Energy-level structure of 131Te from the 130Te(d, p)131Te reaction

I 1.E.1: [

Nuclear Physics A103 (1967) 209--221 ; (~) North-Holland Publishing Co., Amsterdam

2.G

Not to be reproduced by photoprint or microfilmwithout writtenpermissionfrom the publisher

]

ENERGY-LEVEL STRUCTURE

OF 131Te FROM THE 13°Te(d, p)131Te REACTION A. GRAUE, E. JASTAD, J. R. LIEN and P. TORVUND Physics Department, University of Bergen, Bergen, Norway and W. H. MOORE Department of Physics and Laboratory Jot Nuclear Science Massachusetts Institute of Technology Cambridge, Massachusetts, U.S.A. Received 12 June 1967

Abstract: The lZ°Te(d, p)131Te reaction induced by 7.5 MeV deuterons has been studied using the MIT broad-range multiple-gap spectrograph. More than 100 levels ofl~tTe have been identified up to 5.8 MeV excitation energy. Angular distributions have been analysed using DWBA calculations. The number of holes in the neutron shell observed were 1.0 in 2d~, 2.0 in lh~, 0.32 in 3s~_,and 0.12 in 2d}, respectively. These results are in fairly good agreement with pairing theory predictions. E

N u c L E A R REACTION la°Ye(d, p), E = 7.5 MeV; measured (r(Eo, 0); deduced Q. lalTe deduced levels, In, spectroscopic factors. Enriched target.

[

I

1. Introduction The nuclear level structure of several tellurium isotopes has been studied by Jolly 5) using (d, p) a n d (d, t) reactions induced by 14.8 MeV deuterons. The energy resolution was a b o u t 40 keV, and a n g u l a r distributions were taken up to 50 °. I n the present work the energy resolution was approximately 12 keV, a n d a n g u l a r distrib u t i o n data were taken from 7.5 ° to 172.5 ° with respect to the i n c o m i n g deuteron beam. Thus, m a n y of the overlapping levels have been resolved, a n d m a n y new levels have been found. The a n g u l a r distributions have been analysed using D W B A calculations to determine orbital a n g u l a r m o m e n t u m for the transferred particle a n d spectroscopic strengths for a n u m b e r of energy levels of 13 ~Te.

2. Experimental procedure This experiment was carried out using the M I T broad-range multiple-gap magnetic spectrograph 2) in c o n j u n c t i o n with the M I T - O N R Van de Graaff accelerator. The spectrograph allows for a s i m u l t a n e o u s recording of broad-range spectra at t The data described in this work were obtained while one of the authors (A.G.) was a guest of the Laboratory of Nuclear Science, Massachusetts Institute of Technology. The portion of this work that was performed at the High Voltage Laboratory at MIT was supported in part through AEC Contract AT(30-1)-2098 with funds provided by the US Atomic Energy Commission. 209

210

A. GRAUE et al.

25 gaps in 7.5 ° steps from 0 ° to 172.5 ° with respect to the incoming beam. Normally, the 0 ° gap is not used, and a Faraday cup situated in front of this gap records the total amount of exposure. The target consisted of 96_+0.2 % a3°Te, 3.57_+0.2% 128Te and 0.4% 126Te. A thin layer of this mixture was evaporated in vacuum onto a Formvar backing supported by a 2.5 cm diam circular frame. During the exposure this frame was rotated at 200 rpm about an axis normal to the target plane. The beam hit the target off center so that a ring of total area of about 100 times the beam cross section was exposed. In this way, the buildup of surface contamination was smeared out over a larger area and therefore formed a thinner layer than in a nonrotating target. The 13OTe(d ' p)131Te reaction was induced by 7.5 MeV deuterons. The emitted protons were recorded on nuclear-track plates (50 # K o d a k NTB) placed along the focal surfaces of the different gaps. To prevent scattered deuterons from reaching the emulsions, the plates were covered with thin aluminium foils of suitable thickness. The target thickness was measured by elastic scattering of 3.0 MeV deuterons. Since this deuteron energy is well below the Coulomb barrier, approximately 11.5 MeV, the cross section is essentially equal to the theoretically known Rutherford cross section. The measured target thickness was 39.2 pg/cm 2. Absolute measurements of cross sections were therefore made possible. Elastically scattered 7.5 MeV deuterons from 13°Te were studied in a separate exposure in order to check the optical model parameters. The photographic plates were scanned under a microscope in 0.5 m m strips across the exposed zones. The positions of the observed proton groups were referred to a coordinate axis along the plates, and an IBM 7094 computer was used to calculate the reaction Q value as a function of the photographic plate distance with the known incident energy, magnetic field, reaction angle, and spectrograph calibration constants as input parameters.

3. Experimental results A total of 104 levels in 131Te were found below 5.8 MeV excitation energy. The corresponding proton groups are labelled from 0 to 104 in fig. 1 which shows a typical spectrum from the 130Te(d ' p)l 3~Te reaction. The overall energy resolution of the proton groups was approximately 12 keV. The ground-state Q value for the ~3°Te(d, p)131Te reaction was found to be 3.703___0.006 MeV, as compared with the previously reported value 1) of 3.61 +0.1 MeV. The excitation energies of the observed levels in ~31Te are listed in column 1 of table 1. These results are arithmetic averages of energies determined at a minimum of ten reaction angles, and the uncertainties in these excitation energies are estimated as + 5 keV (standard error). The differential cross sections turned out to be strongest at backward angles, and therefore the weakest proton groups were observed generally only at angles larger than about 60 ° . Contaminant proton groups were especially troublesome at forward angles and prevented the taking of angular distribution data

a

"~

I?

0

250

750

A

PLATE

IS

IN

t6,.,

63

131Te {MeV]

..... _%s

- CM

131Te {k~V) 170

- MeV

"~L

ENERGY

ENERGY

ALONG

IN ~52

,:o

.!

35

20

~5

LAB

ANGLE = 105 o

1)0Te ( d , p } 131Te

,6

7tO

,~,IF , I

I

-s

B

= 854g

40

,.....:

4'5 DISTANCE

ALONG PLATE

{CH]

60

55

......./I....=/',./I..-~..-.....~...--,.....A-

, ........._~

~o.o

25

; ,28

170 {G 5 )

x~

2

7~5 - _ _

Fig. 1. Typical proton spectrum. The numbered groups have been assigned to levels in aZ*Te.

~ -...- ..., ...

GAUSS

= 7,500 MeV

Ed

II

PROTON

DISTANCE

,.,70

EXCITATION

gro

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ENERGY

ENERGY . MeV61S

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.

EXCITATION

PROTON

EXPOSURE, ""'~C

8~s

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A. GRAUE et al.

212

TABLE 1 Experimental results f f o m t h e *a°Te(d,p)lalTe reaction Present w o r k

J,t

Sin, j

Jolly *) Ex (MeV)

1.0 2.0 0.31

-:~-+ ~~+

0.25 0.17 0.16

0 0.18 0.29

1 0 2

0.018 0.010 0.12

.1½+ (~+)

0.0045 0.005 (0.02)

1.22

1

0.020

~-, I -

(0.005)

1.46

(1) 1 1 1 3 (1) (1) 1 1

(0.016) 0.034 0.012 0.013 3.5l 0.027 0.037 0.25 0.87

(.~-, ½-)

(0.004)

,½.1,½

(0.0085) (0.003)

1.78

'~ ½'2 .~-

(0.0033) 0.50

2.08 2.27

1

Level No.

Ex (MeV)

In

0 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 51 52

0 0.183 0.297 0.643 0.882 1.043 1.209 1.400 1.471 1.659 1.722 1.786 2.014 2.092 2.278 2.329 2.372 2.512 2.581 2.703 2.752 2.784 2.931 3.002 (3.027) 3.059 3.069 3.141 3.183 (3.203) 3.353 3.377 3.40l 3.412 3.457 3.469 3.505 3.517 3.543 3.563 3.600 3.621 3.663 3.686 3.706 3.736 3.902 3.920 3.935 3.960 3.985 4.028 4.068

2 5 0

(2Jq- 1)St,.,, j

(.~'2 ) ½ )

(0.007)

(~ , ½-) .1% ½ .1 , ½

(0.009) (0.080) (0.22)

0.033

~-, ½-

(0.008)

1

0.38

o

a

(0.095)

2.99

(3)

0.23

G )

(0.03)

3.15

(3)

o.2s

(I)

(0.04)

3

0.18

(.~)

(0.02)

,~

a

2.37 2.49 2.56

3.33 3.38

3.53

(1) (1) 2 1 (2)

0.059 0.049 0.194 0.111 0.249

(,~-, ½ ) (.I , ½-) ~+,~+ ~ , .~-(~+, .1+)

1

0.081

,~-, ½-

1 1 1

0.063 0.146 0.168

-.2a ,½ 3

7,½

3.~,½

3.68

3.93 4.02

213

131Te L E V E I . S

TABLE 1 (continued) Present w o r k Level No.

Ex

In

(2J-l- I )St., j

Stn, J

(MeV)

Jolly 1)

E× (MeV)

53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 8O 81 82 83 84 85 86 87 88 89 9O 91 92 93 94 95 96 97 98 99 100 101 102 103 104

4.092 4.107 4.123 4.135 4.155 4.175 4.187 4.202 4.237 4.249 4.281 4.297 4.323 4.341 4.361 4.440 4.471 4.487 4.519 4.530 4.543 4.558 4.583 4.615 4.644 4.676 4.706 4.730 4.763 4.801 4.823 4.843 4.868 4.892 4.940 4.964 4.998 5.027 5.049 5.103 5.121 5.147 5.170 5.256 5.285 5.348 5.409 5.575 5.631 5.680 5.754 5.780

3--

1--

( ~ , ½-)

1

0.124 0.059 0.053

(1)

0.113

(33-- , ½-)

(1)

0.091

(~3-- ,~1-- )

(o) (o)

0.170 0.131

(~+) (½+)

4.55

(2)

0.173 0.104

(~+, a+) (~ , ½-)

4.63

1

(1)

(1)

3--

4.30

1--

214

A. GRAUE e t al.

for all 24 angles for a number of levels. The main contaminants observed were ~2C and ~60. Both are present in the Formvar backing. Transitions corresponding to the ground state and several excited states of 13 C and a70 were observed with very large intensities. Other weaker peaks corresponding to contaminants of 13C, 14N, 2 3 N a , 32S, 35C1, 37C1, 64Zn, ll4Cd and X28Te w e r e also observed. Identifications of the proton groups from these contaminants were accomplished by observing their

0.4~

~ e -~

I3°Te(d,p)131Te

t

0.04:

":

0.02 ~2

0.01

Level No 8, Ex = 1.471 MeV

~ 0.008 0.006 b 0.3:

~/

0.2

0.1 O.O8_ 0.06~O.04 I F ! ~ 0

r// /

~= 2

~ ~~ ' ~ ,

~---~_._

-

Level No O, Ex = 0

t/

I 50

60

90 120 ~LAS (deg)

150

180

Fig. 2. Observed angular distributions fitted with D W B A cross sections. Shown are typical cases having l -- 0, 1, and 2.

kinematic energy shift with angle. The ~29Te proton groups, however, were identified from the knowledge of the 129Te spectrum 3). Figs. 2 and 3 show some typical angular distributions of proton groups from the 130Te(d ' p ) 1 3 1 T e reaction. The points are experimental cross sections plotted against reaction angle in the laboratory system. Errors arising from statistics and background subtraction are indicated. The D W B A predictions are represented by solid lines.

215

lalTe LEVELS

The orbital angular momentum l. of the transferred neutron was determined for 39 levels by comparing the experimental angular distributions with the calculated angular distributions from the DWBA theory. The results are shown in column 3 of table 1. Except for In = 0, the angular distributions do not show predominant peaks, and they are of the form typical of stripping reactions taking place below the Coulomb barrier. At forward angles, the cross sections are small but increase rapidly within a small angle interval which depends critically on l,. Column 4 of table 1 I

~

I

i

I

T-

I

r

T

I

0.1-0.08'--

0.06 004

//

-

Level No. 3 0 , E× = 3 . 3 5 3 MeV

/ 0.02

0.01 0.008

~f

/

15oTe(d,P) 131Te

,~ 0 . 0 0 6 E

c~

0 004

008 -

b

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h

0,0£o.o4 -

0.01 t 0

V{ ~

r

I 30

l

, 60

T

l

r

90

I 120

I

I 150

,

"-I 180

~L/~8 (deg) Fig. 3. Observed a n g u l a r distributions fitted with D W B A cross sections. S h o w n are typical cases h a v i n g 1 = 3 and 5.

shows the strength function ( 2 J + l)Su, j where J is the total nuclear spin and Su, j is the spectroscopic factor of the 131Te level in question. For weak proton groups where l, assignments have not been possible because of background and contaminant interference at forward angles, columns 3 and 4 of table 1 are left blank. 4. DWBA analysis of the (d, p) data For numerical calculations of the DWBA cross sections, two programs written

216

A. GRAUE et al.

for the G I E R computer at the Niels Bohr Institute, Copenhagen were used. The first program 4), GAP 3, calculates the bound-state wave function for the captured neutron. The potential well for the neutron was taken to be of the Saxon-Woods type. In the calculations, the spin-orbit term of the potential was neglected. The depth of the well was automatically adjusted by the program to give the correct binding energy, Qd. p + 2.2 MeV, for the level in question. Other input parameters are listed in table 2. The second program 4), DWB/SVE (GAP 2), calculates the theoretical DWBA cross section a(ln, Q, Ed, 0)with the bound-state wave functions from the GAP 3 code as input parameters. The zero-range approximation was employed, and the optical wells for both deuterons and protons were taken to be of the surface absorption form: U(r) =-~-V1 + ex + i W d ( l @ e X ' ) d x ' + V c ( r ' r c ) '

(1)

where x = ( r - r o A + ) / a and x' = ( r - r o A' +) / a ", V and W are the depths of the real and imaginary potential, respectively; a and a' are the diffuseness parameters, and TABLE 2 O p t i c a l - m o d e l p a r a m e t e r s for the D W B A calculations ( N o t a t i o n s are e xpl a i ne d in text; units are i n MeV a n d fro)

d eutero ns protons

V

W

r0

r0'

a

a'

rC

98.3 53

61 42

1.15 1.25

1.34 1.25

0.81 0.65

0.68 0.47

1.3 1.25

the Coulomb potential Vc(r, rc) is that of a uniformly charged sphere of radius r c = roc A~. The numerical values of the deuteron parameters used by Jolly 1) in his E d = 14.8 MeV exposure of 13 0Te turned out to give an elastic scattering cross section for 7.5 MeV deuterons, in good agreement with the observed cross section for elastic scattering (see fig. 4). No search for the optical model parameters was performed. The (d, p) calculations were carried out using the same deuteron and proton parameters as those used by Jolly 1), see table 2. These parameters are in turn taken from the works of Perey 5), and Perey and Perey 6). The energy dependence of the optical model parameter has thus been overruled. However, since the reaction is of the Coulomb-stripping type, it is expected that the nuclear forces will have relatively little influence on the wave functions of the relative motions that should be essentially governed by the Coulomb interaction. The calculated DWBA cross sections fit the experimental cross sections quite well, as shown in figs. 2 and 3. The calculations were performed with different values of the lower cutoff rico in the radial integrals. The effect of this turned out to be of importance only in the forward direction.

131Te LEVELS

217

The relation between the experimental cross section, da/df2, and the DWBA cross section o-(l,, Q, E d, 0) is given by dod~2

1 . 5 2 J f + l Stn, jo-(l. ' Q, Ed ' 0). 2J~ + 1

_

(2)

Here Jr and Ji are the angular momenta of the initial and final nuclear states, respectively, and j is the total angular m o m e n t u m of the transferred neutron. The strength functions (2J+l)gt~,j tabulated in table 1 were calculated from eq. (2). iO5

i\

q

I

[

t

I

I ---7

i

~-

IS°Te ( d d ) l S ° T e

\

7.5-MeV Elastic Scattering

104 -

10 3 I

b

\ ~K~\ \

i

'i

x~\\\,..

/

\, 102~ -

Optical

o~el

Rutherford

\ ...,'/

,~

÷ ~-.~

I

~ _

.....

i

:

~ i

!

i

N I I© o

50

60

90 OL.~B

120

~50

SO

(deg)

Fig. 4. Deuteron elastic-scattering data with Rutherford and optical-model cross sections. 5. Level scheme and shell-model assignments Fig. 5 shows the observed level scheme of 131Te. The spectroscopic strengths ( 2 J + 1)S~n,j are plotted for those levels where the value of ln has been determined. According to the simple shell model one expects to find only a few strong transitions corresponding to the transferred neutron's being left in an allowed shellmodel orbit. The residual interaction, however, leads to a fragmentation of the single-particle states, the total strength being split over a number of actual nuclear

218

A, GRAUE et al.

levels. The components of the I, = 1 transition are distributed over an excitation energy interval of 3.8 MeV, indicating a strong residual interaction for the p-states. On the other hand, only one In = 5 transition is observed. In spite of the splitting of the single-particle levels, the simple shell model is a useful tool in assigning total angular m o m e n t u m J to the observed levels and grouping Ex 6.0

MeV

t~=O

£=1

L=2

Z=3

Z=5

10C

5,0

90 8o 70

4.0

6O 5o 40 30

3.0

2.0

1.0

O0

0 (2J+llS Energy

revels

of

131Te

Fig. 5. Observed level structure o f 131Te with distributions o f spectroscopic strengths.

them into single-particle components. The assumed values of Jn and the resulting experimental values of the spectroscopic factors Sin ' j are listed in columns 5 and 6 of table 1. The experimental uncertainties in the observed spectroscopic factors are of the order of 15 % for the strongest transitions and somewhat more for weaker transitions.

13tTe LEVELS

219

The uncertainties 7) implied by the approximations in the DWBA theory are not taken into account. The sum rule for the spectroscopic factors obtained from a (d, p) stripping reaction with a target nucleus of zero angular momentum 8): (2J+ 1)S/,,j = number of vacancies in the (n, l,,j) shell in the target. i

The sum is taken over all final states belonging to the single-particle state (n, In,j), where n is the radial quantum number. Table 3 gives the experimentally deduced values of ~ ( 2 J + 1)S],,y taken from column 4 of table 1. The pairing theory predictions given in table 3 were calculated from the work of Kisslinger and Sorenson 9). TABLE 3 Sum-rule strengths Z ( 2 J + 1)sli, j and quasiparticle energies El."

no. of holes (experimental) no. of holes (pairing theory) Eln ' j (experimental)

J

2d k

1h~k

3s~

2dk

2f~

3p

1.0 1.16 0

2.0 2.5 0.183

0.32 0.09 0.320

0.12 0.11 1.209

4.7 8.0 2.420

3.0 6.0

Also listed in table 3 are the experimental unperturbed single-particle energies (quasi-particle energies) Et, ' j given by

~ S l~,, i j Ei Etn, j --

i

Z Stin, j i 5.1. THE 2d~ SINGLE-PARTICLE LEVEL

A very strong In = 2 transition is observed to the ground state of ~31Te. The 2d~ single-particle level is expected to be filling in the tellurium region; hence, J~ = 3~+2 is assigned to the ground state. This is in agreement with several earlier works 1, 10). The observed strength of the 2d~ level then becomes (2J+ 1)Sln, j = 1.0, indicating one hole in the 2d~ subshell of 13OTe" This result is somewhat surprising from the simple shell-model point of view. According to this, one would expect everything to be paired off in 130Te(J = 0); for example, the 2d~ state should hold an even number of neutrons. However, our result is in good agreement with the value of 1.12 found by Jolly 1). 5.2. THE lh~, SINGLE-PARTICLE STATE Only one In = 5 transition was observed. This was for the first excited state at E x = 0.183 MeV. The 1 . s splitting for the lh levels is of magnitude 5 MeV (ref. 11)) and there is no doubt about assigning J~ = ~ x - to the level found through the In = 5 transition. This again agrees with the value given in earlier publications 1, 1o).

220

A. GRAUEet aL

The total strength for the lh~L level is found to be ( 2 J + 1)Stn, j = 2.0, which means that there are two vacancies in the lf_~ subshell. This result is well understood in terms of the simple shell model. 5.3. THE 3s~ SINGLE-PARTICLE STATE The 3s~ state is assumed to be filling in the tellurium region, and two l, = 0 transitions are found in the lower part of the spectrum at 0.293 and 1.043 MeV excitation energy. The latter level has not been previously observed. The observed total strength of the 3s~ level is 0.32, and the quasi single-particle energy is Eo, ~ = 0.320 MeV. 5.4. THE 2d~ SINGLE-PARTICLE LEVEL The 2d~ single-particle state lies well below (about 1.7 MeV 11)) the 2d~ state, and from simple shell-model theory, no vacancies in the 2d~ subshell are expected. However, an excited J = )+ level is observed in many odd isotopes in this mass region 1,12). In this work, we observed an I, = 2 transition to the level at Ex = 1.209 MeV, and this is believed to be a 2d~ state. This result is in agreement with the systematics of the 2d~ level found by Jolly 1) in several different odd-mass Te isotopes. F r o m the study of both (d, p) stripping and (d, t) pick-up reactions on these isotopes, he was able to resolve the ambiguity of the spin assignment to the 2d levels. The total strength for stripping to the 2d~ state is ( 2 J + l ) S t , , j = 0.12, which indicates that the 13°Te wave function contains a weak component corresponding to a hole in the 2d~ subshell. 5.5. THE 2f~ SINGLE-PARTICLE LEVEL One very strong and three weaker l, = 3 transitions are observed between E x = 2.28 and E x = 3.35 MeV. The l • s splitting of the 2f states is approximately 2.4 MeV (ref. 11)) with the 2f~ state probably being in the unbound region. Assuming J = -~for these four l, = 3 levels, we measured the total strength as ~ ( 2 J + 1)S/n, j = 4.7. From simple shell-model calculations, it is expected that the 2f~ subshell in 13°Te would be completely empty; hence, we expect a value of 8.0 for the total strength of the 2f÷ state. We must therefore expect some of the unassigned levels in the higher part of the spectrum to be components of the 2f~ single-particle state. The quasi single-particle energy of this level is then > 2.42 MeV. 5.6. THE 3p~ AND 3p~r SINGLE-PARTICLE STATES As shown in fig. 5, a wide spread of the In = 1 transition was observed. The 3p~ and 3p~ single-particle states both belong to the seventh major shell, and the l . s splitting for these levels is approximately 1 MeV. Total angular m o m e n t u m assignments for levels formed through In = 1 transitions are therefore very difficult, although it seems reasonable to assume J = 3 - for the lowest lying 3p levels. The

131Te LEVELS

221

total observed 3p strength is only 3.0, so that many levels have been missed in the sum.

6. Conclusions

From the strengths found for the 2d~, lh~, 3s~, and 2d~ single-particle states, the 13°Te ground-state wave function seems to be much more complicated than expected on the basis of the simple shell model. The observed sum of the strengths for these four single-particle states, however, was found to be 3.44. Within the limits of experimental error, this agrees with the value of 4.0 expected from the shell model. No evidence for filling of the lh~ and li~ single-particle states was found in this work. However, if these states are present, they are supposed to be weak and fragmented. Components of these states may therefore be among those weak levels for which we have not been able to assign any In value. On the other hand, several strong levels showing stripping angular distributions (1, = 0 and In = 2) and which are not consistent with the shell model have been found above E~ = 3.6 MeV.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

R. K. Jolly, Phys. Rev. 136 (1964) B683 H. A. Enge and W. W. Buechner, Rev. Sci. Instr. 34 (1963) 155 G. K. Schlegel, B. S. Thesis (1966) MIT, unpublished R. K. Cooper and J. Bang, GIER program library, Niels Bohr Institute, Copenhagen, Denmark F. G. Perey, Phys. Rev. 131 (1963) 745 C. M. Perey and F. G. Perey, Phys. Rev. 132 (1963) 755 L. L. Lee, Jr., J. P. Schiffer, B. Zeidman, G. R. Satchler, R. M. Drisko and R. H. Bassel, Phys. Rev. 136 (1964) B971 M. H. Macfarlane and J. R. French, Revs. Mod. Phys. 32 (1960) 567 L. S. Kisslinger and R. A. Sorensen, Revs. Mod. Phys. 35 (1963) 853 Nuclear Data Sheets, National Academy of Sciences, National Research Council, Printing and Publishing Office, Washington 25, D.C. B. L. Cohen, P. Mukherjee, R. H. Fulmer and A. L. McCarthy, Phys. Rev. 127 (1962) 1678 B. L. Cohen and R. E. Price, Phys. Rev. 121 (1961) 1441