Sound wave propagation in a two-component fermi liquid

Volume 138B, number 1,2,3

PHYSICS LETTERS

12 April 1984

SOUND WAVE PROPAGATION IN A TWO-COMPONENT FERMI LIQUID H. KURASAWA 1 and T. YUKAWA

KEK, Oho-machi, Tsukuba-gun , Ibaraki 305, Japan Received 15 July 1983

We investigate the stability of the sound wave propagation in a two-component (protons and neutrons) Fermi liquid based on the time-dependent Hartree-Fock equation in configuration space. The two types of sound modes appear, which are known as plasmons and phonons. Because of the Coulomb repulsion between protons the plasma mode tends to be stabler than the phonon mode. The isospin mixing caused by the Coulomb interaction cannot be neglected for the excitation modes with wavelength of the order of nuclear size.

In a previous paper [ 1] we investigated the Landau damping in degenerate Fermi liquids on the basis of the time-dependent H a r t r e e - F o c k (TDHF)equation* 1. In contrast to the Landau theory of Fermi liquids, which is considered to be the classical limit of TDHF, we have found a well-defined damped oscillation when the Landau parameter F 0 is positive and smaller than a critical value. For the weak attractive case, however, the Landau damping is so strong that the collective excitation mode is resolved completely to single particle excitation modes, i.e. the zero sound mode does not propagate. We often encounter the negative F 0 for Skyrme type interactions [2]. The overdamping of collective modes will be removed by considering the isospin degree of freedom which introduces the Coulomb repulsive interaction between protons as well as an extra Landau parameter Fd. It produces corrections to the quasi-particle interaction in such a way that the sound wave propagates even if the Landau parameter F 0 is negative. The purpose of this letter is to show how the isospin dependent interaction affects the propagation of sounds. 1 Present address: Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan. ,1 Although the equation which we have employed is the same as the TDHF, the idempotency condition is not imposed. In this sense it may be more adequate to call it the quantum Landau equation. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

The equation of motion is obtained in the same manner as the previous paper [1]. Making the Wigner transformation of the TDHF equation, we obtain

~nrr,/at + (1/ih)exp t2 r! "i~t'~(1)V(2) _ V(1)V(2)~I "k--r p p r ~J X (n(1)e (2) - e(1)n(2))r r, = 0 ,

(1)

where nrr,(r,p, t) and err,(r , p, t) are the Wigner transform of the density matrix and the Hartree-Fock hamiltonian, respectively, and r denotes either proton (p) or neutron (n). By linearizing eq. (1) in terms of the fluctuation 6nrr,(r , p, t) around a stationary and homogeneous distribution n(rO)(p) as

nrr,(r , p, t) = 6rr,n(0)(p ) + 6 n r r , ( r , p, t ) ,

(2)

accordingly

err'(r, e, t) = 8rr, e(0)(p) + ~err,(r, p, t ) ,

(3)

the final equation is expressed in terms o f the Fourier transforms 6ffrr,(k , p, co) and 8"grr,(k, p, co) by +

8ffrr'

=

k/2) - n

0)(p _

k/2)

e(rO)(p + h k / 2 ) - e(rO)(p - ttk/2) - ~co

~ err'

"

(4)

We assume that the effective nuclear force is parametrized by two p a r a m e t e r s f0 and -frO,which are related to the Landau parameters through 1%') ==N(0)f0(') with N(0) being the density of states at the Fermi sur13

Volume 138B, number 1,2,3

PHYSICS LETTERS

p(0)(0)[1 +Cr(r/R)2l ,

face. On this assumption eq. (4) leads to ,~(0) rs~

8Prr' = ~'rr' t~rr ' OCO- f ; ) (~Ppp + 8P'nn) + 2f~ 6p"~r, + 8rr,6rpUC6"~pp I ,

(5)

where =

d3p J(2rrh)

Cp : 0 . 2 0 3 ,

e~°)(p + hk/2)

e~°)~- tikl2) ho~'

(6)

-

-

with the density fluctuation 6~rr,(k, w ) = f

d3p 8"nrr,(k,p, 6o) (27rfi)3

(7)

and the Coulomb force Pc(k) = 4rre2/k 2 .

(8)

While the Coulomb interaction violates the isospin invariance, T 3 is a good quantum number. Eq. (5) therefore splits into two decoupled equations; one for the AT 3 = 0"(r = r') modes and the other for the AT 3 = -+1 (r 4= r') Charge exchange modes. In this paper we consider only the former case. For r = r' eq. (5) is written in the matrix form A \~.~n ] = 0 ,

(9)

A=

_io)x(O)

where 6~"r = 8P'rrand

e(O)(p) = p Z / 2 m + U (U: const.),

(11)

the matrix A is written as A =

1 + ( F 0 +F~ + 2 V c ) M / 4 (F0 - F~)M/4

(Fo-Fo)M/4

) (lO)

Xr"(0) -- ~Crr" (0). N o n - t r i v i a l solutions

1

1 +(F 0 + F~)M/4(l'2)

where [ 1] l 1 ~ y/2 M(y,s)= 1 + 1 -(s-yi2)21nS 2y s + 1 -y/2

1 - ( s + y / 2 ) 2 In s - - 1 + y / 2 2y s + 1 +y/2 '

of eq. (9) exist when detA = 0, which determines the eigenfrequencies o f collective modes. Before studying the dispersion of sound, let us make a comment concerning the Coulomb effects on the ground state density o(rO)(r). So far we have assumed the homogeneity o f the stationary distribution. Since the Coulomb repulsion tends to push the protons outwards, the validity of the homogeneity should be checked. We estimate the density p(0)(r) by means of the T h o m a s - F e r m i approximation and find that the central density p(r0)(0) is appreciably influenced by the Coulomb force. This, however, has little effect on the r-dependence o f p(O), which is approximately given by 14

Cn : 0 . 0 4 2 ,

n(0)~) = 20 [e F - e~0)(p)] ,

(fO - fo)X (0)

1-Cfo +fo)x °)

p(n0)(0) = 0.097 fm -3 ,

where we have used R = 7.1 fm, the Fermi energy = - 8 MeV and the Coulomb potential energy at the center = 24.9 MeV estimated by a uniformly charged sphere with Z = 82 and N = 126. We diagonalize eq. (9) and solve the dispersion equation. For simplicity, we consider a nucleus with N = Z. The AT 3 = 0 modes will not be essentially modified by the neutron excess. Assuming

with (1 - ( f o + f ; + oc)X(pO)

forr
with R being the radius of a nucleus. For example, using the Skyrme I interaction with no surface terms we obtain p(p°)(0) = 0.067 fm -3 ,

(

n(rO)~ + t&/2) -- n(rO)(p -- hk/2)

X

12 April 1984

(13)

with the dimensionless variables y = lik/PF ,

s = mw/(Pvk ) .

(14)

The Coulomb term VC is defined by VC = -~o c N ( 0 ) ~ 0.0325/y 2 ( f o r P F / h = 1.36 f m - 1 ) .

(15)

The matrix A can be diagonalized by the transformation U A U -1 where (cosO U=\-sinO

sinO) cosO

(16) '

with tan20 -- F 0 - F ; / V c .

(17)

Volume 138B, number 1,2,3

PHYSICS LETTERS

Eq. (9) therefore turns out to be

0

I+F_M/2

~a~_

=0,

(18)

where F+_ = ~1 (F 0 + F~) + VC + - [(F 0 - F ; ) 2 + V2) 1/2] , (19) 6~_) =

Vt~pn ).

(20)

The eigenfrequencies of the normal modes are determined by the dispersion equation

1 +F±M(y,s)/2 = 0 .

(21)

In order to find the solutions o f e q . (21) we need to make analytic continuation into the lower half of the complex s-plane [3,4]. Since the analytic structure of eq. (21) is the same as that o f the dispersion equation studied in ref. [1], there are four possible types o f solutions for y < 2, i.e. (i) F+ > F a ,

w: real (undamped oscillation),

(ii) 0 < F_+ < F a ,

The solutions corresponding to a well-defined collective excitation of the system (propagating sound) are for the (i) and (ii) cases. Classifying the normal modes in terms o f whether they propagate or not, the F~ - F 0 plane is divided into three regions for a given y, i.e. the regions I, II and III in which/7+_ > 0, F < 0 < F+ and F± < 0, respectively. These three regions in the F~ - F 0 plane are shown in fig. 1 for y = 0.2. The solid lines, which correspond to/7_+ = 0, are given by [see eq. (19)]

(F0 + VC/2)(F~)+ VC/2 ) = (Vc/2) 2

.

(24)

The region I in which there are two propagating modes extends into the negative F 0 or F~ region due to the Coulomb force. In the region II the normal mode labeled by - does not propagate. It is also seen that the Coulomb force reduces the area o f the region III in which there are no propagating modes. From eq. (17) the isospin character of the normal modes depends on the ratio between the Coulomb and p r o t o n - n e u t r o n force. For FF0 - F~l>> VC (y large), the mixing angle 0 approaches +7r/4 so that the

co: complex (damped oscillation),

(iii)Fb
co: pure imaginary with

Y=0.2

negative sign (strong damping), (iv) F+_ < Fb,

12 April 1984

I

co: pure imaginary with positive

sign (unstable growing) ,

! 2-~ I

(22)

where F a and F b are defined by Fa--

-

2

hm

s ~ l+y/2

M(y, s)

-2/[(l+y/2)1og(l+2/y)

1] -'1

-vc/z

2

F b = -tim

t--,0 M(y, it) -I

=2/(1-y2/41og 11-y/21 - ~

-

1

)

.

(23)

Y The damping results from the overlap o f the spectra of the collective mode and the single particle modes. In such a case the collective mode can decay into particle-hole pairs. The mechanism of damping is known as Landau damping [4]. For the case o f (iii), the collective mode is completely resolved into single particle modes.

-2Fig. 1. Coulomb effects on sound propagations. The solid lines are given by eq. (24). In the region I (F_+ > 0) there exist two propagating sounds, and in the region II (F_ < 0 < /7+) only the mode labeled by + propagates. There are no propagating modes in the region III (F+ < 0).

15

Volume 138B, n u m b e r 1,2,3

PHYSICS LETTERS

12 April 1984

density fluctuations (20) are approximately given b y

a~+ ~ ( a ~ n-+8~p)/,,/2,

for%>%

(6~p ~- 67n)/X/2,

,

:E

(25) 70 = -0.2

for F 0 < F ~ .

g:

In this case the isospin symmetry is retained and the normal modes are classified as the isoscalar and isovector modes. For the opposite limit Vc >> I F0 - F~I (y small), we obtain

6~+ ~ 6~p +0~i~n,

6p_ ~ 6~ n - 0a~p .

3O

20

(26)

ss e

The isospin symmetry breaks down and the normal modes correspond either to the proton density oscillation or to the neutron one. In this limit the dispersion relations are found to be

co+ ~ ( P F k l m ) l F + (1

+

10

co_ ~ (PFk/m)s_(y = 0 ) , Cdpl

(28)

is the proton plasma frequency defined by

6°pX = [(47re 2 1 ; 0 / m ) ] 1/2

'

(29)

P0 = (2/3rr2) (Pv/h) 3 , and s _ ( y = 0) is the solution o f eq. (21) with F _ ( v =0) = ( F 0 + F(~)/2. The frequency co+ is real independently o f F 0 and F(~, while co_ is not necessarily real. The dispersion relation (27) is the well-known relation for the plasmon except the F 0 + F~ term [4]. From eq. (28) the mode corresponding to the neutron density oscillation satisfies the disperion relation o f the zero sound in an uncharged Fermi liquid. In a finite nucleus the wave number k o f the collective modes is o f the order o f 1/R, R being the radius o f the nucleus. The value o f VC defined by eq. (15) is O(1) (for hk = 0.2PF , VC = 0.8125), namely VC is comparable to F 0 - F~ ( F 0 - F~) ~ - 0 . 5 - 1 . 0 for various Skyrme type interactions [2]). This implies that the isospin mixing o f AT 3 = 0 modes may be appreciably large. The numerical example of the dispersion relations is shown in fig. 2 for F 0 = - 0 . 2 and F~ = 0.8. In this case co+ changes from real to complex at the critical value y ~ 0.215, while co_ becomes complex at y 9.4 × 10 -3 and turns to pure imaginary with negative sign at y ~ 0.247. For small y the dispersion relations 16

-I-

~

. e''"

9IF+)

(27) COpl {1 + 3 [1 + s ( F 0 + F~)](PFk/rncopl)2} '

where

Fo': 0.8

i

0

0.1

0.2

0.3

Y Fig. 2. Dispersion relations of the normal modes for F o = -0.2 and F~ = 0.8 with P F ~ = 1.36 fm -1 . The dashed line corresponds to the sound dispersion with co = (PF/m)k. are well-described by eqs. (27) and (28) where the plasma frequency (29) is estimated to be /icopl

~

8 MeV

for

pF/h=

1.36 f m - 1 .

Since the Coulomb force Vc becomes weaker a s y increases, the collective excitations fall into the continuum o f p a r t i c l e - h o l e excitations, i.e. they begin to decay into p a r t i c l e - h o l e pairs when y is greater than the critical value ~ 0 . 2 1 5 for the + mode and ~ 9 . 4 × 10 .3 for the - mode. We note that in spite o f the negative F 0 the - mode is a well-defined excitation mode up to y ~ 0.247, which without the Coulomb force does not propagate for a n y y . Finally we examine strength functions. By the standard method o f the linear response [4], the density fluctuations induced by the external fields Up(k, co) and Un(k , co) acting on protons and neutrons, respectively, are given b y 1 • 6 p p = (COS20X+ + s i n 2 O x _ ) U p + 5 s i n 20(X+ - X _ ) U n ,

6~ n = (sin2Ox+ +cos20x_)u n + i sin 200(+ - ×_)Up, (30) where X_+ are the response functions associated with

12 April 1984

PHYSICS LETTERS

Volume 138B, number 1,2,3

For the long wavelength limit (V c >> IF0 - F~ l) eq. (32) reduces to

0.5

Xp = X. + O(02),

Y = 0.15

Xn = X- + 0 ( 0 2 ) ,

(33)

which are expected from eq. (26). The strength function Sa(k, ~ ) is given by

Sc~(k , w) = -Tr -1 Imx~(k, w + ie) 0

c~ = p , n , v, s, and + .

7'I '

0.S

Y=0.3

0

/

0.6

(e-+ +0)

1.0

$

Fig. 3. Strength functions So~(k,~) f o r y = 0.15 and 0.3 w i t h F o = 0.2 and F~ = 0.8. Scz is p l o t t e d in the unit o f N ( 0 ) = 2mpF/Or~fi3). The arrow (s ~ 1.17) in the upper figure indicates the position of the real solution w+ corresponding to

the undamped mode. Below s = 0.6, Sc~decreases monotonously to zero as s ~ 0.

The numerical examples of Sa as a function of co are shown in fig. 3 f o r y = 0.15 and 0.3 with the Landau parameters F 0 = 0.2 and F~ = 0,8. In the case o f y = 0.15 the approximation (33) is valid. The smallness of Sp (S+) comparing with S n ( S ) is due to the fact that the undamped plasmon located at s ~ 1.17 exhausts a major part of the strength. S n and S_ exhibit a pronounced peak, which represents the Landau damping mode. F o r y = 0.3, co+ as well as w _ becomes complex, which appears as the sharp peak in S+ as s ~ 1.1, whereas w gives the broad peak in S_ at s ~ 0.8. The large mixing of S+ and S_ in Sp and S n is clearly seen in the figure. One of us (H.K.) thanks the Japan Society for the Promotion of Science for financial support.

the normal modes;

References x+(k, co) - _ N ( O ) M(y,s) 4 1 +F+MCv, s)/2 "

(31)

From eq. (30) we obtain the response functions ×p, Xn, Xs and Xv for the proton, neutron, isoscalar and isovector density operators, respectively, Xp = cos20 X+ + sin20 X_ , Xn = sin20 X+ + cos20 X- ,

(32)

Xs = (1 + sin 20)X+ + (1 - sin 2 0 ) X _ , Xv = (1 - sin 20)X+ + (1 + s i n

20)X_

.

[ 1] T. Yukawa and H. Kurasawa, preprint KEK-TH60 (1983). [2] S.-O. BEckman, A.D. Jackson and J. Speth, Phys. Lett. 56B (1975) 209; H. Krivine, J. Treiner and O. Bohigas, Nucl. Phys. A336 (1980) 155; Nguyen Van Giai and H. Sagawa, Phys. Lett. 106B (1981) 379. [3] L. Landau, Zh. Eksp. Teor. Fiz. 7 (1946) 574. [4] D. Pines and P. Nozi~res, The theory of quantum liquids, Vol. 1 (Benjamin, New York, 1966); E.M. Lifshitz and L.P. Pitaevskii, Physical kinetics, Vol. 10 of Course of Theoretical physics (Pergamon, Oxford, 1981).

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