Momentum distribution of nuclei in a one-dimensional nuclear model

Volume 236, number 2

MOMENTUM

PHYSICS LETTERS B

15 February 1990

D I S T R I B U T I O N OF N U C L E I IN A O N E - D I M E N S I O N A L N U C L E A R M O D E L

C. A L E X A N D R O U Paul Scherrer Institute (PSI), CH-5232 Villigen, Switzerland Received 14 November 1989

The momentum distributions of finite nuclei are calculated using Monte Carlo techniques for a one-dimensional model having a two-body potential with a repulsive core and a long-range attraction. The results obtained in mean-field using a density-dependent effective interaction are compared to the exact solution.

1. Introduction

2

V(x1-x2)= ~

v,

i=1~ exp[--(xi--x2)2/0- 2]

Monte Carlo techniques have p r o v i d e d a powerful tool for investigating systems with m a n y degrees o f freedom. Application o f such techniques to the nuclear m a n y - b o d y system enables us to calculate nuclear properties exactly and therefore gives us the possibility o f testing the validity o f the various approximations applied to such systems. Addressing the full nuclear m a n y - b o d y system is, for the moment, not possible due to the problems of treating fermions in more than one spatial d i m e n s i o n with stochastic techniques. However, meaningful tests o f certain approximations used to treat the nuclear system can still be carried out in a p p r o p r i a t e l y constructed models. In a previous work [ 1 ] we studied the ground state energies and densities of finite nuclei and nuclear matter in a simplified many-fermion model having static two-body interactions with strong short-range repulsion and long-range attraction. Since we use Monte Carlo techniques to evaluate the ground-state properties o f many fermion systems, we consider a one-dimensional model with parameters adjusted to resemble as closely as possible the three-dimensional nuclear system. We choose to work with dimensionless quantities. For a saturating system the most physical length scale is given by the saturation density Po and therefore all lengths will be given in units of lo= 1/Po. Energies are measured in units of Eo = h2/ml 2. F o r the two-body potential we take a sum of a short- and a long-range gaussian,

(1.1)

with: V~= - V2= 12, a~=0.2, a 2 = 0 . 8 and where we took m = 1. The saturation density for this model was found in ref. [ 1 ] to be Po= 1.175 + 0.005. There, it was also shown, that an effective mean field with a density-dependent interaction yields a good description for the ground-state energies and densities of finite nuclei. The effective density-dependent two-body interaction considered is given by V~f~(x,,

x2) = {1 + o~[p(x~ ) +p(x2) 1}

× _ v~7 e x p [ - (xl --X2)2/0 "2 ] O'1 N / ~

+

exp[-(x,-x2)2/a~l

,

(1.2)

0"2

where only the short-range part o f Vefr is adjusted to take into account short-range correlations so that ~ r r approaches the bare V at long distances. The three parameters of the short-range effective interaction obtained are: V~ =0.63, 61 = 0 . 3 2 and 0"=5.4 (for a more detailed description o f the model see ref. [ 1 ] ). The purpose of this work is to investigate the mom e n t u m distribution o f finite nuclei within the same model. This quantity is of i m p o r t a n c e in the study o f strongly interacting q u a n t u m fluids and a lot o f work has been done in developing techniques to calculate it reliably [2]. Therefore it is interesting to examine within our model how well mean-field compares to

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125

Volume 236, number 2

PHYSICSLETTERSB

the exact (stochastic) result. We propose to use the initial-value random walk [3] in order to evaluate the exact momentum distribution. This method has proved much more efficient for the calculation of the ground-state energies [ 1 ] than the trace evaluation of the partition function, since a physically motivated wavefunction can be used to optimise the randora walk. However, evaluation of observables other than the energy within this approach needs additional consideration. In this work we show how to apply this method to directly calculate both the momentum distributions and the densities of finite nuclei. Since the initial-value random walk is best formulated in coordinate space, the momentum distribution is calculated by taking the Fourier transform of the one-body density matrix [2]. A similar procedure is adopted for the evaluation of the momentum distribution in mean-field by taking the Fourier transform of the one-body density matrix constructed using the self-consistent single-particle wavefunctions. For the mean-field results, center-ofmass corrections must be taken into account before comparison with our Monte Carlo results can be made. This is done by constructing from the Slater determinant a translationally invariant wavefunction. The outline of this letter is as follows: In section 2 we describe our method for calculating the momentum distribution. Section 3 contains our results and conclusions.

2. Calculation

The ground-state wavefunction can be obtained from a trial function 1¢) by applying the propagator in euclidean time in the limit of very low temperature, lim e x p [ - / / ( H - E ) ]

n(p)=

i ~dye x p ( i p y ) D O ' ) ,

,

(2.1)

(2.2)

--oo

where D(y)=

J Ch',...dxA~((Xl + . . . + X A ) / A ) - oo

x ¢ " ( x , ..... x4)

X~(x, + y - y / A ,

x2-y/A, ...,xA--y/A) ,

(2.3)

and the normalization of the wavefunctions is chosen so that D ( 0 ) = 1. Since our hamiltonian is translationally invariant the ground-state wavefunction generated by the random walk is the intrinsic wavefunction and (2.3) gives the intrinsic D ( y ) . The operator, Ix) ( x + y l , which leads to (2.3) is a non-local operator and therefore it cannot be calculated by staying only in one ordered subspace of coordinate configurations. Some shifts ofx~ to x~ + y will take us out of the ordered subspace. If an odd number of permutations is required to bring us back to the ordered subspace, a negative sign is obtained. D ( y ) is then given by D(y)= N/~

where ¢(x) is positive and non-orthogonal to the ground-state wavefunction bq/) and Eo is the groundstate energy. Eq. (2.1) defines an initial-value problem which can be evaluated stochastically as we now proceed to explain only briefly here. A random walk is generated by breaking e x p ( - fill) into a product of infinitesimal steps exp ( - ~H) and inserting in between these steps a complete set of coordinate states. 126

An initial ensemble of coordinates {x} is generated with probability 0(x) which is then evolved with successive application of e x p ( - ~ H ) . The final ensemble {x} obtained in the large-//limit is distributed according to the ground-state wavefunction. For a many-fermion system in one spatial dimension the minus signs arising from antisymmetry are avoided by working in an ordered subspace x~
1

1¢>

= I~u) (~'lO) e x p [ - / / ( E o - E ) ]

15 February 1990

(--1)P~I*(XI''"'XA--I)

X ~ , / ( Y l , ..., X i " ~ - y , . . . .

XA_ ~) ,

( 2.4 )

where P is the permutation required to bring us into the ordered subspace. I q/) being the intrinsic wavefunction is unaffected by an overall shift o f y / A and depends explicitly only on A - 1 coordinates. N is a trivial normalization factor. In the actual numerical calculation ~u(x~, ..., xA-l) and {u(xl, ..., x,+y, ..., XA_j) are computed in the ordered subspace as (A-1)-dimensional histograms. This limits the

Volume 236, number 2

PHYSICS LETTERS B

m a x i m u m n u m b e r of fermions that can be dealt with. In the present work we have considered systems of up to five fermions. In mean-field D(y) is calculated in the standard way by finding the self-consistent single-particle wavefunctions 0~ (x) of the Hartree-Fock equation using the effective interaction specified in eq. (1.2). The density matrix is then given by

DHv(Y)= ~ f dxO*(x)O,~(x+y).

(2.5)

ot=l

In order to compare the mean-field results with the Monte Carlo results one needs to correct for the center of mass. A possible way to get a translationally invariant wavefunction from the single-particle determinantal wavefunction is to evaluate ~HF(~ 1..... AS)=

dR6(R)~HF(Xl ..... )CA),

(2.6)

where ~HF(X, .... , .tS) is the Hartree-Fock self-consistent solution and R = (X1+...+XA/A) is the center-of-mass coordinate. For a wavefunction which has a separable center of mass, this prescription gives the exact intrinsic wavefunction. The integral in eq. (2.6) can be evaluated stochastically. This is done by sampling an ensemble {x} according to ~HF(X,..... XA), which in an ordered subspace is positive definite, and then shifting all coordinates to the center of mass. The density matrix as defined in (2.3) can then be evaluated and compared to the Monte Carlo result. Since in this approach we can at the same time calculate the exact densities we also show the Monte Carlo results for the densities together with the mean-field results corrected for the center of mass by evaluating

15 February 1990

grams of the density and the wavefunction we use a bin size of0.15 and we taken the time step size e = 0.01 which is sufficiently small also for our model. In fig. 1 we show our Monte Carlo results in comparison with the exact results. The one-body density, shown in fig. la, is in excellent agreement with the exact resuit. The complete agreement between the exact and Monte Carlo results for D ( y ) shown in fig. lb indicates that finite bin effects are negligible. The mom e n t u m distribution obtained from the Fourier transform of D(y) is shown in fig. lc. The Monte Carlo and exact results are in agreement within the statistical errors. This gives us confidence that our error estimators provide a good measure of the statistical fluctuations. For this calculation a total of ~ 10 5 independent events were used. In going from the testcase to the problem of interest we keep e the same and we take a bin size of 0.15,

~

1.50

=

f dR

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

1.00 0.75 0.50 0.25 0 . 0 0

-

0.5

0

1.5 x

1

2

-

-

2.5

3

. . . .

q

(a)

0.5

~"

Z

o.o -0.5

F . . . .

I

0

2

. . . . . .

[

4

6

5

Y

(b) 0.25

p~F(X)

,

,

,

,

,

,

,

,

,

~

,

,

,

~

,

,

,

~

;

,

0.20

¢Lxl...dXA6(R)

0.15

-~

A

× ~ 6(x-xi) lqJnv(xl ..... XA)[ 2 •

(2.7)

i=1

o.lo 0.05 0.00

0

1

2

3

4

5

P

(c) 3. Results and discussion In order to assess the accuracy of our procedure we start by discussing fermions with harmonic two-body interactions where we know the analytic results. We consider a system of three fermions. For the histo-

Fig. I. A three-fermion system with harmonic oscillator interactions. All lengths are given in units of the oscillator parameter. (a) The one-body density p(x). The Monte Carlo results are denoted by diamonds. Error bars are only shown if they are bigger than the symbols.The solid line shows the exact result. (b) The density matrix D(y). The notation is the same as for (a). (c) The momentum distribution n(p). The notation is the same as for (a). 127

Volume 236, number 2

_- . . . .

PHYSICS LETTERS B

i

. . . .

I

. . . .

I

15 February 1990

. . . .

1.0

1.01"5 ~

.

3

particles

0.0

--

1 ....

1 ....

--

1 ....

-

3 particles

0.5

~,'..

0.5

....

0.0

""'~-5"

~>' ° "

~

~

^

¢

-0.5 1.0

~..

1.5

/~.

4 particles

1.0

4 particles

0.5

0.0 0.5

--

0,0

--

1.5

--

~"o."~'~"o ~ - - l . . ~ ' ~ > . ~ . ~

O

-0.5 1.0

1.0 " . . . . . .

5

"

particles

0

particles

2 x

-0.5

3

Fig. 2. The one-body density p(x) for the nuclear model for three, four and give particles, p(x) is in units of l/lo and x in units of lo. The diamonds are the Monte Carlo results, the dotted line is the Hartree-Fock density and the solid line is the Hartree-Fock density corrected for center of mass effects (eq. (2.7) ). 0 . 1 7 5 a n d 0 . 2 6 7 for t h e t h r e e - , f o u r - a n d f i v e - f e r m i o n s y s t e m s t h a t we l o o k e d at, r e s p e c t i v e l y . T h e r e s u l t s are a g a i n c a l c u l a t e d w i t h ~ 105 i n d e p e n d e n t e v e n t s . In fig. 2 we s h o w t h e o n e - b o d y d e n s i t i e s c a l c u l a t e d w i t h t h e i n i t i a l - v a l u e r a n d o m w a l k a n d in m e a n - f e l d with and without center-of-mass corrections. The m e a n - f i e l d s o l u t i o n c o r r e c t e d for t h e c e n t e r o f m a s s p r o v i d e s a g o o d d e s c r i p t i o n for t h e densities. T h e o n l y d i s c r e p a n c y is a n o v e r e s t i m a t i o n o f t h e q u a n t u m d e n s i t y f l u c t u a t i o n s w h i c h is e x p e c t e d s i n c e e x c i t a t i o n s to h i g h e r o r b i t s a r e n o t i n c l u d e d • T h e s e r e s u l t s are in a g r e e m e n t w i t h t h e r e s u l t s f o u n d in ref. [ 1 ] where the exact densities were obtained mainly by evaluating the trace of the propagator using the M e t r o p o l i s a l g o r i t h m . In fig. 3 we s h o w t h e M o n t e C a r l o r e s u l t s for D(y), t h e m e a n - f i e l d r e s u l t DHFO') w i t h n o c e n t e r - o f - m a s s c o r r e c t i o n s a n d D'~v(y) w i t h

,,,l

....

2

I

....

4 Y

l

....

6

Fig. 3. The one-body density matrix DO') for three, four and five particles for the nuclear model. The notation is the same as for fig. 2.

....

I ....

I ....

I .....

0.20 0.15 0.10 0.05 0.00 0.20 4 particles

0.15 ~"

0.10

0.05 0.00

0.2:0 0.15 0.10

0.05

Fig. 4. The momentum distribution n(p) for three, four and five ]~ particles for the nuclear model. The notation is the same as for fig. 2. 128

_

0.0

"'~>" ,.o, ~ ,o I 1

5

0.5

0""

0.5 0.0

~

0.00

2

4 P

6

Volume 236, number 2

°'5 k . . . . 0.4

0.3

PHYSICS LETTERS B

I ....

In this work we have presented a method to efficiently calculate both the m o m e n t u m distributions and densities of nuclei in one spatial d i m e n s i o n using the initial-value r a n d o m walk. In addition to the ground-state energies and densities, our results indicate that the mean-field solution yields, as the n u m ber of particles increases, a good description of the m o m e n t u m distribution of finite nuclei.

I . . . . . . . .

l~ ~ ...~.. ~

...................... . . .

0.2

15 February 1990

Acknowledgement . . . .

0.0

,

~

P Fig. 5. The Fermi gas momentum distribution (dotted line) in comparison with the Monte Carlo momentum distribution for three, four and five particles for the nuclear model denoted with the dott-dashed, solid, and dashed lines, respectively.The Fermi momentum ~Po.

kF=

center-of-mass corrections taken into account according to eq. (2.6). It can be seen that center-of-mass corrections enhance the fluctuations at large distances and, as the n u m b e r of particles increases, the mean-field solution approaches the exact result. The same behaviour is also reflected in the m o m e n t u m distributions shown in fig. 4. In fig. 5 the Monte Carlo results for the m o m e n t u m distribution are collected and shown in comparison with a non-interacting Fermi gas at nuclear matter density. The m o m e n t u m distribution for the finite systems extends beyond the Fermi m o m e n t u m showing the e n h a n c e m e n t of the large-momentum components due to the presence of a strong repulsive core in the two-body interaction.

I would like to thank J.W. Negele for useful discussions and for providing me with some of his computer codes and M. Locher and R. Rosenfelder for critically reading the manuscript. My thanks are also due to Ph. Gaisford for making the computer facilities at PSI optimal for this calculation.

References [ 1] C. Alexandrou, J. Myczkowskyand J.W. Negele, Phys. Rev. C39 (1989) 1076. [2 ] S. Fantoni, Nuovo Cimento A 44 ( 1978 ) 191; E. Manousakis,V.R. Pandharipande and Q.N. Usmani, Phys. Rev. B 31 (1985) 7022; D.S. Lewart and V.R. Pandharipande,Phys. Rev. B 37 ( 1988) 4950; M. Lissia and J.W. Negele, Phys. Rev. D 39 (1989) 1413. [3]J.W. Negele, Proc. Intern. Symp. on Time-dependent Hartree-Fock and beyond (Bad Honnef, June 1982); J. Stat. Phys. 43 (1986) 991; J.W. Negeleand H. Orland, Quantum many-particlesystems (Addison-Wesley,Reading, MA, 1987) Ch. 8.

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