Macroscopic response of the nuclear surface

Nuclear Physics A449 (1986) 446-458 @ North-Holland Publishing Company

MACROSCOPIC

RESPONSE

OF THE NUCLEAR SURFACE

V.I. ABROSIMOF l7re Niels Bohr lnstitufe, University of Copenhagen, DK-2100 Copenhagen 0, Denmark J. RANDRUP** NORDITA,

Blegdumsvej 17, DK-2100 Copenhagen 0, Denmark

Received 14 May 1985 (Revised 27 August 1985) Abstract:

The macroscopic surface response of a semi-infinite Fermi liquid is considered. On the basis of Landau’s kinetic equation, a dispersion relation is derived for the free surface modes. In the

limitof weakquasiparticleinteractionwe find a surfacemode witha purely imaginaryfrequency. This mode is shown to be critically damped if the stiffness is given by the excess surface energy and the friction by the one-body wall formula. By assuming a correspondence with the quantum relation between dissipation rate and response function, the macroscopic surface response function is obtained. The results are applied to inelastic scattering of protons from nuclei. While the qualitative behavior of the nuclear response is reproduced by the semi-infinite model, the decrease with energy is too rapid.

1. Introduction The nuclear surface region is probed in a variety of experiments, particularly inelastic scattering of hadrons to forward angles. Such reactions are of interest because the associated low-momentum transfer explores the collective response of nuclei to external fields. However, the structured collective cross section is obscured by a smooth background which we would like to understand on the basis of macroscopic nuclear properties. For this purpose it is advantageous to consider the surface response of a semiinfinite system. While such a system is free of the many complications arising from the finiteness of real nuclei, it does have the most basic property associated with finiteness, namely a surface breaking the translational invariance characteristic of infinite nuclear matter. The surface response of a semi-infinite nuclear system has been investigated in the random-phase approximation I,*). This is computationally a fairly demanding This work was supported by the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High Energy and Nuctear Physics of the US Department of Energy under contract no. DE-AC03-76SF00098. * Permanent address: Institute for Nuclear Research, Prospect Nauki 119, 252028 Kiev, USSR. ** Permanent address: Nuclear Science Division, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA. 446

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V.I. Abrosimov, J. Randrup / Macroscopic response

approach. Our present interest is in relatively small momentum transfers. For example, for inelastic scattering of 800 MeV protons to an angle of 5”, the momentum transfer is about half the Fermi momentum ‘). For such low momentum transfers we may employ the Landau semiclassical theory of Fermi liquids “). Our study will then proceed as follows. In sect. 2 we consider the free surface modes in semi-infinite nuclear matter described by Landau’s kinetic equation and subject to suitable boundary conditions. After deriving a general dispersion relation, we concentrate on the particular mode for which the frequency is purely imaginary. In sect. 3 we construct the equivalent classical harmonic oscillator and derive the macroscopic surface response function for this mode. In sect. 4 we briefly discuss the comparison with data on inelastic proton scattering. Finally, our study is concluded in sect. 5. 2. Free surface modes We start out by considering the free surface oscillations of a semi-infinite Fermi liquid described by the Landau semiclassical theory4). The system has its surface in the x, y plane and occupies the left half-space z < 0. We assume that the quasiparticle phase-space distribution &r (r, p, t) is governed by the linearized Landau kinetic equation for zero temperature ‘):

$n(r,p,t)+ui

Gn(r,p, t)+S(&-&F)

dp’ -F(fi,,‘)bn(r,p’, 4rm*p,

t)

1

=o.

(2.1)

Here the Fermi momentum PF is related to the Fermi kinetic energy &Fby &F= where m* is the effective quasiparticle mass. The quasiparticle velocity is then u = p/ m* = hk/ m*. The quasiparticles have energies near the Fermi surface and they are assumed to scatter elastically with an angular distribution

pi/2m*,

F($,,‘)=F,SF,p^*p^‘,

(2.2)

where F,, and Fi are the usual Landau parameters characterizing the quasiparticle interaction. In an infinite uniform system, eq. (2.1) has plane-wave solutions proportional to exp (ik * r - id) and, since the equation is linear, any solution is a superposition of such waves. In the present study, we consider a semi-infinite system and are interested in solutions which are localized near the surface (i.e. at z = 0). Such surface waves are superpositions of solutions which are proportional to exp (ik, . r, - iwt) where the subscript I denotes a vector with vanishing z-component (i.e. k, = (k, 510) and r, = (x, y, 0)). Thus, we seek solutions of the form &kl(r,p, t)=-8(&-&F)fk,(z,$)

ei(kL’rA-or),

(2.3)

The functionfkA(z, 3) is the displacement of the local Fermi energy for quasiparticles

with momenta in the direction fi, at the depth z. It follows from (2.1) that this

448

displacement

V.I. Abrosimoq J. Randrup / Macroscopic response

satisfies the equation

+(ik,*u,+u,i

>

[~op(z,k,)+~~p*jj(z,kJ]=O.

i=

Here Si z Fih3/4~m*P~+“, particle number density

Furthermore,

0,l.

AZ, k,) = and the quasiparticle

we have introduced

$SC& - EFlfk,(Z,a

3

(2.4) the quasi-

(2.5)

current density j(z, k,) =

I

~Sw3=Nc,(Z,P*)P.

(2.6)

As the generalized macroscopic coordinate describing the system we shall use the local displacement of the surface Z( r,, t), giving the point of steepest density fall-off, for example. For small amplitudes, the surface displacement is given by

I

dk, Z(r,, t) = ,,Zkl(t)

eikA.rl,

F

(2.7)

where zkl( r) = zkL e -‘@’is the time-dependent surface displacement associated with harmonic distortions with wave number k, and frequency o. The solutions we seek should satisfy the following boundary conditions, fk,(z = O,PI, pz) -fk,(z

=

0, PI,

-Pz)

=-2Uk,pz-2pZ~~j;(Z=0,kl)

(2.8)

and P,,( z = 0, k,) = Uk:i?k ,

(2.9)

(z-+--00).

(2.10)

together with the demand that fk,(z,C+O

Here &A = -imzkl is the (maximum) surface velocity associated with the wave number k, and (T is the specific surface energy of the system (- 1 MeV/fm* for nuclei). Furthermore, Pzz(z, k,) =

I

$(s

- EF)$B&,

6) + ~OP(Z, kdl

(2.11)

is the normal component of the stress tensor. The first condition (2.8) expresses the requirement that the quasiparticles be reflected elastically from the moving surface,

V.I. Abrosimov, J. Randrup / Macroscopic response

449

as seen in the frame following the surface. This boundary condition was also employed for calculating the temperature discontinuity of the interface between liquid 3He and a solid “) and, more recently, for deriving the energy dissipation rate in a deforming nucleus [the wall formula ‘)I. The second condition (2.9) expresses the demand that the normal stress must equal the extra pressure generated by the distortion of the surface. This condition is the same as used in refs. ‘*‘). The last condition (2.10) guarantees that the excitation is localized in the surface region. By multiplying the boundary condition (2.8) by p,an,,/ae - -p,6( E - Ed) and subsequently integrating over p, we obtain the relation (2.12)

jZ(z=O,~J=mP,~k,

where p. = g x&pi/ h3 is the equilibrium nucleon density (g is the spin-isospin degeneracy of the single-particle orbitals). This relation expresses the demand that the quasiparticle current density at the surface, j,(z = 0, k,), must equal the current density associated with the moving system; this latter quantity is mpoUk, since the mass density is mp, and the local velocity is UkL.This boundary condition corresponds to the one used in liquid-drop studies of nuclear vibrations “) and in studies of the nuclear giant resonances based on Landau theory ‘). We wish to find solutions to eq. (2.4), satisfying the boundary conditions (2.810). This problem is analogous to the one considered in refs. loS1l). In ref. lo) it was shown that when the system allows transverse zero sound (which occurs when F, > 6) then there exists a surface mode with an approximately linear dispersion relation. Ref. 11) demonstrated that when the surface wave velocity is much smaller than the Fermi velocity then there exists a strongly damped surface mode (i.e. its frequency w is purely imaginary). We consider a similar problem with no restricting assumption about the wave velocity and for systems where no transverse zero sound exists. The problem can be solved by the Fourier transformation method by extending the functions into the positive half-space and appropriately modifying the conditions at z = 0. This method was proposed by Landau I*) and employed in refs. 6*10).Since our problem is very similar to the one considered in ref. lo), we describe the method only briefly here. First, the displacement function is extended to z > 0 by demanding f(z,

PI, Pz) =f(-z,

PI,

-Pz) .

(2.13)

The kinetic equation (2.4) remains valid after this extension, provided that the term 2pZu,Uk,B(z) is added on the left-hand side, as required in order to comply with the boundary condition (2.8). The entire problem can then be Fourier transformed. The resulting kinetic equation can be reduced to a system of linear equations for p(k) and j(k) which can be solved algebraically. The Fourier transform f(fi, k) = can then be obtained.

co dzfk,(z,fi) I -CC

eikzZ

(2.14)

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V.I. Abrosimov, .I. Randy

/ Macroscopic response

By invoking the boundary condition (2.9), a relation between the frequency w and the wave number k, can be obtained. This dispersion relation contains the normal stress P,,(z, k,) whose Fourier transform is found to be

+s2(s2w-i)+$3w-5(s2w-:)l +;

$3(w+ 1)-30s2w+35s2(s2w-~)]. I

(2.15)

In deriving this result, we have assumed that the quasiparticle interaction is isotropic so that F1 vanishes (see (2.2)); the above expression agrees with the result obtained in ref. lo). In (2.15) we have introduced the dimensionless surface wave velocity c = w/ k, vF. Furthermore, s = W/ kv,,

I 1

(2.16)

A(s) = 1 -&w(s).

(2.17)

s+l w(s) = is In S-l

-l+i&e(jsl-1)

and

Having calculated P,,(k), the normal stress can be obtained by the inverse Fourier transformation Pz,(z = 0, k,) =

I

m dk,

_-oo j$‘zzW

.

(2.18)

The integral can be evaluated by contour integration. In general the velocity c entering in (2.15) is complex but in our further studies we shall restrict our considerations to the special case when c is purely imaginary, c = ic2, so that the surface mode is overdamped. It is convenient to introduce the integration variable K = k,/ k which is related to s by K(s)~= c2/s2-1. The singularities in the integrand in (2.18) occur where A vanishes. The root of the equation A(s) = 0 is denoted s0 and the corresponding value of K is denoted K~. The propagation velocity of the longitudinal zero sound in the infinite Fermi liquid is given by sour. If Fo> 0, as is the case in nuclear matter 13,14),the equation A(so) = 0 has a real root which is larger than unity ‘). Then, since the condition (2.10) demands that c2 < so, we find that Ko=fiJl-c*/S~=fiJl+C:/S;:

(2.19)

451

V.I. Abrosimov, J. Randrup J Macroscopic response

03%

f

Fig. 1. The contour

integration

used for calculating

the integral

in (2.18)

which is purely imaginary. Furthermore, the branchings at s = f 1 correspond to the values K(S = +l) =

*i&T=

*im.

(2.20)

We note that /K~I< 1K / since s,, > 1. The pole in the integrand in (2.18) and the integration contour employed are shown in fig. 1. After calculating the residue, we obtain the normal stress P~~(z = 0, k,) = 3 Uk,POPF 1

XF(~+&-~C;-~S;)** 1

I

ds 0 t

3[16(c;+ s*)]-‘/~

0

0

++l+Fo-2c;-3~2)2+

~(1-s2--16(c:+S*))

I).

(2.21)

Insertion of this result into the boundary condition (2.9) yields the dispersion relation. This relation can be solved numerically to yield o as a function of k, for arbitrary positive values of Fo. We specialize to the case of an almost ideal Fermi gas of quasiparticles (F. + 0). In this case the longitudinal zero-sound velocity is given by ‘) so=l+$e-2/F0,

(2.22)

and the dispersion relation becomes

(2.23)

452

V.I. Abrosimov, J. Randrup / Macroscopic response

The integral is elementary to evaluate and, recalling that icz = c = wfk,nF, we arrive at the result

(2.24) The negative sign has been chosen in order to ensure that the amplitude of the wave decreases with time. We have thus found that there exists an overdamped surface mode in the semi-infinite Fermi liquid, when the quasjparticle interaction is weak. As already mentioned, the same result was found in ref. 11>but under more restrictive conditions. The dispersion relation (2.24) will be employed for deriving the macroscopic response function. 3. Response function In the preceding section we found that there exists in the semi-infinite Landau liquid a free mode with a purely imaginary eigenfrequency w = - ice2= --il+k&,pF. We shall simulate this mode by a damped harmonic oscillator of the general form (3.1) Here the surface displacement is given by the macroscopic coordinate .Z,,( t) introduced in sect. 2, see eq. (2.7). The fact that the frequency w is purely imaginary implies that A 2 oO where A = -yk,/2 Bk, is the damping decrement and o,, = C,J BkL is the undamped eigenfrequency. The stiffness Cki against surface wriggles with wave number k, can be estimated by calculating the additional surface energy generated by such distortions. This yields (3.2) The proportionality to k: reflects the fact that more surface area is generated for larger wave numbers, for a fixed value of the amplitude, The above estimate is essentially classical, valid in the limit where the Fermi wave number k, is large in comparison with the wave number k, characterizing the distortion. When k, = kF, as may occur in cases of practical interest, an accurate estimate must take account of the quanta1 nature of the nucleons making up the medium. This is generally expected to reduce the stiffness so that it effectively vanishes when k, exceeds kF, since such rapid distortions are physically unattainable by the system. The stiffness is also expected to depend somewhat on the surface diffuseness and the temperature. The friction coefficient yk, can be estimated within the one-body dissipation theory which is expected to be valid for the type of system considered. For smali wave numbers k, < kF, the wall formula ‘) is expected to apply, (3.3) In this limit the friction coefficient is independent of k,. However, as for the stiffness parameter, when k, = kF the quanta1 nature of the nucleon constituents is expected

VJ. Abrosimov, J. Randy

/ Macroscopic response

453

to reduce the friction substantially. The dependence of rk, on k,, as well as its sensitivity to surface diffuseness and temperature, have been studied quantitatively in ref. Is) on the basis of linear response theory. The above estimates of c& and rk, are also consistent with recent results of the classical limit of the surface response in Fermi liquids 16). The inertial-mass parameter Bk, is still poorly understood. Estimates based on standard fluid dynamics are not expected to be relevant due to the long mean free path of the nucleons. Studies 19) of inertial-mass parameters in long-mean-free-path systems have been made on a classical basis but no simple understanding has yet emerged. Generally, as for ckl and &,, one would expect a substantial reduction of Bk, for k,a kF due to the finite quanta1 size of the nucleons. In view of the rather uncertain behaviour of the characteristic parameters B,, Yk,r ckl for k, = kF, and since their quanta1 suppressions in this regime are qualitatively similar, we base our further considerations on the estimates (3.2) and (3.3) derived in the limit of a small wave number k,. Caution should then be excercised when applying the results for wave numbers comparable to kF’ A general theory for calculating the macroscopic dynamical properties of moderately excited nuclear systems has been developed on the basis of linear response theory I’)_ Although the focus of that work is on finite systems, the theory is sufficiently general to permit application to the semi-infinite situation as well. It would be interesting to compare the surface response function thus obtained with our present result (3.12). It is interesting to note that for the special mode found in sect. 2, and with the stiffness and friction coefficients employed above, we have

Imo=-~2=_2Cki_ 1;.

(3.4) yk, Here &, = Ck,/ BkLis the square of the undamped eigenfrequency and h = y/2B is the damping decrement. The above relation implies that the oscillator (3.1) is critically damped. Then A.= w. so that o2 = wo. It also follows that the inertial-mass parameter is given by B

_

Yzk, _

kA 4CkL

r:.f.

2uk:'

(3.5)

We are interested in processes in which the nuclear surface is probed by an external agency, such as a high energy proton. It is therefore useful to study the motion in the presence of an external field, the form of which should be chosen so as to best simulate the specific excitation mechanism considered. For elucidating the general features of the nuclear surface response, a suitable external field is of the form V,,,( r, t) = V(z) e’@‘. ‘L-%t~X,rf ) (3.6) where V(z) varies only near z = 0 so that the external force acts on the surface region,

454

V.I. Abrosimov, J. Randrup / Macroscopic response

We therefore

subject

the oscillator

&,&+ and seek solutions

(3.1) to a harmonic

external

Y&G, + C$GL =fo cos (o,,,r)

force,

,

(3.7)

of the form

Zk,(t) =zklcm (%xtt+cp). For such solutions

the square

of the amplitude

(3.8)

for the forced

motion

is given by (3.9)

We note that the rate of energy dissipation or (3.3) is 0 = 72’ so that the time-averaged

in the surface-vibration dissipation rate is

(0 surface)t= 3rw.&,&

oscillator

(3.1)

(3.10)

.

In order to obtain the macroscopic response function we recall the fact that ordinarily, in a quanta1 system, the rate of energy dissipation is directly related to (the imaginary part of) the response function 19), (Q q”antal)r = l%Xt~~“antal(%xt)

*

(3.11)

Here the brackets ( ), denote a time average over a vibrational period and R" is the imaginary part of the quanta1 response function. We now assume that a similar relationship holds for the presently considered surface mode and thus obtain for the associated surface response function

fi =

Wext

yw.‘.x (w~,,-W~)2+4A*w~,t 4u2k4, 2

-fo(&+

= Yzv.f.

wext wi)’ .

(3.12)

In the last line we have used the result (3.5) for Bk, and the fact that A = o. for critical damping. The response function (3.12) is shown in fig. 2 as a function of the field frequency k, = 0.5kF and k, = 0.7kF. w ext, for two fixed values of the transverse momentum: The strength of the external field was chosen as f. = 1 MeV/fm3 and the remaining parameters were given their standard nuclear values, namely u = 1 MeV/fm’ and to w,,~, y = 1.18 MeV/fm4 * lo2 s. The response function rises initially in proportion ak:/hy, and ultimately falls off as wi:. The full has a maximum at w,,~ -&,= width of the peak is approximately 1.2h. At k, = 0.5kF the energy associated with the maximum is about 3 MeV, while at k, = 0.7kF it is about 5 MeV. For energies

V.I. Abrosimov, J. Randrup 1 Macroscopic response

455

2.5

5

IO hw

15

20

25

(MeV)

Fig. 2. Surface response functions (3.12) (solid curve) for the semi-infinite Landau liquid in the external field (3.6), for two different values of the momentum transfer Ak,. The strength fO of the external field is equal to 1 M’eV/fm’. The approximate response function (3.13) is also shown (dashed curve) for k, = 0.5k,.

substantially

above the peak energy we may employ

the asymptotic

approximation

(3.13)

This approximation is also indicated in fig. 2 and it is used for the application in the next section. We note that if the above-discussed quanta1 suppression is the same for all of the three characteristic parameters &,, yk,, ckl, then wO and A are immune to this effect since they are given as ratios. Therefore the result that the considered surface mode is critically damped may remain true up to higher values of k, than might at first have been expected. Furthermore, the response function (3.12) should then be divided by the common suppression factor and would thus receive a substantial enhancement for k, B kF. Therefore, the result (3.12) may represent an underestimate for large wave numbers. Finally, we emphasize that the fall-off as LO;: for high external frequencies is a general feature of our result, independent of the delicate k, dependence of the characteristic parameters.

456

V.I. Abrosimov, J. Randrup / Macroscopic response

4. Application The calculated

surface

response

pared with the experimentally ref. ‘O). At sufficiently external

field is simply

to inelastic

of the semi-infinite

determined

high projectile related

scattering

nuclear

energies,

Landau

response

the wave number

to the observed

scattering

beam energy above 200 MeV suffices.) Furthermore, due high-energy inelastic scattering probes mainly the nuclear distorted-wave impulse approximation, a factorized form differential cross section 2,20). I n accordance with this result, form:

liquid

can be com-

by using the method characterizing

of the

angle 21). (For protons,

a

to the strong absorption, surface region. Using the can be obtained for the we employ the following

(4.1) Here the first factor is the differential cross section for elastic nucleon-nucleon scattering at the same kinematical conditions and Neff is the effective number of target nucleons involved in the reaction. R(w, k) is (the imaginary part of) the response function and Rtot,,( k) = 2fih/ y is the total response as obtained by integratfactor X arises from ing R( w, k) over the energy fiw,,,. Finally, the normalization the fact that the absolute normalization is somewhat difficult to calculate so that an overall adjustment is ultimately made when confronting the data. The external field simulating the inelastic nucleon-nucleus reaction is of the form V&r)

= V(z) e

ik. I

,

(4.2)

where V(z) describes the effect of the absorption of the projectile in the nuclear interior. In scattering on spherical nuclei, the cross section (4.1) receives contributions from a belt around the nucleus located in the equational plane perpendicular to the beam direction.

Therefore

an average

of R(w, k) must be made with respect

to the azimuthal direction of the momentum transfer k, in order to obtain the effective response function R(wext, k) entering in (4.1) [ref. “)I. Using the approximation (3.13) we then find d2a -z5 da dE

c-

a2k4 Yl.f.W3

x.

(4.3)

The constant factor is estimated to be C = 65 mb * fm3/sr * MeV. In fig. 3 we show the measured differential cross section for the inelastic scattering of 800 MeV protons ofi “‘%n to 5” [ref. ‘)I, together with our above approximate result (4.3). The normalization factor X has been fixed by demanding a fit to the data at hi,,, = 10 MeV; this yields X= 5.5. The momentum transfer is about k = 0.5kF The calculated results look roughly similar to the data, but the decrease with energy is too rapid. When making the comparison one should keep in mind the following points: (i) The present calculation includes only the one critically-damped

451

Fig. 3. Measured ‘) and calculated (see text) differential cross sections for the inelastic scattering of 800 MeV protons off “‘Sn to an angle of 5”.

mode obtained in sects. 2 and 3; other modes will probably also contribute, particularly for the higher values of o. (ii) Only spin-isospin scalar excitations have been considered in the present study; in ref. ‘) it was estimated that this channel carries about 70% of the total strength. (iii) The present study concentrates on the macroscopic response which provides a smooth background and does not include special structures, such as giant resonances. 5. Concluding remarks We have studied the surface response of a semi-infinite Fermi liquid described by the semiclassical Landau theory. First, the free surface modes of the semi-infinite Fermi liquid were considered. The imposed boundary conditions correspond to those used in liquid-drop studies of the nuclear isoscalar vibrational modes. A general dispersion relation was then derived. We especially considered the limit of weak quasipa~i~le interactions and demonstrated the existence of a surface mode with a purely imaginary frequency. Estimating the stiffness parameter from the standard surface energy and employing the one-body dissipation theory (the wall formula) for calculating the friction coefficient, we found that the above-mentioned surface mode is critically damped. At the same time, the associated inertial-mass parameter was extracted. Finally, the macroscopic surface response function was obtained by exploiting the expected correspondence between classical and quanta1 mean values. The results were applied to inelastic scattering of high-energy protons off heavy nuclei. The experimentally observed decrease in the surfaces response with increasing energy was qualitatively reproduced, but our calculation leads to a too rapid fall-off.

458

V.I. Abrosimov, J. Randrup / Macroscopic response

The lack of exact agreement limited

scope of the present

In the present spin-isospin situation,

study

scalar when

with the data should

we have assumed

form.

cause no alarm

in view of the

calculation. Our method

the interaction

depends

that the quasiparticle

is readily on spin

applicable and/or

interaction

to the more

isospin.

It would

is of general be of

interest to determine the corresponding surface modes since data exist for the nuclear surface response in the spin’*) and isospin 23) channels. Discussions with H. Esbensen, C. Fiolhais and S. Shlomo have been very helpful. We are grateful to the Niels Bohr Institute and NORDITA for the kind hospitality extended to us during our stay.

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