Quasielastic electron scattering from nuclei

ANNALS

OF PHYSICS

128, 188-240 (1980)

Quasielastic

Electron

Scattering

from Nuclei

R. ROSENFELDER * of Theoretical Stanford University,

Institute

Physics, Department Stanford, California

of Physics,

94305

Received December 14, 1979

Inelastic electron scattering is studied in terms of the “characteristic function” F(t), i.e., the Fourier transform of the response function with respect to the energy transfer to the nucleus. Analytic properties of F(t) are discussed as well as moment and cumulant expansions. The latter are particularly useful in the region of the quasielastic peak where the first few characterize position, width and shape of the peak. The dependence of these observables on ground state properties and the final state interaction between ejected nucleon and residual nucleus is calculated for a variety of models. It is shown that the observed shift of the quasielastic peak is related to the exchange parts of the two-body interaction or equivalently to the nonlocality of the optical potential. Semiclassical methods are used to derive a generalized Fermi gas model for inclusive scattering which includes the final state interaction in a simple way. Numerical results are presented for quasielastic electron scattering from 12C. A similar description within the framework of a relativistic nuclear field theory gives surprisingly good agreement with experiment for medium and heavy nuclei and points out the advantages of a relativistic treatment.

1. INTRODUCTION Electroexcitation of individual nuclear levels is an invaluable tool for studying low-lying collective statesand the giant multipole resonances(for a review of electron scattering see Ref. [l]). At higher energy and momentum transfer q@ = (w, q) a prominent feature commonly called the “quasielastic peak” shows up strongly in the inelastic spectrum. New interest for this part of the spectrum has been generated by recent systematic measurements[2] showing additional strength in the region between the quasielastic peak and the bump at higher excitation energy that corresponds to resonant pion production. It is obvious that an understanding of this long-standing problem needsa reliable and detailed description of the quasielasticpeak. But it is also worthwhile to study the question, What determines quasielastic electron scattering and what can we learn from it? This is especially important in the light of attempts to separatelongitudinal and transverse contributions [3]. * Supported by National Science Foundation Grant NSF PHY-16188. Present address: Institut fur Physik, Universitat Maim, D65 Mainz, West Germany.

188 OOO3-4916/80/090188-53$05.00/O Copyright All rights

0 1980 by Academic Press, Inc. of reproduction in any form reserved.

QUASIELASTIC

ELECTRON SCATTERING

189

The adjective “quasielastic” implies that electron scattering under thesekinematical conditions takes place on individual nucleons that are ejected from the nucleus. Thus for free nucleons with massM which were initially at rest the peak position would occur at an energy transfer w = -q2/2M1 and the observed width of the peak can be attributed to the Fermi motion of the nucleons inside the nucleus. The above picture for quasielasticelectron scattering is most simply incorporated in the Fermi gas model [5] which considers the nucleus as a collection of free nucleons and which has been very successfulin describing experimental data for a wide range of nuclei [6,7]. The Fermi momenta kF extracted from these analyses agree quite well with the values obtained from the ground state density of heavy nuclei (as measuredin elastic electron scattering) and this has been claimed as confirming the essentialvalidity of this simple picture. However, as de Forest pointed out, “Any reasonable model of nuclear structure (including the very simple Fermi gas model) is bound to give a rather impressive fit to the experimental data. As a result it is the detailed shape of the quasielastic peak which reflects the nuclear structure” [8]. indeed it was found necessaryto shift the calculated peak by a small amount 2 (interpreted as average separation energy) to get agreement with experiment. One may therefore suspect that the main deficiency of the Fermi gas model-its neglect of nuclear interaction in bound and continuum states-is hidden in the two fit parameters kF and E. There have been several attempts to include interaction effects in quasielastic electron scattering. Apart from the two- [9] and three-body [IO] systems where the problem can be solved exactly, a variety of approximations has been applied: manybody techniques for infinite nuclear matter [ 111,modifications of the Fermi gasmodel to accommodate a nuclear potential [12] and shell-model calculations [13-171 for finite nuclei. A common feature of the last approach is to use an energy-dependent potential taken as the real part of the empirical optical potential. This is necessaryto reproduce the shift of the quasielastic peak but it carries with it problems of orthogonality and completenessfor the wavefunctions. The present work consistsof two general parts: in the first part sum-rule techniques are usedto study directly the position of the quasielasticpeak and its width in terms of the nuclear interaction and the ground state properties of the nucleus. This turns out to be a quite sensitive and direct method for investigating the interaction effects we are looking for [18] and is similar (although lessfundamental) as the study of moments of the inelastic strucutre function in high-energy physics which give us the most stringent test of strong-interaction physics (see, e.g., [19]). Although the first few moments describe position and breadth of the quasielastic peak reasonably well, a knowledge of higher moments is required to obtain the detailed shapeof the spectrum. However, it becomesincreasingly difficult to calculate higher momentsby the standard sum-rule techniques. Therefore, in the second part of this work, we try to generalize the Fermi gas model in a simple way to include the nuclear interaction. This is achieved by a semiclassicalmethod which is equivalent to the Thomas-Fermi approximation for * Throughout

this work we use fi = c = 1 and the conventions of 141. In particular yz = yo2 - q2.

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R. ROSENFELDER

bound states and the eikonal approximation for scattering states [20]. Since the energy-dependence of the empirical optical potential is mainly due to an intrinsic nonlocality of the optical potential [21], we use a nonlocal potential in the nonrelativistic treatment of one-body knock-out reactions. This also eliminates the theoretical difficulties associated with an energy-dependent (non-hermitian) potential. Recent studies of elastic proton scattering at intermediate and low energies [22] have shown that a relativistic description based on a mixture of (local) scalar and vector potentials is equally successful. This attractive feature of a relativistic field theory of nuclei [23] is implicitly contained in the relativistic “local” Fermi gas model we develop in the last section. It also makes use of the ground state properties of medium and heavy nuclei calculated self-consistently in the same model [24, 251. This paper is organized as follows: Section 2 reviews basic ideas about inclusive electron scattering. Section 3 introduces the generating function for the moments (characteristic function), discusses its properties and studies some solvable models. The first few moments are calculated with a two-body interaction and numerical results for 12C are presented. In Section 4 the concept of Wigner transforms is used to derive a generalized Fermi gas model for inclusive reactions. Section 5 deals with a relativistic description of quasielastic electron scattering and the final chapter contains a discussion and summary of the results.

2. INCLUSIVE ELECTRON SCATTERING

In this section we recall some basic formulas for inclusive electron scattering. In the one-photon exchange approximation (see Fig. 1) the double-differential cross section in the lab system takes the form [26] d20 dQ’ de’

FIG. 1. Inclusive electron scattering from nuclei in the one-photon-exchange approximation.

QUASIELASTIC

ELECTRON

191

SCATTERING

where 01= l/137.036 is the tine-structure constant, 7 Uy= $Tr(k’rUkrV) and the electron mass has been neglected. All the information about the target is contained in the tensor w,,

= (27r)3 c W(P’

- P - q)(P’ / J,(O)l P) p(P’)(P’

I J”(O)\ p>*,

(1)

P'

where the sum runs over a11 final states with four-momentum P’ weighted by the density of these states p(P’). J,(O) is the nuclear current operator in the Heisenberg picture evaluated at the origin. The most general form of W,, compatible with Lorentz invariance, symmetry and current conservation is [27] WI” = - WIG?, q . P,[ g,, - +g] + W2(q2, 4 . P) p _ q *P [ u ___ MT2 q2 qu] [p” - y

CL] ,

(2)

where MT is the target mass. This leads to the famous result d2a YrP-z=

w2+2wltan2$.

(3)

The factor in front is the Mott cross section 4a2ct2

UM = 4 4

cd2cos2 e/2 20 ‘OS 3 = 4E2 sin4 812 ’

(4)

It is convenient to distinguish between longitudinal and transverse structure functions. Upon projecting out the corresponding parts [28] we have m

d2u

=

uM

-$&+

(-&++an2$Sr],

(5)

where

are the longitudinal and transverse structure function, respectively. The above formulas are exact within the one-photon exchange approximation. However, in applications to, nuclear targets we need the explicit form of the current operators acting on the internal nuclear wave function. In the lowest order of a nonrelativistic reduction one finds [29] W’

I P(O)10, p> = (n I PW w,

(7)

W’

I JW 0, P> = =+2MT

(8)

(n I pWl 0) + (n I JW 0)

192

R. ROSENFELDER

with p(q) = 2 e,(q) eia’xi, i=l

representing charge density, convection and magnetization current density, respectively. Note that the recoiling nucleus also contributes to the convection current on the r.h.s. of Eq. (8). In (9) and (10) $ and pi denote the intrinsic coordinate and momentum operators for the individual nucleons xi = xi - f

gl xi;

Pi =

Pi

- a :I Pi

and ei(q) and p&q) are the (three-dimensional) Fourier transforms of charge density and magnetization density of the ith nucleon. It should be pointed out that Eqs. (7)-( 10) are nonrelativistic approximations which are deficient in several ways: (i) Higher-order terms in the reduction procedure may become important when j q 1 becomes comparable to the nucleon mass M. Common practice is to include all terms to order 1/M2, which adds the Darwin-Foldy term to the charge density operator dp(q) = - gl ‘&

[q” - 2iq * (ui X pi)] eiaTx;.

(ii) In deriving Eqs. (7)-(12) it has been tacitly assumed that the energy transfer w to the nucleus is small (of order l/M). This holds for elastic scattering and excitation of low-lying levels but is certainly not the case on the high-energy side of the quasielastic peak. (iii) Retardation effects are not accounted for. In particular, one might expect that the invariant form factors &(q2), F,(q2) for the single nucleons are to be used rather than the three-dimensional Fourier transform of charge and magnetization densities [30]. (iv) Explicit mesonic degrees of freedom are eliminated. Thus, not only is pion production above the pion threshold [5, 311 omitted but also meson-exchange corrections to quasielastic electron scattering [32, 331. All these difficulties are closely related and require in principle a consistent relativistic theory of nuclei. In the following sections we will ignore the subtle and delicate problems of relativistic corrections [34] and take the charge and current operators as

QlJASIELASTlC

193

ELECTRON SCATTERING

given by Eqs. (7)-(10). The inelastic structure functions then reduce to the familiar form

where

is the internal excitation energy and the longitudinal and transverse excitation operators ULIT are given by ~(4 and q x JW q I , respectively. Furthermore the intrinsic structure of the nucleons is taken into account by a common dipole form factor f(q2) = [l - q2/(855 MeV)2]-2 (15) so that ei , pLi in Eqs. (lo), (11) describe charge and magnetization nucleons ei = t(l + 73(i)), Pi

=

HPr,

+

@n)

+

H&I

-

PJ

density of point (16) (17)

T3(i).

It is only in the last section that we will use the full electromagnetic current operator in a relativistic description of quasielastic electron scattering. Finally, we briefly discuss Coulomb corrections which cannot be neglected for heavy nuclei. Although the corresponding Feynman diagrams have been directly estimated [35], we employ the more traditional approach of using distorted electron waves [36]. For high-energy electrons the distorted wave can be approximated by [37] Ii I & h(r) = (k( e

with 1k 1 = 1k I - rc ,

where rc is a mean value of the electrostatic potential of the nucleus (rc w 3&/2R with R = (5/3)1/2 (r2)lj2 for a nucleus with charge 2 and rms-radius (r2)1’2). The net effect is the replacement q2+ q& = w2 + 4(~ - T&E -

Vc - w) sin2 ! L

as argument in the structure function. Note that the amplitude sure that the Mott cross section (4) remains unchanged.

3. SUM-RULE

DESCRIPTION

OF QUASIELASTIC

factor I & j/l k I makes

ELECTRON

SCATTERING

The usefulness of sum rules in nuclear physics is well established for photonuclear reactions, eIectroexcitation of low-lying levels and studies of collective nuclear states (for recent reviews see [38]). In general, sum rules do not provide as much information

194

R. ROSENFELDER

as a detailed (say RPA or coupled-channel) calculation; however, they allow us to study some average properties of the spectrum and its sensitivity to the nuclear interaction in a transparent way. This advantage is particularly important for an investigation of the quasielastic peak as its smooth shape can be characterized by a few moments of the structure function. 3.1.

The Characteristic

Function

For simplicity we assume that the ground-state expectation value of the excitation operator vanishes: (0 I 0 I O} = 0. (19) Thus, e.g., we use the density fluctuation

operator

($ = .f eieiq4 - (0 / f i=l

eieiq”; 1’0)

i=l

rather than the density operator. This simplifies the notation and avoids subtraction of ground-state contributions. In the notation of Ref. [39] the moments of the strength function are defined as m,(q) = (-” dw’ w”Ss(q, a>. JO

In particular (we suppress the q-dependence in the following) m, =

m dw’ S(w) = (0 / LotO ( 0) I0

(21)

is the non-energy-weighted sum rule corresponding to the excitation operator 0. Let us introduce the Fourier transform of the structure function by S(w) = 2

/-‘m dt e-iW’tF(t). m

From Eq. (13) we find that the “characteristic F(t) = t

(22)

function” F(t) has the representation

1 eit('nmFo) I(n I 0 1O)12. n

Using the completeness of nuclear states (“closure”) and Eq. (21) this can be written as (24)

QUASIELASTIC

ELECTRON

195

SCATTERING

where His the (internal) nuclear Hamiltonian and the energy scale has been chosen so that the ground-state energy E, of the nucleus is zero. Note that by construction F(0) = 1. Without detailed calculation we know the following general properties of the structure function from its definition (13) (i) (ii) (iii)

S(w) is real, S(w) = 0 for w’ < 0, S(w) > 0.

How do these restrictions translate into properties of F(t)? The answer follows immediately from Eqs. (22), (23): (i)

As a function of the complex variable t we find F(t)*

= F(-r*).

(25)

In particular, on the imaginary axis I = ip F is real: F(i/!?)* = F@);

(ii)

We have

+m c dt e-iw’tF(t)

= 0

for w’ < 0.

Since all excitation energies are positive2 we conclude from Eq. (23) that F(f) -+ 0 for Im t -+ + co. Thus the integration path in (26) can be closed in the upper half t-plane and it follows that F(t) is analytic for Im t 3 0. From the Cauchy integral representation F(t)=&.$dt’t,

F(f) -t-k’

t real,

we find the dispersion relations Re F(t) =

;p~+=dft~, -33

Im F(t) = - + P j’% dt ’ e -02

.

Thus the real part of the characteristic function can be eliminated and the structure function can be expressed in terms of Im F(t) S(w) = +

&a~‘) lrn dt Im F(t) sin w’t. 0

(28)

by use of Eq. (28)

(29)

2 Note that the ground state is excluded in the sum over final states in (13) or does not contribute due to assumption (19).

196

R.

ROSENFELDER

This result could also have been derived by noting that for w’ > 0

and that according to (25) Re F(t) is even and Im F((t) odd in the variable t. Note that the representation (29) ensures that the structure function vanishes at threshold w’ = 0. (iii) The positivity constraint is formulated most easily on the imaginary axis t = i/3. From Eq. (23) we see that F(ij3) is the Laplace transform of the structure function and hence decreases monotonically from F(0) = 1 to F(co) = 0. More precisely, F($) is completely monotonic [40] in 0 < p < co, i.e., wag”,

akWP)

> o. 9

k = 0, 1, 2....

(30)

Note that the characteristic function on the imaginary axis is given by [41] F(iP) = $ jO= dt h

Im F(t).

Using time-reversal invariance Eq. (24) can also be written as

F(t)= $

(0 I(F+U+eitHU9-)l

O)*

with Y being the time-reversal operator. Since Y+CW = +0+ (the upper sign for longitudinal, the lower sign for transverse excitations) and the nuclear Hamiltonian is assumed to be time-reversal invariant, it follows F(t) = &

(0 1OeitWt j 0). 0

Equations (24) and (32) are frequently written in terms of the excitation operator in the Heisenberg representation U(t) = eitH~e-itH

(33)

and take the form [28] moF(t) = CO I U+(O) U(t)1 0) = (0 I 0(-t)

u+(o)1 0).

(34)

Using 2i Im F(t) = F(t) - F(-t) we can infer from (34) that the imaginary part of the characteristic function is related to a commutator 2im, Im F(t) = (0 /[P(O), C(t)]\ O>,

(35)

QUASIELASTIC

ELECTRON

SCATTERING

197

while the real part involves the corresponding anticommutator. This has the important consequence that Im F(t) is much easier to evaluate than Re F(t). For example, assume that o(t) is a one-body operator. Then Eq. (35) reduces to the simple calculation of the ground-state expectation value of a one-body operator whereas for Re I;(t) one has to deal with a two-body operator. For completeness we mention the relationship between the polarization propagator ~421

(36)

and the characteristic function. It is easy to show that 17(w) = -2~2, lrn dt Im F(t) eiw’t. 0

(37)

Equation (29) is equivalent to the well-known relation S(W) = - $ Im D(W). The polarization propagator is related to the two-particle Green’s function [36] and can be calculated by standard many-body techniques (Hartree-Fock approximation, summation of ring diagrams, etc.) [l 1,421. Finally we give a representation of the characteristic function which shows its close relationship to the partition function used in statistical mechanics. Our notation (t = $3 on the imaginary axis) has anticipated this connection, First we note that by inserting a complete set of nuclear states Eq. (34) becomes

(39 where (40)

is the A-particle density matrix. The invariance properties of the trace can then be used to obtain Wis) = - $ g -WI*=0

3

(41)

where Z(A) = Tr(e-OHo)) is the partition

function for the constrained Hamiltonian ff@) = H-

E,+A$-UpU+.

(42)

198

R. ROSENFELDER

The main advantage of such a formal representation is that numerous approximation methods for the evaluation of the partition function are available in statistical mechanics. We just mention semiclassical expansions [43], variational methods [43,44] and various bounds for the free energy [45]. To obtain the structure function one may analytically continue F(i/3) to the real axis or directly perform the inverse Laplace transformation. 3.2.

Moments and Cumulants

The characteristic function is the generating function for the moments (44)

Such an expansion has only a formal meaning because (i) distance (ii) crucially

the convergence radius of the series may be finite (actually, it is equal to the of the nearest singularity in the lower-half t-plane from the origin), the moments mk may diverge for k > kc . The critical number kc depends on the form of the interaction between the particles.

The second point has been discussed in great detail in [46] where also different expansion schemes have been proposed. We will not dwell on these probleins mainly because only a few moments can be calculated with realistic nuclear interactions. It should be kept in mind, however, that the conclusions based on these moments rely on the assumption that the effect of the higher moments is small. From Eq. (44) we see immediately that Im F(t) is determined by the odd, Re F(t) by the even moments. It has been pointed out repeatedly that the odd moments are simpler to calculate than the even ones. This is due to property (35) for the imaginary part of the characteristic function. The dispersion relations (27), (28) (i.e., analyticity of F(t)) imply that odd and even moments are not independent. In fact, from a knowledge of the imaginary part of the characteristic function one can directly calculate the even moments: mZk = (-)” 9

I+== dt f Im F@k)(t); -co

k = 0, 1, 2 ,....

(45)

Equation (45) shows that, in particular, Im F(t) has to fulfill the condition

I--m+m dt i Im F(t)= rr,

(46)

which according to (27) simply states that F(0) = Re F(0) = 1. A question of great mathematical and practical interest is the following: Assume that a finite number of moments are given. It is obvious that the structure function

QUASIELASTIC ELECTRON SCATTERING

cannot be reconstructed from this information, an integral quantity

199

but is it possible to derive bounds for

S(w), smdw’f(w) 0

where f(w) is a given function ? There is ample mathematical literature on this “problem of moments” [47,48]. Without going into details it may be stated that an optimal approximation for (47) is S(w) 6%P(W) = 5 WiS(W’ - Ei)

(48)

i=l

in the same sense as a Gaussian quadrature formula is the best approximation to a definite integral. The weight factors wi and the mesh points Ei are determined by the moments [48] and it is possible to give rigorous bounds for the integral in (47). A sufficient and necessary condition for the structure function and the excitation energies to be positive (i.e., wi , Ei > 0) is det I mk+z IZ,z=o3 0, n = 0, 1, 2.... det I Q+~+~ ILo

(4%

2 0,

The method of moments has been applied to improve the usual closure approximation [49], to muon capture [50] and to inclusive pion electroproduction from nuclei [51]. In high-energy physics the deep inelastic structure function of the nucleon has been analysed along these lines [52]. Although the method works well for reactions which depend on integral properties of the structure function, an approximation of S’(w) as a sum of (artificial) discrete states is certainly inadequate for quasielastic electron scattering. What is needed for describing the shape of the peak is a (physically meaningful) extrapolation of the moment series to higher moments. For this purpose, it is convenient to introduce the cumulant expansion of the characteristic function3 F(t) =exp

The cumulants or semi-invariants

* See [53] for similar expansions.

l$rgh,/.

h, are given by the recursion relation [54]

(50)

200

R. ROSENFELDER

The first few explicitly read

h, = (H2) - (H)2

= ((H - (H))2),

A, = (H3) - 3 (Hz)(H) A4 =
= ((H - W))4)

+ 2 (H>3 = ((H - (H))3), - 3 (H2>2 + 12
- 3W

- W))2),

and are called mean energy, mean square deviation, skewness and excess, respectively. In fact, the ratio of energy-weighted sum rule (ml) to non-energy-weighted sum rule (m,) is a popular (but of course not unique) definition of the mean excitation energy in various inelastic reactions. It may be also noted that A, provides a quantitative measure of the spread of the nuclear Hamiltonian in the state B ( 0) which is not an eigenstate of H. A Gaussian distribution has vanishing cumulants for k > 2. Thus X, characterizes the deviation of the spectrum from a symmetrical Gaussian distribution centered at w = h, ; in particular, X, > 0 ( X,h, + 2x2 - x,2 >

0, 0, 0, 0,

(52)

which also can be derived directly from the definitions. The cumulant expansion seems to be particularly useful for quasielastic scattering as the smooth shape of the peak can be nicely characterized by peak position (A,), width (proportional to h,) and so on. However, it is in general not possible to use Eq. (50) retaining only a few terms because we would violate the general properties of F(t) derived in the previous section. To illustrate this, consider

This approximation has the wrong behavior for Im t -+ + 03. We can circumvent this difficulty by taking the imaginary part Im F(t) m sin th1e-t2’e’2

(53)

and inserting it into (29). The result for w’ > 0 is S(w)

c-a (2Ty;)1,2

k+‘--@‘2~~

-

(w’ 3 -(J).

(54)

QUASIELASTIC

ELECTRON

SCATTERING

201

By this procedure we have generated (via the dispersion relation) a characteristic function with the correct large-t-behaviour. However, it may be noted immediately that m, in Eq. (54) is not identical with the integrated structure function. This is due to the fact that (53) yields 1’” dt f lm F(t) = 77erf -m

f&j

thus violating the normalization condition (46). It is only for a large momentum transfer that the Gaussian result (54) becomes meaningful because &, h, --f q2 (see Section 3.2) and condition (46) is asymptotically satisfied. Thus, it seems to be nontrivial to find parametrizations of F(t) which satisfy the general constraints and correctly give the first few moments.4 An example for such a parametrization is based on the [I, II-Pad6 approximant for In F F(t) a exp

I

ith,

1 - fritA,/Al

I ’

(55)

Recalling the inequalities (52) an essential singularity in the lowerhalf t-plane is found at t, = -2&/h, and it can be verified that F(i/3) is indeed completely monotonic. The corresponding structure function is given by

where I1 is the modified Bessel function of the first kind. It is easy to show that (56)has the correct moments m, , m, , m2 plus an infinite number of higher moments all expressible by the first three. A final comment concerns the asymptotic behaviour of In F. It may be argued that large values of t (or 8) correspond to long interaction times (or low temperatures) and are thus related to the low-lying part of the spectrum. As this region is dominated by the giant dipole resonance (GDR) one may conjecture lirntes In F(t) = iECDR t. If this is the case the [2, I]-PadC approximant for In F(t) would give a rough, but consistent, description of resonance and quasielastic region (compare [55]). 3.3.

One-Body Models

In this section we study some exactly solvable one-body models. The first one is characterized by the absence of any interaction (Fermi gas model) whereas in the following two examples the particles move in a local (square well) potential and a nonlocal (separable) potential, respectively. As the potential origin is fixed in space we neglect center-of-mass effects in the following. Thus we do not distinguish between w and W’ in (14) and we take xi = xi , pi = pi in (11). Also in most cases we consider longitudinal excitations only. 4 The ansatz F(t) w Cz, w, exp(it&) conesponding Eq. (46) reads in this case x:i wi = 1.

to (48) is, of course, such a parametrization.

202

R. ROSENFELDER

Let us start with a one-body Hamiltonian

in the energy representation

H = c Eiaitai

(57)

and assume a one-body excitation operator 0 = C O,ja,+aj . i,j

From (33) it follows that the operator in the Heisenberg picture is also a one-body operator

and, as was noted earlier, one has to evaluate the ground-state expectation value of a one-body operator for the imaginary part of the characteristic function. The result is simply m, Im F(t) = C @$O,,(k 1p 1.j) sin t(~ - 4, i,i,k

where P = C I m>
(59)

m
is the one-body density matrix which is diagonal in the representation (57). Recalling Eq. (29) the structure becomes for w > 0 s(w)

=

c

6$oik(k

1 p lj>

&J

-

(%

-

Ek))

-

tw-

-w)

(604

i,i,k

= I,&

(6Ob)

The second line could have been obtained more easily in any shell-model calculation that correctly incorporates the Pauli principle. It is instructive to see how the “Pauli blocking” term emerges in the present formalism: Eq. (60a) tells us that for every blocked excitation k -+ i there is also a corresponding deexcitation i -+ k which is blocked. 3.3.1. Fermi Gas Model

For quasielastic electron scattering at high-momentum transfer the excited states are high in the continuum and are therefore frequently approximated by plane waves. The extent to which this is a valid assumption will be discussed later in more detail when we deal with the energy-dependence of the optical potential. Let us assume here

QUASIELASTIC

ELECTRON

203

SCATTERING

merely that for high-momentum transfer the free Hamiltonian (57)-(60). Then (60a) becomes in the momentum representation

can be used in Eqs.

- (w --f -w)

with a corresponding sum over spin and isospin variables implicitly assumed. Note that the density matrix is no longer diagonal unless the ground state is built up by single-particle plane waves (“pure” Fermi gas model). Recalling Eq. (9) we have for longitudinal excitations (P I 0 I P’>

=

%P -

q -

P’).

As the neutrons do not contribute (we neglect the electric form factor of the neutron) we obtain Srton(w)

= J-d3p n,(p) 6 (u -

(p2+M9)2 + &)

- (0 -+ -w),

(61)

where n,(p) = Tc,(p

I p I P>

=

1

I

I2

(62)

i
is the (proton) momentum normalization condition

distribution

inside the nucleus. From (59) we have the

s

d3p n,(p) = Tr p = Z.

For spin-zero nuclei the momentum distribution p. Equation (61) can then be further reduced to

depends only on the magnitude

(63) of

with (65) It should be pointed out that the upper limit of integration in the above equation is due to the analyticity constraint imposed on the characteristic function. According to our previous discussion this constraint leads to the correct blocking of unphysical excitations. Equation (64) correctly vanishes for q = 0 and becomes the familiar Fermi gas result [l, 261 if a step-like momentum distribution is inserted. Although this would be consistent with using the free .Hamiltonian the present derivation shows how the “pure” Fermi gas model can be easily generalized to accommodate a more realistic momentum distribution. We also note that in this case (64) includes

204

R.

ROSENFELDER

Pauli blocking in a more satisfactory way than the ad hoc exclusion factor 1 - n (p + q)/n(O) which is commonly used but may become negative. At high-momentum transfer there is essentially no restriction for the struck particle this corresponds to replacing the upper integration limit in (64) by infinity. As the integrand is positive we find that the structure function becomes maximal at p- = 0, i.e., q2 for q --f co. wmax = 2~ (66)

The peak position obeys I Q I SLo4%

Wmax) = const

More generally at high-momentum

= 27~~4 loa4 p&4.

transfer 1q I SL(q, W) is a function of

MI P-=91,&j only. This peculiar scakng property of the structure function (called “y-scaling” in [28]) has been tested recently [56]. Ordinary “x-scaling” can also be studied but is of only academic interest [57] in the momentum and energy region in which Eq. (64) is supposed to be valid. The first relativistic correction (12) to the charge operator is easily evaluated and leads to a multiplicative factor

I--&

WJ

- 1)

on the r.h.s. of (64). For the transverse structure function spin and isospin variables have to be treated more carefully, but it is straightforward to obtain

M +2=m

m dpp&9 Is ?I-

p2 ,:-”

-(co-+-w)

I

.

033)

The second line is the contribution of the convection current while the first line is due to the magnetization current where the neutrons also contribute (the neutron momentum distribution is normalized to J d@z,(p) = N). It is obvious that the magnetization current dominates at high-momentum transfer and that the peak position in this limit is still given by (66). Neglecting the convection current contribution the expression for the peak height and the concept of “y-scaling” remain valid by taking out the appropriate q-dependent factors [56]. Relativistic kinematics for the particles can be included easily but we postpone that for a more complete relativistic treatment of quasielastic electron scattering in the last chapter.

QUASIELASTIC

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205

SCATTERING

Let us study the moments of the structure function in this model. For simplicity we consider the longitudinal part only. It can be easily shown that the characteristic function on the imaginary axis has the form

where pg = E (p, i 1q I). Thus F(‘(i/3)is . indeed completely monotonic power-series expansion in p we find the moments mk = 27r (-$$)k/om

dp PP s:’

In particular, the Fermi gas Coulomb

Mp>

and from the

+ P
sum rule is given by

m, = 4n

s 0m 4 pp,n( p>

(70)

starting linearly with 1q 1 and approaching for large-momentum

transfer

lim m, = 2.

(71)

q-m

The latter property is model-independent whereas the low-momentum should be quadratic not linear (see section 3.4.). The first moment reads m, =4rJ-$-/” 0

behaviour

q2 = Z 2M

4w

n(p)

for all values of q. Equation (69) gives for the second moment explicitly m,=h

(5)”

j-m 4-w< 0

(P>’ + ;P<~) n(p)

gzg&:

+&]

9

U’3b)

where T = <0 1 T 10)/Z is the mean kinetic energy per particle (proton) in the ground state. Similarly m3 = Z (&)’

[4(T)

+ &]

(74)

for all values of q. We again observe that the odd moments have a particularly simple structure. It is interesting to consider the cumulants A, in the high-momentum transfer limit. The mean energy (75)

206

R. ROSENFELDER

is identical with the peak position (66). For the mean square deviation the leading q2-contributions of (H2) and (H)2 cancel and we obtain

X,-%;(T)&.

(76)

This cancellation is complete for the skewness A, =% 0,

(77)

showing that the quasielastic peak as described by Eq. (64) is symmetric at highmomentum transfer. Figure 2 shows the longitudinal structure function for 12C with a harmonic oscillator momentum distribution b(p)

2b3 ebz9z (1 + ; b2p2) = a2e

for various momentum transfers. Also plotted is the approximation (56) based on the moments m, , m, , m2. We note a reasonable agreement with the exact result although the peak position is somewhat shifted to lower excitation energies. This is because (55) provides an extrapolation to higher cumulants. In particular, a nonvanishing skewness Aa = @,2/h, is generated which leads to an additional negative shift ---CT> at highmomentum transfer (see Eq. (80)).

w [MN] FIG. 2. Longitudinal structure functidn for ‘T in the Fermi gas model (solid line) and the approximation (56) (dashed line) based on the first three moments. A harmonic oscillator momentum distribution with b = 1.63 fm was used. Note the different scale for the excitation energy in the lower half.

QUASIELASTIC

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207

We emphasize that Eqs. (71)-(77) could have been obtained by standard sum-rule techniques without knowledge of the exact solution and that the first few moments already tell us position, width and shape of the quasielastic peak at high-momentum transfer. In the next sections we will study how these expressions are modified due to interaction effects. 3.3.2. Local Potential

In our discussion the first moment-the energy-weighted sum rule-plays a prominent role: in accordance with its interpretation as a mean energy the peak position is approximately given by wmaX FZ h, = (H)=+m,.

(78)

We have seen that (78) is fulfilled in the Fermi gas model and it is natural to ask whether it also holds when the struck particle moves in a local potential V(r). In order to calculate m, for this case we use the t -+ O-limit of Eq. (35) to get the well-known result ml = $(O I[O+, [H, O]]l 0).

(79)

Thus a local one-body potential does not contribute to (79) as it commutes with the charge operator. By evaluating the double commutator with the kinetic energy operator we obtain m, = Zq2/2M identical with the Fermi gas result (72). This is essentially the Thomas-Reiche-Kuhn sum rule [38] for the full operator exp(iq * x) which reduces to the dipole operator in the low-momentum limit. Thus the prediction of Eq. (78) is that in a local potential model the quasielastic peak has its maximum at the same energy transfer as in a free model. To check this prediction we have performed a numerical calculation assuming that the struck particle initially was bound in a square-well potential with depth I’, . For simplicity a 1s bound state with binding energy Ed = 20 MeV was taken. A partial wave decomposition then yields Sdq, w) = 8MB(w

- ~,j)p f

(22 + 1) R12

1=0

with p = (2M(w - .~n))l/~. The radial integral is given by

Rz= J*m dr r2#o(r)j2(l

q / r) JfQ$.

0

Herejl is the spherical Bessel function of order I, #o the normalized bound state wave function and yl(p, r) the radial wave function which behaves asymptotically like sin(pr - h/2 + 8,). Analytic expressions for iJI0, yz and the phase shift & can be found in every elementary book on quantum mechanics; however, the radial integral, the partial wave summation and the determination of the maximum of the structure function have to be done numerically. 595/128/I-14

208

R. ROSENFELDER

A popular concept is that the observed shift of the quasielastic peak is given by the average binding energy [6]. It is evident from Fig. 3 that a local potential does not show this behavior: bound state and continuum state interaction seem to cancel nearly completely. The remaining negative shift (relative to (78)) can be explained in terms of higher cumulants: including the skewness in the cumulant expansion (50) one finds at high-momentum transfer5 the following improvement over Eq. (78)

Again we repeat that this relation relies on the assumption that the higher moments are small. As the second moment can be written as m2 =(O)

D+D IO)

w

with D = W, 071,

W)

it is not modified by a local potential and Eqs. (73b) and (76) apply as well. It is only in the third moment m3 = $(O IiD+, [H,Dlll that a local potential

makes a contribution

01

(83)

[45, 501

,,,(P@ = z-4. 'AVj. 3 12M2'

(84)

FIG. 3. Position of the quasielastic peak in a square-well potential model for different values of the potential depth V,. The particle was initially bound in a Is-state with binding energy
5 Remember that in this limit non-analyticity no role.

effects in the truncated cumulant expansion play

QUASIELASTIC

Combining

ELECTRON

SCATTERING

209

Eqs. (76), (77), (80) and (84) we finally get

In the square-well potential model the additional term amounts to -4.8 and -8.6 MeV for the two different choices of potential depths. This is in fair agreement with the values we see in Fig. 3. It may be surprising that a positive skewness (i.e., a high-energy tail) induces a small negative shitt. However, it can be easily seen that this situation must occur if the lower moments m, , m, , m2 have to be preserved. Using a harmonic oscillator potential we may estimate the negative shift to be roughly --Q(T). The present example clearly demonstrates the usefulness and elegance of the moment method especially when compared with the lack of insight inherent in a numerical calculation. 3.3.3. Nonlocal Potential

As we have seen in the previous section a local one-body potential does not affect the first moments (m, , m, , m2) explicitly; properties of the potential only enter indirectly via the ground-state wave function. This has the consequence that at highmomentum transfer the quasielastic cross section has its maximum at (nearly) the same excitation energy as that predicted by a model without interactions--contrary to the empirical tinding [6, 71. This is, of course, not surprising as the optical potential is known to be nonlocal and/or energy-dependent [21]. As for the energy-dependence it has the unpleasant feature of destroying orthogonality and completeness (which can be approximately restored by some recipes [SS]). Hence we will concentrate on nonlocal potentials. Let us examine how the energy-weighted sum rule is modified by a nonlocal interaction. From (79) we find a contribution ml

(PW

=

_ J

d”r d3r’ p(r, r’) V(r, r’)[l - cos q . (r - r‘)]

(85)

due to the nonlocality of the potential. Here p(r, r’) is the mixed density, i.e., the coordinate representation of the one-body density matrix. Note that (85) indeed vanishes for a local potential V(r, r’) = V(r) 6(r - r’). At low-momentum transfer Eq. (85) is proportional to q2 and can be combined with the classical result (72) to give ml=Z

92 2M*

for q -+ 0.

(86)

It is a well-known result that the nuclear interaction may be incorporated in this limit by using an effective mass M *. For the other extreme case we find from (85) ml

(Pot)

~-Kc ---tz<=

s

d3r d3r’ p(r, r’) V(r, r’)

(87)

210

R. ROSENFELDER

unless the nonlocal potential has a special singularity structure.6 It thus seems that a nonlocal one-body potential can account for the empirical observations at both lowand high-momentum transfer. A particular simple nonlocal interaction is provided by a separable potential v = h I gxg I

@ < 01,

(88)

which allows for closed analytical solutions. The relevant formulas are collected in Appendix A together with the longitudinal structure function for the Yamaguchi potential [59] (pig)

This parametrization

=---

B112 =

1

(89)

P2 + S”.

gives a p-s-falloff for the momentum

distribution

thus rendering

m3 and higher moments infinite. As a consequence neither (78) nor (80) is applicable

for the position of the quasielastic peak: although the mean energy indicates an asymptotic shift of twice the binding energy (Eq. (AlO)) a close inspection of the exact solution (A6) shows that the quasielastic peak is displaced just by the binding energy. This example illustrates how critical Eqs. (78) and (80) depend on the assumption that higher cumulants are finite and small. In real nuclei, however, the situation is different from the one we encountered in the above toy model. Recent calculations of the momentum distribution (including correlations) show that its asymptotic behaviour is exponential rather than powerlike [60]. Thus we have some confidence that a sum-rule description of quasielastic electron scattering is meaningful. 3.4. Two-Body

Interaction

The previous section considered the particles moving in a given mean field. We now turn to a more microscopic picture in which the nucleons interact via a two-body interaction and study the moments of the structure function in this case. Furthermore we include center-of-mass effects neglected up to now. They enter in two places: first the internal excitation operator as given by Eqs. (9)-(11) has to be used and second we must correct for the usual lack of translational invariance in the many-body ground-state wavefunction. The latter problem is most easily handled by assuming harmonic oscillator single-particle wavefunctions which, for example, give the usual center-of-mass (CM) factor for the form factor [26]. Let us begin with the zero-order moment (Coulomb sum rule)

m. = C I+ I Q I WI2 nio = (0 1C e,efeiP’(x+;)

[ 0) - Z2F02(q),

iA B For example, a momentum dependent potential V = ipeV(r) i.e., Eq. (86) holds for all values of q.

(90)

+ h.c. leads to ,\Pot) = -Zq*< V>,

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211

SCATTERING

where Jxq)

=

-&

(0

/ 1

e&?-;

1 0)

z

is the elastic form factor. Splitting the double sum in (91) into a one- and a two-body piece we arrive at m, = 2 - Z2&2(q) + (0 1 c e,e,e’Q’(-) i#i

j O),

(92)

the last term being a genuine “correlation function.” At high-momentum transfer it vanishes-irrespective of how complicated the structure of the ground state is. It is this observation (together with the obvious fact that lim,,, F,,(q) = 0) which leads to the model-independent limit (71). For low q only dipole excitations contribute to (90). Thus wl(cr) = aq2 + w14L (93) where the coefficient a can be related to the dipole photo-sum-rule

D-~ [61]

For dipole transitions an effective charge for protons and neutrons has to be used; therefore we expect CM corrections to be most important at low-momentum transfer. This is indeed borne out in a shell-model evaluation of m, for the light nuclei 4He: 12C, 160, details of which are given in Appendix Bl. From (B4), (B5) we find a 50 “/, reduction without CM corrections B a= Zb2 x (95) with CM corrections i in the value of the coefficient a. With Z = A/2 and an oscillator parameter b = 1A116 fm the dipole sum rule uel can be estimated from (95) to be u-I M 0.036A4/3 fm2 in fair agreement with the data. Figure 4 shows a comparison of m, in 12C for different models: Fermi gas (Eq. (70)) with a Fermi momentum kF = 221 MeV/c and harmonic oscillator with and without center-of-mass correction. Apart from the long-wavelength region the differences are small and the calculations in Ref. [64] also show that short-range correlations affect the Coulomb sum rule on the 55 % level only. Let us now turn to the energy-weighted sum rule ml given by Eq. (79). The classical part of the sum rule is evaluated by calculating the double commutator of intrinsic kinetic energy with the intrinsic charge operator. This is easily done and gives

212

R.

ROSENFELDER

FIG. 4. Non-energy-weighted sum rule with (solid line) and without (dashed line) center-of-mass corrections in the harmonic oscillator model. Also shown is the Fermi gas result (dot-dashed line) with a Fermi momentum kF = 221 MeV/c.

Again we find that CM corrections are important

for long wavelengths (98)

where they give the correct dependence on neutron and proton number (analogous to (95)). For high-momentum transfer (97) becomes

showing that the recoil energy of the whole nucleus must be subtracted to get the internal excitation energy. To evaluate the potential contribution to the energy-weighted sum rule we have to specify the two-body interaction. We take the following finite-range7 central potential,

Vij(r) = W(r) + B(r) Prj - H(r) P& - M(r) PZP,‘, ,

where Pl” , Pi2 are the usual spin and isospin exchange operators. The latter do not commute with the isovector charge operator leading to the result [63] ml (‘Ot) = a (0 1 C (H(rij) i#j

+ M(r,,) PG)(l - cos q * rJ($i)

*I

-

am

T&))\

0).

(101) 7 This is essential as a zero-range potential gives a vanishing contribution to (101). It may also be noted that a Skyrme-type potential (zero range but momentum dependent) leads to Eq. (102) for all q [50]. This is because it basically is the low-momentum, limit of the two-body interaction (100) [65]. Consequently such a potential may be used for processes such as muon capture but is inadequate for high-q-reactions.

QUASIELASTIC

ELECTRON

As in (85) and (86) the potential contribution lengths and can be combined with (98) to give NZ q2 t171= 7 rM*

213

SCATTERING

is proportional

NZ q2 = 7 2~ (1 +

to q2 for long wave-

for q-to.

K)

(102)

Here K = M/M* - 1 is the famous enhancement factor of the classical sum rule. For high-momentum transfer Eq. (101) tends to a constant

t?7(pot) =c z.z= ) (0 / c (H(r,,)+ 1

MCrij)

fYjNT(iJ

’ %.i) -

73(i) T,(j))1 0).

(103)

i#j

We recall that the structure function was Fourier-transformed w - q2/2MA so that (78) should read more precisely

with respect to W’ =

Thus, according to (71), (99) and (103) the position of the quasielastic peak is given by 2

for q 4 co,

showing that the center-of-mass corrections cancel in this limit. We note that the potential contribution to m, involves a two-body operator in (101). In this respect the energy-weighted sum rule is no more fundamental than other moments and depends on the two-body correlation function as well [38]. Appendix B2 gives the details of an evaluation for 12C and Table I shows the enhancement factor and the asymptotic shift for different effective interactions and correlation functions.* It can be seen that the numbers we get are all very similar with the exception of hardcore potentials which give an asymptotic shift near the experimental value Z = 25 MeV [6,7]. Figure 5 shows the q-dependence of ;I(@ = Xl - A;ermi gas, i.e., the shift compared to the Fermi gas model. It is interesting to note that i: approaches the giant dipole energy at low-momentum transfer indicating a nice interpolation between resonance and quasielastic region. It may be argued that the correlation functions used are not consistent with the two-body interaction. Although a more retied calculation (especially for hard-core potentials) is desirable, we do not feel that the above results would change drastically. More important, in our opinion, is the neglect of the tensor force in the two-body interaction (100). This is evident from the calculated dipole enhancement factor K m 0.4-0.5 as compared to the value K = 0.86 from photoabsorption measurements 8 Due to a small computational error in

[18] these values

are

slightly different.

214

R.

ROSENFELDER

TABLE

I

Enhancement Factor K and Asymptotic Shift Z of the Quasielastic Peak in ‘*C for Different Two-Body Interactions”

Radial shape of potential

Correlation

function

K

(ML)

Gaussian [67]

ii CK

0.42 0.45

11.4 12.3

11.4

Yukawa 1681

ii CK

0.34 0.38

11.4 12.4

21.1

Square well [66]

ii CK

0.45 0.50

12.7 13.8

15.6

Exponential

ii CK

0.37 0.41

12.2 13.3

15.0

Square well with hard core [66]

ij CK

0.52 0.57

19.6 21.2

32.3

Exponential

ii CK

0.49 0.54

24.9 27.0

53.7

[66]

with hard core 1661

ajj-coupling [62] and Cohen-Kurath (CK) [63] correlation functions are employed. In addition, the parameter S {describing the broadening of the peak due to the interaction) is given for jj-coupling wavefunctions.

01 O

100

”

200

1 300

”

400

1’

500

I-

1: 1 b’@“Cl

FIG. 5. Shift of the quasielastic peak with respect to the Fermi gas as a function of the momentum transfer. A Gaussian two-body effective interaction [67] with Cohen-Kurath (solid line) and J& coupling (dashed line) correlation function was used. The result with a hard-core square-well potential [66] is denoted by the dot-dashed curve.

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215

SCATTERING

up to the pion threshold [69]. There is convincing evidence [70] that the tensor force plays a decisive role in making up the difference and it would be very interesting also to study its effects in the high-momentum limit of the energy-weighted sum rule. Some cautionary remarks should be made in addition. (i) An analysis using a more realistic momentum distribution than the Fermi gas step function might change the “experimental” number .Z = 25 MeV. (ii) We have not considered transverse excitations, which are at least as important as the Coulomb part under the kinematical conditions of the experiment in Refs. [6, 71. Having different spin, isospin and momentum dependence, the car responding excitation operators should give, in principle, different results for (79). However, we have seen that the magnetization current dominates for high q and the approximate SU(4)-symmetry of the ground state can then be used to relate axial vector excitations to vector (longitudinal) excitations. (iii) One might ask whether a nonrelativistic potential theory remains valid in the high-momentum limit. Again we relegate this problem to the last section, where we try to find a relativistic description of the quasielastic process. (iv) The contribution of higher cumulants to the peak position has to be considered. Taking the example in Section 3.3.2 as an indication one would expect an additional negative shift of approximately --g(T) w -5 MeV. Thus ,ipo”) should be even larger and the argument for a strong tensor force contribution becomes more convincing. This remains speculation as long as the higher moments have not been calculated with a realistic interaction. As a step in this direction we have evaluated the moment m2 with the same two-body potential (100) as that used for the energy-weighted sum rule. It is shown in Appendix B3 that for high-momentum transfer (compare (73b)) m2 =+Z($)“(l

-+,‘+fz(Q&+2Za&(l-+)

+z5

(105)

where 5 is related to the ground-state expectation value of a three-body operator. Using Eqs. (99) and (103) we again find that the leading term cancels in the mean square deviation

Aa”;(

+ 62

with a2 = 5 - 22.

Since X2 can be related to the width of the quasielastic peak, Eq. (106) implies a broadening of the peak due to the two-body interaction. Consequently the mean kinetic energy (T) determined from the experimental width will be smaller than the value derived by a Fermi gas fit. Significantly lower values of the Fermi momentum kF have indeed been obtained in the work of Brieva and Dellafiore [ 121. These authors have used a momentum-dependent potential in the single-particle spectrum of the Fermi gas model. In this context we note that the special nonlocal potential we con595/41-15

216

R. ROSENFELDER

sidered in Section 3.3.3. gave the same result as Eq. (106) with 8 = 2E (see Eqs. (AlO), (Al 1)). It can be seen from Table I, where values of 6 for the different interactions are listed, that this relation gives the correct order of magnitude. From an experimental point of view the broadening of the quasielastic peak may be best investigated by a careful measurement of the peak height. This is because the area under the peak is preserved by the non-energy-weighted sum rule m, = 2 so that a broader distribution necessarily implies a reduced peak height. Finally it should be mentioned that similar energy-weighted sum rules for electron scattering have been considered by Drell and Schwartz [71] and by Inopin and Roshchupkin [72]. The main difference to our approach is that these sum rules are to be compared directly with the integrated cross section. However, pion production and transverse contributions in the spectrum make a clear experimental test very difficult [731. 4. SEMICLASSICAL

METHODS

We have seen in the previous section that a study of the first few moments of the structure function can give us detailed information about the final state interaction of the ejected nucleons. In particular, we found that a local (energy-independent) potential cannot account for the empirically observed shift of the quasielastic peak at highmomentum transfer. However, it became increasingly difficult to evaluate higher moments with a nonlocal potential or a two-body exchange interaction. On the other hand we observed that the leading q-dependence of moments and cumulants was given in most cases by the Fermi gas model (see Eqs. (99), (103), (105) and (106)), the interaction effects constituting a small (albeit non-negligible) correction to it. It thus seems desirable to find a simple generalization of the Fermi gas model including (nonlocal) interactions. This is achieved by a semiclassical method based on the Wigner transform of a quantum-mechanical operator [74-771. It was shown in [20] that such a description leads in a natural way to the Thomas-Fermi approximation for bound and the eikonal approximation for scattering states. 4.1. Wigner Transform

In this section we recall some properties of the Wigner transform of a quantummechanical (one-body) operator A, A,@, p) = /d3x (r - $1 A ) r + $)

The reverse transformation

is Weyl’s quantization

(r ( A ( r'> = (27r)" /&I

eiD”.

(107)

prescription for a classical quantity

Aw (J-$q

p) eg.(r-r'l.

QUASIELASTIC

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SCATTERING

217

According to (107) the Wigner transform of a local potential is simply VW(r) whereas a nonlocal potential gives rise to a momentum dependence. For example the parametrization used by Perey and Buck [58],

leads to V,(r, p) = V(r) eC82P2’4. A fundamental formula relates the Wigner transform of a product of operators to the individual Wigner transforms9 (AB)w (r, p) = A,@, p) e*inzBw(r, p).

(111)

ii = 5, * 0, - G, * ii,

(112)

Here

is the Poisson bracket operator. Indeed from (111)

CA,av = WAw , B,) + O(ti2), where (A, , Bw} is the Poisson bracket of the classical dynamical variables A, , Bw . For further application we need Tr(AB) = Tr,, Sd3r (r I AB 1r) = (2~r-~ Tr,, Jd3r d3p (AB)w (r, p), where Tr,, denotes the partial trace with respect to the spin and isospin variables. It is easy to show that this reduces to Tr(AB) = (2.rr)-3 Tr,, !“d3r d3p A&,

p) B,(r, p).

(113)

The main advantage of the new (r, p)-representation is that Eq. (111) suggests a simple semiclassical expansion in powers of fi. We note that in lowest order the Wigner transform of a function of an operator can be replaced by the function of the corresponding classical quantity. 4.2. Local Fermi Gas Approximation

Equation (110) shows that the Wigner transform of the nonlocal potential (109) vanishes for high momenta of the particle. It is precisely this weakening of the interaction in the continuum which leads to the shift of the quasielastic peak at highs Here we explicitly showthe &dependenceof the exponent.

218

R. ROSENFELDER

momentum transfer. Furthermore we have seen that the behavior of a nonlocal potential is similar to that of a two-body exchange interaction (compare (87) and (103)) and that center-of-mass corrections seem to be unimportant at high-momentum transfer (Eq. (104)). All these properties indicate that the nuclear interaction in quasielastic scattering can be well described by a real, nonlocal potential. This choice also eliminates the difficulties encountered when using an energy-dependent potential. As the imaginary part of the empirical optical potential describes the loss of flux from the elastic channel into all the inelastic channels which are included in an inclusive reaction, we identify this one-body potential with the real part of the empirical optical potential. This usual assumption has been critically examined in [78]. Adopting it here too we can make use of the many empirical and theoretical [21, 791 investigations of the optical potential. For the particular parametrization (109) a Woods-Saxon form V(r) = V,[I + e(7-R)la]-1

(114)

is usually taken and the nonlocality range is found to be /3 a 0.9 - 1 fm [58, SO]. Let us now write down the structure function in the (r, p)-representation. Our starting point is Eq. (39), which reads in the one-body case n@(t) = Tr(O+eitHOe-itHp). Here p is the one-body density matrix [81] p = O(E, - H)

(115)

and EF the Fermi energy. Using (113) we can write n@(t) = (2?~)-~ Tr,, /d3r d3p (OteitWe-itH)w

(r, p) p&r, p).

It is easy to show that (e-iq.rAeiq.r)w (r, p) = &(r, Thus for longitudinal

M’dt)

p + q).

excitations we have

1+ = GW3 Tr,, 2

73

]d3r d3p pdr, P)

x [(eitH)w (r, p + q) eiG2(e-i”H)w (r, p)],

(116)

which is still an exact (although formal) expression. In the lowest order of the semiclassical expansion we neglect the Poisson bracket

QUASIELASTIC

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219

SCATTERING

operator and approximate exp(itH)w by exp(irHw). Equation (116) then becomes for spin-independent potentials ,77,~L(t)

and the longitudinal

=

/d3r d3p pw(r, p) Pw(rsD

2p-3

‘Q) e-ifHw(rzP)

structure function is given by

&(q, w) = 2(27~-~ j-d”r d3p p&r, p) S(w - JW,

P + q) + &k

P)) - (a - -w). 1117)

To the same order of approximation the Wigner transform of the density matrix (115) can be replaced by the Thomas-Fermi approximation [Sl] (I181

pw(r, P) = &% - Hw(r, 1.9 = &k,(v) - p), where kr(r) is the local Fermi momentum have a local Fermi gas model $(q, co) = 2(27i--3 ld3r d3p S(0~

- fh&,

defined by H,(r, kr(r)) = EF . We then

p +

9) +

f&4,

PI)

Wdr>

- p> - (w

-

-w>

(119) im.,posing energy and momentum conservation at each phase space point (r, p). The result may also be called a local density approximation as Eq. (118) implies

p&>= Tr& Ip Ir>= TW3

jd3p pdr,

P> =

kF3(r) I

7

(120)

In fact, in the Thomas-Fermi approximation the ground state of the system is obtained by filling particles in plane wave states up to a Fermi momentum determined by the density at that point. Note that (117) correctly vanishes for q = 0 showing a consistent treatment of bound and scattering states. It is also remarkable that a local potential cancels in the energy-conserving a-function leading to the previous Fermi gas result (61). Here the relation n,(p)

r

Tr,
I P I pi

=

&W3

Jd3r

pdr,

P)

has been employed. This result fits nicely into the picture we obtained from the detailed analysis of the moments of the structure function in the previous chapter. In fact, the first moment of (117) reads ml = Z &

+ 2(2~)-~ ld3r d3p [Vw(r, p + q) -

Vdr, p)] p&, p),

220

R.

ROSENFELDER

the second term being exactly the Wigner transform of (85). Thus, our approximation preserves the energy-weighted sum rule. In addition, it is easy to show that

I??,=% z \(&)”

+ k (&J-l

[$ (k - l)(T)

- (V)]

+ 0 (gJ’/

(121)

assuming that limp+, Vw(r, p) = 0. Thus not only is the asymptotic shift of the quasielastic peak generated by this model (note i: = -( V} = -Tr p V/Tr p), but also the broadening of the peak. In particular, one finds Eq. (106) valid with lo

62= (V2) - ( vy.

(122)

Equation (119) can be reduced to

which is a two-dimensional integral once the t-integration has been performed by means of the a-function. We note that a nonlocal potential leads to an implicit equation for t(v, p, w) and to an additional factor h(t) = tlM

aH,/at

=

tlM

tp4

(124)

+ av,lat *

This “Perey factor” [SS] is just the ratio of the ordinary velocity of the particle to the generalized velocity in the presence of a momentum-dependent potential. The latter also enters the convection current Jdr,

This is necessary to maintain wpw

=

[H>

plw = [J!!.

P; 4) = (5

+ v,V,(r,

W5)

p)] eiq’r.

gauge invariance + v,

(c, p + ;) -

v,

(r, p - $1

eiasr=

q . Jw

in lowest order. It should be remembered that the semiclassical expansion requires Vw(r, p) to be a smooth function of p, i.e., the nonlocality should be of short range

WI. lo This requires p&r, p) > 0 which in general is not true. Thus, in order to be consistent, only the Thomas-Fermi approximation (118) should be used.

221

QUASIELASTIC ELECTRON SCATTERING

The transverse structure function can be evaluated similarly generalization of Eq. (68)

ST= &-&2&”

to give the foIlowing

+ p,2&n}

+ 2(27r-3 j-d”r d3pp2M;A~;p’fS

S(w - f&h,

p + @ + Hw(r, 10)

- (w * -co).

(126)

Again the first term is the contribution of the magnetization current, the superscripts p, II referring to proton and neutron Fermi momenta i$Y(r) = (37r2pP*“(r))1/3. The second line is due to the convection current with the generalized velocity responsible for the factor hV. In deriving the above expression derivatives arising from WP)W

(r, P> = t&4- + ~3 &Jr,

PI

have been neglected since they are of the same order of magnitude as the quantum corrections from the Poisson bracket operator. We expect them to be small for heavy nuclei where the Thomas-Fermi approximation should be valid. However, in principle all these corrections can be evaluated systematically [77]. Note that the DarwinFoldy correction for the longitudinal structure function is again given by (67).

w [MN] FIG. 6. Quasielastic cross section in the “local” Fermi gas approximation with the nonlocal potential (109), (114). The parameter values V0 = -94.7 MeV, R = 2.51 fm, a = 1 fm have been fixed by performing a Thomas-Fermi calculation for the ground state. The nonlocality range /? = 0.9 fm was taken from optical potential analyses [58, SO]. No Coulomb corrections were applied and the nucleon form factor was assumed to have the dipole form. Experimental data are from Ref. [7].

222

R. ROSENFELDER

We have performed a numerical calculation of quasielastic electron scattering from 12C with all parameters of the nonlocal potential fixed by ground-state properties (binding energy, rms radius, form factor minimum). A detailed account is given in Appendix C. Figure 6 shows that the calculated cross section compares well with the experimental data without adjusting any parameters. The modification of the convection current due to the nonlocal potential was found to be of little importance whereas the Darwin-Foldy term cannot be neglected (I q 1 m 450 MeV/c at the peak position). This type of calculation can be easily extended to heavier nuclei where the ThomasFermi description of the ground state is expected to work better than in a light nucleus such as 12C. A more stringent test, especially for the chosen form (110) of the nonlocal potential, would be quasielastic scattering under different kinematical conditions. From empirical analyses [22,79,82] the energy-dependence of the real part of the optical potential has been found approximately linear (logarithmic) at low (high) incident energy of the particle. In particular, there is evidence that Re V,,,t becomes repulsive above 300-400 MeV, indicating that the parametrization (110) is inadequate at these energies. 5. RELATIVISTIC

MODELS

We have pointed out repeatedly in the previous sections that a relativistic description of quasielastic electron scattering is desirable. Relativistic effects play an important role in several respects: (i) At high-momentum has to be used:

transfer relativistic kinematics for the ejected nucleon

E, = (p” + IV~)~/~.

(127)

(ii) The transformation properties of a single-particle are different from those for the nonrelativistic case

wavefunction u&*.X

u, D=cE,+M1‘I2 t 1 2E,

(0 . PYKI

+ M> 1 x

(128)

o( is a two-component Pauli spinor). (iii) The full electromagnetic vertex between two on-shell nucleons is

(129) rather than the nonrelativistic reduction we have used up to now. (iv) If the nucleons inside the nucleus obey a relativistic wave equation one has to specify the transformation properties of the potentials. For example, it is well known that a combination of a scalar potential and the zero component of a vector potential

QUASIELASTIC

ELECTRON

223

SCATTERING

gives rise to important effects even at low energies: a strong nuclear spin-orbit force [83] and an energy-dependence of the real part of the optical potential [22]. A simple but consistent framework which contains all the properties listed above is provided by a nuclear field theory due to Walecka [23]. We will use it in the next two sections in a description of quasie!astic scattering which follows closely the Fermi gas and the “local” Fermi gas model of the previous sections. 5.1. Infinite Nuclear Matter The original model [23] consists of a baryon field (#) of mass M coupled to a scalar meson field (4) of mass m, and a vector field (V,) of mass mv . The field equations read (cl + m*“> 4 = &J;$4 (0 + m”> VLL= .wJ~~~, Ga - $TVF” + 884 - M) # = 0.

(130) (131) (132)

Due to rotational invariance only the zero component of the vector field contributes in nuclear matter or spin-zero nuclei. In the high-density limit the sources on the r.h.s. of (130), (131) become large and the quantized meson fields can be replaced by their classical (mean) values +0 , V0 . This basic “mean field” approximation allows for an explicit solution in nuclear matter with the coupling constants and masses fitted to the equilibrium properties of nuclear matter. Equation (132) then becomes for # = u,e-ip’x (P - gv%l~lJ - M*) u, = 0,

(133)

where the effective mass M* is given by M* = M - g&i.

(134)

Equation (133) is just a free Dirac equation with an energy spectrum l D = gvv,

f E,* = g, V, & (p” + M*2)V

(135)

and solutions ua identical to (128) with the nucleon mass M everywhere replaced by M*. We note from (135) that the vector potential cancels in energy differences and that the whole effect of the nuclear interaction can be incorporated into an effective mass. Thus we immediately can use the results of the relativistic Fermi gas model [5, 321 Wib” = g

F

j.*

PY

6(k,p - p) -&

S(w P

P+Q

Ep+q

+

CJMP

+

4,I-J)

(136)

224

R.

for protons and similarly calculate L(P’,

ROSENFELDER

for neutrons. An interesting question now arises when we P) = TrW

+ M*) J,(P + M*) JJ.

(137)

Should we also use the effective mass M* in the current operator (129) or not? In the first case we would have an enhancement of the anomalous magnetic moment of the nucleons inside the nucleus. Figure 7 shows that with this choice it is impossible to get agreement with the experimental data because the magnetic scattering becomes too large. Indeed, it is a well-known result that an effective mass approximation (to be more precise: an approximation where M is replaced everywhere by M*) only makes sense at low-momentum transfer. However, if we adopt the point of view that the effective mass is an approximate way to take into account interaction effects in the nuclear wave function and that the current operator is the same as for free nucleons (in the spirit of the impulse approximation), we can describe quasielastic scattering very well (Fig. 8). Of course, this numerical agreement does not provide a convincing justification of our choice. The main difficulty is that, in order to answer the question definitely, we must be able to calculate the electromagnetic form factors Fli2 of the nucleon. This is clearly beyond the range of the simple field theory we have used as it assumes pointlike nucleons.ll For convenience we give the longitudinal and transverse structure function following from (136)

- (w + -w), ST =2773

(138)

M* p/kv cc iv* s ~E; d,p) I2T:

T2

+ M*2 (P’ -

(

co2 f 2wE,* - q2 2lql -

- (u ---f -co),

(13%

with the relativistic “scaling variable” p- = ; /I q 1- w (1 -. F)1’2

and the combination

[

of form factors T,*(q2) = - 42 4M*2

iMq2)

+ g

4(41)‘,

(141)

T2(q2) = F12(q2) - 42 4M2 F 2Q2). I1 To be more precise, the nucleons become “dressed” by a surrounding cloud of scalar and vector mesons but this is not sufficient to explain the observed nucleon structure (see for example Ref. [86]).

QUASIELASTIC

ELECTRON SCATTERING

w

225

[MN]

FIG. 7. Quasielastic cross section for calcium with the electromagnetic current modified inside the nucleus. The relativistic Fermi gas model (Eqs. (138), (139)) with a Skyrme-III-momentum distribution [84] was used. No Coulomb corrections were applied but the full nucleon electromagnetic form factors of [85] were used. Experimental data are from Ref. [7].

w

[Mev]

FIG. 8. Same as in Fig. 7 but with the free electromagnetic current. The effective mass approximation is restricted to the wavefunction as an effect of the nuclear interaction.

In Eqs. (138), (139) (which hold for protons and neutrons separately) we have generalized the Fermi gas step function to an arbitrary momentum distribution in the same way as in Section 3.3.1. Note that a localized wavefunction necessarily contains negative energy components [4] which have been neglected here. For M* = M, Fl = F, , w, ; q 1< M we recover the nonrelativistic results (64), (68). Figures 9-l 1 show the best-fit results for W, 40Ca and zoePb. A realistic momentum distribution from density-dependent Hartree-Fock calculations [84] and the electromagnetic form factors in the parametrization (8.2) of Hiihler et al. [85] have been employed. There is a difference in the cross sections of typically 5 % if the less precise

226

R. ROSENFELDER

12c E ~500 Mev, M”=0.?45M

I

150 w

11.L

200

o-60”

250

[Mev]

FIG. 9. Best fit to the carbon data of [7]. The free electromagnetic current, a density dependent Hartree-Fock momentum distribution and the full nucleon form factors are used. In addition Coulomb corrections with ac = -4.1 MeV are applied. The fit was restricted to the circled data points.

w

[MeV]

FIG. 10. Same as in Fig. 9 but for calcium (pc = -9.6 MeV).

dipole form factor (I 5) is used instead. The neutron momentum distribution has been taken as n,(p> = N/Z nF(p) and center-of-mass corrections have been neglected (we suppose them to be small from previous discussions). Coulomb corrections have been applied according to Eq. (18). It is remarkable how well this one-parameter fit describes the experimental data especially when compared with the two-parameter fit of Moniz et al. [6] where the Fermi momentum and the shift of the peak have been adjusted. Here we have kept the momentum distribution of the ground state fixed and varied the effective mass as a measure of the final-state interaction. The resulting

QUASIELASTIC

ELECTRON

w

SCATTERING

227

[MeV]

FIG. 11. Same as in Fig. 9 but for lead (rc = -24.9 MeV).

values of M* are in reasonable agreement with what is used at low-momentum transfer but significantly lower than M* = 0.56 M obtained by solving the mean field equations (130)-(132) in nuclear matter [23]. This discrepancy is resolved when we consider linite nuclei in more detail in the next section. It is interesting to analyse the position of the quasielastic peak in the present relativistic description. This is nontrivial because Eqs. (138), (139) depend in a complicated way on the relativistic invariants w = q * P/M, and q2. Nevertheless we may estimate the maximum of the integrals to be occurring when p- = 0, i.e., for

In terms of the nonrelativistic

variables w, q2 Eq. (143) reads wmax m (q%+ M*2)1/2 - M*.

This is simply the kinetic energy of a nucleon of mass M* which was at rest before it acquired a momentum q from the virtual photon. The shift of the quasielastic peak with respect to the nonrelativistic Fermi gas position q2/2M is thus c(q) = (92 + &f*2)W - M* - 2,

(144)

valid at high-momentum transfer (1 q 1 >, 2k,). This is plotted in Fig. 12 for different values of M*/M. Note that there is no asymptotic saturation as in the nonrelativistic models. Experimentally the momentum dependence of the peak position has been studied in [87] for the light nuclei 6Li, ‘Li, vBe:12 Equation (144) should be more valid for heavier nuclei where it predicts the la We note that the result for BLi is at variance with RefS. [6,7].

228

R.

ROSENFELDER

loo!-

M” =0,56M

,L._-

pl.-.----.. 400

600 l$CMev/cl

800

FIG. 12. The asymptotic shift of the quasielastic peak according to Eq. (114). The two points at I q 1 = 450 MeV/c are from the analyses of Ref. [7] with Coulomb and Coulomb + mesonexchange corrections applied, respectively.

shift to be independent of the mass number. This is indeed the case if Coulomb corrections are taken into account. It can be shown from (18) that these amount to subtracting 0.87 ZA-l13 MeV (145) from the values determined in Ref. [7]. Figure 13b demonstrates that c = 30 MeV is fairly constant across the nuclear table except for the light nuclei where surface effects may be important. Using this value in Fig. 12 we arrive at M* w 0.7 M at I q I M 450 MeV/c, which is in good agreement with the numbers we obtained from our individual fits. In Fig. 13c we also have tried to subtract the effects of meson-exchange contributions. According to the calculations in [32, 331 they become more prominent for heavier nuclei; in fact, a more detailed analysis shows that the corresponding shift varies like kFs from -2 MeV in 12C to -6.6 MeV in 2osPb. By applying these corrections we actually observe an extended mass range where the shift of the quasielastic peak is constant. Similarly meson exchange corrections may also be responsible for the differences between fit and experimental data we observe in Fig. 11 for lead. 5.2. Finite Nuclei

For finite nuclei the coupled mean field equations (130)-(132) have been solved in the Thomas-Fermi approximation [24,25]. The resulting local effective mass M*(r) is plotted in Fig. 14 for calcium and lead. It is evident that only in the interior of the nucleus M*/M is near the nuclear matter value 0.56 but approaches one near and beyond the nuclear radius. We may thus conclude that the fitted value M* w 0.7M we obtained in the last section represents a suitable average over the nucleus. The easiest and most natural way to perform such an average is the “local” Fermi gas model we derived in Section 4.2, According to this model (136) can be generalized by replacing s

d3p B(k, - p) .** -+ const

I

d3r d3p 0(k,(r) - p) a..,

QUASIELASTIC

ELECTRON

40:.

: i

301. 20:.

i

1

I

229

SCATTERING

i

1

i i

(a)

IO I

m >

30.

g

20

i

w

(b)

IO -1 P 30; 20 IO I

FIG. 13. (a) Shift of the quasieIastic peak for various nuclei as determined in Ref. [7]. (b) Coulomb corrections subtracted. (c) Meson-exchange contributions 132, 331 are taken out in addition. The dashed lines denote the average value of the shift for heavy nuclei.

FIG. 14. Effective mass M*(r) approximation) [24,25].

= M - g&,(r)

in Walecka’s mean field theory (Thomas-Fe.rmj

230

R.

ROSENFELDER

Since it is local the vector potential cancels again in the energy-conserving S-function and we obtain for protons and neutrons separately SL

=

V77F3~d3~d3p

&$dr)

-p)

E*c~~$)crj

B

x /7’,*(r) $ + Tz ( Ez(;*&u’2 S, = 2(W3

6(w

-

Df(l

E,*,,(r)+

I,*)

,“I - (w + -co),

/d3r d3pB&.(r) - p) E*;;!& S(w - E;+,,(r) + I,*) B II+9

x 2Tf(r) + T, ” -&,fz’qz 1

1 - (w -+ -0)

with E;(r) = (p” + A4*2(r))1’2,

2 T,*(r) = - 4M*“(rJ (4 +

(148)

M*(r) F2 2. M )

(14%

As in Section 4.2. the expressions for SL , Sr can be reduced to a two-dimensional integral. From a technical point of view they are simpler as no implicit equation has to be solved to perform the integration over the &function. Figures 15 and 16 show the results for quasielastic electron scattering from 40Ca and 208Pb. The densities p*,“(r) (to determine kiln(r)) and the effective mass M*(r)are taken from Thomas-Fermi calculations which successfully describe the ground-state properties of these nuclei [24,25]. Thus there is no free parameter left to be adjusted to the experimental data. The surprisingly good agreement with experiment is due to 40 Ca &=500MeV.

w

B=6d

[ Mev]

FIG. 15. Quasielastic cross section for calcium in the relativistic “local” Fermi gas model (Eqs. (1463, (147)). Densities and fields are taken from ground-state calculations 124, 251. The calculation uses the free electromagnetic current (129) with form factors from Ref. [85]. The Coulomb distortion of electron waves is taken into account with pc = -9.6 MeV. Experimental data are from Ref. [7].

QUASIELASTIC

ELECTRON

1

40

80

I

120

I

I60 w [

231

SCATTERING

200

240

280

Mev]

FIG. 16. Same as in Fig. 15 but for lead. The solid line includes Coulomb corrections (rc = -24.9 MeV) whereas the dashed line does not.

the built-in features of a relativistic theory we mentioned at the beginning of this section. To be more specific, a Foldy-Wouthuysen transformation of the nuclear Dirac equation gives

(150) showing that a simple Dirac Hamiltonian with a mixture of scalar and vector potentials contains a variety of effects on the nonrelativistic level: a central potential (second and third terms), a Darwin correction (fourth term) and a spin-orbit potential (last term). In particular

represents a nonlocal interaction similar to that in the Hartree-Fock Hamiltonian with density-dependent interactions [65]. It is this nonlocality which is responsible for the shift of the quasielastic peak as was discussed in Section 4. Of course, Eq. (150) is only valid for p < A4 whereas at high energies the equivalent real potential can be shown to be linearly energy-dependent [22]. This (empirically too strong) energydependence has the effect of extending the high-energy tail of the quasielastic peak as seen in Figs. 15 and 16. In this region Czyz and Gottfried [36] originally proposed to look for correlation effects which subsequently were found to be small [88]. This is consistent with our sum-rule analysis and the observation that the momentum distribution is modified only for momenta greater than 2 fm-l [60] which play no significant role for quasielastic electron scattering.

232

R. ROSENFELDER

6. SUMMARY We have studied in detail inelastic electron scattering in the region of the quasielastic peak with an emphasis on the effects of the nuclear interaction on position and width of the peak. The investigative tool was the “characteristic function” I the general properties of which have been discussed. Being the generating function of the moments, F(t) is especially suited for a description of quasielastic electron scattering at high-momentum transfer as a few moments characterize the smooth spectrum. In particular, the first moment was shown to determine the peak position to a good degree of accuracy. In this respect quasielastic electron scattering is complementary to the study of photoabsorption sum rules: in the latter case we investigate the longwavelength limit of the energy-weighted sum rule whereas the former explores the high-momentum limit. Consequently the observed shift of the quasielastic peak could be related to exchange forces in the two-body interaction-an interpretation which also has been proposed in [8]. On the one-body level the nonlocality of the effective interaction (seen, e.g., as an energy-dependence of the empirical optical potential) affects quasielastic electron scattering in a very similar way. By means of a semiclassical expansion we have shown that nonlocal interactions can be incorporated very naturally in a “local” Fermi gas model. Together with a Thomas-Fermi calculation of groundstate properties this model has been successfully used to calculate the inclusive cross section from lzC without free parameters. Quasielastic electron scattering at j q 1 hv 500 MeV/c crosses the borderline where relativistic effects become important. In the nonrelativistic calculation this is reflected by ambiguities in the reduction scheme and by the fact that “correction” factors are no longer small. We have applied a simple relativistic model, Walecka’s nuclear field theory in the mean field approximation [23], to quasielastic electron scattering and obtained good agreement with the experimental data in a parameter-free calculation. The few coupling constants and masses in this model are fixed to describe static properties of nuclei and nuclear matter. It is an attractive feature of this calculation that not only are the correct kinematics and the full electromagnetic vertex implemented but also the mixture of scalar and vector potentials provides a strong spin-orbit force and an energy-dependence of the real part of the optical potential. Thus, from a phenomenological point of view the basic properties of the nuclear interaction in bound and continuum states are built into the relativistic model in a very economical way. An open question in this model is whether the form of the electromagnetic current is modified by the presence of other nucleons or not. Our numerical calculations seem to indicate that the free operator should be used but a more conclusive answer to the fascinating question “Do nucleons inside a nucleus behave like free ones?” has to await a comprehensive theory which also includes pions and p-mesons. Such an extension of the original model has been worked out recently and has met with some success in explaining properties of finite nuclei [24, 251. A systematic study of the nuclear structure function with explicit consideration of mesonic degrees of freedom seems worthwhile even before we enter the region of explicit pion production with its many open problems.

233

QUASIELASTIC ELECTRON SCATTERING

APPENDIX

A: LONGITUDINALSTRUCTUREFUNCTIONFORASEPARABLEPOTENTIAL

The normalized given by

bound

state wave function

for the separable potential

(88) is

where EB > 0 is the binding energy determined from the equation -- ; = C(-EB)

642)

with (A3) For the bound state problem

the in in the denominator

of (A3) is irrelevant. The

scattering solution is 1 1 + XC(k2/2M) .

(p 1 a+!$‘) = S(p - k) + 2M;d(p’ ;;‘F;ck’

Thus the structure function can be written as S(q, w> = /d3P %w =

s

d3P

-

1 &
EB -

-

q)/’

p2/2M)

6(w

-

/ &dP

-

EB -

p2/2M)

9)

-

1 +A;&pf;~,

+

$

Irn

I (&

(

+ 1 +

xccw

) q) /’

EB , q) _ EB)

)

(A4)

with (A5)

The first term in (A4) is just the expression obtained when the continuum states are taken as plane waves The second term contains the distortion effects and exactly cancels at zero momentum transfer the contribution arising from the non-orthogonality of bound state and plane wave. Note that C(W - %) and I(w - cB , q) are real below threshold so that S(q, w) = 0 for w < EB. For the Yamaguchi potential (89) the corresponding integrals can be performed analytically. For simplicity we take the special case EB = p2/2M. Then (A4) becomes (y

=

/ q l/p,

x

=

(+B

-

l)““)

234

R. ROSENFELDER

where C = 16/7ren , D = (1 + y2 - x2)” + 4x2 and the elastic form factor is FO(y) = (1 + JJ~/~)-~ It can be seen that the first term dominates at high-momentum transfer and that in this limit Eq (A6) has its maximum at x = y, i.e., for o = q2/2M + cB. The first moments are given by

m,(u) = 1 - KJ2(Y), ml(Y) = bb2 + 2 - 2MJ7)], mz(v> = dB2[Yy2 + Y4 + + y4F,(

(A7)

’

648)

y)].

(A%

Equation (A8) gives an effective mass M* = &Mat low ( q 1 and lim h1 = & cl+m

+ 2Eg .

(Al’9

Similarly (Al 1) which is consistent with Eq. (76) ((T) = +J but with an additional nonlocal interaction.

APPENDIX Bl.

B: EVALUATION

term due to the

OF MOMENTS

m,

Let us introduce the vector (V) and axial vector (A) correlation muon capture [62] DYsA(ql , q2) = (0 / 1 I+ i#j

T-(j)(l,

functions used in

$0(i) * o(j)) eiql+iqz’x; 10).

031)

For p-shell nuclei and harmonic oscillator radial wave functions one finds D(q, q) = D(q) = @(c,, + c,Y +

C2Y2)

032)

where y = +(qb)2 and b is the oscillator parameter For vector excitations one has c,” = Z and c1v = 0. Furthermore c2” = 0, 8/9, 2 for *He, 12C, la0 in a pure jjcoupling model [62]; configuration mixing with Cohen-Kurath wave functions gives c2” = 1.09 in 12C [63]. In the shell model the correlation term in Eq. (92) can be split into a direct and an exchange term (0 I C eiejeiq.(xi-xj) 10) = P(F&, i#i

- D”(q).

(B3)

QUASIELASTIC

ELECTRON

Note that the direct term cancels the ground-state corrections are neglected:

4l(q)= z - DW

235

SCATTERING

contribution

if center-of-mass

(no CM corrections).

034)

As the two-body term is unaffected by CM corrections while the shell-model factor picks up the usual correction factor exp( y/2A) we obtain m,(q) = z - D”(q)

- Z2(F&,

1)

(e’i-4 -

form

(with CM corrections).

(B5)

The shell-model form factor for 4He, 12C and I60 is given by (FJSM = e-VI2 (1 - 9

y) .

036)

B2. ml In an N = Z shell-model ml

(pot)

=

_ s

nucleus Eq. (101) can be directly evaluated to give

~3P[~tP) - @(P + ci, - !&P

-

mw

+ wo

D”(P) + QMWP)l, (B7)

where H(r) = HV(r), M(r) = MV(r) and V(p)

=

(24-3

pr

V(r)

eip.r.

038)

For hard-core potentials the lower integration limit starts at the hard-core radius to simulate the defect wave function [66]. The 12C correlation function DA(q) has the form (B2) with coefficients C/ = (38/9) (5.47), clA = 32/27 (0.35), c2” = (8/9) (1.09) forjj-coupling [62] (Cohen-Kurath configuration mixing [63]). The harmonic oscillator parameter is taken as b = 1.63 fm. B3. m2 We calculate the second moment by means of Eq. (81) with

- ; (1 - 6&&j

+ &P,p,)($j)

x 7(i))3] .

W

Let us write D+ = &l {-->+ e-*aex:. In the q -+ co limit only terms with j = I survive in D+D. Thus we have to evaluate the ground-state expectation value of one-,

236

R. ROSENFELDER

two- and three-body operators. It is easily seen that for spin-zero nuclei the result is given by (105) with

Assuming an isospin zero Slater determinant for the ground state (BlO) can be written as

where m2A(q, FQJ = ZF,(q, - q,) - ~“%l, is the .generalization form Nly

99,)

VW

of (B4). For p-shell nuclei the correlation functions (B 1) take the

q2) = 2e-(Yl+Y~)'2U + 441 - ivl)(l

- fy2) + ~(Y,Y,Y~

pdx) + 4y,y,P,(x))

(B13) with yl12 = $&b2, x = q1 * q2/l q1I j q2I . In particular, for 12C: day = 2, d,” = 4/3, d2v = 219 and doA = 10/9,dlA = 413, d2A = 26181. It can be checked that for q1 = q2 Eq. (B13) reduces to the expressions given in Appendixes Bl, B2.

APPENDIX

C: THOMAS-FERMI CALCULATION WITH A NONLOCAL POTENTIAL

Using the semiclassical expression (118) for the one-body density matrix the total energy

1+ G&l = Tr (& P) + TrV’p/p)+ 2 can be expressed as a functional of the Fermi momentum is taken as V,(r)

73

W Vcp)

(Cl)

kr(r). The Coulomb potential

= (1 - &) CYj-d3x , p’“),

/

(C2)

so that the ejected proton moves in the field of (Z - 1) protons and the last term in (Cl) becomes the direct part of the Coulomb energy. Equation (Cl) should be varied under the constraint jd3r p(r) = sd3r 3

= Z

(C3)

QUASIELASTIC

for protons and similarly

ELECTRON

237

SCATTERING

for neutrons. The resulting Thomas-Fermi EF = g

+ V(r, kF(r))

equation reads

+ Vc(r).

At the classical turning point r,, the Fermi momentum vanishes. Thus (C4) becomes at that point

(C4)

and consequently the density

w We solve Eqs (CZ)-(CS) iteratively series

by expanding the density in a Fourier-Bessel

f b,q,2j,,(q,r) f(r) = J$ ?7 0 “=I This procedure has the following reads

with qv = z.

nice features: the normalization

condition

(C3)

5 (-)y+l b, = I v-1

and the Coulomb potential is V,(r)

= (’

- 1)a r.

for

r < r, .

The elastic form factor is simply given by

F,(q)= jotIqI ro>$ t-Y+1 bv& Y

"=l

m

and (C6) can be inverted to give b, = g

,,” dr r2p(r)jo(q,r)

= 2l;b(qJ.

Also kinetic, potential and Coulomb energy in (Cl) can be worked out analytically. We start with a guess for the Fermi energy EF < 0 and a set of coefficients b$‘), calculate the classical turning point r, from (C5), solve the Thomas-Fermi Eq. (C4) for k+(r) and use (ClO) to get a new set of coefficients b,(r). The normalization condition is checked until (C7) is fulfilled to better than one part in 104. The main technical problem is the solution of Eq. (C4). As the Coulomb potential is slowly varying we use the set br-l’ in the nth iteration step to calculate it via (CS). With a nonlocal potential of the form (110) the problem can then be reduced to solving the transcendental equation y = xe-u

(x > 0).

(Cl 0

238

R. ROSENFELDER

The same problem arises in calculating Eq. (123) although with greater values of X. The power series y(x)

=

f

(-)*+I

g

xk

(CW

k=l

however, is only convergent for x < e (note: u(e) = 1). In the range 0 < x < 1000 we found the following polynomial approximation (from a Chebyshev fit) most useful y(t) = t(0.999884 + 4.24131 * 10-3t - 2.71508 10-=Q2 + 3.01801 * 10-2t3 - 1.1742 10-2t4 + 2.33058 * 10-3t5 10-V + 9.92016 . 10-V) + E l

l

-

2.37995

l

(C13)

with t = 5 ln(l + 3x + 8~x3 and 1E ] < 8 x 10-5. ACKNOWLEDGMENT I would like to thank J. D. Walecka, B. D. Serot and T. W. Donnelly for many helpful discussions and suggestions.

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