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Coriolis anti-pairing theory of nuclear rotations nuclear meissner-effect
Volume 34B, number 7
CORIOLIS
PHYSICS
ANTI-PAIRING NUCLEAR
LETTERS
THEORY
OF
12 April 1971
NUCLEAR
ROTATIONS.
MEISSNER-EFFECT
B. L. B I R B R A I R Ioffe Physico-Technical Institute, Leningrad, USSR Received 25 February 1971
It is shown that the ground-state rotational bands of spherical nuclei are formed by a gradual alignment of the angular momenta of valence nucleons along the rotation axis. The predicted spin and excitation energy values of the first rotational state in 212po nucleus are J = 16+ and E = 2.91 MeV which is in reasonable agreement with the experimental data (J = (18+), E = 2.93 MeV).
T h e m i c r o s c o p i c t h e o r y of n u c l e a r r o t a t i o n s [1-4] is b a s e d on the M o t t e l s o n - V a l a t i n c o n c e p t [5] of the C o r i o l i s a n t i - p a i r i n g e f f e c t . A s shown in r e f s . [1] and [2] this t h e o r y g i v e s a s a t i s f a c t o r y d e s c r i p t i o n of r o t a t i o n a l s t a t e s in d e f o r m e d nuclei. A s p o i n t e d in r e f . [5] t h i s a p p r o a c h is e q u i v a l e n t to the p r o b l e m of a s u p e r c o n d u c t o r in the m a g n e t i c f i e l d . So it s e e m s n a t u r a l to e x p e c t the e x i s t e n c e of a n u c l e a r a n a l o g of the M e i s s n e r e f f e c t . ( T h e r e e x i s t two c r i t i c a l v a l u e s H 1 and H 2 of a m a g n e t i c f i e l d in the s u p e r c o n d u c t o r . F o r H < H 1 the e n e r g y gap A is the s a m e a s that in the a b s e n c e of a m a g n e t i c f i e l d (A = AO) w h i l e the m a g n e t i c i n d u c t a n c e B is z e r o . F o r H 1 < H < H2, A b e c o m e s l e s s than Ao and B b e c o m e s n o n z e r o w h e r e a s f o r H > H 2 the m a g n e t i c f i e l d d e s t r o y s the s u p e r c o n d u c t i v i t y (A = 0).) We s h a l l s h o w that s u c h an a n a l o g y e x i s t s in the c a s e of the r o t a t i o n of a s p h e r i c a l n u c l e u s . F o l l o w i n g [1-4] we c a l c u l a t e the m i n i m u m of the e n e r g y g = (H) (H is the n u c l e a r H a m i l t o n i a ~ at a f i x e d v a l u e of the a n g u l a r m o m e n t u m J = ( J x ) . U s i n g the L a g r a n g e m u l t i p l i e r m e t h o d , w e o b t a i n the p r o b l e m with the H a m i l t o n i a n H ' = H - co ~fx" We u s e the G r e e n f u n c t i o n m e t h o d f o r the s o lution. H o w e v e r , in c o n t r a s t to [1, 2] we do not u s e the q u a s i c l a s s i c a l l i m i t . T h e e q u a t i o n s f o r G r e e n f u n c t i o n s g and f a r e :
(1)
w h e r e fz is the s i n g l e - p a r t i c l e H a m i l t o n i a n and A is the e n e r g y gap. T h e q u a s i p a r t i c l e e n e r g i e s E i a r e p o l e s of g a n d f a s f u n c t i o n s of ~, t h u s b e i n g s o l u t i o n s of the h o m o g e n e o u s s y s t e m c o t 558
P P = ~1" u k X u k x ' gxx'
k\£-Ek+i
h h VkxVkx' ~ • 6 + E+Ek-i6 }'
(2) p p Vkx u k x ,
h h uk~. VleX,
A s f o l l o w s f r o m the c o n n e c t i o n b e t w e e n the G r e e n f u n c t i o n and the d e n s i t y m a t r i x [6] a l l the a v e r a g e s a r e e x p r e s s e d t h r o u g h the h o l e r e s i d u e s up)t, vhix" I n s e r t i n g eq. (2) into eq. (1) we get:
(E i - e x ) u i x - w~. . j ~ x , uix, - A v i x = 0 it' vix,
- Auix
(3)
= o
w i t h the o b v i o u s o r t h o n o r m a l i z a t i o n c o n d i t i o n
~t
(C -f~ +¢0/x)g + AS : 1,
(c+h+wfx) f + Ag= O.
r e s p o n d i n g to eq. (1). U s i n g the p r o p e r t i e s of the a n g u l a r m o m e n t u m o p e r a t o r J x it can be shown that e a c h s o l u t i o n with c = E i > 0 c o r r e s p o n d s to the one with E = - E l . F o r t h i s r e a s o n g a n d f can b e w r i t t e n in the f o l l o w i n g f o r m (it is c o n v e n i e n t to u s e the r e p r e s e n t a t i o n X in w h i c h h is d i a g o n a l : fz q0x = EX q~X, E x b e i n g s i n g l e - p a r t i cle e n e r g i e s c o u n t e d f r o m the c h e m i c a l p o t e n t i a l ~):
(uix Uk X + v i x Vk X) = 5ik"
(4)
H e r e the h o l e i n d e x "h" is o m i t t e d . T h e p a r t i c l e r e s i d u e s uPx, vPk o b e y the s y s t e m l i k e ( 3 ) b u t w i t h the o p p o s i t e s i g n of co : the C o r i o l i s i n t e r a c t i o n is t i m e - o d d so its s i g n is d i f f e r e n t f o r p a r t i c l e s and h o l e s .
P H Y S I C S LETTERS
Volume 3433, number 7
a l l the E j m a r e n e g a t i v e thus the s u m s in e q s .
U s i n g w e l l known r e l a t i o n s of the m a n y - b o d y t h e o r y [6] we g e t : =
i,kk'
]kk, vi~vik,
+
(5, 6) r u n o v e r E j m s o l u t i o n s . So we o b t a i n :
;
A2
=
(5)
2 i~. 2G
12 April 1971
1 =G.~
2j+l
2Ej ;
5
N=~" (2j+I) E ~ E : j ; 1
= ~ . (~j + ~ ) ( 2 j + 1) E j - £ j A
= G ~uixvix;
N= ~v 2 "
i,k
i,k
(6)
i~ '
G b e i n g the p a i r i n g c o n s t a n t . It is c l e a r that e q s . (6) s h o u l d b e s o l v e d s e p a r a t e l y f o r p r o t o n s and n e u t r o n s , w h i l e both p r o t o n s and n e u t r o n s s h o u l d b e i n c l u d e d in e q s . (5). In the c a s e of a d e f o r m e d n u c l e u s , f o r r o t a t i o n a r o u n d the x - a x i s w h i c h is o r t h o g o n a l to the s y m m e t r y a x i s z, the p e r t u r b a t i o n t h e o r y is a p p l i c a b l e f o r s m a l l w. T h e n e q s . (3-5) g i v e
:
2E:
A2 -
(11)
J=O ;
i . e . , the s a m e r e l a t i o n s a s t h o s e in the a b s e n c e of the r o t a t i o n . T h u s , A = A o f o r w < w 1 w h e r e the f i r s t c r i t i c a l v e l o c i t y w 1 is d e f i n e d by
eq. (10). C o n s i d e r ¢o > w 1. In t h i s c a s e nj s u b s t a t e s with
Ejrn > O,
m = j , j -1 . . . . .
j + l - nj
(12)
and 2j + 1 - n j s u b s t a t e s with
J= Jw ;
~= ~ o + ½Jw2 ;
(7)
/41-x 2
E3.
E}t,+ A2
-
i . e . , the u s u a l I(I+ 1) law f o r the r o t a t i o n a l b a n d e n e r g i e s w i t h the c o r r e c t e x p r e s s i o n f o r the m o m e n t of i n e r t i a . Now l e t us apply e q s . (3-6) to s p h e r i c a l n u c l e i . In t h i s c a s e t h e r e is no d i f f e r e n c e b e t w e e n x and z a x e s , s o e q s . (3) a r e :
(E-ej-com)ujm
- AVjm = 0
;
(8)
+
E j m > O,
m = j , j - 1 . . . . . nj - j
(13)
c o r r e s p o n d to e a c h j - s t a t e . The s e t of nj c o r r e s p o n d i n g to a g i v e n w v a l u e can be e a s i l y o b t a i n e d f r o m eq. (9). P u t t i n g e q s . (9) into e q s . (5), (6) and s u m m i n g o v e r the s e t (12), (13) we get 2 j + l - 2nj 1 =G~
j
2Ej
'
(14)
N - ]~nj = ~ ( 2 j + l - 2nj) Ej - ~ j . j 2Ej '
(E + ~ j - w r n ) V j r n - AUjm = 0 ,
J =~½nj(2j + 1- nj) ;
j and m b e i n g the s i n g l e - p a r t i c l e s t a t e a n g u l a r m o m e n t u m and its z - c o m p o n e n t ( o t h e r q u a n t u m n u m b e r s a r e o m i t t e d ) . S o l v i n g e q s . (8) w e o h fain:
; : ~(~j+ ~)[(2j+ I-2nj)
J +
+2
-
- 2
Ej - £j
:
2E:
+
+
-
-
; u:mv:m:-
A
.,
j;
(9) Ej =
2+A2;
Ejm = - Ej,.m
W e m u s t put t h e s e s o l u t i o n s into e q s . (5, 6). A s s e e n f r o m eel. (2) the s u m s o v e r " i " in e q s . (5) and (6) r u n o v e r s o l u t i o n s of eq. (3) with E i > 0. C l e a r l y w h e n ¢OJmax < m i n {Ej}
(10)
E:
(15) A2
+ nj] - 2 G
- m > 0 n the a l i g n e d We c a l l the s u b s t a t e s with E_j s t a t e s " s i n c e the t o t a l a n g u l a r m o m e n t u m J is the l a r g e s t one w h i c h is p o s s i b l e f o r a g i v e n s e t of nj. So if the a l i g n m e n t b e g i n s at l a r g e j v a l u e s the t o t a l a n g u l a r m o m e n t u m J is a l s o l a r g e . T h u s the f i r s t r o t a t i o n a l s t a t e of a s p h e r i c a l n u c l e u s can h a v e v e r y l a r g e a n g u l a r m o m e n t u m v a l u e in c o n t r a s t to the c a s e of d e f o r m e d n u c l e i . A s s e e n f r o m eq. (9) the ~ j nj i n c r e a s e s with i n c r e a s i n g ¢o so we o b t a i n a s e q u e n c e of a l i g n e d s t a t e s . F o r s o m e w = 0)2 v a l u e N = ~ j nj, i . e . , a l l the v a l e n c e n u c l e o n s b e c o m e a l i g n e d . A s s e e n f r o m eq. (14) A = 0 i n t h i s c a s e . So n = A o f o r ¢o< ¢o1, A < A o f o r w 1 < ¢o< w 2 a n d A = 0
559
Volume 34B, number 7
PHYSICS
f o r w > 0)2 in a c o m p l e t e a n a l o g y w i t h t h e Meissner-effect. A s s e e n f r o m eq. (15), J i s i n t e g e r f o r e v e n a n d h a l f - i n t e g e r f o r odd So the s o l u t i o n s w i t h odd ~ c o r r e s p o n d to odd n u c l e i w h i l e t h o s e w i t h e~ren c o r r e s p o n d to e v e n
~j nj
nj
~j nj.
~j nj
ones. Let us consider the 212po nucleus as an example. Here the states with minimal Ej values are the lh9/2 and proton state and the 2g9/2 n e u t r o n one. T h e y a r e a l i g n e d s i m u l t a n e o u s l y s i n c e the c o r r e s p o n d i n g values are very c l o s e . So w e o b t a i n J = 16 + f o r the a n g u l a r m o m e n t u m v a l u e . A s s e e n f r o m eq. (14) t h i s s t a t e i s a n o r m a l one s i n c e N n = nn(2g9/2) = 2, Np = n p ( l h 9 / 2 ) = 2. F o r t h i s r e a s o n t h e e x c i t a t i o n e n e r g y i s e q u a l to the s u m of p r o t o n and n e u t r o n o d d - e v e n m a s s d i f f e r e n c e s as can be easily obt a i n e d f r o m eq. (15). T h i s e n a b l e s us to m a k e a d i r e c t c o m p a r i s o n with e x p e r i m e n t . The p r o ton 2 P p a n d n e u t r o n 2 P n o d d - e v e n m a s s d i f f e r e n c e s a r e 1.45 M e V a n d 1.46 M e V [7] s o E1the°r. = 2.91 M e V . T h e c o r r e s p o n d i n g e x p e r i 6+ m e n t a l v a l u e s a r e J = (18+), E = 2.93 M e V [8]. A n o t h e r e x a m p l e p r o v i d e s the 2 1 1 p o n u c l e u s . H e r e the r o t a t i o n a l s t a t e i s f o r m e d by the a l i g n -
Ej
560
12 April 1971
LETTERS
ment of the lh9/2 proton state thus giving J = 25/2 + for the angular momentum value (the ground-state spin is 9/2+). The excitation energy should be equal to the proton odd-even mass difference 2Pp = 1.45 MeV. This also agrees with the experiment [8] (J= (25/2+), E = 1430 keV). Calculations of Meissner bands in nuclei with more valence nucleons are in progress.
References [1] Yu. T. Grin and I. M. Pavlitchenkov, Zh. Eksp. i Teor. Fiz. 43 (1962) 465. [2] Yu. T. Grin and A. I. Larkin, Yadern. Fiz. 2 (1965) 40. [3] K. Y. Chan and J. G. Valatin, Nucl. Phys. 82 {1966) 222. [4] K. Y. Chart, Nucl. Phys. 85 (1966) 261. [5] B. R. Mottelson and J. G. Valatin, Phys. Rev. L e t t e r s 5 (1960) 511. [6] A. A. Brikosov, L. P. Gor'kov and I. E. Dzyaloshinsky, Quantum field-theoretical methods in statistical physics (Moscow, 1962). [7} V. A. Kravtsov, Atomic m a s s e s and nuclear binding e n e r g i e s (Moscow, 1965). [8] C. M. L e d e r e r , J. M. Hollander and I. P e r l m a n , Table of isotopes (John Wiley and Sons, New York,
1967).
CORIOLIS
PHYSICS
ANTI-PAIRING NUCLEAR
LETTERS
THEORY
OF
12 April 1971
NUCLEAR
ROTATIONS.
MEISSNER-EFFECT
B. L. B I R B R A I R Ioffe Physico-Technical Institute, Leningrad, USSR Received 25 February 1971
It is shown that the ground-state rotational bands of spherical nuclei are formed by a gradual alignment of the angular momenta of valence nucleons along the rotation axis. The predicted spin and excitation energy values of the first rotational state in 212po nucleus are J = 16+ and E = 2.91 MeV which is in reasonable agreement with the experimental data (J = (18+), E = 2.93 MeV).
T h e m i c r o s c o p i c t h e o r y of n u c l e a r r o t a t i o n s [1-4] is b a s e d on the M o t t e l s o n - V a l a t i n c o n c e p t [5] of the C o r i o l i s a n t i - p a i r i n g e f f e c t . A s shown in r e f s . [1] and [2] this t h e o r y g i v e s a s a t i s f a c t o r y d e s c r i p t i o n of r o t a t i o n a l s t a t e s in d e f o r m e d nuclei. A s p o i n t e d in r e f . [5] t h i s a p p r o a c h is e q u i v a l e n t to the p r o b l e m of a s u p e r c o n d u c t o r in the m a g n e t i c f i e l d . So it s e e m s n a t u r a l to e x p e c t the e x i s t e n c e of a n u c l e a r a n a l o g of the M e i s s n e r e f f e c t . ( T h e r e e x i s t two c r i t i c a l v a l u e s H 1 and H 2 of a m a g n e t i c f i e l d in the s u p e r c o n d u c t o r . F o r H < H 1 the e n e r g y gap A is the s a m e a s that in the a b s e n c e of a m a g n e t i c f i e l d (A = AO) w h i l e the m a g n e t i c i n d u c t a n c e B is z e r o . F o r H 1 < H < H2, A b e c o m e s l e s s than Ao and B b e c o m e s n o n z e r o w h e r e a s f o r H > H 2 the m a g n e t i c f i e l d d e s t r o y s the s u p e r c o n d u c t i v i t y (A = 0).) We s h a l l s h o w that s u c h an a n a l o g y e x i s t s in the c a s e of the r o t a t i o n of a s p h e r i c a l n u c l e u s . F o l l o w i n g [1-4] we c a l c u l a t e the m i n i m u m of the e n e r g y g = (H) (H is the n u c l e a r H a m i l t o n i a ~ at a f i x e d v a l u e of the a n g u l a r m o m e n t u m J = ( J x ) . U s i n g the L a g r a n g e m u l t i p l i e r m e t h o d , w e o b t a i n the p r o b l e m with the H a m i l t o n i a n H ' = H - co ~fx" We u s e the G r e e n f u n c t i o n m e t h o d f o r the s o lution. H o w e v e r , in c o n t r a s t to [1, 2] we do not u s e the q u a s i c l a s s i c a l l i m i t . T h e e q u a t i o n s f o r G r e e n f u n c t i o n s g and f a r e :
(1)
w h e r e fz is the s i n g l e - p a r t i c l e H a m i l t o n i a n and A is the e n e r g y gap. T h e q u a s i p a r t i c l e e n e r g i e s E i a r e p o l e s of g a n d f a s f u n c t i o n s of ~, t h u s b e i n g s o l u t i o n s of the h o m o g e n e o u s s y s t e m c o t 558
P P = ~1" u k X u k x ' gxx'
k\£-Ek+i
h h VkxVkx' ~ • 6 + E+Ek-i6 }'
(2) p p Vkx u k x ,
h h uk~. VleX,
A s f o l l o w s f r o m the c o n n e c t i o n b e t w e e n the G r e e n f u n c t i o n and the d e n s i t y m a t r i x [6] a l l the a v e r a g e s a r e e x p r e s s e d t h r o u g h the h o l e r e s i d u e s up)t, vhix" I n s e r t i n g eq. (2) into eq. (1) we get:
(E i - e x ) u i x - w~. . j ~ x , uix, - A v i x = 0 it' vix,
- Auix
(3)
= o
w i t h the o b v i o u s o r t h o n o r m a l i z a t i o n c o n d i t i o n
~t
(C -f~ +¢0/x)g + AS : 1,
(c+h+wfx) f + Ag= O.
r e s p o n d i n g to eq. (1). U s i n g the p r o p e r t i e s of the a n g u l a r m o m e n t u m o p e r a t o r J x it can be shown that e a c h s o l u t i o n with c = E i > 0 c o r r e s p o n d s to the one with E = - E l . F o r t h i s r e a s o n g a n d f can b e w r i t t e n in the f o l l o w i n g f o r m (it is c o n v e n i e n t to u s e the r e p r e s e n t a t i o n X in w h i c h h is d i a g o n a l : fz q0x = EX q~X, E x b e i n g s i n g l e - p a r t i cle e n e r g i e s c o u n t e d f r o m the c h e m i c a l p o t e n t i a l ~):
(uix Uk X + v i x Vk X) = 5ik"
(4)
H e r e the h o l e i n d e x "h" is o m i t t e d . T h e p a r t i c l e r e s i d u e s uPx, vPk o b e y the s y s t e m l i k e ( 3 ) b u t w i t h the o p p o s i t e s i g n of co : the C o r i o l i s i n t e r a c t i o n is t i m e - o d d so its s i g n is d i f f e r e n t f o r p a r t i c l e s and h o l e s .
P H Y S I C S LETTERS
Volume 3433, number 7
a l l the E j m a r e n e g a t i v e thus the s u m s in e q s .
U s i n g w e l l known r e l a t i o n s of the m a n y - b o d y t h e o r y [6] we g e t : =
i,kk'
]kk, vi~vik,
+
(5, 6) r u n o v e r E j m s o l u t i o n s . So we o b t a i n :
;
A2
=
(5)
2 i~. 2G
12 April 1971
1 =G.~
2j+l
2Ej ;
5
N=~" (2j+I) E ~ E : j ; 1
= ~ . (~j + ~ ) ( 2 j + 1) E j - £ j A
= G ~uixvix;
N= ~v 2 "
i,k
i,k
(6)
i~ '
G b e i n g the p a i r i n g c o n s t a n t . It is c l e a r that e q s . (6) s h o u l d b e s o l v e d s e p a r a t e l y f o r p r o t o n s and n e u t r o n s , w h i l e both p r o t o n s and n e u t r o n s s h o u l d b e i n c l u d e d in e q s . (5). In the c a s e of a d e f o r m e d n u c l e u s , f o r r o t a t i o n a r o u n d the x - a x i s w h i c h is o r t h o g o n a l to the s y m m e t r y a x i s z, the p e r t u r b a t i o n t h e o r y is a p p l i c a b l e f o r s m a l l w. T h e n e q s . (3-5) g i v e
:
2E:
A2 -
(11)
J=O ;
i . e . , the s a m e r e l a t i o n s a s t h o s e in the a b s e n c e of the r o t a t i o n . T h u s , A = A o f o r w < w 1 w h e r e the f i r s t c r i t i c a l v e l o c i t y w 1 is d e f i n e d by
eq. (10). C o n s i d e r ¢o > w 1. In t h i s c a s e nj s u b s t a t e s with
Ejrn > O,
m = j , j -1 . . . . .
j + l - nj
(12)
and 2j + 1 - n j s u b s t a t e s with
J= Jw ;
~= ~ o + ½Jw2 ;
(7)
/41-x 2
E3.
E}t,+ A2
-
i . e . , the u s u a l I(I+ 1) law f o r the r o t a t i o n a l b a n d e n e r g i e s w i t h the c o r r e c t e x p r e s s i o n f o r the m o m e n t of i n e r t i a . Now l e t us apply e q s . (3-6) to s p h e r i c a l n u c l e i . In t h i s c a s e t h e r e is no d i f f e r e n c e b e t w e e n x and z a x e s , s o e q s . (3) a r e :
(E-ej-com)ujm
- AVjm = 0
;
(8)
+
E j m > O,
m = j , j - 1 . . . . . nj - j
(13)
c o r r e s p o n d to e a c h j - s t a t e . The s e t of nj c o r r e s p o n d i n g to a g i v e n w v a l u e can be e a s i l y o b t a i n e d f r o m eq. (9). P u t t i n g e q s . (9) into e q s . (5), (6) and s u m m i n g o v e r the s e t (12), (13) we get 2 j + l - 2nj 1 =G~
j
2Ej
'
(14)
N - ]~nj = ~ ( 2 j + l - 2nj) Ej - ~ j . j 2Ej '
(E + ~ j - w r n ) V j r n - AUjm = 0 ,
J =~½nj(2j + 1- nj) ;
j and m b e i n g the s i n g l e - p a r t i c l e s t a t e a n g u l a r m o m e n t u m and its z - c o m p o n e n t ( o t h e r q u a n t u m n u m b e r s a r e o m i t t e d ) . S o l v i n g e q s . (8) w e o h fain:
; : ~(~j+ ~)[(2j+ I-2nj)
J +
+2
-
- 2
Ej - £j
:
2E:
+
+
-
-
; u:mv:m:-
A
.,
j;
(9) Ej =
2+A2;
Ejm = - Ej,.m
W e m u s t put t h e s e s o l u t i o n s into e q s . (5, 6). A s s e e n f r o m eel. (2) the s u m s o v e r " i " in e q s . (5) and (6) r u n o v e r s o l u t i o n s of eq. (3) with E i > 0. C l e a r l y w h e n ¢OJmax < m i n {Ej}
(10)
E:
(15) A2
+ nj] - 2 G
- m > 0 n the a l i g n e d We c a l l the s u b s t a t e s with E_j s t a t e s " s i n c e the t o t a l a n g u l a r m o m e n t u m J is the l a r g e s t one w h i c h is p o s s i b l e f o r a g i v e n s e t of nj. So if the a l i g n m e n t b e g i n s at l a r g e j v a l u e s the t o t a l a n g u l a r m o m e n t u m J is a l s o l a r g e . T h u s the f i r s t r o t a t i o n a l s t a t e of a s p h e r i c a l n u c l e u s can h a v e v e r y l a r g e a n g u l a r m o m e n t u m v a l u e in c o n t r a s t to the c a s e of d e f o r m e d n u c l e i . A s s e e n f r o m eq. (9) the ~ j nj i n c r e a s e s with i n c r e a s i n g ¢o so we o b t a i n a s e q u e n c e of a l i g n e d s t a t e s . F o r s o m e w = 0)2 v a l u e N = ~ j nj, i . e . , a l l the v a l e n c e n u c l e o n s b e c o m e a l i g n e d . A s s e e n f r o m eq. (14) A = 0 i n t h i s c a s e . So n = A o f o r ¢o< ¢o1, A < A o f o r w 1 < ¢o< w 2 a n d A = 0
559
Volume 34B, number 7
PHYSICS
f o r w > 0)2 in a c o m p l e t e a n a l o g y w i t h t h e Meissner-effect. A s s e e n f r o m eq. (15), J i s i n t e g e r f o r e v e n a n d h a l f - i n t e g e r f o r odd So the s o l u t i o n s w i t h odd ~ c o r r e s p o n d to odd n u c l e i w h i l e t h o s e w i t h e~ren c o r r e s p o n d to e v e n
~j nj
nj
~j nj.
~j nj
ones. Let us consider the 212po nucleus as an example. Here the states with minimal Ej values are the lh9/2 and proton state and the 2g9/2 n e u t r o n one. T h e y a r e a l i g n e d s i m u l t a n e o u s l y s i n c e the c o r r e s p o n d i n g values are very c l o s e . So w e o b t a i n J = 16 + f o r the a n g u l a r m o m e n t u m v a l u e . A s s e e n f r o m eq. (14) t h i s s t a t e i s a n o r m a l one s i n c e N n = nn(2g9/2) = 2, Np = n p ( l h 9 / 2 ) = 2. F o r t h i s r e a s o n t h e e x c i t a t i o n e n e r g y i s e q u a l to the s u m of p r o t o n and n e u t r o n o d d - e v e n m a s s d i f f e r e n c e s as can be easily obt a i n e d f r o m eq. (15). T h i s e n a b l e s us to m a k e a d i r e c t c o m p a r i s o n with e x p e r i m e n t . The p r o ton 2 P p a n d n e u t r o n 2 P n o d d - e v e n m a s s d i f f e r e n c e s a r e 1.45 M e V a n d 1.46 M e V [7] s o E1the°r. = 2.91 M e V . T h e c o r r e s p o n d i n g e x p e r i 6+ m e n t a l v a l u e s a r e J = (18+), E = 2.93 M e V [8]. A n o t h e r e x a m p l e p r o v i d e s the 2 1 1 p o n u c l e u s . H e r e the r o t a t i o n a l s t a t e i s f o r m e d by the a l i g n -
Ej
560
12 April 1971
LETTERS
ment of the lh9/2 proton state thus giving J = 25/2 + for the angular momentum value (the ground-state spin is 9/2+). The excitation energy should be equal to the proton odd-even mass difference 2Pp = 1.45 MeV. This also agrees with the experiment [8] (J= (25/2+), E = 1430 keV). Calculations of Meissner bands in nuclei with more valence nucleons are in progress.
References [1] Yu. T. Grin and I. M. Pavlitchenkov, Zh. Eksp. i Teor. Fiz. 43 (1962) 465. [2] Yu. T. Grin and A. I. Larkin, Yadern. Fiz. 2 (1965) 40. [3] K. Y. Chan and J. G. Valatin, Nucl. Phys. 82 {1966) 222. [4] K. Y. Chart, Nucl. Phys. 85 (1966) 261. [5] B. R. Mottelson and J. G. Valatin, Phys. Rev. L e t t e r s 5 (1960) 511. [6] A. A. Brikosov, L. P. Gor'kov and I. E. Dzyaloshinsky, Quantum field-theoretical methods in statistical physics (Moscow, 1962). [7} V. A. Kravtsov, Atomic m a s s e s and nuclear binding e n e r g i e s (Moscow, 1965). [8] C. M. L e d e r e r , J. M. Hollander and I. P e r l m a n , Table of isotopes (John Wiley and Sons, New York,
1967).