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Collective coordinates for the description of nuclear rotations
Volume 31B, number
PHYSICS
5
COLLECTIVE
LETTERS
COORDINATES OF NUCLEAR
FOR THE ROTATIONS
2 March 1970
DESCRIPTION *
V. GUPTA ** and R. SKINNER University
of Saskatchewan, Received
Saskatoon,
Canada
11 January 1970
We introduce collective and intrinsic coordinates in terms of which there exist many-body wave functions senarable into one factor that describes the rotation of the system as a whole and another the residual intrinsic structure.
An approximate calculation of nuclear rotational intrinsic structure, say be deformed determinantal states, gives a wave function that describes a deformed nucleus bound to a fixed spatial direction. The violation of rotational invariance can be removed by a technique similar to that used for the center -of -mass problem [ 1,2]: the arguments of the wave function are taken to be intrinsic coordinates relative to axes determined by the dynamics of the problem and not arbitrarily chosen to be, say, the z-direction. The problem of introducing collective coordinates for the description of nuclear rotations reduces to finding the definition of the intrinsic axes. We have developed a general method for deriving the transformation from the laboratory coordinates to a set of collective and intrinsic coordinates appropriate for the wave-function factorization associated with a collective band structure in which the band states are described by wave functions consisting of two factors; one, labelled by the eigenvalues of some operator (the angular momentum in the rotational case), varies throughout the band and the other, fixed throughout the band, describes the residual intrinsic structure of the states. The technique has been applied to the rotational problem only for the two-dimensional case, but some of the results can be easily extended to the threedimensional case. The method [3] has been applied also to the description of the collective vibrational motions discussed by Goshen and Lipkin [4] and to a purely quantum-mechanical * Work supported in part by grants in aid of research from the National Research Council of Canada. ** Present address: Nuclear Research Centre, University of Alberta, Edmonton, Canada.
256
procedure for introducing the center -of -mass transformation. A minimum condition for a description of model collective states is that one be able to separate the wave function into the two factors described above. True band behaviour results if, in addition, the Hamiltonian separates into a sum of corresponding terms, as in the centerof -mass case. The feasibility of the separation of the Hamiltonian can be investigated in terms of the general features of the coordinate transformation derived on the basis of the wavefunction separability. The operator whose eigenvalues label the collective states, in this case the angular momentum, is used as a generator for the collective transformation. The transformation U(o) = exp (icuL/A) generated by L corresponds to a Fotation of the system, u(cr)xa ~-l(o) = R(o). xa. The operator U(o) is a displacement operator so we may introduce a derivative representation for L. The derivative is with respect to an angle 0, a collective coordinate associated in some manner with all the particles and the argument of the collective factor in the wave function. Under the transformation u(a), 0 - Q + cy so, in part, the transformation equations must have the form X a = R(e)*eat where the Q ‘s are intrinsic coordinates not affected by the transformation operator u(a). The number of collective and intrinsic coordinates is one greater than the number of laboratory coordinates so there exists one constraint $(Ql, * *. 9 @A) = 0, where the function I$ is to be determined. The constraint and the relation xa = = R(O). pa determine completely the transformation from the laboratory coordinates to the collective and intrinsic coordinates.
Volume 31B, number 5
PHYSICS
T h e t r a n s f o r m a t i o n law m u s t r e l a t e the m o m e n t u m o p e r a t o r - i E S / S x a to the d e r i v a t i v e s w i t h r e s p e c t to 0 and the ~ 's. D e r i v a t i v e s with r e s p e c t to a l l the ~ ' s cannot be i n t r o d u c e d u n l e s s we a l l o w the ~ ' s to v a r y i n d e p e n d e n t l y of t h e c o n s t r a i n t . T h e r e f o r e , we i n t r o d u c e a n e w l a b o r a t o r y c o o r d i n a t e X, d e f i n e d by X ----- ~ ( ~ 1 , . . . ~ A ) , w h i c h w i l l be z e r o in the e x t e n d e d c o n f i g u r a t i o n s p a c e on the s u b s p a c e of physical interest. The configuration-space volu m e e l e m e n t t r a n s f o r m s a s 5(32)dXI-I(d 2 Xa) = = [ 5(X, x)/5(O, ~)] 6(0) d0 I ~ (d2 Pa)- T h e 5-func tion e n s u r e s t h e e l i m i n a t i o n of any s p u r i o u s s t a t e s [5]. T h e J a c o b i a n O(x,X)/8(~, 0) o f the t r a n s f o r m a t i o n is p r o p o r t i o n a l to G(~) = 2 - ) 8 0 / 8 0 a , w h e r e (0a, Pa) a r e the i n t r i n s i c p o l a r c o o r d i n a t e s of ~ a. T h e t r a n s f o r m a t i o n is c o n s i s t e n t only if G(Q) ¢ 0. T h e g e n e r a l s o l u t i o n of t h i s e q u a t i o n i s 0 = ~ k a O a + F(OI-O 2, . . . , 01-ÙA; Pl, • • • PA) w h e r e F is an a r b i t r a r y f u n c t i o n of i t s a r g u m e n t s and the k ' s a r e c o n s t a n t s s a t i s f y i n g ~ k a = 1. T h i s a r b i t r a r y f u n c t i o n can b e e l i m i n a t e d , and t h e f o r m u l a e t h e r e b y s i m p l i f i e d , by t h e t r a n s f o r m a t i o n 0 a ~ 0 - F, 0 ~ 0 + F, Pa ~ Pa. (For a two-dimensional angular momentum e i g e n f u n e t i o n , t h i s t r a n s f o r m a t i o n r e s u l t s in a p h a s e c h a n g e of exp [ i m F ] of the i n t r i n s i c w a v e f u n c t i o n . ) It m a y be d e s i r a b l e in a p r a c t i c a l c a l c u l a t i o n of r o t a t i o n a l s t a t e s to s e l e c t that F w h i c h , within the a p p r o x i m a t i o n u s e d , r e s u l t s in the least variation among the wave functions for t h e l o w e s t I y i n g i n t r i n s i c s t a t e s of the r o t a t i o n a l band. T h e a b o v e t r a n s f o r m a t i o n y i e l d s ~b = ~ k a Oa, w h e r e the~k a a r e c o n s t a n t s s u b j e c t o n l y to the c o n d i t i o n 2 J k a = 1. T h e d e r i v a t i v e o p e r a t o r b e comes 2
fla :
02a/O ,
f .a : -kaOla/O
.
T h i s c o m p l e t e s that p a r t of t h e d e r i v a t i o n of the t r a n s f o r m a t i o n f r o m t h e l a b o r a t o r y to t h e c o l l e c t i v e and i n t r i n s i c c o o r d i n a t e s w h i c h can b e d e d u c e d f r o m t h e e x i s t e n c e of s e p a r a b l e w a v e functions. T h e d e t e r m i n a t i o n of the s e t (ka} of c o n s t a n t s is a d y n a m i c a l p r o b I e m . F o r e x a m p l e , f o r center-of-mass motions, our procedure applied to c o l l e c t i v e s t a t e s c h a r a c t e r i z e d by t h e t o t a l l i n e a r m o m e n t u m l e a d s to the t r a n s f o r m a t i o n Xa = Oa + R, ~ k a Oa = O, w h e r e t h e k a ' S a r e c o n s t a n t s s u b j e c t only to ~ k a = 1. T h e u s u a l
L E T TE RS
2 March 1970 r
r e s u l t , k a = m a / M , f o l l o w s f r o m the i m p o s i t i o n of t h e c o n d i t i o n that no c r o s s - t e r m s (O/OR)~_8/Spa) a p p e a r in the k i n e t i c - e n e r g y o p e r a t o r -)--) (fi2/2ma) 82/8x2a . T h a t the C o r i o l i s c r o s s - t e r m s (8/aO)(a/SPia) in the r o t a t i o n a l c a s e cannot be e l i m i n a t e d f r o m the k i n e t i c - e n e r g y o p e r a t o r can be shown d i r e c t ly: T h e ~/SXia'S , w r i t t e n in t e r m s of an u n d e t e r m i n e d qS, can b e s u b s t i t u t e d into the k i n e t i c e n e r g y o p e r a t o r and the c r o s s - t e r m s e q u a t e d to z e r o . T h e d i f f e r e n t i a l e q u a t i o n s f o r qb o b t a i n e d in t h i s way a r e n o n - i n t e g r a b l e , p r o v i n g that t h e C o r i o l i s t e r m cannot be e l i m i n a t e d by a t r a n s f o r m a t i o n of the f o r m x a = R ( o ) ' p a , 0 ( 9 ) = 0. T h e p r e s e n t c a l c u l a t i o n y i e l d s only t h e t w o dimensional rotational-intrinsic Hamiltonian, but t h i s H a m i l t o n i a n c o r r e s p o n d s to the t h r e e dimensional Hamiltonian H = - ~ ( ~ 2 / 2 m a) 0 2 / o p 2 a i,a
a~ L a' . l l-1 a. L' +
+½ L"121"c-~ [-~aLa" 1-11a * L'. I 2-1]" L i +~L.I
2
I.
L + V(O1,...,OA)
in t h e s a m e way that L 2 / 2 I f o r a r i g i d p l a n a r f o r m c o r r e s p o n d s to the H a m i l t o n i a n ½L. I - 1 . L f o r a r i g i d body. T h e d i a g o n a l i z e d i n e r t i a l d y a d i e s [ l a , [ 2a h a v e e l e m e n t s (Ina)ii
ma(O2a+O2a)/(kia)n;i,j,k, cyclic,
n = 1,2.
L a = - ifi~a × 0/0 ~ a and the q u a n t i t i e s without l e t t e r i n d i c e s d e n o t e the s u m , o v e r a l i p a r t i c l e s , of the c o r r e s p o n d i n g q u a n t i t i e s with such i n d i c e s . T h e H a m i l t o n i a n h a s the g e n e r a l f o r m d i s c u s s e d by V i l l a r s [6] so a l l his c o n c l u s i o n s a p p l y to it. Coordinate transformations involving noni n t e g r a b l e c o n s t r a i n t s h a v e b e e n u s e d in the p a s t [7, 8] to o b t a i n the r i g i d - b o d y m o m e n t of i n e r t i a . T h i s r e s u l t can b e o b t a i n e d in the t w o d i m e n s i o n a l e a s e f r o m the c o n s t r a i n t 0 = = ~ a k a 0 a by p r o c e e d i n g a l o n g l i n e s d e s c r i b e d by V i l l a r s [6] and u s i n g the e s t i m a t e (Oa2}0 ~ Oa2, an a p p r o x i m a t i o n v a l i d f o r a m a c r o s c o p i c pl~inar object. We a r e g r a t e f u l to P r o f e s s o r s E. L. T o m u s i a k and L. E. H. T r a i n o r f o r i n f o r m a t i v e d i s c u s s i o n s and to t h e N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a for supporting this investigation. We also app r e c i a t e t h e c o n s t r u c t i v e c r i t i c i s m s of the r e f eree. 257
V o l u m e 31B, n u m b e r 5
PHYSICS
Reference s 1. s. G a r t e n h a u s and C. S c h w a r t z , P h y s . Rev. 108 (1957) 482~ 2. R. S k i n n e r , Can. J. P h y s . 34 11956) 901. 3. V. Gupta and R. S k i n n e r , in: C o n t r . I n t e r n . Conf. on P r o p e r t i e s of n u c l e a r s t a t e s (Les P r e s s e s de l ' U n i v e r s i t ~ M o n t r e a l , 1969) p. 40.
258
LETTERS
2 M a r c h 1970
4. S. G o s h e n and H. J. Lipkin, Ann. P h y s . (N.Y.) 6 (1959) 301. 5. J . P . Elliott and T. H. R. S k y r m e , P r o c . Roy. Soc. {London) A232 {1955) 561. 6. F . M . H . V i l l a r s , Nucl. P h y s . 74 (1965) 353. 7. F. V i l l a r s , Nucl. P h y s . 3 (1957) 240. 8. D . J . Rowe, p r i v a t e c o m m u n i c a t i o n {1969).
PHYSICS
5
COLLECTIVE
LETTERS
COORDINATES OF NUCLEAR
FOR THE ROTATIONS
2 March 1970
DESCRIPTION *
V. GUPTA ** and R. SKINNER University
of Saskatchewan, Received
Saskatoon,
Canada
11 January 1970
We introduce collective and intrinsic coordinates in terms of which there exist many-body wave functions senarable into one factor that describes the rotation of the system as a whole and another the residual intrinsic structure.
An approximate calculation of nuclear rotational intrinsic structure, say be deformed determinantal states, gives a wave function that describes a deformed nucleus bound to a fixed spatial direction. The violation of rotational invariance can be removed by a technique similar to that used for the center -of -mass problem [ 1,2]: the arguments of the wave function are taken to be intrinsic coordinates relative to axes determined by the dynamics of the problem and not arbitrarily chosen to be, say, the z-direction. The problem of introducing collective coordinates for the description of nuclear rotations reduces to finding the definition of the intrinsic axes. We have developed a general method for deriving the transformation from the laboratory coordinates to a set of collective and intrinsic coordinates appropriate for the wave-function factorization associated with a collective band structure in which the band states are described by wave functions consisting of two factors; one, labelled by the eigenvalues of some operator (the angular momentum in the rotational case), varies throughout the band and the other, fixed throughout the band, describes the residual intrinsic structure of the states. The technique has been applied to the rotational problem only for the two-dimensional case, but some of the results can be easily extended to the threedimensional case. The method [3] has been applied also to the description of the collective vibrational motions discussed by Goshen and Lipkin [4] and to a purely quantum-mechanical * Work supported in part by grants in aid of research from the National Research Council of Canada. ** Present address: Nuclear Research Centre, University of Alberta, Edmonton, Canada.
256
procedure for introducing the center -of -mass transformation. A minimum condition for a description of model collective states is that one be able to separate the wave function into the two factors described above. True band behaviour results if, in addition, the Hamiltonian separates into a sum of corresponding terms, as in the centerof -mass case. The feasibility of the separation of the Hamiltonian can be investigated in terms of the general features of the coordinate transformation derived on the basis of the wavefunction separability. The operator whose eigenvalues label the collective states, in this case the angular momentum, is used as a generator for the collective transformation. The transformation U(o) = exp (icuL/A) generated by L corresponds to a Fotation of the system, u(cr)xa ~-l(o) = R(o). xa. The operator U(o) is a displacement operator so we may introduce a derivative representation for L. The derivative is with respect to an angle 0, a collective coordinate associated in some manner with all the particles and the argument of the collective factor in the wave function. Under the transformation u(a), 0 - Q + cy so, in part, the transformation equations must have the form X a = R(e)*eat where the Q ‘s are intrinsic coordinates not affected by the transformation operator u(a). The number of collective and intrinsic coordinates is one greater than the number of laboratory coordinates so there exists one constraint $(Ql, * *. 9 @A) = 0, where the function I$ is to be determined. The constraint and the relation xa = = R(O). pa determine completely the transformation from the laboratory coordinates to the collective and intrinsic coordinates.
Volume 31B, number 5
PHYSICS
T h e t r a n s f o r m a t i o n law m u s t r e l a t e the m o m e n t u m o p e r a t o r - i E S / S x a to the d e r i v a t i v e s w i t h r e s p e c t to 0 and the ~ 's. D e r i v a t i v e s with r e s p e c t to a l l the ~ ' s cannot be i n t r o d u c e d u n l e s s we a l l o w the ~ ' s to v a r y i n d e p e n d e n t l y of t h e c o n s t r a i n t . T h e r e f o r e , we i n t r o d u c e a n e w l a b o r a t o r y c o o r d i n a t e X, d e f i n e d by X ----- ~ ( ~ 1 , . . . ~ A ) , w h i c h w i l l be z e r o in the e x t e n d e d c o n f i g u r a t i o n s p a c e on the s u b s p a c e of physical interest. The configuration-space volu m e e l e m e n t t r a n s f o r m s a s 5(32)dXI-I(d 2 Xa) = = [ 5(X, x)/5(O, ~)] 6(0) d0 I ~ (d2 Pa)- T h e 5-func tion e n s u r e s t h e e l i m i n a t i o n of any s p u r i o u s s t a t e s [5]. T h e J a c o b i a n O(x,X)/8(~, 0) o f the t r a n s f o r m a t i o n is p r o p o r t i o n a l to G(~) = 2 - ) 8 0 / 8 0 a , w h e r e (0a, Pa) a r e the i n t r i n s i c p o l a r c o o r d i n a t e s of ~ a. T h e t r a n s f o r m a t i o n is c o n s i s t e n t only if G(Q) ¢ 0. T h e g e n e r a l s o l u t i o n of t h i s e q u a t i o n i s 0 = ~ k a O a + F(OI-O 2, . . . , 01-ÙA; Pl, • • • PA) w h e r e F is an a r b i t r a r y f u n c t i o n of i t s a r g u m e n t s and the k ' s a r e c o n s t a n t s s a t i s f y i n g ~ k a = 1. T h i s a r b i t r a r y f u n c t i o n can b e e l i m i n a t e d , and t h e f o r m u l a e t h e r e b y s i m p l i f i e d , by t h e t r a n s f o r m a t i o n 0 a ~ 0 - F, 0 ~ 0 + F, Pa ~ Pa. (For a two-dimensional angular momentum e i g e n f u n e t i o n , t h i s t r a n s f o r m a t i o n r e s u l t s in a p h a s e c h a n g e of exp [ i m F ] of the i n t r i n s i c w a v e f u n c t i o n . ) It m a y be d e s i r a b l e in a p r a c t i c a l c a l c u l a t i o n of r o t a t i o n a l s t a t e s to s e l e c t that F w h i c h , within the a p p r o x i m a t i o n u s e d , r e s u l t s in the least variation among the wave functions for t h e l o w e s t I y i n g i n t r i n s i c s t a t e s of the r o t a t i o n a l band. T h e a b o v e t r a n s f o r m a t i o n y i e l d s ~b = ~ k a Oa, w h e r e the~k a a r e c o n s t a n t s s u b j e c t o n l y to the c o n d i t i o n 2 J k a = 1. T h e d e r i v a t i v e o p e r a t o r b e comes 2
fla :
02a/O ,
f .a : -kaOla/O
.
T h i s c o m p l e t e s that p a r t of t h e d e r i v a t i o n of the t r a n s f o r m a t i o n f r o m t h e l a b o r a t o r y to t h e c o l l e c t i v e and i n t r i n s i c c o o r d i n a t e s w h i c h can b e d e d u c e d f r o m t h e e x i s t e n c e of s e p a r a b l e w a v e functions. T h e d e t e r m i n a t i o n of the s e t (ka} of c o n s t a n t s is a d y n a m i c a l p r o b I e m . F o r e x a m p l e , f o r center-of-mass motions, our procedure applied to c o l l e c t i v e s t a t e s c h a r a c t e r i z e d by t h e t o t a l l i n e a r m o m e n t u m l e a d s to the t r a n s f o r m a t i o n Xa = Oa + R, ~ k a Oa = O, w h e r e t h e k a ' S a r e c o n s t a n t s s u b j e c t only to ~ k a = 1. T h e u s u a l
L E T TE RS
2 March 1970 r
r e s u l t , k a = m a / M , f o l l o w s f r o m the i m p o s i t i o n of t h e c o n d i t i o n that no c r o s s - t e r m s (O/OR)~_8/Spa) a p p e a r in the k i n e t i c - e n e r g y o p e r a t o r -)--) (fi2/2ma) 82/8x2a . T h a t the C o r i o l i s c r o s s - t e r m s (8/aO)(a/SPia) in the r o t a t i o n a l c a s e cannot be e l i m i n a t e d f r o m the k i n e t i c - e n e r g y o p e r a t o r can be shown d i r e c t ly: T h e ~/SXia'S , w r i t t e n in t e r m s of an u n d e t e r m i n e d qS, can b e s u b s t i t u t e d into the k i n e t i c e n e r g y o p e r a t o r and the c r o s s - t e r m s e q u a t e d to z e r o . T h e d i f f e r e n t i a l e q u a t i o n s f o r qb o b t a i n e d in t h i s way a r e n o n - i n t e g r a b l e , p r o v i n g that t h e C o r i o l i s t e r m cannot be e l i m i n a t e d by a t r a n s f o r m a t i o n of the f o r m x a = R ( o ) ' p a , 0 ( 9 ) = 0. T h e p r e s e n t c a l c u l a t i o n y i e l d s only t h e t w o dimensional rotational-intrinsic Hamiltonian, but t h i s H a m i l t o n i a n c o r r e s p o n d s to the t h r e e dimensional Hamiltonian H = - ~ ( ~ 2 / 2 m a) 0 2 / o p 2 a i,a
a~ L a' . l l-1 a. L' +
+½ L"121"c-~ [-~aLa" 1-11a * L'. I 2-1]" L i +~L.I
2
I.
L + V(O1,...,OA)
in t h e s a m e way that L 2 / 2 I f o r a r i g i d p l a n a r f o r m c o r r e s p o n d s to the H a m i l t o n i a n ½L. I - 1 . L f o r a r i g i d body. T h e d i a g o n a l i z e d i n e r t i a l d y a d i e s [ l a , [ 2a h a v e e l e m e n t s (Ina)ii
ma(O2a+O2a)/(kia)n;i,j,k, cyclic,
n = 1,2.
L a = - ifi~a × 0/0 ~ a and the q u a n t i t i e s without l e t t e r i n d i c e s d e n o t e the s u m , o v e r a l i p a r t i c l e s , of the c o r r e s p o n d i n g q u a n t i t i e s with such i n d i c e s . T h e H a m i l t o n i a n h a s the g e n e r a l f o r m d i s c u s s e d by V i l l a r s [6] so a l l his c o n c l u s i o n s a p p l y to it. Coordinate transformations involving noni n t e g r a b l e c o n s t r a i n t s h a v e b e e n u s e d in the p a s t [7, 8] to o b t a i n the r i g i d - b o d y m o m e n t of i n e r t i a . T h i s r e s u l t can b e o b t a i n e d in the t w o d i m e n s i o n a l e a s e f r o m the c o n s t r a i n t 0 = = ~ a k a 0 a by p r o c e e d i n g a l o n g l i n e s d e s c r i b e d by V i l l a r s [6] and u s i n g the e s t i m a t e (Oa2}0 ~ Oa2, an a p p r o x i m a t i o n v a l i d f o r a m a c r o s c o p i c pl~inar object. We a r e g r a t e f u l to P r o f e s s o r s E. L. T o m u s i a k and L. E. H. T r a i n o r f o r i n f o r m a t i v e d i s c u s s i o n s and to t h e N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a for supporting this investigation. We also app r e c i a t e t h e c o n s t r u c t i v e c r i t i c i s m s of the r e f eree. 257
V o l u m e 31B, n u m b e r 5
PHYSICS
Reference s 1. s. G a r t e n h a u s and C. S c h w a r t z , P h y s . Rev. 108 (1957) 482~ 2. R. S k i n n e r , Can. J. P h y s . 34 11956) 901. 3. V. Gupta and R. S k i n n e r , in: C o n t r . I n t e r n . Conf. on P r o p e r t i e s of n u c l e a r s t a t e s (Les P r e s s e s de l ' U n i v e r s i t ~ M o n t r e a l , 1969) p. 40.
258
LETTERS
2 M a r c h 1970
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