4 reveals that individual measurements on 98% of the particles with momenta up to 5 BeV are limited by scattering to an accuracy of 90%. Beyond 5 BeV the limitation is due to the inability to measure a deflection to better than 3 millradians. It might be stated that 266 out of the 330 particles considered had apparent momenta less than 5 Bev. -uDISCUSSION OF RESULTS A total of 330 cases are considered. 197 of these are positive particles, 132 are negative, and the sign of one particle could not be determined. These particles are enumerated in the order of increasing momentum in Table II. The excess of positives divided by the total gives .2. This positive excess lies intermediate to the values reported by previous workers (see Table I). It is considerably higher than the average. This difference could be attributed to statistical error until the number of cases is increased. If this large excess is not of statistical or instrumental* origin, it might be due to a unique selection characteristic of the device. That is, the present instrument differs from most single chamber types in that particles are received through a narrow and well defined aperture. The aperture defined by the two counters is 8.67 x 10-2 cm2 • steradians. Only a few particles inclined more than about 5 degrees with the vertical can be detected. If the positive excess of energetic particles is a function of the zenith angle of entry, this work is consistent with a larger excess for nearly vertical incidence. A second major instrumental difference is the method of reducing the soft component. Single chamber techniques depend mostly on lead absorbers to limit the soft component. The present technique cuts out all particles with momenta less than .J BeV by means of the magnetic field. Only parti- * Instrumental errors could be due to a malalignrnent of the Geiger counters or of the illuminating beams. Such dissymmetries could favor the detection of positive or negative particles. An earlier run with an entirely independent setting of the counters and the lights considered "" 100 cases. These gave a positive excess of - .2. - 18 - cles above 1.3 BeV "see" an aperture essentially equal to the no field aperture (see Figure 2). It is interesting to note that those workers who used lead to filter the soft component obtained the highest positive excess values. Figure 5 presents a block curve of the differential momentum spectrum of all particles. This is a plot of the number of particl es found in each .5 BeV momentum interval. The smooth curve is derived from the slope of the integral spectrum* for the particles. the gross trends of the block curve. It appears to represent fairly The instrumental maximum occurs at about .85 BeV; the instrumental maxima of other workers are given in Table I. Figure 6 is a plot of the differential spectrum on log-log coordinate paper. This indicates that the high energy portion ( P> 2.5 BeV) of the p-2.J spectrum is well approximated by a function of the form I(P) • 0<. Beyond 2.5 BeV, the relative aperture of the instrument does not di_ffer from unity by more than 3% (see Figure 2). Thus this power law is not sub- ject to appreciable correction due to aperture effect. The power law ex- ponent -2.1 is intermediate to those derived from the data of other workers (see Table I). The values listed for other experimenters are derived in In some most cases from a replotting of data presented in the journals. cases the determinations may have been crude due to the inaccuracies of transplanting data from the small figures of the publications. The com- parison must be regarded as very approximate. If the "tail" of the spectrum is to assume the form of a power law with the same exponent holding to infinite energies, the exponent must be * The integral spectrum is a direct plot of the data in Table II. - 19 less than -2. otherwise the integral for the total energy carried by the radiation diverges . The results _o f the present work would be consistent with a power law behavior to infinite momenta. Several workers (see Table I) obtain data inconsistent with this possibility. It should be stated that a power law behavior with an exponent less than -2 is by no means a necessity. It is entirely possible that at some high momentum the character of the spectrum could change or cut off so as to give a convergent integral beyond this momentwn. For lesser momenta any or no power law could hold. Figure 7 is a plot of the differential spectra for the positive and the negative particles. These are shown in block form; they exhibit sim- ilar behavior in their gross trends. There exists a probability that the few particles which show small deflections (and consequently appear to have large momenta) are in reality low momentum particles which have been scattered. The chance that a particle with true momentum P will scatter so as to yield an apparent momentum P ' = ff P or greater is given by (10). Let us denote this by C ; whence p' -4 )~ C :: l8 ( P'-P (ll) • /0 If the true differential spectrum of the particles is the number of particles between into the range c12J 00 I P, and = ,.a· ,o◄ ~ ' , then "1, ( P >~ >P,) that will scatter > P > P' is " ~ Y--,,(S (;;) dP - 20- If we assume that the integral spectrum obtained is approximately correct, we may compute n from it. to have momenta greater than 25 BeV. For example, 12 particles appeared By use of (12) we compute that there is a chance that 1/10 of a particle from the remaining 318 with momenta less than 25 BeV could have been scattered in such a way as to be mistaken for one of the twelve. Indeed the scattering would have to be in a plane nearly normal to the magnetic field to be SQ interpreted; thus, the chances are even smaller. We see from Figure 4 that a particle of 16 BeV momentum could be mistaken for a 25 BeV particle due to the error on measurement in ff . Particles with lesser momentum could not be misidentified as 25 BeV particles. Thus the existence of single particles with momenta and energy of 16 BeV is quite certain on the basis of these direct measurements. If the true spectrum cuts off at some momentum, this cut-off must be greater than 16 Bev. The particles for which the spectra have been determined are most probably mesotrons. No heavily ionizing tracks were observed; so it is not likely that protons were present with momenta less than .7 BeV. This is the value of momentum below which it is possible to distinguish mesotrons and protons by specific ionization. The possibility of an appreci- able electron component is slight as these particles tend to produce showers; only single particles were considered. Rossi (8) states, "For the determination of the momentum spectrum of mesons at sea level from cloud chamber pictures of cosmic ray particles in magnetic fields, one is justified in considering as mesons all penetrating particles which cannot be shown to be different from mesons by their specific ionization." - 21 REFERENCES (1) c. D. Anderson ands. H. Neddermeyer, Int. Conf. Physics, London, (1934), 177 (2) L. Le Prince-Ringuet, J. de Phys. et Rad., (1937), 8, 207 (3) P. M. S. Blackett, Proc. Roy Soc., (1937), Al59, 1 (4) H. Jones, Rev. Mod. Phys., (1939), 11, 235 (5) D. J. Hughes, Phys. Rev., (1940), 57, 592 (6) J. G. Wilson, Nature, (1946), 158, 415 (7) L. J.kossy, Cosmic Rays, Oxford Press, (1948), 98 (8) B. Rossi, Rev. Mod. Phys., (1948), 20, 537 - 22 - APPENDIX A THE APERTURE OF THE INSTRUMENT It has been mentioned that the presence of the magnetic field introwith duces selective properties into the instrument. Particles lower than a ~ critical momentum are completely excluded from detection. Particles with higher momenta are received through a smaller effective aperture. This selectivity is, of course, an instrumental bias which must be removed in the final accounting of the data. Figure 8 presents a series of sketches which illustrate the effect of the magnetic field on particles of different momenta or radii of curvature I' . Figure 8(a) shows the trajectory of a particle with the minimum radius _r'0 • Particles with a lesser radius cannot be detected; hence ;00 is a "cut-off" value for ;<:J • Sketches 8(b), 8(c), 8(d) describe particles of radius somewhat greater than .Po . It is evident that such particles can penetrate the top counter in a limited region and be detected. Let ~ be the distance from the left end of the top counter and the point of penetration as shown in 8 (c). }. may assume values between zero (see 8 (b)) and )° • f is the maximum value of x possible for particles with the radius considered' (see 8(d)). Also indicated is the angle, .a&, , between the extreme directions of entry at a station A • 8(c) shows the nature of the limit- ations imposed on the angles of entry, ~ , at each station X • Figures 8(e), 8(£), and 8(g) demonstrate the behavior of particles with a radius .,,o, . A particle with this particular radius enters the top counter vertically at its left end and emerges from the right end of the bottom counter (see 8(g)). The geometrical constraints limiting e; have - 23 the same character for ,P between l°o and /', . For ;a greater than I° 1 , a different type of constraint is imposed on &, . Some typical trajectories for particles with such a radius are shown in 8(h), 8(i), 8(j), 8(k), and 8(1). For X between zero and the extremities of the bottom counter indirectly limit 8 (i), 8 ( j)). At 'X • a.. Gl. , G, (see 8(h), , a particle entering with a maximum e-; , emerges vertically from the left end of the bottom counter (see 8(j)). For X greater than Gt , the limitations on &, resume the form imposed on particles with ;(b ~;C~~ (see 8(j), 8(k), 8(1)). Figure 8(m) shows the behavior of particles with an infinite radius of curvature. The integration of equation (1) with respect to dA and d!l.. is carried out over the top counter. Referring to the dimensions shown on Figure 1, d fl and d lL are given by ~ &; (13) dA:: ,d-x, (14) d.I2.:l:A&, D is given as a function of X below the sketches of Figure 8. The dependence given applies to the class of radii pictured in the accompanying sketch. Also, the limits within which -X may vary are written on Figure 8. .6. <9j and the limits on X are computed from the first order approximations of some rigorous equations. Quantities of order higher were neglected in reducing the precise equations. The aperture integral is (15) JdA d.n.. ~ ~ and - 24 - Above X, and X~ are the appropriate limits of integration for any particular momentum or radius considered. The relative aperture, I V is obtained on dividing (15) by ( ~r) ;,hich is the no-field aperture for all particles. (16) I The results of the computation for y are presented below. I'; ~ I° ~ ,.,0 I For -r - 6E°t I I and(l7) For /0, ~p~ I r = The quantities € , ~ I- 2. z ( ~ ) ( "/ - 1) ( 2 7+I) 00 ~ ( 3 - 2 E) (;') , ,P, , and "? are defined by '- (18) E -= (19) ~ (20) t° I ::. DL '2.W (21) 12-= I+ 2E : 0 2 D L. ( 2.D -1..) 8W (2. ;, -1) -,.. E'2... The following properties of equations (16) and (17) are noted: (a) ...!... ➔ '( (b) -11'I -- ...!..2. +.l.f 3 I (c) t (d) d(¥) ➔ d('°) P, as /0 '➔ r' 0 for f' -c.,.C, I as ,0 ➔ ::. 0 for ,,0 :: /'6 0 oc:, and r' ➔ 06 - 2.5 (e) The slopes of the equations (16) and (17) are equal at the point ( I' =I', ) separating their regions of applicability. This is 2. I - :, E Figure 2 is a plot of the relative aperture computed by inserting the dimensions of Figure 1 into (16) and (17). It may be noted that the equations for ~ asswne a simple form for L =D • That is, for the case of the counters placed directly above and below the field. This corresponds to the maximum attainable aperture. The equations, of course, are valid only when the vertical length of the field is much greater than its width and breadth. - 26 - . APPENDIX B DERIVATION OF THE MOMENTUM FROM THE DEFLECTION AND ------ - - ----- ---DETEfiliIINATION OF THE DEFLECTION CONSTANT The momenta of the particles are determined from their observed deflections caused by a magnetic field. It is assumed that the field is unidirectional; that is, everywhere parallel to a horizontal line. It is further assumed that the field strength is solely a function of a vertical space coordinate. The field has been explored by Mr. Donald Glaser, and this assumed behavior was found to be approximately correct. Under these conditions, the magnitude of field strength at a depth 1 below a datum level may be expressed as B<!) . P Figure (9) shows a particle of momentum and charge ..Q. moving in a plane normal to such a field. The field above and below the two dotted lines is zero; thus the particle trajectory in these regions is rectilinear. Let ~ be the angle between the direction of motion at depth 1 and the vertical as shown in the figure; <9; and Crz. being the values assumed by ~ in the regions void of magnetic field. The radius of curvature, j) , of the trajectory at depth .R is given by (22) but (23) I to Substituting (23) in (22), there results _ ~ e-- d& -: ~ 8(/)d.R - 27 - Solution of the above for ..E. yields JL p f Bc_t) ol.P -:. K or (24) where K, the "deflection constant" of the field, has the units of gauss centimeters in the e.m.u. system. Thus the momentum or magnetic curva- ture of a particle is directly calculable from <9, , <""z. and the deflection constant t( • ~ and ~ are obtained from the cloud chamber photographs. K.. is a function of the magnetic field strength and hence depends upon the magnet current, I , which produces the field. f((r) could be determined from a complete knowledge of the magnetic field strength as it depends on J) for each value of I . However, it is the integral of the magnetic field that is of interest here, rather than a detailed knowledge of its distribution. The following technique is devised to deter- mine this integral. First it is noted that a perfectly flexible wire carrying a current . and under tension T assumes a radius of curvature ;o such that . <- (25) _J_ ::. fJ <, T BC/) But (25) is precisely equation (22) with the ratio i:=- appearing in place of ~ . Hence it is evident that if such a wire enters the magnetic - 28 ~ field at an angle and emerges at an angle the deflection con- ~2. , stant of the field may be obtained by means of K(I) (26) ~ '. ( ~ 0-; -~~~) (, An actual determination of K(I.) is made using the arrangement shown in Figure 10. A current, . , is made to flow through the thin wire circuit ABCD. ~ Points A and Dare held fixed. Weights, W, are hung at points C and B. - The branch -BC is loosely coiled wire; the branches AB and DC thus emerge vertically from the magnetic field. The arrangement yields data for two - determinations of the deflection constant, one for each branch, AB and DC. I The angles &, and &,' are measured from photographs. The above technique neglects the mass and rigidity of the wire. mass of the wire is one-thousandths part of W. The The change in internal stress energy of the wire due to its curvature is computed. This is found to be less than one forty-thousandths of the change in potential energy i . Thus the assumptions of Was the current is increased from zero to of a massless and perfectly flexible wire are not unreasonable. Copper wire .005 inch in diameter is used in the experiment. Wis 100 grams; i. is about . OJ amperes. The arithmetic mean of the two determinations of a function of I in Figure 11. tive average magnetic field. l<(I) is plotted as A separate ordinate scale shows an eff ecThis is obtained by dividing K. by 40 centi- meters, the vertical length of the pole faces. The curve, as expected, has the form of the "knee" of a hysteresis curve. If the error in K. is measured in terms of the dispersion of points about the curve, this error is about 1% in the region above the 11lmee 11 • It is estimated that the - 29 greatest source of error in the determination of K. is due to the measurement of &, from the photographs. It should be mentioned that the technique is suitable for the determination of magnetic field strengths in other apparatus. Where the cloud chamber is immersed in a magnetic field, the actual curvature of the wire may be photographed and measured.* In all cases, the desired information is obtained from a "sample particle trajectory" in the form of a wire carrying a known current and under a !mown tension. * This has been done by R. V. Adams for a one-chamber instrument with results in good agreement with other techniques. - 30 - APPENDIX C MEASUREMENT OF THE DEFLECTION FROM PHOTOGRAPHS The particle shown in Figure 1 moves in a helical path through the region of the approximately uniform magnetic field. Let a rectangular coordinate system be centered at the point of the particle's entry into this field. The ~ a.xis is taken along the direction of the field; in Figure 12, this axis is normal to the plane of the paper. The ?C and ~ axes are oriented horizontally and vertically respectively as shown in the figure. (28) (29) The equations of the helix are 'a,- :: /0 [ ~ ~, - ~ &-J [ u - ~ ~]. -t In the above, ,,o is the radius of the cylindrical surface containing the helix. v-- is the speed of the particle. t: is the time measured from the instant of entry into the field. t r ~ o is the component of lr along the direction of the field; thus € is the pitch angle of the helix. , as before, is the angle between the tangent to the projected &- curve (projected on X a-' plane) and the 'a' axis. 9- and -t are related according to (30 t: The inclination of the trajectory with the X~ plane is measured by (31) - 31 Since (9,- is small, ~ is given approximately by (32) Thus the trajectories of the particles detected are very nearly plane curves. The planes are parallel to the X axis. jectory plane and the X 'r plane is The angle between a tra- o , the pitch angle of the helix. In the experiment performed, it is the component of a particle's momentum parallel to the .ic ~ plane which is determined. &; and ~ may be determined from photographs of the particle's trajectory in the two cloud chambers. ~ and ~ are then substituted into equation (24) to compute the desired momentum component. The total momentum is obtained by dividing this component by u-ci o . Since 8 cannot exceed 2.0•l0-2 radians (see Figure 1), the total momentum cannot differ by more than 2.0•10- 2 percent from the computed component. Each cloud chamber is photographed by a pair of identical cameras. The cameras are held fixed with their films coplanar; the common plane being perpendicular to the magnetic field. a thin wire held vertically. All four cameras photograph The image of this wire establishes a true vertical reference on each film. ~ is determined from a pair of photographs of the top cloud chamber. The "pinhole" camera image of a straight line is the intersection of a plane determined by the line and the "pinhole" with the plane of the camera's film. The actual angle measured in each photograph is that be- tween the images of the fine vertical reference wire and of the track (see Figure 3). The angle of interest, &, , is that between the projec- tions of the vertical wire and the track upon the film. The measured angle and the desired angle are not generally the same. This is due to - 32 the two distinct methods of generating the directions in space - one by pinhole photography (i.e., projecting through a point) and the other by direct perpendicular projection onto a plane. In any case we are interested in comparing angles between directed lines established by different constructions. A common feature of the two constructions is that each involves the intersections of certain planes with the film plane. A second feature in common is that all of the planes which intersect the film plane contain the track. As far as establishing directions in space we may as well consider the intersections in a plane (plane P in Figure 13 (a)) parallel to the film and passing through the front nodal points of the two cameras. Let us denote these nodal points as o, and 0 2 (see Figure 13 (a)). Let T be the point of intersection of the extended track with plane P . Line TV is the intersection of a vertical plane through the track with plane P . Line W is the intersection of the plane through the track with P which is normal to P . The desired angle, ~, , is <9-I = L v,D The angles we measure on the film are "YI = LV'T0 1 "\JI ::.. and 1 t. L V"T O~ "" - 33 If S - be the distance o, Oz. , then it is easily shown (see Figure 13(b)) that to-- I [ §J. • ~ ("f, ~ "/'~) ] C.0-0 --Y,.._ Y, -,.. (33) s-, : or (34) (9,,- - Y.'?. +- ~ _, [ I In (33) and (34) S 1 = o, Q 1 and intersection of a line through ;~. ~ CY. -'¥-"l..)J ~ t; <\ = O?. Q2 • Q 1 is the point of 0 1 and normal to To, a similar point for camera Ot • of measuring s with I D • ~ t. is (33) and (34) would provide two means ~ in terms of available angles and distances if cS, and cS 2. could be established. If E, , El are the distances of closest approach of the image of the track to the film centers and f is the distance from the nodal points to the film, then (35) e-, ::: or (36) e;- -:. - Q, ~, and ~'l. ~,. ta_-• [ €,&,. ~tY.-Ytl "I'; .... U1 ~,._ -f & '"'t'"l ta_ ~ I [t:1. "£ 1. ' ~ ('¥'. .. Y3,) 't- -Fs are perpendicular to of the track intersected by Q, i:-, ] , ~t. 'ei • P ; <:..+-,, 'If, J ~, , ~-z. are the points In the above ::. It is noted that Y, and are observable from film @ • £, are observable from film 0 ~, or ; '1ft. , E'i t 2 may be obtained by utilizing data - 34 from both films simultaneously (such as by stereoscopic projection). f and S are constants. Let the pictures be projected through the same lenses used in photography. is The film is f- distant from the rear nodal points and the screen £ • distant from the front nodal points of the lenses. tances ¢0 1> Then the dis- ~ of closest approach of the two projected track images are ~o d, So I ':::. ?Jo't. -- ~t,. ~2. t, r, The error introduced in &; by using found by taking the differential of 6 o, in (33) instead of 6, is $; in (JJ). This is evaluated under the most pessimistic conditions; it is d9-1 ~ 10-5 radians. Hence one is completely justified in using c5 o I for 6, in (33). This means that simple projection of each camera's picture will provide ample data to determine ~, to 10-5 radians. The necessary data are "r, , Y1. , The error in equating ~ to "r, is found from (33). under the most pessimistic conditions is .0015 radians. &o I and S . This evaluated This may be ig- nored until cloud chambers with lesser distortion are available. INSTRUMENTAL MAXIM1JM OF DATE, WORKERS AND REFERENCE RESOLVING POWER (BeV) NUMBER OF CASES CONSIDERED DIFFERENTIAL SPECTRUM (BeV) APPROXIMATE P01/IER POSITIVE LAW EXPONENT EXCESS FOR HIGH MOMENTA MEANS OF LIMITDm SOFT COMPONEN'I . 5 78 ~.85 -1.7 • 08 1937 LE PRINCE-RINGUET (2) - 20 ~ 300 ,., 1.0 --- .26 Lead 1937 BLACKErT (3) ~ 20 ~ 800 -- .85 -1.9 .01 ---- 1934 ANDERSON AND NEDDERMEYER ,v ----:.:a - (1) I U1 1939 JONES - 10 923 ,... .9 -1.9 .13 Lead 1940 HUGHES --- 16 674 - 1.0 -2 .. 5 .1 Lead 1946 WIISON ,... 20 - 600 - .75 -1.8 --- Lead - 40 330 - .85 -2.1 .2 (4) (5) (6) 1949 (PRESENT WORK) Strong Magnetic Cut-off TABLE I I - 36 TABLE II ENUMERATION OF PARTICLES IN THE ORDER OF INCREASING MOMENTUM PARTICLE NUMBER 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 MOMENTUM (BeV) .351 .390 .423 .438 .490 .500 .532 .540 .543 .568 .578 .600 .600 .606 .610 .630 .640 .643 .647 .650 .657 .663 .667 .674 .674 .680 .700 .703 .729 . 732 .732 .740 -758 .764 .764 .764 .773 .773 .783 .792 .823 .828. .835 .845 .857 .863 .868 .876 .881 .887 PARTICLE SIGN . . .. + . . 1' -t 1' .. 1' .. .. . .... . . 1' 1' . . NUMBER MOMENTUM (BeV) 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 13 74 15 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 .893 .893 .906 .906 .919 .940 . 948 .962 .985 .990 1.01 1.02 1.03 1.06 1.07 1.07 1.10 1.10 1.11 1.11 1.11 1.11 1.12 1.14 1.14 1.15 1.18 1.18 1.20 1.20 1.21 1.22 1.24 1.24 1.24 1.25 1.26 ·1.26 1.26 1.26 1.30 1.30 1.33 1.33 1.33 1.34 1.34 1.34 1.34 1.34 SIGN . -t .. ... . .. 1' 1' .. -t . . -t . .. ... -t -t "t "t .. . "t - 37 TABLE II (continued) PARTICLE NUMBER MOMENTUM 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 1.50 1.34 1.34 1.37 1.38 1.38 1.40 1.40 1.40 1.42 1.42 1.43 1.43 1.45 1.46 1.46 1.48 1.48 1.52 1.52 1.52 1.54 1.57 1 • .57 1.60 1.62 1.64 1.66 1.68 1.70 1.70 1.7.5 1.77 1.77 1.77 1.77 1.83 1.83 1.83 1.83 1.8,S 1.85 1.85 1.85 1.88 1.88 1.88 1.91 1.94 1.94 1.94 (Bev) SIGN ~ + 1' + + -t + + + + .,. .,.-t + -t + + + . . + + -t + . . + + -t + + . -t . + PARTICLE NUMBER MOMENTUM (BeV) SIGN 151 152 153 154 155 156 157 158 1.59 160 161 162 163 164 165 166 167 168 169 • 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 18.5 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 1.97 1.97 1.97 1.97 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.03 2.03 2.06 2.06 2.10 2.14 2.14 2.21 2.21 2.21 2.21 2.21 2.29 2.29 2.33 2.38 2.38 2.38 2.43 2.47 2.47 2.47 2.52 2.52 2.52 2.57 2.57 2.57 2.57 2.57 2.57 2.62 2.62 2.62 2.62 2.62 2.68 2.68 2.68 + + + -t + + + .,. + . -t + + .,. ... ..,. . + + -t .. .,.1' .,. .,. .,. . .. . . + + - 38 TABLE II (continued) PARTICLE MOMENTUM (BeV) NUMBER 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 2.68 2.68 2.68 2.74 2.74 2.80 2.80 2.80 2.93 2.93 3.00 3.00 3.01 3.01 3.01 3.15 315 3.15 3.15 3.23 3.32 3.32 3.32 3.32 3.41 3.41 3.41 3.41 3.50 3.50 3.50 3.60 3.60 3.60 J.60 3.60 3.71 3.71 3.71 J.71 J.82 3.82 3.94 3.94 4.07 4.07 4.07 4.07 4.20 4.20 SIGN 1' . . 't + 't 1' 1' 1' 1' 1' 1' 1' -t 1' 't 't 1' . 't' 1' ... 1' 1' 1' 1' 1' PARTICLE MOMENTUM NUMBER (BeV) 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 . 297 298 299 JOO 4.20 4.35 4-35 4.35 4.35 4.50 4.67 4.67 4.85 4.85 4.85 5.0 5.0 5.0 5.0 5.0 5.2 5. 2 5.7 5.7 5.7 6.0 6.o 6.0 6.3 6.3 6.3 6.6 6.6 6.6 6.6 7.0 7.4 7.4 7.9 1.9 7.9 7.9 7.9 7.9 8.4 8.4 8.4 9.0 9.7 9.7 9.7 10.5 11.5 11.5 SIGN . + + + + 1' 1' . 1' 't -t ... .,. 1' 1' 1' 1' ... 1' + 1' 1' .. ... ... ... 't , - 39 TABLE II (continued) PARTICLE MOMENTUM NUMBER (BeV) 301 302 11.5 13 13 13 13 13 14 303 304 30.5 ,306 307 308 309 .310 311 312 .31.3 314 .315 316 .317 318 .319 320 321 322 32.3 324 325 .326 327 328 329 330 14 14 14 14 16 16 16 18 18 18 18 25 25 25 25 25 32 42 42 42 >42 1>42 t:-42 SIGN + . + -t . -t + . -t + + . + -t -t -t -t ? -40- G,, G2 : GEIGl'-"t Tt1a6 /fV COI/IICIDENCE. c,,c 2 : CLouo CHRMBE.R.s T~l9 ✓ECTO~Y OF Pl'IRT/C'-.£ D L I 'I I I I I : I I I I I I I ', R.EGIO/V OF MAGNETIC ( F1ELD L.~-~i_---~~ .J D-:::./62 CM W-:: 15" CM T: • 3./8CM L -:c 40cr,,r , ScNEMnr,c ~1w oF APPl'IRATtlS Sc,,L.s:: }la s1~E. /Ou,, w 1 FtGVRE /. - 41 - J J• .8 --f)o Ill .6 ~J,... F1GVRE. 2 \- IU Cl .4 ~ o,____,_____________,,_---.....-----------------0 0 1 .3 3 I 6 ffl) 3 I RELl'1TIVE ( Fo,q APE.RTVAE. 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