Strength distributions

Volume 25B. n u m b e r 2

PHYSICS

STRENGTH

LETTERS

7 August 1967

DISTRIBUTIONS*

J. B. FRENCH and L. S. HSU Department of Physics and Astronomy. Universily of Rochesler. Rochester. New York, USA R e c e i v e d 23 J u n e 1967

Methods a r e given f o r s t u d y i n g the s t r e n g t h d i s t r i b u t i o n s f o r o p e r a t o r s which act in o r b e t w e e n v e c t o r s p a c e s of f e r m i o n s d i s t r i b u t e d o v e r a finite set of s i n g l e - p a r t i c l e s t a t e s . S o m e e l e m e n t a r y a p p l i c a t i o n s to p a r t i c l e , hole and h o l e - p a r t i c l e e x c i t a t i o n s a r e d i s c u s s e d .

We c o n s i d e r the t r a n s i t i o n s induced by an excitation o p e r a t o r which acts inside an m - p a r t i c l e v e c t o r space o r between two such s p a c e s . In p a r t i c u l a r we deal with spaces of m f e r m i o n s d i s t r i b u t e d over N s i n g l e - p a r t i c l e s t a t e s ; an example Yn would be the s t a t e s of (ds) , m p a r t i c l e s in the n u c l e a r (ds) s h e l l (for which N= 24), and we might be c o n s i d e r i n g E2 t r a n s i t i o n s in this space or p a r t i c l e e x c i t a t i o n s produced by adding a d or s p a r t i c l e t0,.,.it. The m - p a r t i c l e space will have altogether ~ ) s t a t e s , a n u m b e r which may be v e r y l a r g e . Under the influence of the H a m i l tonian which is effective in the space, these may span a c o n s i d e r a b l e domain of energy, (probably 100 MeV for (ds)12). Consequently our excitation s t r e n g t h s will be expected to have a wide d i s t r i bution with r e s p e c t to the e n e r g i e s of both the i n i tial and final s t a t e s involved in the t r a n s i t i o n s . T h e r e a r e two s i n g l e - v a r i a b l e d i s t r i b u t i o n s which a r e of p a r t i c u l a r i n t e r e s t . Roughly speaking these focus on the q u e s t i o n s 1) W h e r e in the s p e c t r u m is the s t r e n g t h c o n c e n t r a t e d , if at all ? 2) Is m o s t of the s t r e n g t h given up to t r a n s i tions between s t a t e s which a r e n e a r n e i g h b o r s or ones which a r e f a r a p a r t ? In other words, is the t r a n s i t i o n e n e r g y s m a l l or l a r g e ? More c o m p l i c a t e d d i s t r i b u t i o n s asking, for example, whether the t r a n s i t i o n s involving s m a l l e n e r g y t r a n s f e r lie high or low in the v e c t o r space a r e also of i n t e r e s t . In the p r e s e n t p a p e r we c o n s i d e r the d i s t r i b u t i o n s , d e s c r i b i n g them in t e r m s of t h e i r e n e r g y m o m e n t s . We give methods for evaluating the l o w - o r d e r m o m e n t s , and, as e l e m e n t a r y e x a m p l e s , some r e s u l t s * S u p p o r t e d in p a r t by the U . S . A t o m i c E n e r g y C o m mission.

d e r i v e d t h e r e b y for p a r t i c l e , hole, and h o l e - p a r ticle excitations. The methods which we use r e q u i r e no s t a t i s t i c a l a s s u m p t i o n s about the s t r e n g t h s and a r e applicable to spaces of a r b i t r a r y complexity. T h e r e do a r i s e however, the u s u a l questions c o n c e r n i n g the r e l a t i o n s h i p between the " a l g e b r a i c m a n y - b o d y p r o b l e m " which we c o n s i d e r h e r e and the t r u e m a n y - b o d y p r o b l e m . C o n c e r n i n g these we make only the r e m a r k s that (i) the d i s t r i b u tions a r e of obvious i n t e r e s t i n s o f a r a s m o d e l s a r e c o n c e r n e d and have s p e c i a l p e r t i n e n c e to s t a t i s t i c a l studies, and (ii) if our space has been s e n s i b l y chosen, by c o n s i d e r i n g the "low-lying" s e g m e n t s of the d i s t r i b u t i o n (and m o r e g e n e r a l l y those f e a t u r e s of the d i s t r i b u t i o n which a r e i n v a r i a n t u n d e r e x t e n s i o n s of the space), we would expect to make f a i r l y d i r e c t contact with e x p e r i m e n t a l o b s e r v a t i o n ; this however, r e q u i r i n g as it does the u s e of r e l a t i v e l y high m o m e n t s , is not our m a i n p u r p o s e at p r e s e n t . So much for g e n e r a l d i s c u s s i o n . Suppose now we have an excitation o p e r a t o r O which o p e r a t e s in a v e c t o r space S in which there is a H a m i l tonian H. F o r the t r a n s i t i o n a -~/3, the s t r e n g t h 9, a function of the two e n e r g i e s Ec~ and Eft, is defined ~ as ] ( f [O[ot)l 2 which, to within a n o r m a lization, would be, in the e x a m p l e s m e n t i o n e d above, the BE2 value and the s p e c t r o s c o p i c f a c tor. We c o n s i d e r the s t r e n g t h d i s t r i b u t i o n in t e r m s of its m o m e n t w h e r e p --- ½(E~ +Eot) and q ~ ( E l - E c ~ ) , the s p e c i a l c a s e s where p=O or q= O, then f o c u s s i n g on the two questions which we have m e n t i o n e d e a r l i e r , a s i s c l e a r f r o m eq. (1) below.

See footnote next page.

75

Volume 25B, number 2

PHYSICS

We h a v e , by m a k i n g a b i n o m i a l e x p a n s i o n f o r the f i r s t f o r m , and a s e q u e n t i a l u s e of

~(otiO +] ~)(fllKia) (Efl:t:Ea) =(a iO+ [ H K ] ± I s ) f o r the s e c o n d that

9P, q (s) -_

=d- l(s)2-P~ <[3i O ia ) (E~ +E~ )P (E~ -Ea )q= ot,fl = 2-1) ~ A I (P, q) (HIo+HP+q- t o ) S

:

(1)

t

LETTERS

7 August 1967

9P'q;+(m) ± J(m)ClP'q;-= 2-P([K+(P)' K-(q)]±)m

(2)

w h e r e K + a r e the m u l t i p l e c o m m u t a t o r s o c c u r r i n g in eq. (1), the (+, -) r e s u l t s b e i n g s i g n i f i c a n t * when O+O i s of (odd, even) m a x i m u m p a r t i c l e rank. T h e (0, 1) m o m e n t d e t e r m i n e s t h e a v e r a g e e n e r g y ~ t r a n s f e r r e d in t h e e x c i t a t i o n , w h i l e (0, 2) d e t e r m i n e s the s p r e a d of t h e s e e n e r g i e s (or i t s r . m . s . v a l u e ) . T h e (1,0) m o m e n t f i x e s 3~, the a v e r a g e e n e r g y in the s p a c e at which the e x c i t a tion t a k e s p l a c e (defined f o r a s i n g l e t r a n s i t i o n a s the m e a n of the i n i t i a l and f i n a l e n e r g i e s ) ; (2, 0) t e l l s us s o m e t h i n g about the l o c a l i z a t i o n . T h e other moments give further information. We h a v e , w h e r e % Z a r e the d i s p e r s i o n s

: 2 -p ([H... [H[HO+]+]+...]+ [H... [H[HO]_]_... ] _ ) S . •

¢

p-fold a-commutator

•

t

9 0 0 d = 9 01,

q-fold commutator 900)42= 9 1 0 ,

T h e l a s t f o r m m a y be w r i t t e n in m a n y o t h e r w a y s . H e r e At(p,q) ~s the c o e f f i c i e n t of x t in the e x p a n s i o n of (1 +~5) (1 -x) q, d(S) i s the d i m e n s i o n a l i t y of S and ( ) ~ d e n o t e s the e x p e c t a t i o n v a l u e a v e r a g e d o v e r the s t a t e s of S, w h i c h in the p r e s e n t c a s e we c a n w r i t e a l s o a s ( ) m . T h e s e a v e r a g e s of c o u r s e a r e t r a c e s but we s h o u l d d i s t i n g u i s h b e t w e e n t h e c a s e s w h e r e the s e t s {~} and {fl} a r e i d e n t i c a l (as in E2 t r a n s i t i o n s ) and w h e r e t h e y a r e d i s j o i n t (as in p a r t i c l e e x c i t a t i o n ) . In the f i r s t c a s e , by a c y c l i c r e o r d e r i n g of t h e o p e r a t o r s w h i c h c h a n g e s the o r d e r of O +, O we s e e t h a t the o d d - q m o m e n t s v a n i s h , t h i s of c o u r s e b e c a u s e v e r y t r a n s i t i o n i s c o u n t e d t w i c e with o p p o s i t e e n e r g y d i f f e r e n c e s (a point w h i c h m u s t be b o r n e in m i n d in any a n a l y s i s ) . In t h e s e c o n d , s u c h a r e o r d e r i n g p r o d u c e s a t r a c e in a d i f f e r e n t s p a c e . In t h i s c a s e a l s o a r e l a t i o n s h i p b e t w e e n t h e s t r e n g t h d i s t r i b u t i o n s f o r a p a i r of a d j o i n t e x c i t a t i o n s O ± a c t i n g on the s a m e s e t of s t a t e s i s of i n t e r e s t . In p a r t i c u l a r we h a v e $ If, as will be usual, O c a r r i e s angular momentum or isospin, we sum over the components, O+O then becoming a scalar product (or m o r e generally, a z e r o coupled tensor). In this case there is no interference between the different tensors in an operator of mixed rank. In later papers, when we discuss multipole decompositions, m o r e general products will be appropriate. Note also that we do not eliminate the (f~=ot) "transitions" which would contribute to the distributions of number-conserving even-parity excitations. We have however found in some particular cases that their inclusion does not greatly affect things, and of course they do not contribute at all to the {q ¢ 0) m o ments.

76

(900) 2 (72

900902 = ()00)2~2 = 900920

(901) 2 ' _ (910)2.

(3)

C o n s i d e r now the e v a l u a t i o n of the m o m e n t s . We h a v e d i s c u s s e d e a r l i e r [2,3] m e t h o d s f o r e v a l u a t i n g the s c a l a r a v e r a g e of an o p e r a t o r of m i x e d p a r t i c l e r a n k ** z,. One m e t h o d u s e s the f a c t that ( ) m is a p o l y n o m i a l in m of o r d e r v and e x p r e s s e s the a v e r a g e in t e r m s of the a v e r a g e s f o r (v + 1) p a r t i c u l a r v a l u e s of m. U s i n g t h i s , we h a v e f r o m eq. (5) of r e f . 2 the r e s u l t that

c)P,q ) J ( m ) = ( v 2 +1 x (

~ (-1) t - v l /=0

x

+

kin-v 1

1! \ v 2 + l /

m +(V1+1)

(t)

N ×

~

(-1)t-v1

+N(N_m ) "N-I "

×

t=N-v 1 m - -v 2 - 1) X ( Nt --l m

t (Vl+l)

-1 9~,)q

(4)

* The notation O~ is particularly appropriate when O÷ corresponds to adding one or more particles to the system. The interesting commutator in eq. (2) (the one encountered in dealing udth spectral functions) is that which reduces the maximum particle rank of the integrand operator. A special case of these operators has recently been used by Rowe [1] in studying " equations of motion" methods in nuclear problems. ** See footnote next page.

Volume 25B, number 2

PHYSICS LETTERS

w h i c h e x p r e s s e s t h e m - p a r t i c l e m o m e n t s of i n t e r e s t to u s in t e r m s of t h e s a m e m o m e n t s f o r certain systems which are simpler in that they i n v o l v e s m a l l e r n u m b e r s of p a r t i c l e s o r h o l e s . W e h a v e h e r e t h a t v= (v 1 + v 2 + 1). S u p p o s e n o w t h a t O+O i s of m a x i m u m p a r t i c l e r a n k k (k = 1,2,4 r e s p e c t i v e l y f o r (i) p a r t i c l e o r h o l e e x c i t a t i o n , (ii) h o l e - p a r t i c l e , t w o - p a r t i c l e o r t w o - h o l e e x c i t a t i o n , a n d (iii) o r - p a r t i c l e e x c i t a t i o n ) . T h e n , f o r t h e u s u a l c a s e t h a t H i s of m a x i m u m p a r t i c l e r a n k 2, t h e (p, q) m o m e n t r e q u i r e s t h e a v e r a g i n g of a n o p e r a t o r of m a x i m u m r a n k (2p + q + k). F o r m a n y p u r p o s e s t h e m o s t c o n v e n i e n t c h o i c e of t h e s e p a r a t e p a r a m e t e r s in eq. (3) (the c h o i c e w h i c h m i n i m i z e s t h e n u m b e r of h o l e s o r p a r t i c l e s n e e d e d f o r t h e i n p u t t r a c e s ) i s v 1 = v 2 = p + ½ ( q + k - 1 ) if (q + h) i s o d d , v 1 = = v 2 + 1 (or v 2 = v 1 + 1) =p +½ (q+k) if (q + k) i s e v e n . F o r e x a m p l e t h e m - p a r t i c l e l i n e a r (1, 0) moment for a particle or hole excitation derives f r o m a k n o w l e d g e of t h e o n e - p a r t i c l e a n d o n e h o l e m o m e n t s w h i l e t h e (0, 4) m o m e n t f o r t h e

same process needs the one- and two-particle and one- and two-hole moments. Eq. (4) expresses m-particle moments in terms of a set of "input" moments, which in some simple cases may be conveniently written in terms of input energies or Hamiltonian matrix elements. As minor examples consider the linear moments (01, I0) for adding a particle in an orbit p. The (input moment -* input energy) relationships are easily worked out and we find (N-1)~+(m)

=m[ W(N)-W(p-1) ]+ (N-m - I )[ W(O)-W(p) ]

2 ( N - l ) (N-2) ~4Y+(m)= = r e ( N - 2 ) W(N) + 2Nm ( N - m - l )

-

mn(N-2m) W ( p -1) + ( N - m - l )

- (2m-l) ( N - m - l ) (N-2) W(0)

W(1) +

7 August1967

for example a single-hole energy). Both ~ and a r e p o l y - n o m i a l s in m r a t h e r t h a n r a t i o s of s u c h polynomials because, since the excitation annihilates the N-particle state, all moments must c o n t a i n a f a c t o r (N-m), t h e (00) m o m e n t t h e n b e i n g s i m p l y a m u l t i p l e of t h i s q u a n t i t y ~. F o r t h e s i n g l e - s h e l l c a s e ( t a k i n g W(O) = W(p) = O) we h a v e

C+(m ) = m w ( 2 ) , ~ + ( m )

= ½m~w(2).

F o r a X - m u l t i p o l e e x c i t a t i o n ~$ (e.g. h o l e - p a r ticle or electromagnetic) ~ X = ~rs)Zr~s we c a n r s

OZX

d e f i n e a n " e x c i t a t i o n H a m i l t o n i a n " H(e) = a n d t h e n ( f o r X ¢ 0) we h a v e f r o m e q s . (3, 4)

.~X)

"~(m) = -(m-2) (N-m-l) N(N-2)-I(HH(e)) 1 + + ½ ( m - l ) ( N - m - 1 ) N ( N - 1 ) ( g - 2 ) - I 0 V - 3 ) - I ( Hg(e)>2+ + ( m - l ) ( m - 2 ) NOV-2) -1 (Y-3) -1 ( H H ( e ) ) g - 1

(6)

w h i c h m a y a g a i n b e e a s i l y e v a l u a t e d e x p l i c i t l y . It g i v e s f o r t h e o n e - s h e l l c a s e (with W(0) = W(1) = 0) ]42(m) = ½ ( m - l ) ( m - 2 ) ( N - l ) ( Y - 3 ) -1 W',2) +

+ 2(m-l) (N-m-l) (N-2)'I(N-3)-I(-)X(2X+I)½fiX (7) where fix is the X-multipole coefficient for the interaction [4]. Apart from the monopole interactions, only the single k-multipole part of H contributes to c}42(m), a result which is true in the general case. Going beyond these rather uninteresting linear moments, we shall shortly present extended results for excitations in the (ds) shell and for other cases as well. It is clear also that there will be considerable interest in the decompositions of the strength distribution according to configurations, isospin, and angular momentum (and other groups which may be pertinent in given cases). We intend to discuss these matters later.

( N - 2 m - 2 ) W(p) +

(5)

while the hole-excitation res.ults follow from Pq;+ q P q', ( r e + l ) 9(m)' ' = (-1) (N-m) 9 ( ~ - ~ 1 ) . H e r e . t h e W s a r e a v e r a g e e n e r g i e s a s i n d i c a t e d (W(p-l)) b e i n g ** A k - p a r t i c l e o p e r a t o r has p a r t i c l e r a n k k. An e x tension to n o n - n u m b e r - c o n s e r v i n g o p e r a t o r s is a p p r o p r i a t e . In general we shall say that an o p e r a t o r of the f o r m (A)r(B) s where A is a c r e a t i o n o p e r a t o r and B = A + h a s p a r t i c l e r a n k ½(r+s), but of c o u r s e only o p e r a t o r s with r=s have a n o n - v a n i s h ing a v e r a g e . The notions h e r e could be unified by introducing the t e n s o r i a l r a n k with r e s p e c t to the U(N) group.

~- The reduction f r o m the obvious value (2P + 2q + k) c o m e s f r o m the c o m m u t a t o r s in eq. (1). In fact we write the c o m m u t a t o r f o r m only in o r d e r to e s t a blish this reduction of rank. We would calculate with the binomial f o r m . Note too that when the s i g n i ficant combination of m o m e n t s is taken for a pair of adjoint excitations (eq. 2) the r a n k of the integrand $ o p e r a t o r is f u r t h e r reduced by unity. This in fact enables one to write i m m e d i a t e l y the C~+ r e s u l t of eq. (5). F o r a h o l e - p a r t i c l e excitation c a r r y i n g n o n - z e r o angular m o m e n t u m note that all m o m e n t s involve the f a c t o r rn(N-m) so that once ~ aKain'}42(m) is given by a polynomial. " ~ s is the Racah unit t e n s o r of r a n k k which t r a n s f e r s a p a r t i c l e f r o m o r b i t s to orbit r . We a s s u m e the n o r m a l i z a t i o n >2. (or X )2 = 1 ~'S

rs

"

77

Volume 25B, n u m b e r 2

PHYSICS

References

EFFECT

OF

7 August 1967

F.S. Chang, J . B . F r e n c h and K . F . R a t c l i f f . Phys. L e t t e r s 23 (1966) 251; J . B . F r e n c h , P r o c . Dacca S e m i n a r (1967). F o r a special case see also J . M . Pasachoff. Phys. Rev. 150 (1966) 47. 4. J . B . F r e n c h , P r o c . Varenna S u m m e r School 36 (Academic P r e s s . New York, 1967}. See eq. (5.72).

1. D . J . R o w e . private communication. 2. J . B . F r e n c h . Phys. L e t t e r s 23 (1966) 248. 3. H. B a n e r j e e and J. B. F r e n c h . Phys. L e t t e r s 23 (1966) 245;

THE

LETTERS

THREE-BODY

FORCES

IN

3H*~

C. P A S K Applied Mathemalics Department. University of New South Wales. Kensinglon. N.S. W.. Australia Received 26 June 1967

Results ~f calculations of the contribution of t h r e e - b o d y forces to the binding energy of 3H a r e reported. T h e s e suggest that such forces may be of c o n s i d e r a b l e i m p o r t a n c e in nuclear physics.

It h a s l o n g b e e n r e a l i s e d t h a t w h e n t h e s c a t t e r i n g b e h a v i o u r of t w o n u c l e o n s a n d t h e p r o p e r t i e s of t h e d e u t e r o n c o u l d b e a c c o u n t e d f o r , it m i g h t b e possible to investigate three nucleon nuclei with a v i e w to d e t e r m i n i n g w h e t h e r o r n o t t h r e e - b o d y f o r c e s e x i s t . In t h i s w o r k we h a v e m a d e u s e of t h e t h r e e - b o d y w a v e f u n c t i o n s (in t h e c l a s s i f i c a t i o n of D e r r i c k a n d B l a t t [1] a n d K a l o t a s a n d D e l v e s [2]) o b t a i n e d b y D a v i e s [3] on t h e b a s i s of t h e H a m a d a - J o h n s t o n p o t e n t i a l [4], to i n v e s t i g a t e meson theory three-body forces. In D a v i e s ' f i n a l r e s u l t s , t h e w a v e f u n c t i o n c o n t a i n s a m i x t u r e of S, S' a n d D s t a t e s , t h e a m p l i tude for the P states being zero. Using a numeric a l i n t e g r a t i o n p r o c e d u r e , w h i c h we a l s o u s e , t h e g r o u n d s t a t e e n e r g y of 3H w a s f o u n d to b e --< - 5 . 8 6 2 M e V . D a v i e s a l s o f i n d s C h a r g e r a d i u s of 3H : R = 1.92 f m .

(^)

(1)

T h e l a t e s t e x p e r i m e n t s [5] g i v e R = 1.70 ~- 0.05 fm. T h r e e - b o d y p o t e n t i a l s a r e p r e d i c t e d by t h e meson theory which has successfully given part of t h e t w o - b o d y p o t e n t i a l . T h i s l a t t e r f a c t s u g g e s t s t h a t we s h o u l d i n v e s t i g a t e t h e s i m p l e s t , l o n g * R e s e a r c h s p o n s o r e d in part by the A i r F o r c e Office of Scientific R e s e a r c h , Office of Aerospace R e s e a r c h , United States A i r F o r c e , under AFOSR c o n t r a c t grant n u m b e r AF-AFOSRO 685-66. J- This work f o r m s part of a P h . D . t h e s i s to be s u b m i t ted to the U n i v e r s i t y of New South Wales. 78

iiiii (¢)

(D) Fig. 1.

range, meson theory three-body potentials. These are derived from processes involving two pions. We considered two such potentials. The simplest three-body potential is perhaps the one obtained from a straightforward perturbat i o n t h e o r y . If o n e i n c l u d e s a l l t h e 4th o r d e r p e r turbation terms, a total cancellation occurs and V3 = 0, to t h a t o r d e r . T h i s c a n c e l l a t i o n a g r e e s w i t h c a l c u l a t i o n s by W e n t z e l [6].