An angular correlation test of time reversal invariance

2.B

[

Nuclear Physics A199 (1973) 401 --412; (~) North-Holland Publishing Co., Amsterdam

[

Not to be reproduced by photoprint or microfilm without written permission from the publisher

AN ANGULAR CORRELATION TEST OF TIME REVERSAL INVARIANCE M. J. t-[OLMESt, W. D. HAMILTON and R. A. FOX School of Mathematical and Physical Sciences, University of Sussex, Brighton, Sussex, UK Received 13 July 1972 (Revised 8 September 1972) Abstract: The angular correlation between the 300 keV and 600 keV groups ofT-rays occurring in 192pt following the E-decay of 192Ir has been measured from a cryogenically polarised, source. The source was prepared by dissolving iridium into iron and cooling to 50 mK in a 3I-[e-4He dilution refrigerator. A series of measurements were made and from the observed asymmetries a phase difference oft/ ~ (4~5) × 10-3 was found between the electric quadrupole and magnetic dipole components of the 3 + --2 + transitions. Thus no evidence is found for electromagnetic T-violation.

E[

I

RADIOACTIVITY 192pt [from ~gqr(n,),)]; measured 77(0) from cryogenically polarized nucleus; deduced limit on T-violation in e.m. interaction. 1. Introduction

The application o f physical symmetries and their related conservation laws has been frequently used to simplify physical problems. Since the observation 1) that parity was not conserved in nuclear/3-decay, the investigation o f the symmetry laws satisfied by the nuclear Hamiltonian has received increased attention. Of particular interest recently has been the study o f parity (P) and time-reversal (T) symmetries displayed by a nucleus undergoing electromagnetic transitions. It is known that there is no gross violation o f the symmetries in strong or electromagnetic interactions as has been observed in the parity-violating weak interaction, however even a small admixture o f parity or time-reversal violating terms in the nuclear Hamittonian would be of interest. The general method for studying the magnitude o f such terms, if they exist, is to search for the occurrence o f a quantity which changes sign under a particular symmetry operation. Appropriate quantities for tests of parity and time-reversal invariance have been discussed by J a c o b s o h n and Henley 2 ) a n d have been tabulated by Boehm 3). The present experiment observes the 7-7 directional correlation from a polarized nucleus. Such a correlation is sensitive to the phase difference, r/, between the electric and magnetic c o m p o n e n t s o f a mixed multipole electromagnetic transition with mixing ratio given by 6ei'k This phase difference is identically zero 4) if time-reversal invariance holds. If r/ is non-zero and the nuclei are polarized, then the expression Present address: Institut fiir Experimentelle Kcrnphysik, Karlsruhe, W. Germany. 401

402

M . J . H O L M E S e t al.

for the angular distribution of radiation contains the term ( j ' k l x k2) (k t • kz) where j is the initial-state nuclear spin and the k are photon momenta for photons emitted in cascade. This term is invariant under P but changes sign under the operation T. Consequently the observation of such a term will prove the existence of a P-even T-odd part of the nuclear Hamiltonian. It can be seen that, in contrast to the pseudoscalar term (j • k) which tests parity violation, the present experiment requires the determination of three quantities rather than two. It is this fact which limits the accuracy of time reversal as compared with parity experiments. The 7-7 directional correlation from an initial state polarised by capture of a polarised neutron has been measured several times in an attempt to measure sin r/ [refs. s-v)]. Additionally a coincidence asymmetry has been sought in the fl-~-;~ correlation from an unpolarised nucleus. This method is equivalent to a polarised-nucieus-y-2 correlation because the fl-particle is used to specify t h e nuclear spin direction 8-10). In general, methods employing dynamic polarisation have been limited statistically by the low rate of accumulation of events. The directional correlation from a cryogenically polarised nucleus has the advantage that the initial nuclear spin direction j0 is known, which reduces the number of required measurements to two. This experiment has been reported briefly before 21). Since, however, it represents the first experimental test of time-reversal invariance using thermal equilibrium nuclear polarisation, a more detailed account is presented here.

2. Principle o f the m e t h o d

The directional correlation from an assembly of polarised nuclei has been explicitly derived by Krupchitsky and Lobov ~1). The exact form of the expression depends on the nature of the polarising process. In the case of low-temperature polarisation the coincidence distribution for a 7-~ correlation in a plane perpendicular to the direction of polarisation is given by F

1

W(O) = 1 +

' /-3Bl(J)Jo'(kl "" 2~~ ~ 4A~(2) / Pk(cos 0)Bk(1)--2\, × k2) l (1+161l)(1+1621) k=Z, /2k+l

× am (al)ea(cos o)tkt +

Y~ k -

-]

L, + 1, j, ,Jo)j,

(1)

where B I ( j ) characterises the degree of polarisation lz), Pk (COS 0) is the Legendre polynominal of order k, P~(cos 0) is its first derivative with respect to cos 0, and F]k(L 1 , LI + 1, j~, Jo) is the triple correlation coefficient la). The complex mixing ratio of the first transition is 6~e~"' and di2eiq-" refers to the second radiation. The terms Jo, kl, k2 are unit vectors directed along the initial-state spin direc-

TIME REVERSAL INVARIANCE

403

tions, and photon emission directions respectively. In addition, B~(1) =

A~.(2)

=

Fk(L1, L I , J0, J l ) - 2 Re 161[Fk(L1, L1 + 1, Jo, Jl) +[6al2Fk(L, +l, L, +l,jo,j~),

Fk(L2, L 2 ,J2 , j l ) + 2

Re

(2)

1621Fk(L2,L2 + I,j2,j, ) +I6212Fk(L2+l,

LE+I,jE,j~).

(3)

The following point should be noted as it strongly affects a T-invariance test based on the detection of an imaginary part of 6. The imaginary part of the first radiation mixing ratio enters the eq. (1) through lm(ht) = 16xlsin qt,

(4)

while the phase angle of the second radiation does not, since it enters only through Re(fi2) -- lfi2lcos '72.

(5)

Eq. (1) can be written in a simplified form which displays the term of interest if it is assumed that the phase difference r/is small for all transitions and that the first radiation is a mixed E2:M1 transition with mixing ratio 6ei~:

W(O, q) = l-t-a2ea(cos O)+a4ea(cos O) -3x/gB,(j)F22(1, 2,j~ ,jo)A2(2) ~

sin 20 sin r/. (6)

This may be represented by

W(O, rl) = W(O)+ V(O, ~1),

(7)

where W(O) is the "normal" directional correlation function and V(O, q) is the "irregular" part. An experiment to search for V(O, rl) should compare coincidence counts at angles 45 ° and 135 ° from polarized nuclei in order to maximize sin 20 and give cancellation of the regular part of the correlation function. For this choice of angle the modulus of the irregular part is

V(O, q ) = 3,/:,Bl(j)F22(I, 2, Jl, jo)A~(2) (]--+6--62) sin r/.

(8)

In eq. (8), 6 refers to the first transition and is specified with the sign convention of Krane and Steffen x4). Inspection of eq. (8) reveals that it is quite possible for an accurate measurement of V(O, rl) to yield an inaccurate estimate of sin r/, if the angular momentum coupling coefficients are unfavourable. In particular, even if a high degree of polarisation is achieved and 6 is also of the order of one, the spin sequence of the cascade may still yield a low value of the triple F-coefficient. It may be seen in table 1 that this coefficient tends to have small values for spin sequences

404

et aL

M. J. H O L M E S TABLE l

T h e coefficient F~2(I , 2, J l , ] o ) w h i c h o c c u r s in eq. (6) Intermediate state Jt

Initial s t a t e Jo

0 1 2 3 4 5

0

1

2

0 0 0 0 0 0

0 --0.273 --0.070 - - 0.059 -- 0.050 -- 0.043

3

0 0.241 0.091 -- 0.069 -- 0 . 0 6 2 - - 0.053

4

5

0 0.179 0.141 0.045 -- 0.059 -- 0.057

0 0.140 0.118 0.099 0.027 0.050

0 0.114 0.098 0.087 0.076 0.018

0.178 0.143 0.116 0.034 -- 0 . 0 5 4 - - 0.051

0.142 0.115 0.101 0.086 0.022 --- 0.046

0.118 0.096 0085 0.077 0.068 0.015

I n t e r m e d i a t e s t a t e Jl

½ Initial s t a t e Jo

~5 ,~

0 0 0 0 0 0

0.365 0.146 --0.073 --0.063 -- 0.054 - - 0.046

0.239 0.178 0.062 --0.063 - - 0.058 - - 0.051

which are commonly found. The term 6/(1 -I-62) has a maximum value of 0.5 when 6 = 1. The magnitude of A~(2) coefficients range between 0 and about 1. The degree of polarization obtainable depends on the magnetic moment of the parent nucleus, the hyperfine field of the host material, assuming a suitable one may be found, and the temperatures that are available. It is improbable that strongly fed cascades exist in nuclei with a convenient half-life where the coefficient of sin r/is much greater than about 0.1. 3. The nucleus 19ZIr After an extensive search of the literature the decay of 1 9 2 I r w a s chosen. The ground state of a 92Ir has a moment of 1.41 n.m., and readily dissolves in iron where it L,-=gZ]r177d I

_•'/. +

;

2%

~3 +

~ X ~

~

!

11

TIT

T

T

598

588

I

I

295

i

lz00

"-T--'zl

612

316

igz PI

Fig. 1. Schematic representation of the decay of ~92Ir showing the four observed cascades in z92pt"

TIME REVERSAL INVARIANCE

405

experiences a hyperfine field of about 1.5 x 106 MOe [ref. 15)]. Hence Bz(j ) ~ 0.8 may be obtained at 50 mK. The fl-decay populates excited states of 192pt" Fig. 1 shows a schematic representation of the decay showing the four cascades, which have one radiation at about 600 keV and another at about 300 keV. Cascade I is a fairly good example. It is strongly fed and has a suitable mixing ratio, ~5 = - 2 . 1 . The importance of good detection efficiency necessitates the use of NaI detectors which implies that the other three cascades are also observed. In this case the coefficient of sin r/ is reduced from 0.076BI(j ) to approximately 0.031BI(j ).

4. The experiment The source material 192ir was obtained from the Radiochemical Centre, Amersham in the form of hexachloriridic acid in dilute HCI. This was evaporated onto small pieces of 99.999 ~ pure iron and raised to 1600 °C in a reducing atmosphere. The resulting solution of iridium in iron was cast into needles about 0.5 cm long and 0.07 cm diameter and cooled rapidly. The cylindrical shape was desirable both for good definition of the source position and for small demagnetization factors in an applied field. The radioactive strength of the source was limited by the heat extraction rate of the refrigerator and

Hognetic Fietd 0own 1st

{klxk z porottet to 51

Counting Position

2rid Counting Position g(30OkeV]

AI600 keVl " :"

T

'~'~ Al6~keV] ks

~

0 " { 300keV}

]

_ks

kj ~,~,

kz

®

C " 1300keVl

-7 8 1600keV] C(300 keV)

II (600keVJ

Mognetic Fietd Up (k lxk2OntipOrOuel to I ) D{30g keY) A[600 keY)

A (600 keV]

,,r Q

j

k~ ~

o

_k, " \ J r

~ [30OkeVI

Q

kz

C (300keY)

~z

B 1600 keVI C1300keYl

B(600 keVI

Fig. 2. T h e four-detector geometry o f the time-reversal test. T h e d a s h e d lines s h o w the ideal situation while the solid lines indicate the actual positions o f the detectors, offset to minimise the effects o f the intermediate state perturbation.

406

M.J. HOLMES et aL

the resolving time of the coincidence electronics, both of which implied a restriction to about 200/~Ci. The source was soldered with non-superconducting eutectic Cd-Bi solder into a cavity in a copper rod which screwed directly onto the bottom of the mixing chamber of an Oxford Instruments Mark IV dilution refrigerator. Thermal contact to the 3He-4He mixture was provided by 1000 cm 2 of copper sinter. The source lay on the axis of a vertical superconducting Helmholtz pair which could supply up to 6 KOe at the sample. The iron needles were found to saturate in about 1.5 kOe, so 2 kOe was used as a polarising field. Four N a I detectors were arranged on a correlation table outside the refrigerator. This correlation table consisted of two concentric annuli which could be independently rotated. Mounted on the inner one were two detectors gated to accept radiation at 300 keV and separated by 90 ° from each other. On the outer annulus two further detectors, accepting 600 keV radiation and separated by 90 °, were positioned such that each made an angle of either 45 ° or 135 ° with one of the inner pair (fig. 2). Rotation of the outer pair through 90 ° allowed a second counting position to be formed in which the angle between each 45 ° pair became 135 ° and vice versa. A function F of the four observed coincidence rates in the two counting positions may be formed such that Ct2 C34. C'14C~3 _ VW(4L'/17)I4" F C14C23 C'12C34 LW(kn)J ' (9) and all solid-angle, detector-efficiency and coincidence-efficiency factors cancel. In the expression, C~j refers to the coincidence count rate between detector i and j, and the dash indicates the second counting position. The formation of expressions of this type in four-detector systems has been considered by Baker and Hamilton a 6) who show that F = 1+ 8 A , ,

(10)

where A, is known as the asymmetry which is equal in this case to a I = 0.031Bl(j)(sin rh + (0.19) sin t/iv ),

(11)

where cascades I and [V contribute. It is assumed that the random-to-true ratio is small, so that accidental coincidences do not affect the result. 4.1. THE PERTURBATION OFFSET The function F only suppresses the even terms of the correlation function for which W(¼~) = W ( ~ ) . This is not the case if the correlation is perturbed. The half-lives of the intermediate states of the observed cascades are such that an appreciable perturbation asymmetry is observed in the internal magnetic field. In the case of magnetic field perturba-

T I M E REVERSAL I N V A R I A N C E

407

tion expression (9) should be replaced by F = LW(¼n-~z)J "

(12)

Clearly the effect of the intermediate state perturbation may be removed if the counters are physically offset by an angle equal to the effective ogz. Precise calculation of the appropriate offset angle is not straightforward because the toz for the two states 8

~.

XlO 3

The Perturbofion Asymmetry

-[

\~\

7 Asymmetry 6

i't

s 3

\ \

2 1 0

\ 0 !

1 I

-1

-2 -3

2 I

3 I

~ t

\5/

6 I

7

8

c

I

9 I

Offset ongle [Degrees]

\ \ \ \

-5 -6 -7 -8

\ \

Fig. 3. Experimental results for the asymmetry due to the perturbation of the intermediate state, as a function of the offset angle of detectors A and B (fig. 2) from the ideal positions.

through which the observed cascades pass are not sufficiently well-known. Estimates indicate that it is in the region of 5 °. In addition the method of sample preparation may affect the result. Measurements of the asymmetry at warm temperatures (0.5 K) were made for a range of offset angles (fig. 3) and the appropriate angle was taken to be 5 °. Any residual asymmetry due to the intermediate state perturbation was eliminated in the actual runs by comparing all cold counts (50 mK) with warm counts (0.5 K) taken under identical circumstances. 4.2. S O U R C E M I S C E N T E R I N G

The regular part of the correlation may also enter the expression F if the source is miscentered. In this case although the cancellation of solid angle in the formation of the asymmetry expression provides a first-order correction, a residual miscentering

408

M.J. HOLMES

et al.

effect remains. This is due to the distortion of the counting angle by the source displacement 17). A computer programme was constructed which simulated the geometry of the experiment and the miscentering asymmetry was calculated for a range of source positions (fig. 4). The calculation indicates that the source should be

Source Miscentering Asymmetry +Z,+8

*'21

+t,2

l-kl

//k_ z

ylcml

o.,\_ x~

:' 5 \

\

xlcml

Fig. 4. The results of a computer simulation of source miscentcring. centered to 0.1 ~ . Before each run the source was centered to about 1 part in 10 a. In practice the source centering was observed by continuously monitoring the moving detector single rates, and found to be slightly worse than this. However, a miscentering asymmetry may only falsify the experimental results if a systematic change in source position occurs between cold and warm counts. Experimental results showed that no systematic movement of the source occurred during runs. The same programme allowed the effects of various other detector misalignments to be investigated. In particular asymmetries of up to 10 -3 may be generated by small (1 °) displacements of the detectors from their required positions. While such effects are eliminated by the control runs, they must nevertheless be kept to a minimum as it is not desirable to search for a small effect against a background of a much larger one. To reduce effects of this type the true axes of the Na[ detectors were determined using a strong but spatially well-defined 22Na source before the start of the experiment. 4.3. COINCIDENCE DETERMINATION A shared element coincidence system was used to determine coincidences. This system has been described by Baker and Hamilton la). The output from the four detectors was fanned into two channels which drove a TAC. A gate on the time spec-

TIME REVERSAL INVARIANCE

409

trum was effectively the master coincidence unit, selecting coincidence events between all pairs. This rate was counted in four 30 MHz scalers which were in turn gated by routing pulses generated by slow coincidence units in parallel to the master unit. In addition two single-channel analysers selected only events with the correct energy requirements and provided a gate for the TAC output. A fifth Nal detector was positioned under the refrigerator and along the axis of the field. This detector, which was equipped with feedback high-voltage stabilization, allowed a continuous record of the sample temperature to be made by observing the anistropy of the 468 keV 2 + ~ 0 + transition in tgzPt. 4.4. E X P E R I M E N T A L

RUNS

The duration of runs was affected by the requirements of the refrigerator. The refrigerator was serviced at 24 h intervals, followed by 10 h of cold counts (50 mK) and 10 h of warm counts (0.5 K) During these runs the detectors were automatically interchanged from position 1 to position 2 every 100 sec and the accumulated data The Asymmetries Heosured in the Time Reversol Experiment

Asymmetry

3

-3

XlO

z 1

-3

I ITrITIL TIT IT

Fig. 5. T h e difference between cold a n d w a r m a s y m m e t r y for thirteen runs. T h e arrow shows the relative field direction for each run. T h e a s y m m e t r i e s have been normalised to allow for changes in the m a g n i t u d e and. direction o f the nuclear polarisation.

was punched on paper tape. A computer programme calculated the asymmetry for each observation and the overall asymmetry for the run. The programme also performed various statistical tests and made a trend reduction of the data 28). The asymmetry from each warm run was subtracted from the corresponding cold run. After each complete experimental cycle (24 h) the magnet field direction was reversed. Fig. 5 shows the collected results. Reversing the field interchanges the detector angles

410

M.J. HOLMES

et al.

and consequently the sign of the asymmetry. In the figure the sign of the asymmetry has been changed for the case where the field was up. The degree ofpolarisation B1 (J) which is temperature dependent ranged between 0.9 and 0.7 for different runs. The asymmetries displayed in fig. 5 have been normalised to the highest-temperature case so that they are directly comparable. 4.5. TEMPERATURE DEPENDENCE OF THE CORRELATION FUNCTION The interpretation of the asymmetry measurement requires some care. Although the fact has not received much attention in the literature it is known that the general expression for a symmetry conserving ("regular") directional correlation contains terms which depend on even powers of the orientation parameters. The correlation function from an oriented nucleus has been presented by Kaye, Read and Willmott 19). The specialization of the general formulae to particular cases relevant to symmetry tests of this type is discussed by Coutinho and Ridley 20). Fortunately at temperatures in the region of 50 inK, the orientation parameter B 2 (nuclear alignment) is small, and the effects of alignment are also small. Furthermore changes in the magnitude of the even terms of the correlation do not directly affect the measured asymmetry, but they may do so when combined with departures from an ideal experimental arrangement such as source miscentering. Further protection is offered by changing the field direction after every run. Alignment dependent effects do not change sign when the field is reversed. A calculation for the experimental case indicated that an upper limit for the asymmetry due to nuclear alignment effects is about 5 × 10- 5. This will not seriously affect the present experiment but does present a limit for the method unless a cascade could be used with an initial spin of ½. In this case alignment dependent effects vanish since B 2 is identically zero. 5. Results and conclusions

The measurement of the asymmetry and temperature during the runs, together with a knowledge of the other terms occurring in eq. (11), allows an estimate of the phase difference to be made. The errors quoted are based on the observation of over 6 x 108 coincident events per detector pair and are calculated from the distribution of the approximately 1500 observations of the asymmetry made during the experimental runs. The third moment of the distribution of data points (skewness) was used as a criterion for data rejection. Data were rejected if the skewness exceeded 0.5 and some instrumental malfunction was suspected. The fourth moment of the distribution (kurtosis) had an average value of 2.7 which corresponds approximately to a normal distribution of data. A product of correlated variates of the type measured in this experiment will not, in general, give rise to a normal distribution even if the variables themselves are normally distributed 16). We find sin r / = 4-I-5 x 10 - 3 which supercedes our previously reported value zl). The result represents an increased accuracy compared with the fl-),-), correlation

411

TIME REVERSAL INVARIANCE TABLE 2 Collected results to date for angular correlation measurements of sin r/ Method pol. neutron 7-Y(O) pol. neutronT-7(0) pol. neutronT-7(O )

fl-y'-~(O) /~-~,-~,(0) fl-7-"t(O) pol. nucleusT-~(0)

Source 49Ti 36C1 36C1 l O6Rh l°6Rh 56Mn 192ir

(sin ~/) × 103 -- 170 --'250 4 = 12 - - 0 . 9 + 2.9 30 __4-250 4 4- 18 4 -- 26 4 ~: 5

Ref. 5) 6) 7)

(1966) (1968) (1972)

s) (1964) 9) (1968) xo) (1969) present work

method. Recent results for various measurement of sin q are presented in table 2. It may be seen that present measurements provide no evidence for time-reversal non-invariance in nuclear 7-decay neither do they rule out the possibility of such violation. Attempts have been made to estimate sin r/ [ref. 22)] but it must be stressed that such calculations are highly model dependent. Alternatively Coutinho and Blin-Stoyle 23) have used the present experimental results to determine the upper limit for the coupling constant of a particular T-violating model based on a potential given by Huffman 24). It is highly desirable that the measurements of sin r/be extended by a further order of magnitude in order that theories might be more sensitively tested. The status of the present experimental techniques does not indicate that this will be straightforward. Angular correlation tests are promising, especially if a really favourable decay scheme could be found. They do not share with M6ssbauer techniques the problem of Faraday rotation. Hannon and Trammel 25) have pointed out that effects originating in the atomic electron shells can introduce, via the internal conversion process, a phase difference r/, irrespective of T-violation. These effects are of the same order as the most accurate M6ssbauer measurements z6), which examine transitions having an energy less than 100 keV. The effect can be expected to be much smaller in the case of the higherenergy transitions used in correlation experiments where internal conversion coefficients are typically several orders of magnitude smaller, and it is not considered likely to represent any significant contribution in the present experiment. This work was supported by the Science Research Council to whom M. J. H. is grateful for a Research Studentship. The authors wish to thank F.A.B. Coutinho, P. A. Ridley and Professor R. J. Blin-Stoyle for stimulating and useful discussions. References 1) C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes and R. P. Hudson, Phys. Rev. 105 (1957) 1413 2) B. A. Jacobsohn and E. M. Henley, Phys. Rev. 113 (1959) 234 3) F. Bochm, in Hyperline structure and nuclear radiation, ed. E. Matthias and D. A. Shirley (North-Holland, Amsterdam, 1968)

412 4) 5) 6) 7)

8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26)

M . J . HOLMES et aL S. P. Lloyd, Phys. Rex,. 81 (1951) 161 J. Kajfosz, J. Kopecky and J. Honzatko, Phys. Lett. 20 (1966) 284 J. Eichler, Nucl. Phys. A120 0968) 535; A127 (1969) 693 M. I. Bulgakov, A. D. Gulko, G. V. Danilyan, I. L. Karpikhin, P. A. Krupchitsky, V. V. Novitsky, V. S. Pavlov, Yu. A. Oratovsky, E. I. Turkovsky and S. S. Trastin, Institute of theoretical and experimental physics, Moscow, preprint, 1972 E. Fusehini, V. Gadjokov, C. Maroni and P. Veronesi, Nuovo Cim. 33 (1964) 1309 R. B. Perkins and E. T. Ritler, Phys. Rev. 174 (1968) 1426 M. Garrell, H. Frauenfelder, D. Ganek and D. C. Sutton, Phys. Rev. 187 (1969) 1410 P. A. K.rupchitsky and G. A. Lobov, Atomic Energy Review VII (1969) 91 R. J. Blin-Stoyle and M. A. Grace, Handbuch der Physik 42 (1957) 555 J. Kajfosz, University of Rez-Praha, technical report UJV 1789 (1967) K. S. Krone and R. M. Steffen, Phys. Rev. C2 (1970) 724 P. G. E. Reid, M. Scott and N. J. Stone, Nucl. Phys. A129 (1969) 273 K. D. Baker and W. D. Hamilton, Nuel. Instr. 86 (1970) 77 E. Breitenberger, Phil. Mag. 45 (1954) 497 D. W. Cruse and W. D. Hamilton, NucL Instr. 57 (1967) 29 G. Kaye, E. J. C. Read and J. C. Willmott, Tables of coefficients (Pergamon Press, 1968) F. A. B. Coutinho and P. A. Ridley, University of Sussex, internal report, 1972 M. J. Holmes, W. D. Hamilton and R. A. Fox, Phys. Lett. 37B (1971) 170 C. F. Clement and L. Heller, Phys. Rev. Lett. 27 (1971) 545 F. A. B. Coutinho and R.J. Blin-Stoyle, to be published A. H. Huffman, Phys. Rev. D1 (1970) 882 J. P. Harmon and G. T. Trammell, Phys. Rev. Lett. 21 (1968) 726 M. Atec, B. Chrisman, P. Dcbrunner and H. Frauenfelder, Phys. Rev. Lett. 20 (1968) 691