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Triaxiality in nuclear rotational states
Volume
34B. number
7
PHYSICS
TRIAXIALITY
IN NUCLEAR
LETTERS
12 April
ROTATIONAL
STATES
1971
z
K. TANABE $$ and K. SUGAWARA-TANABE Physics
Department,
Yale
University, Received
New Haven,
Connecticut
06520,
USA
10 January 1971
Non-axially symmetric deformation of a nucleus is introduced to account for the admixture of the collective rotational bands and the distribution of the yrast levels. A new rotational level scheme and an approximate formula for the yrast are derived.
Some specific properties of the yrast cascades have been observed in the recent studies of the gamma-ray deexcitation of compound-nucleus reaction products from (HI,xn) reactions, and the characteristic level distribution in the yrast region is reasonably accounted by the admixted collective bands [l]. This implies the possibility of the non-axially symmetric deformation as pointed out by Mottelson [2]. On the other hand, in odd mass nuclei, there have been experiments suggesting the importance of the band mixture [3]. The prupose of present note is to show that these characteristics are qualitatively described by assuming the non-axially symmetric deformation of a nucleus, when its angular momentum is at least greater than the observed maximum value belonging to the ground state collective band. We assume a rigid rotator for simplicity, then the Hamiltonian for an even-even nucleus is given by
c
H=
z;/(2ai)
i=x,
y,
=12/(2$y
-tcl/a,-l/a,,Z~-ac1/a,-1:a,,z~
,
2
where.2 ‘s are the moments of inertia defined along the principal axes. We assume the following relation,
a, G3a,‘a,
(2)
*
We express the angular momentum operator in terms of Bose operators in analogy to the HolsteinPrimakoff transformation [4] Z+ = I,+ iZy = -(2Z)-9a+[l-ii/(2Z)]+ Z_ = I, - iZy = -(21)-i I,
=
[l-%/(2Z)]+a
z-ii
with n = a+a, where a and a+ satisfy the commutation relation of Bose operator. Physically this replacement is applicable only to the state whose eigenvalue n for the number operator k does not exceed 21. We expand the Hamiltonian in powers of n/(21) up to the first order. In the axially (i.e. 9% = ay) this expression is equivalent to the exact Hamiltonian. To diagonalize the Hamiltonian we further introduce the linear transformation
symmetric
limit
a = q+b +q_*b a+
= q-b
+q+*b
(4)
$Research (Report Yale 2726-590) supported by the U.S. Atomic Energy Commission under Contract AT (30-l) 2726. $$Present Address: Department of Physics, Saitama University, Urawa, Saitama, Japan.
575
Volume 34B, number 7
PHYSICS L E T T E R S
12 April 1971
with
I.+I 2 - d . _ l 2 = 1
,
(5)
where b and b+ a r e new Bose o p e r a t o r s . Then, f r o m the r e q u i r e m e n t that the coefficients of bb and b+b + vanish we find a set of the following r e a l solutions for , ± up to the f i r s t o r d e r in i/(21).
,± = ½(f~)
3
I i +~i(f
~ fl)2 I
(6)
with the definition f - : [ ( 1 / ~ z - 1 / ~ y ) / ( 1 / ~ x -
1/~x)]¼ .
The diagonal p a r t of our H a m i l t o n i a n is given by
H
-
I ( I + 1) 2~ z
2I[B-
1 2I(A-B)](k+½)
+Ak 2
,
(7)
where k = b+b and
A = ¼(2/~z - 1/~x - 1/~y)
(8) B = ½[(1/~z - 1 / ~ x ) ( 1 / ~ z
- 1/~y)]½ .
Putting K = I - k , where k is the eigenvalue of k, we obtain a new level scheme E ( I , K) = 2(A - B)C [I - ½(K/C - 1)] 2 + [A - (A - B ) / ( 2 C )] K2 - ~ ( 1 / ~ X + 1 / ~ y )
.
(9)
where
C = 1 + }( 1 / ~ x + 1 / ~ y ) / ( A - B)
(10)
In the axially s y m m e t r i c l i m i t , (9) r e d u c e s to the u s u a l e x p r e s s i o n I ( I + 1)/(2~ x) + [ 1 / ( 2 ~ z) - 1/(2~x) ] K2. It is i n f e r r e d that the levels for positive and negative K a r e split and the lowest level for a fixed I is not n e c e s s a r i l y of K= 0. The lower bound of the y r a s t r e g i o n is given by E y r a s t ( I ) = [1/(2 ~z) - B 2 / A ] 1(I + 1) + ~1 (A - B 2 / A )
,
(11)
which is an approximate e x p r e s s i o n for the y r a s t line itself. It may be e x p r e s s e d that in an odd m a s s n u c l e u s the last nucleon c o n t r i b u t e s to enhance the n o n axially s y m m e t r i c d e f o r m a t i o n much m o r e c o m p a r e d with an e v e n - e v e n n u c l e u s , b e c a u s e the r e s t o r t i n g force to the axially s y m m e t r i c shape such as the p a i r i n g force is not strong enough to g u a r a n t e e such a s y m m e t r i c deformation. To take into c o n s i d e r a t i o n the c o n t r i b u t i o n of the last nucleon to the d e f o r m a tion, we decompose the total a n g u l a r m o m e n t u m into the spin s , and the a n g u l a r m o m e n t u m of the r e s t s y s t e m R including the o r b i t a l a n g u l a r m o m e n t u m of this nucleon. Here we introduce such a model a p p r o p r i a t e to the H a m i l t o n i a n obtained by^replacing 1 by R = I - s in (1). The eigenvalue of R 2 i s uniquely given by either (I + ½)2 + (I + ½) or (I + ½)z _ (I + ½) according to the s p i n - p a r i t y selection rule. Thus the energy eigenvalue for the odd m a s s case is obtained by r e p l a c i n g I in (9) simply by e i t h e r I + ½ or I - ½ according to the I value and p a r i t y of this state. Then, denoting the energy value by E (/) we can show that the quantity defined be D (I) = [E (I) - E ( / - 1)]/(2 I) take s a l t e r n a tingly l a r g e r or s m a l l e r value with i n c r e a s i n g I, even if I does not belong to ½-band, as follows
D I ) = 2(A - B ) ( 2 C - K / I ) n(I ) = 0
forR =I+½ forR = I - ½ •
(12)
In these equations expected at large I, ~ ' s a r e different f r o m the m o m e n t s of i n e r t i a for e v e n - e v e n n u c l e u s , but b e a r the c o n t r i b u t i o n f r o m the final nueleon. Now we c o n s t r u c t the wave function. The i n v e r s e t r a n s f o r m a t i o n of (4) is e x p r e s s e d in the f o r m b = e x p s . a - exp(-s) with the a n t i - h e r m i t i a n o p e r a t o r defined by s = - (a+a + - aa), where x = a r c o s h , + / 2 >/ 0. The energy eigenstate is d e g e n e r a t e with r e s p e c t to M which is the eigenvalue of I Z in the space fixed s y s t e m , and is given by 576
(13)
PHYSICS L E T T E R S
Volume 34B, number 7
link)
=
12 Apri[ 1971
1 ,b+b+...b+ t (k!)-~ k I°) b ,
(14)
where the v a c u u m 10)b is defined by blO} b = O. On the other hand for the s a m e I and M the eigenstate of I z in the body fixed s y s t e m is given by
" " - ½ 'a+a+'''a+ ]0 )a' [ I M n ) = [n.) n
(15)
where the v a c u u m [0>a i s defined by a 10}a The state lIMk> can be expanded by
=
0. In eq. (15) the states a r e s p u r i o u s i f n exceeds 2I.
2I
II Mk )
=
E ]IMn>Gnk(X) n=O
(16)
•
To e l i m i n a t e the c o n t r i b u t i o n s of the s p u r i o u s state in the above expansion we have a s s u m e d that any m a t r i x e l e m e n t r e l a t e d to the s p u r i o u s states v a n i s h e s [5]. In o r d e r to d e r i v e the explicit functional f o r m of the quantity Gnk(X ) we make use of a c o m m u t a t i o n r e l a t i o n [d/d~ ,~ ] = 1 for a s c a l a r v a r i a b l e ~. Because of this the r e l e v a n t expectation value can be c a l c u l a t e d by r e p l a c i n g a, a + and ket v a c u u m by d / d ~ , ~ and unity r e s p e c t i v e l y , and by putting ~ = 0 a f t e r all the d i f f e r e n t i a l s a r e explicitly c a r r i e d out, i.e.
(ntk!)-½ [ dn d ~nLexp{ -x(~ 2
d2
The power of ~ can be expanded in t e r m s of the o r t h o n o r m a l set of the h a r m o n i c o s c i l l a t o r wave I 1 7/ I functions Un(~) = N n exp (-~ 2) H n ( ~ ) , with N n = [ 2 ~ / ( ~ 2 n!)]-~, where Hn(~) is the H e r m i t e p o l y n o m i a l of n - t h o r d e r . In the expansion =
(i8)
Nz
l=O the coefficient Ikl i s given by min(k ,l)
1
Ikl = n ~ k I l !
~
m =0
2m/2 2k_m
[(n - m)/2] ![(k - m)/2]
,
(19)
where the s u m m a t i o n extends over the v a l u e s of m which give even i n t e g e r s for both n - m and k-re. Using eqs. (17), (18), (19) and the defining p r o p e r t i e s of the H e r m i t e p o l y n o m i a l [6] we can evaluate the coefficients for the s p e c i a l c a s e s as follows, 1
G00 (x) = 77+2
_3
and
G11 (x) = ~/+2
(20)
The coefficient Gnk(X) for any n and k is obtainable f r o m the two v a l u e s in (20) and the r e c u r s i o n r e lations d e r i v e d d i r e c t l y f r o m the definition in (16). It is m a r k e d that Gnk(X) = 6nk in the s y m m e t r i c l i m i t (i.e. x = 0). When the m a t r i x e l e m e n t s of the p h y s i c a l t e n s o r T f z " .. a r e known in the body-fixed s y s t e m , we can c a l c u l a t e the c o r r e s p o n d i n g t e n s o r TFG" .. in the laboYatory s y s t e m u s i n g the r e l a t i o n
T F G . . " = k F f ~ G g . . . Tfg .. "
(21)
Since the m a t r i x e l e m e n t s of the d i r e c t i o n c o s i n e s ~'s between the states in (15) have been obtained by C r o s s et al. [7], the expectation value of (21) is r e a d i l y evaluated by making use of the expansion in (16). A s an example, we calculate a m a t r i x e l e m e n t of the quadrupole t e n s o r
q(2) = (23_)½
0
QZZ
_ 6- ½ F=X, Y, Z
QFF 577
Volume 34B, number 7
PHYSICS
LETTERS
12 April 1971
a s f o l l o w s . If the c h a r g e d e n s i t y is p r o p o r t i o n a l to the m a s s d e n s i t y in t h i s r i g i d body, the q u a d r u p o l e m o m e n t is g i v e n by 2
Ze
~ x + ~y
21
3(1- n) 2 - I(I+ 1)
n=O [ I ( I + 1) ( 2 I - 1) (2I+ 3)]2
Gnk(X) 62
w h e r e c-~ is the m a s s of the n u c l e u s . T h e a u t h o r s w i s h to thank P r o f e s s o r J. C. R a s m u s s e n f o r v a l u a b l e d i s c u s s i o n s . One of t h e a u t h o r (K.T.) w i s h e s to t h a n k P r o f e s s o r V. W. H u g h e s f o r the h o s p i t a l i t y e x t e n d e d h i m . A n o t h e r (K.S.T.) w i s h e s to thank P r o f e s s o r R. E. B e r l i n g e r f o r the h o s p i t a l i t y e x t e n d e d h e r . T h e y a l s o thank Dr. G i n o c c i o f o r r e a d i n g the m a n u s c r i p t and v a l u a b l e c o m m e n t s .
R ere 7"enc e s [1] J. o. Newton, F. S. Stephens, R. M. Diamond, W. H. Kelly and D. Ward, Nuclear Physics A141 (1970) 631. This paper contains the references to the pertinent experiments. [2] B.R. Mottelson, Leigh Page Lectures at Yale University {1970). [3] S.A. Hjorth, H. Ryde, K.A. Hagemann, G. LcCvhcCiden and J. C. Waddington, Nuclear Physics A144 (1970) 513; I. Rezenka, S. Hu[tberg, H. Ryde, J . O . Rasmussen, F.M. Berntha[ and J. Alonso, to be published. [4] T. Holstein and H. Primakoff, Phys. Rev. 58 (1940) 1098. [5] F . J . Dyson, Phys. Rev. 102 (1956) 1217 and 1230. [6] A. Erdelyi, Higher Transcendental Functions (McGraw-Hi|l, 1958), Vo[. 2, p. 192. [7] P . C . Cross, R.M. Hainer andG. W. King, J. Chem. Phys. 12 (1946) 210.
578
34B. number
7
PHYSICS
TRIAXIALITY
IN NUCLEAR
LETTERS
12 April
ROTATIONAL
STATES
1971
z
K. TANABE $$ and K. SUGAWARA-TANABE Physics
Department,
Yale
University, Received
New Haven,
Connecticut
06520,
USA
10 January 1971
Non-axially symmetric deformation of a nucleus is introduced to account for the admixture of the collective rotational bands and the distribution of the yrast levels. A new rotational level scheme and an approximate formula for the yrast are derived.
Some specific properties of the yrast cascades have been observed in the recent studies of the gamma-ray deexcitation of compound-nucleus reaction products from (HI,xn) reactions, and the characteristic level distribution in the yrast region is reasonably accounted by the admixted collective bands [l]. This implies the possibility of the non-axially symmetric deformation as pointed out by Mottelson [2]. On the other hand, in odd mass nuclei, there have been experiments suggesting the importance of the band mixture [3]. The prupose of present note is to show that these characteristics are qualitatively described by assuming the non-axially symmetric deformation of a nucleus, when its angular momentum is at least greater than the observed maximum value belonging to the ground state collective band. We assume a rigid rotator for simplicity, then the Hamiltonian for an even-even nucleus is given by
c
H=
z;/(2ai)
i=x,
y,
=12/(2$y
-tcl/a,-l/a,,Z~-ac1/a,-1:a,,z~
,
2
where.2 ‘s are the moments of inertia defined along the principal axes. We assume the following relation,
a, G3a,‘a,
(2)
*
We express the angular momentum operator in terms of Bose operators in analogy to the HolsteinPrimakoff transformation [4] Z+ = I,+ iZy = -(2Z)-9a+[l-ii/(2Z)]+ Z_ = I, - iZy = -(21)-i I,
=
[l-%/(2Z)]+a
z-ii
with n = a+a, where a and a+ satisfy the commutation relation of Bose operator. Physically this replacement is applicable only to the state whose eigenvalue n for the number operator k does not exceed 21. We expand the Hamiltonian in powers of n/(21) up to the first order. In the axially (i.e. 9% = ay) this expression is equivalent to the exact Hamiltonian. To diagonalize the Hamiltonian we further introduce the linear transformation
symmetric
limit
a = q+b +q_*b a+
= q-b
+q+*b
(4)
$Research (Report Yale 2726-590) supported by the U.S. Atomic Energy Commission under Contract AT (30-l) 2726. $$Present Address: Department of Physics, Saitama University, Urawa, Saitama, Japan.
575
Volume 34B, number 7
PHYSICS L E T T E R S
12 April 1971
with
I.+I 2 - d . _ l 2 = 1
,
(5)
where b and b+ a r e new Bose o p e r a t o r s . Then, f r o m the r e q u i r e m e n t that the coefficients of bb and b+b + vanish we find a set of the following r e a l solutions for , ± up to the f i r s t o r d e r in i/(21).
,± = ½(f~)
3
I i +~i(f
~ fl)2 I
(6)
with the definition f - : [ ( 1 / ~ z - 1 / ~ y ) / ( 1 / ~ x -
1/~x)]¼ .
The diagonal p a r t of our H a m i l t o n i a n is given by
H
-
I ( I + 1) 2~ z
2I[B-
1 2I(A-B)](k+½)
+Ak 2
,
(7)
where k = b+b and
A = ¼(2/~z - 1/~x - 1/~y)
(8) B = ½[(1/~z - 1 / ~ x ) ( 1 / ~ z
- 1/~y)]½ .
Putting K = I - k , where k is the eigenvalue of k, we obtain a new level scheme E ( I , K) = 2(A - B)C [I - ½(K/C - 1)] 2 + [A - (A - B ) / ( 2 C )] K2 - ~ ( 1 / ~ X + 1 / ~ y )
.
(9)
where
C = 1 + }( 1 / ~ x + 1 / ~ y ) / ( A - B)
(10)
In the axially s y m m e t r i c l i m i t , (9) r e d u c e s to the u s u a l e x p r e s s i o n I ( I + 1)/(2~ x) + [ 1 / ( 2 ~ z) - 1/(2~x) ] K2. It is i n f e r r e d that the levels for positive and negative K a r e split and the lowest level for a fixed I is not n e c e s s a r i l y of K= 0. The lower bound of the y r a s t r e g i o n is given by E y r a s t ( I ) = [1/(2 ~z) - B 2 / A ] 1(I + 1) + ~1 (A - B 2 / A )
,
(11)
which is an approximate e x p r e s s i o n for the y r a s t line itself. It may be e x p r e s s e d that in an odd m a s s n u c l e u s the last nucleon c o n t r i b u t e s to enhance the n o n axially s y m m e t r i c d e f o r m a t i o n much m o r e c o m p a r e d with an e v e n - e v e n n u c l e u s , b e c a u s e the r e s t o r t i n g force to the axially s y m m e t r i c shape such as the p a i r i n g force is not strong enough to g u a r a n t e e such a s y m m e t r i c deformation. To take into c o n s i d e r a t i o n the c o n t r i b u t i o n of the last nucleon to the d e f o r m a tion, we decompose the total a n g u l a r m o m e n t u m into the spin s , and the a n g u l a r m o m e n t u m of the r e s t s y s t e m R including the o r b i t a l a n g u l a r m o m e n t u m of this nucleon. Here we introduce such a model a p p r o p r i a t e to the H a m i l t o n i a n obtained by^replacing 1 by R = I - s in (1). The eigenvalue of R 2 i s uniquely given by either (I + ½)2 + (I + ½) or (I + ½)z _ (I + ½) according to the s p i n - p a r i t y selection rule. Thus the energy eigenvalue for the odd m a s s case is obtained by r e p l a c i n g I in (9) simply by e i t h e r I + ½ or I - ½ according to the I value and p a r i t y of this state. Then, denoting the energy value by E (/) we can show that the quantity defined be D (I) = [E (I) - E ( / - 1)]/(2 I) take s a l t e r n a tingly l a r g e r or s m a l l e r value with i n c r e a s i n g I, even if I does not belong to ½-band, as follows
D I ) = 2(A - B ) ( 2 C - K / I ) n(I ) = 0
forR =I+½ forR = I - ½ •
(12)
In these equations expected at large I, ~ ' s a r e different f r o m the m o m e n t s of i n e r t i a for e v e n - e v e n n u c l e u s , but b e a r the c o n t r i b u t i o n f r o m the final nueleon. Now we c o n s t r u c t the wave function. The i n v e r s e t r a n s f o r m a t i o n of (4) is e x p r e s s e d in the f o r m b = e x p s . a - exp(-s) with the a n t i - h e r m i t i a n o p e r a t o r defined by s = - (a+a + - aa), where x = a r c o s h , + / 2 >/ 0. The energy eigenstate is d e g e n e r a t e with r e s p e c t to M which is the eigenvalue of I Z in the space fixed s y s t e m , and is given by 576
(13)
PHYSICS L E T T E R S
Volume 34B, number 7
link)
=
12 Apri[ 1971
1 ,b+b+...b+ t (k!)-~ k I°) b ,
(14)
where the v a c u u m 10)b is defined by blO} b = O. On the other hand for the s a m e I and M the eigenstate of I z in the body fixed s y s t e m is given by
" " - ½ 'a+a+'''a+ ]0 )a' [ I M n ) = [n.) n
(15)
where the v a c u u m [0>a i s defined by a 10}a The state lIMk> can be expanded by
=
0. In eq. (15) the states a r e s p u r i o u s i f n exceeds 2I.
2I
II Mk )
=
E ]IMn>Gnk(X) n=O
(16)
•
To e l i m i n a t e the c o n t r i b u t i o n s of the s p u r i o u s state in the above expansion we have a s s u m e d that any m a t r i x e l e m e n t r e l a t e d to the s p u r i o u s states v a n i s h e s [5]. In o r d e r to d e r i v e the explicit functional f o r m of the quantity Gnk(X ) we make use of a c o m m u t a t i o n r e l a t i o n [d/d~ ,~ ] = 1 for a s c a l a r v a r i a b l e ~. Because of this the r e l e v a n t expectation value can be c a l c u l a t e d by r e p l a c i n g a, a + and ket v a c u u m by d / d ~ , ~ and unity r e s p e c t i v e l y , and by putting ~ = 0 a f t e r all the d i f f e r e n t i a l s a r e explicitly c a r r i e d out, i.e.
(ntk!)-½ [ dn d ~nLexp{ -x(~ 2
d2
The power of ~ can be expanded in t e r m s of the o r t h o n o r m a l set of the h a r m o n i c o s c i l l a t o r wave I 1 7/ I functions Un(~) = N n exp (-~ 2) H n ( ~ ) , with N n = [ 2 ~ / ( ~ 2 n!)]-~, where Hn(~) is the H e r m i t e p o l y n o m i a l of n - t h o r d e r . In the expansion =
(i8)
Nz
l=O the coefficient Ikl i s given by min(k ,l)
1
Ikl = n ~ k I l !
~
m =0
2m/2 2k_m
[(n - m)/2] ![(k - m)/2]
,
(19)
where the s u m m a t i o n extends over the v a l u e s of m which give even i n t e g e r s for both n - m and k-re. Using eqs. (17), (18), (19) and the defining p r o p e r t i e s of the H e r m i t e p o l y n o m i a l [6] we can evaluate the coefficients for the s p e c i a l c a s e s as follows, 1
G00 (x) = 77+2
_3
and
G11 (x) = ~/+2
(20)
The coefficient Gnk(X) for any n and k is obtainable f r o m the two v a l u e s in (20) and the r e c u r s i o n r e lations d e r i v e d d i r e c t l y f r o m the definition in (16). It is m a r k e d that Gnk(X) = 6nk in the s y m m e t r i c l i m i t (i.e. x = 0). When the m a t r i x e l e m e n t s of the p h y s i c a l t e n s o r T f z " .. a r e known in the body-fixed s y s t e m , we can c a l c u l a t e the c o r r e s p o n d i n g t e n s o r TFG" .. in the laboYatory s y s t e m u s i n g the r e l a t i o n
T F G . . " = k F f ~ G g . . . Tfg .. "
(21)
Since the m a t r i x e l e m e n t s of the d i r e c t i o n c o s i n e s ~'s between the states in (15) have been obtained by C r o s s et al. [7], the expectation value of (21) is r e a d i l y evaluated by making use of the expansion in (16). A s an example, we calculate a m a t r i x e l e m e n t of the quadrupole t e n s o r
q(2) = (23_)½
0
QZZ
_ 6- ½ F=X, Y, Z
QFF 577
Volume 34B, number 7
PHYSICS
LETTERS
12 April 1971
a s f o l l o w s . If the c h a r g e d e n s i t y is p r o p o r t i o n a l to the m a s s d e n s i t y in t h i s r i g i d body, the q u a d r u p o l e m o m e n t is g i v e n by 2
Ze
~ x + ~y
21
3(1- n) 2 - I(I+ 1)
n=O [ I ( I + 1) ( 2 I - 1) (2I+ 3)]2
Gnk(X) 62
w h e r e c-~ is the m a s s of the n u c l e u s . T h e a u t h o r s w i s h to thank P r o f e s s o r J. C. R a s m u s s e n f o r v a l u a b l e d i s c u s s i o n s . One of t h e a u t h o r (K.T.) w i s h e s to t h a n k P r o f e s s o r V. W. H u g h e s f o r the h o s p i t a l i t y e x t e n d e d h i m . A n o t h e r (K.S.T.) w i s h e s to thank P r o f e s s o r R. E. B e r l i n g e r f o r the h o s p i t a l i t y e x t e n d e d h e r . T h e y a l s o thank Dr. G i n o c c i o f o r r e a d i n g the m a n u s c r i p t and v a l u a b l e c o m m e n t s .
R ere 7"enc e s [1] J. o. Newton, F. S. Stephens, R. M. Diamond, W. H. Kelly and D. Ward, Nuclear Physics A141 (1970) 631. This paper contains the references to the pertinent experiments. [2] B.R. Mottelson, Leigh Page Lectures at Yale University {1970). [3] S.A. Hjorth, H. Ryde, K.A. Hagemann, G. LcCvhcCiden and J. C. Waddington, Nuclear Physics A144 (1970) 513; I. Rezenka, S. Hu[tberg, H. Ryde, J . O . Rasmussen, F.M. Berntha[ and J. Alonso, to be published. [4] T. Holstein and H. Primakoff, Phys. Rev. 58 (1940) 1098. [5] F . J . Dyson, Phys. Rev. 102 (1956) 1217 and 1230. [6] A. Erdelyi, Higher Transcendental Functions (McGraw-Hi|l, 1958), Vo[. 2, p. 192. [7] P . C . Cross, R.M. Hainer andG. W. King, J. Chem. Phys. 12 (1946) 210.
578