THE DME AS AN APPROXIMATION TO THE WIGNER TRANSFORM

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THE DME AS AN APPROXIMATION TO THE WIGNER TRANSFORM

J. Martorell, E. de Guerra

To cite this version:

J. Martorell, E. de Guerra. THE DME AS AN APPROXIMATION TO THE WIGNER TRANS- FORM. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-199-C6-204. �10.1051/jphyscol:1984623�.

�jpa-00224224�

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T H E DME A S AN APPROXIMATION T O THE WIGNER TRANSFORM

J. M a r t o r e l l and E . Moya d e Guerra*

Departamento de F'csica Atomiea y Nuelear, Universidad de Palma de Ma 2 Lorca, Spain

* ~ e ~ a r t a m e n t o de F[sica Atomiea y NueZear, Universidad de Extremadura, Badajoz, Spain

~ é s u m é - La DME sans t r o n c a t i o n donne une transformée de Wigner approchée qui r e p r o d u i t correctement tous l e s moments de l a transformée exacte. ^A l ' a i d e d'un modèle d ' o s c i l l a t e u r pour noyaux magiques on Gtudie analytique- ment c e s moments e t f a i t l a comparison avec l a DME tronquée.

Abstract - The DME without t r u n c a t i o n l e a d s t o a,Wigner transform with t h e same moments a s t h e e x a c t transform. Using a harmonic o s c i l l a t o r mode1 f o r t h e magic nuclei we obtain a n a l y t i c expressions f o r t h e s e moments and compare with those of various truncated DME's.

As an a l t e r n a t i v e t o conventional s e m i c l a s s i c a l approaches we want t o d i s c u s s here t h e s u i t a b i l i t y of t h e Density Matrix Expansion (DME) of Negele and Vautherin (1) a s an approximation t o t h e Wigner transform of t h e one body d e n s i t y of s p h e r i c a l s p i n - s a t u r a t e d n u c l e i . As remarkeci by Campi and Bouyssy ( 2 ) , t h e r e i s a very simple d e r i v a t i o n f o r the DNE in terms of the Wigner transform. Following s t a n - dard n o t a t i o n s ( 3 ) , we w r i t e

and

P (Fl ,F2 ⁾⁼^cp(F1 u ,F2a) + + + u

where: fi ⁼ (?1+t2)/2, s = r l - r 2 . We d e f i n e a l s o t h e moments of t h e Wigner transform f o l lowing t h e usual conventions :

and t h e r e f o r e : iig = p ( % ) , ~2 = r ( R ) -+ - ' v2p($) a s i s well known. Since we a r e only i n t e r e s t e d l n ground S t a t e s of spf'n s a t u r a t e d nuclei odd moments a r e always zero.

The approximation t o ;(fi,<) and f (fi,;) given by t h e DME i s then obtained a s f o l - lows ( 2 , 4 ) :

a ) F i r s t , p(n,<) i s approximated by i t s average over t h e angle between fi and z:

[4 1 This can be al so w r i t t e n a s :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984623

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C6-200 JOURNAL DE PHYSIQUE

A + ^-f

where p(Û) = p(Z,;=0) and b ⁼i/l(Vl-7,) b ) The forma1 expansion:

( v a l i d f o r a r b i t r a r y k, and -1 5 y / k 2 1 ) leads n a t u r a l l y t o the DME ansatz f o r p:

A

w i t h 6 r ICI ^and

where t h e a i a r e t h e c o e f f i c i e n t s o f t h e 2n+l Legendre polynomial, and t h e +n are the moments o f t h e exact Wigner transform.

c ) From pDME, u s i n g t h e d e f i n i t i o n [11 t h e corresponding DHE expression f o r the Wigner t r a n s f o r m i s obtained:

This i s t h e approximation t o t h e Wigner t r a n s f o r m whose s u i t a b i l i t y we discuss i n what f o l l o w s :

i ) As expected, keeping o n l y t h e f i r s t term i n the s e r i e s leads t o t h e w e l l known Thomas Fermi approximation. ( T h i s f o l l o w s from the f a t t h a t t h e f i r s t term i n pDFIE. eq. [ I l , i s j u s t t h e S l a t e r approximation t o p ( i , i ) ) .

i i ) Due t o the f a c t o r ~ ( p - k ) , fOME(p) i s a d i s t r i b u t i o n . Thus t h e r e i s no reason t o expect t h a t f D q E ( p ) a t f i x e d R should g i v e a good v i s u a l f i t t o f ( p ) , and as we w i l l skow i t does n o t . Rather we expect t h a t o n l y i n t e g r a l proper- t i e s o f f, l i k e i t s moments, can be a c c u r a t e l y g i v e n by ~ D M E . Indeed, if we use eq. [3] t o i n t r o d u c e t h e moments, $nE, o f fDME, i t can be e a s i l y shown (4) t h a t

eDME _2n = p 2 n ( ~ ) , f o r a l i n .

which t h e r e f o r e proves t h a t , f o r i n t e g r a l p r o p e r t i e s , fDbIE i s a very good approximation t o f.

As g i v e n i n Eqs. [7 - 91, DME i s o f li t t l e use however, since i t r e q u i r e s adding up an i n f i n i t e number o f ternis. Therefore a d d i t i o n a l approximations a r e needed and u s u a l l y the s e r i e s i s t r u n c a t e d a f t e r a few terms, choosing t h e parameter k so as t o expect a f a s t convergence. The s i m p l e s t o f these approximations are:

a ) That o f Negele and Vautherin (NV): two terms are r e t a i n e d , and

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For expansions t r u n c a t e d a t n =rima, one can prove ( 4 ) t h a t eq. [IO] i s s t i l l v a l i d f o r n 5 n b u t the h i g h e r moments then do n o t agree any more. This was recog- n i z e d by ?%ii and Bouyssy, and l e d them t o t r y t o f i n d improved versions o f t h e i r approximation. For t h a t , they found i t u s e f u l t o d e f i n e s u i t a b l y renormalized moments M3,,(R) as f o l l o w s :

which f o r a Thomas Fermi type o f Wigner t r a n s f o r m ( f T F ( p ) = 40(p-k)) g i v e s :

i.e. a constant value independent o f n. Remarking t h a t f o r f ' s t h a t a r e l e p t o - dermous i n p one expects a l i n e a r dependence o f M on n, they w r i t e

and assuming a reasonably small, t h i s leads t o

and a corresponding expression f o r f . However Campi and Bouyssy were a b l e t o clieck t h e v a l i d i t y o f t h e i r r e s u l t s o n l y i n v e r y l i g h t n u c l e i ( 1 6 0 and ''Ca) f o r t h e harmonic o s c i l l a t o r model (h.0.). We have r e c e n t l y ( 4 ) developed a n a l y t i c expressions f o r f i n t h a t model f o r a l 1 magic n u c l e i . I n a d d i t i o n we have shown t h a t the most s a l i e n t f e a t u r e s o f Uigner transforms obtained from Density Depen- dent Hartree Fock c a l c u l a t i o n s a r e w e l l reproduced by t h e h.0. model even i n heavy n u c l e i . Here we use these a n a l y t i c r e s u l t s t o complete t h e a n a l y s i s o f r e f . ( 2 ) and extend i t t o heavy double magic n u c l e i . The e x p l i c i t r e s u l t s f o r t h e moments are:

16 2 2

0) = 32 N e-R Ib ( - 2 1 3 1 7 ) 40 - , 2 2

v2,,( Ca) = 32 N e -R Ib (41:-317 + 1 2 l n ) 2 p2,,( 48 Ca) = 1 6 N e -R 2l b [ 4 I o - 2 I 2 n n + ^{- 1}l n + ^-4

1 5 2 7 1;

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v z n ( Z r ) 90 = 16 N e -R2/b2[.71n _O+ G? ₃ 11" ₁ - ^ZIn₂⁺^.^!₇ ^In₃ ⁺ 9 ! ! 4 ^2-¹ⁿ

2 A

+ - (41n - -8 ln + +- In 16 n ] 3 0,l 7 1,l 63 2 , l ) ' 945 '0,2 p 2 n ( 2 0 8 ~ b ) = 16 N e

405 n 7 2 32 In + +- ^In

+ -1 _{l l !}_{5 1}+-in ₃ ₆+ t (61;,~ - ^-₇ _1,l 21 2 , l

- - '6 1" ^+%II" ) + 8 ( 2 1 " - A I " + L I " )

23! 3,l .. ^4,l 15 21 0,2 77 1,2 429 2,2

+ ⁶⁷²⁰

7!!.13!! 1:,3]

where

4 = ( 2 n ) - 3 / 2 ( ~ b ) - 2 n - 3

l e a d i n g t o t h e f o l l o v ~ i n g e x p l i c i t expressiotis f o r t h e Mk: ( x F R/b), ( k = 2,4,6,. . . ^);

M ~ (= ~

Iw

~ ~ ⁽ⁱ) ( k + l ) + x 2 ) / ( k + x 2 ) ] ' I k / b 3.2

40 (k+3)!! (& (k2+2k+5) i2 1 kx 2 t2 1 x 4) / ( 8 + 2 5 1 4 l l k x 11 / b Mk( Ca) = [-

3.2k'2

Mk( 48 Ca) = [--- 1 4

(k+3) !! (4-2(k+3) +r (k+3) (k+5) trOS (k+3) (k+5) (k+7) 3 . z k l 2

4 8 2 4 16 4 32 6

+ ( - 4 - 3 (k+3) +- ( k + 3 ) ( k + 5 ) ) ~ + (5+15 ( k + 3 ) ) x + - x }/(5+4x 4

15 105

32 6 l/klb + - x 105 11

90 (k+3) !! 22 1

Mk( Z r ) = (-7 +- (k+3)-2(k+3)(k+5) +7 ( k + 3 ) ( k + 5 ) ( ~ + 7 ) +

3.2 3

1 44 8 2 8

+ - (k+3) ...( K+9) +(-3--7j (k+3) - 3 (k+3)(k+5) +a ( k + 3 ) ( k + 5 ) ( k + 7 ) ) ~ 2 189

4 8 4 8 32 6 16 8

+ ( - 8 - - (k+3) 3 + T 1 ( k + 3 ) ( k + 5 ) ) ~ + (7+63 ( k + 3 ) ) ~ + 189 x )

2 4 8 6 16 8 'lk

/ ( 5 + 1 0 ~ - 4 x + T X + W X ) ] / b

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Fig. 3

P

F i g . 2

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With these expressions one can then check eqs. [7] - [ 9 ] f o r the DME. I n f i g . 1 we present the r e s u l t s f o r t h e angle averaged p(R,s) using t h e DME expansion

eq. [7] f o r 40Ca and "Ca w i t h t h e p r e s c r i p t i o n k = kF(R). I n both cases R = 1 fm and the v a r i a b l e i s s k ~ . Only t h e o u t e r p a r t o f p(R,s) the c o n t r i b u t i o n s o f terms beyond the f i r s t one o r two become noticeable,confirming t h a t the t r u n c a t e d DME i s a good approximation f o r p(R,s) when t h i s i s t o be used w i t h s h o r t ranged i n t e r a c - t i o n s . The corresponding expansion, eq. [ 9 ] f o r f i s shown i n f i g . 2 f o r "Zr. As expected, t h e r e i s no convergence towards t h e exact f. I t can be f u r t h e r shown ( 4 ) t h a t as more terms are added t o t h e expansion, t h e ~ D M E becomes more and more o s c i l l a t i n g , and i t s o s c i l l a t i o n s increase i n amplitude.

We t u r n now t o t h e i n t e g r a l p r o p e r t i e s o f f , and discuss t h e renormalized moments, Mk, given by each approximation. The a n a l y t i c expressions f o r t h e NV and CB moments a r e obtained from the s u i t a b l y t r u n c a t e d expression [91, and are:

M:

: = M2 , independent o f n

(as expected from a m o d i f i e d Thomas Fermi Wigner transform), and

N o t i c e t h a t , as expected, a l s o i n t h i s case M!' = M2.

I n f i g . 3 we show f o r 4 magic n u c l e i the Mk as a f u n c t i o n o f k f o r d i f f e r e n t values o f R, and compare them f o r various approximations. One can make t h e f o l l o w i n g remarks:

a) The v a r i a t i o n o f t h e Mk i n t h e range o f k considered i s a t most 20-30%, i n t h e l i g h t e r n u c l e i , much l e s s i n t h e heavier ones, so even a constant Mk (Campi- Bouyssy p r e s c r i p t i o n ) i s n o t a bad f i r s t approximation, p a r t i c u l a r l y f o r heavy nucl e i .

b) The dependence on k i s l i n e a r f o r R a t t h e surface (and beyond) o f t h e n u c l e i b u t n o t so i n s i d e t h e nucl eus. This i s c l o s e l y r e l ated t o t h e f a c t t h a t t h e exact f shows s t r o n g o s c i l l a t i o n s as a f u n c t i o n o f p f o r small R, b u t these disappear when R increases. Thus i n t h e l a t t e r case f ( p ) i s more iepkoder- mous and t h i s leads t o l i n e a r i t y of t h e Mk. However a l i n e a r approximation l i k e t h a t proposed i n eq. [14] does n o t appear t o be a p p r o p r i a t e f o r t h e i n - t e r n a l region: a t R = O i n most n u c l e i i t gives a much poorer f i t t o 1.1 than t h e o r i g i n a l Campi-Bouyssy p r e s c r i p t i o n MCB = M2. It appears thus t b us t h a t t h e r e i s no simple way t o f i t b e t t e r t h e Mk 8nd irnprove on t h e already s a t i s - f a c t o r y 'cruncated DME's o f NV and CB.

References

(1) J. Negele, D. Vautherin, Phys. Rev. C5 (1972) 1472.

( 2 ) X. Campi, A. Bouyssy, Phys. L e t t . 73Ë8(1978) 263, and Nukleonika fi (1979) 1.

( 3 ) P. Ring, P. Schuck, "The Nuclear Many Body Problem", Springer Verlag 1980.

(4) ^J. M a r t o r e l l , E. Moya de Guerra, Ann. o f Phys. t o be published, and Proc.

I n t . Workshop on "Mathematical Methods i n Nuclear Physics", Granada 1983, t o be published i n the s e r i e s "Lecture notes i n Physics", Springer Verlag.