The theory of n-d scattering with tensor forces

I

eBo

2 L ] Nuclear Physws 11 (1959) 432---443, (~) North-Holland Publ~shzng Co, Amsterdam I

Not to be reproduced by photoprmt or mmrofilm mlthout written permission from the pubhsher

T H E T H E O R Y OF n - - d S C A T T E R I N G L

M

DELVES* and D

WITH TENSOR FORCES BROWNt?

Phgsws Department, Auckland Umvers*tg, Auckland, N Z Received 8 J a n u a r y 1959 A b s t r a c t : The scattering of n e u t r o n s b y d e u t e r o n s a t energies below 3 5 MeV has been investigated in an a p p r o x i m a t i o n including the effects of b o t h the t e n s o r force and the distortion of t h e d e u t e r o n b y the n e u t r o n The distortion is f o u n d to be small a t these energies in t h e d o u b l e t state, b u t n o t m t h e q u a r t e t s t a t e Angular distribution a n d polarization curves for the n e u t r o n are given as a f u n c t i o n of energy, a n d the second order spin m o m e n t of t h e d e u t e r o n calculated a t 3 25 MeV n e u t r o n energy Agreement with e x p e r i m e n t s where these exist is reasonable Polarizations of the order of 10 o~ are predicted

1. I n t r o d u c t i o n The scattering of nucleons by deuterons is perhaps the simplest example of a problem involving forces between more than two nucleons, and thus might be hoped to yield more reformation on these forces than the two-body data However, at low energies progress has been Inndered by lack of suitable approximation methods Previous calculations 1, 3, a) have assumed that the deuteron IS not distorted by the incoming neutron, and have neglected all effects of spin-orbit couphng in the nucleon-nucleon force It is hard to estimate the validity of these approximations, so that conclusions on the nucleon-nucleon force based on these calculations can only be tentative The reasonable agreement wlth experimental angular dlstrlbutlons found at energies greater than a few MeV gives hope that the errors introduced are small in tins region, but there is some evidence 4) that at lower energies this is not so Moreover a calculation of the neutron polarization induced by the scattering would be of great interest, and this involves introducing explicitly a nucleon-nucleon spin-orbit coupling force These have been introduced ~, 8) in Born approximation, which leads to zero polarization with real potentials, and a recent paper by Bransden et al ~) sets up the problem including tensor forces, in a form suitable for calculation, but neglecting the distortion of the deuteron It would seem useful to have available a simple method which will give an estimate of tins distortion at low energies, a~ d of the characteristic effects of including the tensor force such as the polarization of the neutron We give such a method here, based on a semi-classical picture of the scatter* N o w a t t h e Clarendon L a b o r a t o r y , Oxford, E n g l a n d *¢ P r e s e n t address

A E R E , Harwell, E n g l a n d 432

THE

THEORY

OF

n--d

SCATTERING

WITH

TENSOR

FORCES

~

mg which is expected to be reasonable at energies less than a few MeV. Tins picture leads to an effective two-body neutron-deuteron potential, the wave equation for winch can be solved numerically

2. Equivalent

n--d

potential

Labelling the incoming neutron and the two particles in the deuteron 1, 2, 3 respectively, and writing rl2 = r 1, r =

r13 : rl+R

r 2, =

1"23 :

2R,

r~--R,

(1)

those parts of the interaction containing 1 are

VI~+Vls,

(2)

where V,j will in general contain spin and exchange operators m addition to r,~ At low neutron energies we expect physmally an adiabatic approximation to hold: for a given r the neutron sees an effective potential Vnd---- <~diV~+Vlal~d)

(3)

where Cd lS the wave-function of the deuteron in the presence of a neutron at distance r from its centre of mass Tins is not known. The usual approximation made is to set Cd equal to the free-state wave-function of the deuteron, winch is thus not distorted by the presence of the neutron. Ttns IS only expected to be good at high energies, where one can reasonably expect some sort of impulse approximation to be valid We here assume a form for 4o containing an adjustable parameter which is a measure of the average distortion experienced over the colhslon, and we shall fit this parameter by reqmrmg a fit to the zero-energy scattering lengths. We neglect the small d-state probability of the deuteron and take ¢d = fl~[0 04893 exp(--0 0170/~r~)+0.1968 exp(--0 0911fl~r 2) + 0 41555 exp(--0 4323flg"rg)]

(4)

where r ----- 2R is measured in units of 10-xa cm This wave-functlon with fl = 1 represents the free state wave-function of the deuteron to wltinn 5 ~/o to several deuteron radn a) The nucleon-nucleon potential assumed has the form V, = --V0[~(I+P~)+~s ,

.s~+TS,j]exp(--ar ~)

(5)

Such a potentml differs from recent fits to the two-body data m having no repulsive core and no term of the form L • S The effects of such features on the results is discussed later A potential of the form (5) can give a reasonable fit to the low energy two-body data for a range of parameters Vo, ,t, a, ~ ll)

434

L

M

DELVES

AND

D

BROWN

With tins form for V~ we have

~½ ~ 2 ( a + 2 b ) t e x p (

2abr2 a+2b}

(6)

and :

7~½S1".Sdo e x p - -

( 2abrZ~

(7)

The exchange terms can be evaluated approximately by expanding P~ = exp(vrr~j ×p,j)/?~ as a power series in r~j × p~j l~), rewriting r,~ and p,j in terms of r and p relative to the deuteron centre of mass, regrouping the terms, and neglecting terms in R × P higher than the first Such terms contribute nothing at zero energy; at higher energies they are small owing to the spherical s y m m e t r y of the deuteron The result, valid at low energies, is

-- 2(a+2b)~:t½P~exp I

a+2b--2ab r 2j-I

(8)

The tensor terms can be evaluated by expanding S12 = -

3

(S 1 " rl)(s~, r l ) - - % "%

(9)

and noticing that the averages over % and R are independent Tins gives

Sldr2 J]"~ 0 r, +R2R2 [e-*-R)'--e-~(~-R)'Je-ZbR* dR

(10)

where Sxd has the same form as (9) with % replaced by the deuteron spin Sd. The expression (10) ,s zero for r = 0, as would be expected from considerahons of symmetry. For large r >> b it is Sld [';71:(a -3I- 2b )] ½ V 2abr~7 4[(a+2b)~+#] exp [_ a+2b_l

( 10a)

and for small r << b it is 2(a+2b)½ exp (--at ~)

(10b)

For mtermechate values of r, (10) must be integrated numerically. The hmlt of a point deuteron (b -+ Qo) Is of interest, in this case we have at once <$diSxse-a't'+Slae-"2tkba> = Side -a',

(10c)

T H E T H E O R Y OF n - - d

S C A T T E R I N G W I T H T E N S O R FORCES

4~5

3. N u m e r i c a l C a l c u l a t i o n s The effective neutron deuteron potential derived in the previous section contains non-zero matrix elements between the quartet and doublet states of the system, as well as between states of different angular momenta Thus S 2 Is not a good quantum number in n - - d colhsmns with tensor forces, In contrast to nucleon-nucleon scattering Nevertheless we shall neglect terms leadang to a change in channel spin, the remaining matrix elements of S1d are then given by

S~d~S~

Sxd ~

= 0,

M- H m

= -- [(2 j - 3 ) / 4 j ] ~ j ~ . , +{ 3 ( 2 j - 1 ) (2j + 3) }½/4j~.i+~,t, t Sld 6~'dJ,/+t,| -- [ ( 2 j - 3 ) / 4 j ] ~ j + ~ . t + { 3 ( 2 j - 1 ) ( 2 j + 3 ) } ½ / 4 j ~ . j _ l .

(11)

Sldq/~l_~, ! = [ ( 2 J + 5 ) / 4 ( J + l ) ] ~ ' ~ , 1 _ ~ , t + { 3 ( 2 J - 1 ) ( 2 J + 3 ) } ½ / 4 ( J + l ) a d ~ , . v + L

t,

"

-

M

Sld ql r.]+t,! = - [ (2J + 5) / 4 ( J + l )] ~ , i + o , t + { 3 ( 2 J - 1 ) (2J + 3) }½/ 4 ( J + l )ql~,l_t. l, where ~ . z . , is the usual normalmed spin-angle function. Note that the nucleon spin operators used in the definition of S,~ chffer m normallzahon from the Pauh spin matrices by a factor The constants of the mohon are then S 2, J , ~, and the doublet states of the system are single channel states, the Schrodmger equahon for these can be integrated without trouble to give the (real) channel phase-shifts The quartet states for J va ½ are two-channel states in which channels with orbital angular momenta l and l + 2 , say, are coupled together Two such channels are characterized by two real elgenphase-shifts *8~j~ and 4¢5pl. and a mixture parameter 4ej~, In the notation of Blatt and Bmdenharn 13.1,) We shall usually drop the suffixes J , ~ and the spin index. The functions cot ~5~, cot 6p and tan e can be expanded in terms of the wave-number k as follows 13. i4): 1

k*~+l cot ~ = [(2/+ Z)!t]2/a~' I -

T,

cot 6, = L

+½,~k*+...], 1 ap

+½,,**+ ],

(12)

t a n , = qok*+qxk4+ .. , and the coefficients qo, qx, a~, a a, r~, r a can be calculated from a knowledge of the wave-functions at zero energy. The necessary relations are given in ref 14) for the state with l = O, which xs here the J = {, ~ = + 1 quartet state The extensmn to states with l v~ 0 is straightforward, the results are

436

L

M

D E L V E S AND D

BROWN

r~ ---- 2 %2Z+2 .I~°{ oU~002 --0U~ 2 --0W~ 2 - - %4 / + 2 ( 2 / + 1 ) --2 r --2Z)d r, 2ta 2z+e (00)" U002-]- W 002 U 2 W 2 a41+lOt2l+~5-2r-(2t+4)'tdr, rD---- / e Jo to p o p - - o . ~ - - o ~ - - ,e ~ / j-

ql = ( , 2~_(2z+I)T ,-iz+lj, _ 5 , a= -v

(13) rr

00 r r

to'--'= o'-'p

. . . . .

oo

.

.

.

.

.

.

.

.

.

.

.

-l-oVV~ oVV~ - - o ( - ; = o U p - - o ~ o l / V p

~2z+1up ~21+5/.,---uo: ~ t . - l - o~] ( 2 l + 5)-1 r-(V+4)qo } dr, oU~, oW~ and oUp, 0WB are particular sets of solutions of the coupled equations for the s t a t e at zero energy. T h e i r definitions, t o g e t h e r with those of the o t h e r t e r m s in (13) and the results for % , a B and qo are given in refs 13) a n d la). The single channel states can be included u n l f o r m l y w l t h the two-channel states b y defining for t h e m a second phase-shift (~pand a m i x t u r e p a r a m e t e r e which are b o t h identically zero, the eigenphase-stuft ~ is t h e n the phaseshift for the channel as usually defined The p a r a m e t e r 4%1" for J __ ~,3 • -- + 1 is the q u a r t e t scattering length, a n d there is similarly a d o u b l e t scattering length 2a~t+ E x p e r i m e n t a l l y these are n o t k n o w n uniquely, the two sets (A) 2a = (0 8 + 0 3) × 10 -i3 cm,

4a ----- (8 34-0 3) × 10 -13 era,

(B)

' a = (2.44-0 2) × 10 -13 cm

2a =

(6 24-0 1) × 10 -13 cm,

(14)

b o t h fitting the available d a t a , and there has been some discussion theoretically as t o which of these sets is correct 1,15-1s). T h e scattering lengths 2a and 4a have been calculated n u m e r i c a l l y for nucleon-nucleon potentials of the form (5) of intrinsic ranges from 1 6 to 3 3 × 10 -13 cm, a n d values of the d e u t e r o n distortion p a r a m e t e r fl from 1 (no distortion) to c~ (point deuteron) N o t e t h a t the picture of the colhslon used does n o t allow values of fl less t h a n one for a t t r a c t i v e potentials, the distorting n e u t r o n will always t e n d to compress the d e u t e r o n It was not possible to get a g r e e m e n t with set (A) for a n y values of fl a n d force constants This is due to the v e r y small doublet scattering length for this set, winch corresponds to a partial well more t h a n twice as deep as is required for the t w o - b o d y d a t a Thus, set (A) m a y be ruled out I t was possible, however, to fit set (B) for all nucleon-nucleon potentials in the range tried, the calculated d e u t e r o n distortions are not v e r y sensitive to this range, a n d are given in table 1 for r e p r e s e n t a t i v e nucleon intrinsic ranges These results indicate t h a t neglect of the d i s t o r t i o n of the d e u t e r o n is a good a p p r o x i m a t i o n even at v e r y low energies in the d o u b l e t state, b u t is not valid in the q u a r t e t state T h e value 0 8 for fl m the doublet state with intrinsic range 2 6 × 10 -13 cm is p r o b a b l y n o t significantly different from 1, however, the results would seem to rule out a nucleon-nucleon force of range as long as 3 3 × 10-13 cm; such a range is a l r e a d y in disagreement with other d a t a n )

THE THEORY OF n--d SCATTERING WITH TENSOR FORCES

437

To estimate the polarization of the neutron in the scattering, the elements of the scattering matrix have been calculated for neutron energies from 0 to 3 25 MeV assuming a nucleon-nucleon force with an intrinsic range of 2 6 × 10-la cm and parameters Vo = 31 58MeV,

oc= O, 7 =

393

(15)

TABLE 1 D e u t e r o n d i s t o r t i o n p a r a m e t e r fl Nucleon-nucleon intrinsic range

~d

~q

16 26 33

1 05 08 0 67

19 2 13 2 08

A neutron distortion as given in table 1 was assumed, independent of energy For the one-channel states the calculation is straaghtforward, for the two-channel states the effective n - - d force has been replaced by a square well giving the same scattering lengths and the coupled equations integrated numerically at zero energy to give the first two terms in the expansions (12) of the d~, dp, e States up to J = { have been included, the numerical results are given In table 2, together with those for the single-channel states TABLE 2 E x p a n m o n c o e t h c m n t s for t h e s c a t t e r i n g rnatrLx S

J

~

½

+

½

+

t ½ ½

+ -

t j

+ +

aa x 1013

7 0 4 0 --5 --0 2 --9 --4 --1

52 21 419 767 55 78 73 77

ap x 1013

r a x 1013

0 0 0 0 0 0 2 45 2 62 241 3 58

2 28 0 0 772 0 - - 2 217 --35 7 1 09 - - 1 2 04 - - 0 351 --

2 04

r B x 10 x3

0 0 0 0 0 0 --1 1 --0 --0

33 145 686 857

qo x 1015

qz x 10 5°

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 477 0 000403 0 0000195 0 986

0219 0663 00399 114

These results have been used to calculate the angular distribution and polarization of the neutron over this range of energies The angular distribution m a y be written da/dg2 = ~ a L P L (cos O) L

(16)

438

L

M

DELVES AND D

BROWN

Figs 1 to 4 give plots of 4~a 0 -- O'tot, al/ao, a2/a o and aa/a 0 agmnst neutron energy in the laboratory system The experlmental points have been calculated from the experimental angular distributions 19-23), the full line drawn through the experimental points for atot is taken from ref ~4), while for alia o, aJa o the smooth curve labelled "experimental" has been drawn through the experimental points by eye for comparison purposes. There is also plotted the theoretical results for a nucleon-nucleon force of the same Intrinsic range but with no tensor force, 1 e 7 = 0, winch glves the same n - - d scattering lengths, the amount of tensor force is not determined

~3 ~2

I

,

o'2

o'.,

'

!

,o ~ En (McV)

I

,i

i

,o

Fig 1 A graph of the total n - - d cross-section against neutron energy m the laboratory system Curve A experimental from ref 24), B

theory, no tensor force,

C theory, including tensor force The open mrcles are experimental points from refs lg-zs)

o

/ /

i

i

; |

OI

m

I,,/ I

I

02

04

I

!

i

IO 2 E n (McV)

i

4

I

I

I0

Fig 2. A graph of aJao against neutron energy m the laboratory system Curve A experimental, B theory, no tensor force, C theory including tensor force The open circles are exper,mental points from refs ,,-2s)

T H E T H E O R Y OF n - - d

12

!

i

S C A T T E R I N G W I T H T E N S O R FORCES

i

,

!

i

i

439

i

A

08

O4 B (~)'1

02

04

I0 ~) En (MW)

4

tO

Fig 3 A graph of az/a e agalnst neutron energy in the laboratory system Curve A experimental, B theory, no tensor force, C theory, including tensor force The open circles are experlmenta] points from refs zD-ts)

OS

4 0 0

O

o,

•

o

0'2 En (M~)

Fig 4 A graph of as/a o agamst neutron energy in the laboratory system Curve B theory, no tensor force. C theory, including tensor force The open circles are experimental points from refs z,-ts)

umquely b y the two-body data but hes between 7' = 0 and 7 = 3 93 n) Figs 5 and 6 gzve angular distributions and the energy vanatzon at 90 ° c m. of the neutron polarization, measured along rtne×rse~ttered The variation with energy zs rather rapzd Polarlzat]on measurements have been carrmd out b y Brullman et al. ~) at 3 MeV, and b y Whzte, Chisholm and Brown ~6) in this laboratory at 2 26 and 3 1 MeV The two results are not conszstent, that of BruUmanbemg ( + 3 + 6 ) % a n d t h a t of White ( + 4 8 + 5 ) % at 2.26 MeV and ( + 4 0 4 - 2 0 ) % at 3.1 MeV. Note figs. 5 and 6 differ from the results quoted m ref 2e) owing to the correctmn of a numerical error. A recent measurement at 2 MeV b y Cranberg 2s) also indicates a small ( < 7 %) polanzatmn. Sege1.7) has carried out measurements of the

440

L

M

DELVES

AND

D

BROWN

polarization of protons scattered from deutermm at 3 MeV, and obtained (--12-4-7) % The results of Brullman and Segel are in agreement with the theorehcal predictions, those of White et al are not The experiment is bemg repeated with neutrons in this laboratoryt

-I0

~ I

J

I I

En (MeV)

Fig 5 A g r a p h of n e u t r o n polarization a t 0 = 90 ° c m , $ = 0, a g a i n s t n e u t r o n e n e r g y m the l a b o r a t o r y s y s t e m The polarization is m e a s u r e d ~n the d l r e c t m n of r i n e × r s c a t t e r e d i

i

%-s

o

~

ooll

t~o

Fxg 6 A n g u l a r d l s t r l b u t l o n s of n e u t r o n polarization a t ~ ~- 0

The second spin moments of the recoll deuteron are plotted m fig 7 at 3 25 MeV neutron lab energy It is seen that the effects are large, but this does not seem a prachcable way of preparing ahgned deuterons owing to the low intensities available ¢ N o t e a d d e d *n p r o o / B r u l h n a n et al h a v e recently a0) r e m e a s u r e d t h e polarization at 53 ° c m , obtaining ( + 1 5 + 8 ) %

THE

THEORY

OF

n--d

02

!

b o

A

SCATTERING

!

i

WITH

i

TENSOR

FORCES

441

i

c

-02

-04 9cm

F i g 7 A g r a p h of t h e second order s p i n m o m e n t s of t h e d e u t e r o n a t 3 25 MeV n e u t r o n e n e r g y in t h e l a b o r a t o r y s y s t e m T h e z-axis is t a k e n a l o n g t h e a n g l e of s c a t t e r i n g of t h e d e u t e r o n C u r v e A ld~,

B (T22)lda, C /da

4. D i s c u s s i o n The theory of n - - d scattering with tensor forces presented here has been rather severely slmphfied In fitting the distortion of the deuteron at zero energy to the observed scattering lengths we have neglected the effects of the Pauh prlnclple, the coupling between the doublet and quartet states, and the D-state probability of the deuteron The last effect IS expected to be small for the energies we are interested In, the neutron will see only the gross structure of the deuteron At higher energies further slmphflcatlons have been made The effective range expansion for the coupled channels is valid only so long as higher terms in the series are not important For nucleon-nucleon colhslons this is not until an energy of 10--15 MeV, but for n - - d scattering they are already appreciable at less than 1 MeV Above this energy the results become increasingly unreliable owing to this case Moreover, we have lmphcltly assumed that the distortion of the deuteron is independent of energy and of angular moment u m Tkls as unlikely to be so, the distortion is expected to tend to zero at high enough energies, and to do so more quickly with increasing angular momentum Inclusion of this effect, and of momentum-dependent terms omitted from the Majorana exchange Integrals, leads to an explicitly energydependent effective potential, at is then no longer vahd to use effective range expansions in the form (12)29) Nonetheless at would appear from the measure of agreement with experimental angular distributions that the most important features of the problem have been retained In particular, the results for the angular distribution

442

L M DELVES AND D BROWN

show that in the absence of any tensor force the contributions from angular momenta greater than zero are far too small, while the nucleon-nucleon potential parameters (15) reproduce the observed curves of al/a o fairly well The main discrepancy IS in the prediction of too large an effective range parameter r~ for the quartet J = 3, z ~ _ 1 state, giving an l -- 1 resonance in the total cross-section and too sharp a rise in the curve for alia o Polarization calculations are more sensitive to changes in the scattering matrix, and the results shown are not expected to give more than a qualitative account of the polarization to be expected The angular distribution of polarlzatxon is less sensitive than the total magnitude to small changes in the force parameters, and measurements for a range of angles and energies would be of great interest It would also be of interest to repeat the calculations for a more realistic nucleon-nucleon potential, including a term of the type L . S, and a repulsive core A spin orbit force of this nature will not change the fit to the scattering lengths, but will contribute to the calculated asymmetry and polarization Inclusion of a repulsive core on the other hand m a y change the calculated chstortlon significantly The major contribution to the effective n - - d potential comes from the region of the deuteron wavefunction near the orlgln, with a repulsive core the wave-function is zero m this region. On the other hand the nucleon-nucleon potential is then much deeper outside the core to fit the two-body data. It is therefore likely that calculated chstortlons will be smaller than found with a smoothly varying potential, but the scattering matrix will be more sensitive to the distortion than with a smoothly varying potential such as that assumed here. The effect of distorting the deuteron is to make the equivalent n - - d potential deeper and of a shorter range than for an undistorted deuteron, these two effects partially cancel

5. C o n c l u s i o n s The methods used give a reasonable account of the angular distribution and absolute cross-section of neutrons scattered from deuterium at low energies, and enable the scattering lengths to be determined umquely. Neglect of the chstortlon of the deuteron by the neutron is a good approximation in the doublet state, but m the quartet state the deuteron is strongly distorted at low energies The polarization of the neutron in the scattering Is expected to be of the order of --2 % at 3 MeV. References 1) R A C h n s t m n a n d J L G a m m e l , P h y s l~ev 91 (1953) 100 2) R A B u c k i n g h a m a n d H S W M a s s e y , P r o c R o y Soc A 179 (1941) 123 3) R A B u c l a n g h a m , S J H u b b a r d a n d H S W Massey, Proc R o y Soc A 211 (1952) 183

THE

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)

THEORY

O F n---d S C A T T E R I N G

WITH

TENSOR

FORCES

443

A Werner, Nuclear Physics 1 (1956) 9 T Y Wu and J Ashkm, Phys Rev 73 (1948) 973 B H Bransden, Proc Roy Soc A 209 (1951)380 B H Bransden, K Smith and C Tate, Proc Roy Soc A 247 (1958) 73 L C Bledenham and J M Blatt. Nuclear Physics 6 (1958) 359 J L Gammel and R M Thaler. Phys Rev 107 (1957) 1337 P S Slgnell and R E Marshak, Phys Rev, 106 (1957) 832 J M Blatt and V Wmsskopf, Theoretical Nuclear Physics (J Wiley and Sons, 1952) Chapters II and IV J Wheeler. Phys Rev 50 (1936) 643 J Iv[ Blatt and L C Bledenharn, Phys Rev 8b (1952)399 L C Bledenharn and J M. Blatt, Phys Rev 93 (1954) 1387 A Troesche and M Verde, Helv Phys Acta 24 (1951) 39 M M Gordon, Ph~s Rev 80 (1950)111 F G Prohammer and T A Vqelton, Quarterly Report ORNL 1005 As Davldov and G T Ffllppov, J E T P. (USSR) 4 (1957) 267 R K Adair, A Okazakl and M Walt, Phys Rev 89 (1953)1165 J D Seagrave, Phys Rev 97 (1955)1757 J C Allred, A H Armstrong and L Rosen, Phys Rev 91 (1953) 1165 J D Seagrave and L Cranberg, Phys Rev. 105 (1957) 1816 I Hamouda and G de Montmoulhn, Helv Phys Acta 25 (1952) 107 D J Hughes and J A Harvey, Neutron Cross-secUons BNL 325 (1955) ~i Brullman, H J Gerber and D Miner, Helv Phys Acta 31 (1958) 318 R E White, A Clnsholm and D Brown, Nuclear Physics 7 (1958) 233 R Segel. private commumcatlon L Cranberg. Bull Am Phys Soc 3 (1958)365 L M Delves, Nuclear Physics 8 (1958) 358 M Brullman el al, Helv Phys Acta 31 (1958) 580