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A GENERAL ITERATION SCHEME FOR THE
CALCULATION OF LEVEL DENSITIES, AND
RESULTS USING A SEMICLASSICAL
APPROXIMATION
A. Blin, B. Hiller, R. Hasse, P. Schuck
To cite this version:
A GENERAL I T E R A T I O N SCHEME FOR THE C A L C U L A T I O N OF L E V E L D E N S I T I E S , AND RESULTS U S I N G A S E M I C L A S S I C A L A P P R O X I M A T I O N
A . H . B l i n , B . H i l l e r , R.W. as se* and P. ~chuck**
Centre dlEtudes Nucléaires de Grenoble, Laboratoire de Physique Nucléaire, 85 X, 38041 Grenoble Cedex, France
* ~ n s t i t u t Laue-Langevin, 38042 Grenoble Cedex, France
* * ~ n s t i t u t des Sciences Nucléaires, 38026 Grenoble Cedex, France
Résumé
-
Un schéma général e s t d é r i v é pour c a l c u l e r des d e n s i t é s de niveaux àm p a r t i c u l e s e t n trous des fermions pour t o u t Hamiltonien à p a r t i c u l e s indé- pendantes en t e n a n t compte du p r i n c i p e de P a u l i . Cette technique e s t appliquée pour o b t e n i r l e s d e n s i t é s de niveaux de l ' o s c i l l a t e u r harmonique i s o t r o p i q u e
à t r o i s dimensions semiclassiquement dans l'approximation de Thomas-Fermi. Abstract
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A general scheme i s derived t o c a l c u l a t e m-particle n-hole fermion l e v e l densi t i e s f o r any s i n g l e p a r t i c l e Hamil tonian taking i n t o account Pauli exclusion. This technique i s applied t o o b t a i n l e v e l d e n s i t i e s of t h e t h r e e dimensional i s o t r o p i c Harmonic Oscil l a t o r semicl a s s i c a l l y i n theThomas- Fermi approach .
1
-
INTRODUCTION-
I t has been shown r e c e n t l y 111 t h a t t h e usual Thomas-Fermi (TF) approach C21,
which i s a s e m i c l a s s i c a l treatment, y i e l d s average 1 - p a r t i c l e 1-hole and 2 - p a r t i c l e 2-hole level d e n s i t i e s , o f an i s o t r o p i c Harmonic O s c i l l a t o r , which a r e i n a very good agreement t o t h e e x a c t quantum mechanical l e v e l d e n s i t i e s . We develop a gene- r a l scheme which permits t o c a l c u l a t e . e x a c t l y a r b i t r a r y m-particle n-hole e x c i t a - t i o n s f o r any type of s i n g l e p a r t i c l e Hamiltonians. We e x p l i c i t l y take i n t o account t h e Pauli exclusion p r i n c i p l e f o r fermions. Encouraged by t h e above mentioned TF r e s u l t s , we use our scheme t o c a l c u l a t e a l 1 m-particle n-hole l e v e l d e n s i t i e s u p t o m,n = 3 .
Level d e n s i t i e s with equal number o f p a r t i c l e s and holes a r e o f i n t e r e s t f o r instan- ce i n the study o f precompound r e a c t i o n s and of t h e s p r e a d i n g w i d t b o f g i a n t resonan- ces [3,41. O n t h e o t h e r hand, t h e l e v e l d e n s i t i e s a s s o c i a t e d w i t h unequal number of p a r t i c l e s and holes f i n d t h e i r a p p l i c a t i o n in t h e c a l c u l a t i o n o f s h e l l mode1 o p t i c a l p o t e n t i a l s [21.
In t h e next s e c t i o n we d e r i v e an i t e r a t i o n formula t o c a l c u l a t e t h e m-particle n-hole level d e n s i t i e s . In s e c t i o n 3 we apply t h i s scheme t o t h e case o f an i s o t r o - pic t h r e e dimensional Harmonic o s c i l l a t o r . We o b t a i n a n a l y t i c a l r e s u l t s i n the T F approach.
2
-
DERIVATION OF AN ITERATION FORMULA FOR MANY-PARTICLE MANY-HOLE LEVEL DENSITIES-
The d e f i n i t i o n of the e x a c t m-particle n-hole l e v e l d e n s i t y f o r fermions as a func- t i o n o f t h e energy E i s :Pm h n
where the min f o l d sum o u t s i d e of t h e Dirac d e l t a function 6 extends over al1 p a r t i - c l e s t a t e s ( l a b e l l e d by t h e indexes p l through pm) with t h e r e s t r i c t i o n p l < .
.
.<pm,C6-232 JOURNAL DE PHYSIQUE
and s i m i l a r l y f o r t h e hole s t a t e s h . . The r e s t r i c t i o n s p l < .
.
.< p,
and h l < . . . < h na r e t o avoid m u l t i p l e counting of t i e same terms and t o honor t h p Pauli exclusion p r i n c i p l e . The E~ a r e the eigenvalues of the s i n g l e p a r t i c l e Hamiltonian H .
Equation (2.1) can be r e w r i t t e n as :
pm 'n 1-im 'n
nh(E) = C
< u l
,...
pm,v,,...
v n l IT B(H - E ~ ) T B ( E ~ - H ~ ) ~ ( E - c E f C E ~ )p l < . ..<Pm
u=u
1 Fi v=v1 u=u1" v=viV I < .
.
.<vnBy introducing t h e proper s t e p functions 0 , t h e p a r t i c l e s t a t e s a r e guaranteed t o l i e above the fermi energy E F , and t h e hole s t a t e s below. We replaced t h e summation i n d i c e s pi and h . by pi and v . t o i n d i c a t e t h a t each of them runs now over a l 1 par- t i c l e and hole s t a t e s . We introduced a l s o t h e product e i g e n s t a t e s with t h e property
Hi 1 ~ 1 . . . u m 5 V I . . . V n > = E i ] p l . . .Pm, V I . .
.Cn>
where i i s of the s e t
{ u l , .
.
. p m , v ~ , . . . v n } .A convenient r e p r e s e n t a t i o n o f t h e level d e n s i t y i s obtained by applying a Fourier transform ( E + t , with the d e f i n i t i o n : = ) t o e q . ( 2 . 2 ) :
00 h
i . e . t h e resul t f a c t o r i z e s i n t o a p a r t i c l e component
and a hole component
Note t h a t i f t h e sums i n e q s . ( 2 . 5 ) and ( 2 . 6 ) were u n r e s t r i c t e d , one could w r i t e
A",
and 8 simply a s t r a c e s . However, i t i s p o s s i b l e t o express ,,A, and R i n terms of t r a c e g , by observing the following. A r e s t r i c t e d sum over two indice? o f a function f can be w r i t t e n as an u n r e s t r i c t e d sum minus a sum over t h e diagonal terms and d i - vided by two t o take i n t o account double counting. To s i m p l i f y Our n o t a t i o n , we de- f i nef ( u ) < ~ I B ( H ~ - E ~ ) e-BH1i~u> ( 2 . 7 )
and note t h a t e q . ( 2 . 5 ) f o r two p a r t i c l e s can then be w r i t t e n as
S i m i l a r l y , i n t h e case o f t h r e e i n d i c e s , we have
p,>~i, e t c . , which occur 3 ! times f o r t h r e e i n d i c e s .
Note, t h a t i n e q s . ( 2 . 8 ) and ( 2 . 9 ) t h e sums on t h e r i g h t hand s i d e a r e now u n r e s t r i c - t e d and t h e r e f o r e can be w r i t t e n a s t r a c e s . For an a r b i t r a r y number of i n d i c e s we o b t a i n rom comb t o r i c s t h e following i t e r a t i o n formulae f o r a new s e t o f func- t i o n s A b ) and B!l7(defined as and s i m i l a r l y f o r BA')): Bh')(B) = B j e ) ( $ ) . ( B i o ) ( 8 ) ) n - 1
-
'
(n)
Bn-k+l ('+'-')(8) f o r n 2 k=2 with t h e i n i t i a l d e f i n i t i o n s A t r { B ( H - E ~ ) e-
( ~ + l ) BHj
The Fourier transformed level d e n s i t y i s then 1 ( 0 ) 1
Gmpnh(B) = Am(B) .Bn(B) = Am ( 8 ) .
5
BA0)(8)The functions A(') and B(') a r e completely determined by t h e i t e r a t i o n procedure, e q s . ( 2 . 1 0 ) t h r o t g h (2.147, as i s t h e l e v e l . d e n s i t y , e q . ( 2 . 1 5 ) . Note t h a t l e v e l den- s i t i e s o f t h e form gopnh and gmpoh a r e obtained by s e t t i n g Ao(8) = 1 o r Bo(@) = 1 , r e s p e c t i v e l y , as fol lowç immediately frorn eq. ( 2 . 1 ) .
The above equations a r e v a l i d f o r l e v e l d e n s i t i e s of any ( s i n g l e p a r t i c l e ) Hamilto- nian. We emphasize t h a t i n t h e case L? = O , t h e sums which a r e s u b t r a c t e d i n
e q s . ( 2 . 1 1 ) and ( 2 . 1 2 ) a r e t h e c o r r e c t i o n terms due t o t h e Pauli exclusion p r i n c i p l e (because they s u b t r a c t terms with equal i n d i c e s , compare with eqs. ( 2 . 8 ) and (2.9)). We now proceed t o apply these equations t o a p a r t i c u l a r case.
3 - SEMICLASSICAL L E V E L DENSITIES OF THE HARMONIC OSCILLATOR
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We consider t h e t h r e e dimensional i s o t r o p i c Harmonic Oscil 1 a t o r Hamil t o n i a n .
The semiclassical (Thomas-Fermi) approximation [21 i s obtained by replacing the o p e r a t o r H . by i t s c l a s s i c a l c o u n t e r p a r t and t h e t r a c e s i n e q s . ( 2 . 1 1 ) - (2.15) by
i n t e g r a t i o a s over phase space, i . e . .
C6-234 JOURNAL DE PHYSIQUE
I n t h e s e m i c l a s s i c a l a p p r o x i m a t i o n eqs. (2.13) and (2.14) become
Then t h e l e v e l d e n s i t i e s g (E) a r e o b t a i n e d w i t h t h e h e l p o f i t e r a t i o n formulae eqs.(2.11) and (2.12) and mpnh by an i n v e r s e F o u r i e r t r a n s f o r m o f eq.(2.15), which can be done a n a l y t i c a l l y , s i n c e i t i n v o l v e s o n l y terms o f t h e t y p e ,-uBcF B-V
transform t0 ~ ( E - V E ~ ) ( E - ~ E ~ ) " - ' / ( v - ~ ) ! ( v , ~ i n t e r g e r s , v>o, u=o, ti,.
.
.).
For t h e case o f o n e - p a r t i c l e one-hole and t w o - p a r t i c l e two-hole e x c i t a t i o n s t h e l e v e l d e n s i t i e s have been d e r i v e d i n r e f . [Il:: a l ready. F o r h i g h e r o r d e r l e v e l den- s i t i e s t h e a n a l y t i c a l e x p r e s s i o n s become p r o h i b i t i v e l y l e n g t h y t o be c a l c u l a t e d by hand. T h e r e f o r e we made use o f t h e computer code REDUCE [5], which a l l o w s f o r ana- l y t i c a l t r e a t m e n t o f l e n g t h y e x p r e s s i o n s .We now d i s c u s s some o f t h e r e s u l t s . We a r e n o t g o i n g t o compare t h e s e m i c l a s s i c a l l e v e l d e n s i t i e s t o t h e e x a c t quantum mechanical ones, s i n c e i t has been shown i n r e f s . [ll and [ 6 ] t h a t a l r e a d y f o r glplh and gZPzh t h e o v e r a l l agreement o f t h e semi- c l a s s i c a l r e s u l t s w i t h t h e e x a c t ones i s e x c e l l e n t .
3.1. The Role o f t h e P a u l i E x c l u s i o n P r i n c i p l e -
F i r s t we show t h e r o l e o f t h e P a u l i e x c l u s i o n p r i n c i p l e . I n F i g . 1, t h e l o w e r c u r - ve shows t h e complete l e v e l d e n s i t y g, 3 h as a f u n c t i o n o f t h e energy, whereas t h e upper c u r v e r e p r e s e n t s t h e d i r e c t t e r g s
,
i . e . i s n o t c o r r e c t e d f o r t h e P a u l i p r i n c i p l e (see t h e remark a t t h e end o f S e c t i o n 2 ) . The d e v i a t i o n i s o f t h e o r d e r o f 35% a t E/tiw = 5, and o f 17% a t E/3w = 50 i n t h i s case. I t g e t s s m a l l e r f o r h i g h e r energieP. I t i s a l s o s m a l l e r f o r O l o w e r o r d e r e x c i t a t i o n s (e.g. f o r gZpzh i ti s 7 % a t E/fiwo = 5 and 5% a t E/-Awo = 5 0 ) .
Fig.l-
The e f f e c t o f t h e P a u l i p r i n c i p l e on t h e s e m i c l a s s i c a l 3 - p a r t i c l e 3 - h o l e l e - v e l d e n s i t y g, as a f u n c t i o n o f energy E, i n u n i t s o f Ruo, which i s t h e Harmonic o s c i l l a t o r c o n g t a n t . The upper c u r v e i s c a l c u l a t e d w i t h o u t t a k i n g P a u l i e x c l u s i o n i n t o account, t h e l o w e r one i s t h e f u l l l e v e l d e n s i t y o b e y i n g t h e P a u l i p r i n c i p l e . The curves i n a l 1 f i g u r e s a r e f o r 40 fermions (no i s o s p i n c o n s i d e r e d ) .F i g . 2. AS s h o u l d be expectei!lhthe l e v e l d e n s i t y eurves d i f f e r by o r d e r s o f magnitude, s i n c e t h e a v a i l a b l e phase space i n c r e a s e s d r a s t i c a l l y w i t h t h e o r d e r o f p a r t i c l e - h o l e e x c i t a t i o n s
.
$ipnhxhw.
and glxno.
F i .2
-
The e x c i t a t i o n l e v e l d e n s i t ~ e s glpIh ( c r o s s e s ) , gZpzh ( d o t s ) and g,p3h &rmost c u r v e ) . The l o w e s t c u r v e 1 n d i c a t e s t h e o n e - p a r t i c l e l e v e l densi t y giD e p o s i t i n g a g i v e n amount o f energy E i n a system o f many p a r t i c l e s i n a p o t e n t i a l e x c i t e s a l 1 n - p a r t i c l e n - h o l e s t a t e s w h i c h a r e above t h e i r r e s p e c t i v e t h r e s h o l d s . T h e r e f o r e we show t h e sum
i n Fig.3 and compare i t t o t h e w e l l - known s t a t i s t i c a l f o r m u l a [ 7 1
where we c a l c u l a t e t h e Fermi energy E~ ~ e m i c l a ~ ~ i c a l l y f r o m t h e p a r t i c l e number N i n t h e Harmonic O s c i l l a t o r
113
E~ = (3N) Tuo (3.7)
The s t a t i s t i c a l p r e d i c t i o n l i e s above t h e s e m i c l a s s i c a l sum f o r h i g h e r e x c i t a t i o n s , supposedly because we d i d n o t t a k e i n t o account e x c i t a t i o n s h i g h e r t h a n 3 - p a r t i c l e 3 - h o l e . Al though t h e diagram shows o n l y e x c i t a t i o n s below 4 tlw,, where t h e r e a r e no 4 - p a r t i c l e 4 - h o l e s t a t e s a l l o w e d by quantum mechanics, we shouTd n o t e t h a t t h e semi- c l a s s i c a l r e s u l t s e x h i b i t an average b e h a v i o r . T h e r e f o r e t h e gqpqh l e v e l d e n s i t y does have a t a i l below i t s e x a c t t h r e s h o l d and s h o u l d be i n c l uded i n t h e sum (3.5), which we d i d n o t .
The number o f s t a t e s f o r an n - p a r t i c l e n-hole e x c i t a t i o n i s g i v e n by :
E
Nnpnh(E) =
j
dE1gnpnh(El)JOURNAL DE PHYSIQUE
Fig.3
-
The s t a t i s t i c a l e x c i t a t i o n level d e n s i t y ( s e e e q . ( 3 . 6 ) ) , d o t t e d l i n e , and the g i p l h + g ~ p , h + g3P3h) f u l l curve.Since N l p i h and N~~~~ a r e already shown
N 3 P 3 h f o r reference ( F i g . 4 ) . i n Ref. [ 1 ' 3 ~ 3 ~ - 1 we i n c l ude t h e graph f o r Fi .4 The number o f . s t a t e s f o r 3+httlcle 3-hole e x c ~ t a t l o n s , obtained by i n t e g r a t i n g s , , , , ~ ( E ) , e q . ( 3 . 8 ) . 3.3. P a r t i c l e Level Densities
-
We proceed t o consider t h e s i n g l e p a r t i c l e l e v e l d e n s i t y , defined i n anology t o e q . ( 2 . 1 ) as
gi(E) = C 6 ( E - c i )
t h e minus s i g n being necessary s i n c e hole e n e r g i e s a r e negative. The term g o p l h ( - E ) c o n t r i b u t e s i n t h e energy range o . . . E
,
whereas g l o h ( E ) c o n t r i b u t e s from c F t o i n f i n i t y . They match of course exac!ly a t t h e fepm~ energy E The curve o f gl i s included i n Fig.2 ( i t i s t h e lowest one). In anology t o eq. Fi3.10) we can d e f i - ne g2 and 93 which a r e shown i n Fig.5. They decrease s h a r p l y a t 2cF and 3 c F , r e s - p e c t i v e l y , t h e matching point of t h e two components of g2 and o f g 3 . Although t h e l e v e l d e n s i t y g l f o r s i n g l e p a r t i c l e s i s considerable a t z F ( s e e f i g . 2 ) , t h e phase space f o r two p a r t i c l e s o r two h o l e s , and t h r e e p a r t i c l e s o r t h r e e holes a t c F per p a r t i c l e o r hole i s d e p l e t e d , due t o momentum conservation. The energy of g2 and g3 extends from O t o i n f i n i t y .g2%no.
and g3x7=o,
2250
Fi . 5
-
The 2 - p a r t i c l e and 3 - p a r t i c l e . l e v e 1 d e n s i t i e s g2 (dashed l i n e ) and g 3 ( f u l l3%.
The points on t h e energy a x i i w i t h two times and t h r e e times t h e Fermi ener-gy c F a r e i n d i c a t e d ( s e e t e x t f o r d i s c u s s i o n ) . Level d e n s i t y gl i s shown i n Fig.2.
3 . 4 . Level Densities with Unequal Number o f P a r t i c l e s and Holes
-
F i n a l l y , i n Fig.6, we show m-particle n-hole l e v e l d e n s i t i e s ( f o r m f n and m , nfO), which a r e o f i n t e r e s t i n - o ~ t i c a l mode1 c a l c u l a t i o n s [ 2 1 . Again, we include gmpnh(E) and g (-E) i n one curve. The r e s u l t s a r e defined i n the range
-
c E < +m, -ana t h e 1$6@ d e n s i t i e s have again t h e i r minimum a t t h e i r r e s p e c t i v e matching p o i n t , which l i e s a t ( m - n ) . ~ F'4
-
CONCLUSION -We have derived a general i t e r a t i o n procedure t o c a l c u l a t e m-particle n-hole level densi t i e s of any s i n g l e p a r t i c l e Hamil t o n i a n , i n c l uding the Pauli p r i n c i p l e . We have used t h i s approach t o derive a n a l y t i c a l expressions f o r t h e semiclassi cal level den- s i t i e s u p t o 3 - p a r t i c l e s 3-holes o f t h e i s o t r o p i c Harmonic O s c i l l a t o r , i n t h e Thomas -Fermi approximation.
JOURNAL DE PHYSIQUE
Fig.6
- Level d e n s i t y gmpnh f o r m # n and m, n # o. a ) gZplh(E)+
glpzh(-E) w i t h minimum a t E = EF b, ~ , p i h ( ~ ) + g l p 3 h (-E) w i t h minimum a t E = 2~ F
below an e x c i t a t i o n energy o f 4 Ruo, i n a s e m i c l a s s i c a l c a l c u l a t i o n .
F i n a l l y we have seen t h a t t h e a v a i l a b l e phase space a t t h e Fermi s u r f a c e i s d e p l e t e d f o r p a r t i c l e - h o l e e x c i t a t i o n s w i t h m # n ( e x c e p t f o r t h e s i n g l e p a r t i c l e ) .
We thank Mme C. P e t i t v e r y much f o r t y p i n g t h e m a n u s c r i p t .
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