Theoretical challenges in neutron scattering

Ph~,sica 137B (1986) 148-156 North-Hnlland, Amsterdam

T H E O R E T I C A L C H A L L E N G E S IN N E U T R O N S C A T T E R I N G Stephen W. LOVESEY Rutherford Appleton l/ahoratorv, ('hilton. O.~Jbrds'hire. OXI I OQX. ['K

Topics in the interpretation of neutron scanering experiments from paramagnets and q u a n t u m fluids are used to illustrate the puissance of the technique in condensed matter research, and to record some fund,,mental shortcomings in the available theory of many-particle systems.

I. Prologue A key virtue of neutrons, as a probe of condensed matter, is that the neutron-matter interaction is so weak that first-order perturbation theory (Fermi's Golden Rule) is wholly adequate to account for the neutron scattering cross-section. In other words, neutron scattering provides information on the chemical and physical properties of matter that is undistorted by the neutron probe. An interpretation of the measured cross-section is not hindered by uncertainty about the nature of the radiation-matter interaction or correct specification of the functional form of the cross-section. Van Hove's formulation of the cross-section factorizes it into terms that specify the strength of the interaction (the square of the appropriate scattering length) and correlation functions that depend only on the static and dynamic properties of the target particles. Calculation of the correlation functions for realistic models of the target particles is the primary objective of theory. The framework of theory behind the interpretation of neutron scattering experiments is linear response theory, and Van Hove's reduction of the cross-section to a correlation function is a specific example of the fluctuation-dissipation theorem (for recent reviews see ref. 1 and 2). Linear response theory is derived by retaining the first term in the expansion of the observed response of a sample in powers of the external perturbation. The approximation is equivalent to the use of Fermi's 0378-4363/86/$03.50 ". Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Golden Rule for transition rates, and the first Born approximation in scattering theory. The fluctuation-dissipation theorem shows that the linear response is expressible in terms of the spectrum of spontaneous fluctuations in the appropriate target sample variable. For example, the coherent neutron cross-section is an observed (linear) response that is expressible in terms of the spatial and temporal Fourier transform of the autocorrelation function of the target particle density: this function is usually denoted, following Van Hove, by S(k. va) where k and ~ are the changes in the neutron wave vector and energy (h = 1) that occur in scattering. "['he goal of theory is to calculate the spectrum of spontaneous fluctuations, e.g. S(k, ~) for coherent neutron scattering. Exact calculations of the cross-section are possible for some idealized models; examples are scattering off perfect Fermi and Bose fluids, a particle confined by a harmonic potential, and linear hydrodynamic models. For the most part idealized models are only useful guides in the interpretation of measurements, and the theorist seeking a comprehensive interpretation must accept the challenge of the subtle features of non-trivial many-particle models. The few research topics discussed here reflect, to some extent, my own interests, and no attempt is made to catalogue theoretical challenges in neutron scattering. Even so, the selected topics should serve to illustrate both the puissance of the neutron scattering technique is condensed matter re-

S. IV. Lovesey / Theoretical challenges in neutron scattering

149

search, and the very fundamental nature of questions posed by confrontations of theory and experiment.

propriate expression for the response function is

2. Paramagnets

f~

For a perfect paramagnet, in which there are no interactions and no correlations between the ions, magnetic neutron scattering is strictly elastic, spatially isotropic, and independent of the temperature. Scattering from exchanged coupled (Heisenberg) paramagnets is distinctly different, even in the limit of infinite temperatures when the short-range spin correlation vanishes. The difference is revealed in the frequency moments of the spin response function [2]. For the mean-square energy width is found to be proportional to the square of the exchange parameter and the wavevector dependence reflects local symmetry, is k 2 for k--, 0, and maximum widths are achieved at the zone boundary. Higher-order frequency moments imply that the response function approximates to a Lorentz curve near the zone centre. For high temperature and large wave vectors, theory and computer simulations [3-5] of the spin response function show a peak at finite frequencies which is evidence of a spin oscillation mode. The gross features of the temperature dependence of the paramagnetic scattering are obtained by noting that the normalized frequency moments are inversely proportional to the wave-vector dependent susceptibility. A phase transition is heralded by a strong increase in the susceptibility at the wave vector that characterizes the ultimate, ordered state. It follows that, for such wave vectors, there is a pronounced decrease in the inelasticity of the scattering as the phase transition is approached by decreasing the temperature. The spin response function observed in neutron scattering experiments is usefully decomposed into the spin relaxation function [1,2], isothermal susceptibility x ( k ) and the detailed balance factor, {l+n(to)}=

{1-exp(-to/T)}-',

where the equality defines n(to) and T is the temperature (Boltzmann's constant = 1). The ap-

S(k, to) = x ( k ) t o ( 1 +n(to)}F(k, to).

(2.1)

Here, F(k, to) is the relaxation function which is an even function of to, and normalized to unity, dto F ( k , to) = 1.

2.1. Long wavelengths In the limit of long wavelengths the susceptibility approximates to

x ( k ) = A / ( t c 2 +k2), where t¢ is the inverse correlation length, which vanishes at the phase transition, and A is only weakly temperature dependent. For small k and long times (small to) the spin density obeys a diffusion equation, and the corresponding form for F( k, to) is

r( k, to) = F( k ) / { to2 + [ e ( k ) ] 2 ) . Above the transition (at T = T~)

l'(k)=kZD,

T> T~,

where D is the spin diffusion constant. For ferromagnetic systems various theories predict a temperature dependence Dax 1/2, while for T = T~ the width function F ( k ) is finite and proportional to k 5/2 (see [6,7] and references therein). These predictions are in accord with experiments. For an antiferromagnet critical slowing down is observed in the relaxation of the staggered magnetization and the prediction 1"c0¢3/2 is also in accord with experiments. The results for F(k) differ from the conventional theory of critical slowing down, proposed by Van Hove, because the transport coefficients diverge at the critical temperature [6,7]. Three types of theory have been used to study dynamic critical phenomena, namely, coupledmode theories, dynamic scaling and renormalization group techniques [6]. The latter provides systematic perturbation theory for correlation functions near T~ and critical exponents [8,9] and hence some foundation for the phenomenological dynamic scaling and coupled-mode theories. Coupled-mode theories are very successful. They have been developed by numerous authors [3,9]

150

.S'. H ~ l.ot'es~ 3' / T h o ~ r e t t c a l c h a l l e n v , e v m ?lelllrotl .~( a t l e r l n R

and all theories lead to essentially the same equation for the spin correlation function. The latter is consistent with dynamic scaling for both ferroand antiferromagnetically coupled Heisenberg magnets. Moreover, the computed F(/,-. ca) for ferromagnetic coupling is consistent with experiments for all wave vectors and temperatures. The known shortcomings of the coupled-mode equation for t.'(/,. 0~) are failure to predict a three-peaked response for antiferromagnetic systems, observed near the critical temperature in neutron scattering experiments and computer simulations, and the absence of long lived spin oscillations in one-dimensional magnets [101. For the one-dimensional Heisenberg magnet the critical temperature is T = 0, when F(k, oo) for classical spins is exhausted by undamped spin waves. Long lived spin waves have been observed in scattering off numerous highly anisotropic rnagnets that exhibit properties expected of one-dimensional spin systems, at least in some restricted. but readily accessible, temperature range [11]. Loveluck and Windsor [15] conclude, on the basis of a comparison between theoretical and computer simulation studies, that there is no wholly adequate theory of the spin dynamics of classical spin Heisenberg chains. Measurements of critical scattering from ferroand antiferromagnetically coupled systems differ in the ordered phase. Whereas a central, diffuse peak is observed in the antiferromagnets MnF~ and RbMnF~ no such peak is evident in scattering from Fe, Ni or EuO just below the transition temperature [12]. No theoretical explanation of this observation exists [8,9].

2.2. Short wat:elengths Measurements on the simple ferromagnets EuO and EuS [12,13] reveal distinct structure in F(k, ~c ) at finite ~0 for large k. near the zone boundary, and temperatures well above T,.. The structure is consistent with the predictions of coupled-mode theories [3], and it is also observed in computer simulations [5] at T = zc where there is no shortrange order between the spins. The spin response of EuO at the zone boundary shows a sharp spin wave at T = 7 ~ - 10 which

remains a distinct feature at T = 2T~, albeit weaker and more heavily damped. Spin waves at small k soften and coalesce, to form a peak at ~e --0, as the temperature is raised through 7~. Such a behaviour is observed for Eu() halfway to the zone boundary for the (111) direction at which k = 1.2 A ~. Recent neutron scattering experiments on paramagnetic iron and nickel show a similar cross-over from spin diffusion behaviour at small k to heavily damped spin waves at large k [14]. For nickel (77 = 631 K) the cross-over wave vector at T = 1.06T~. is about 0.25 A 1 which is small compared to the maximum allowed wave vector (2.0A i). Thus far there is no theory for the spin response of metallic ferromagnets that is as successful as the coupled-mode theory of insulating ferromagnets legitimately described by a Heisenberg exchange interaction.

3. Quantum fluids In this section we consider neutron scattering off ~tle, aHe and electrons in simple metals and semiconductors. These topics represent a well established research subject (4He), a tribute to the ingenuity of experimentalists (:~He) and a burgeoning field of research made possible with advanced spallation neutron sources (electrons in metals and semiconductors). Neutron-nuclear scattering from a fluid consisting of a single isotope measures the weighted sum of two correlation functions. We denote the single-atom coherent and incoherent cross-sections by ~. and o., respectively. If the N scattering nuclei are located at positions R,,, and possess spin angular momentum i,, the response function is [2] .~( k. ~ ) = ( l / 2 ~ r N ) f ~ [ ~ d t e x p ( - i ~ o , )

× Y] { ~.Y,,,,(/,, t ) + { o,/i(i + 1)} x (exp( - i k . R , , ) i , . i , , ( t ) e x p { i k . R , , ( t ) } ) } . I3.11

S. W. Lovesey / Theoretical challenges in neutron scattering

where

Y,,~(k. t)

= ( e x p ( - i k . R,,)exp{ik. R b ( t ) } ). (3.2)

Here, ( . . . ) denotes a thermal average of the enclosed quantity, and R~(t) and i,,(t) are Heisenberg operators. Neutron scattering from 4 He is purely coherent, whereas for 3He the single-atom incoherent cross-section is approximately onequarter of the total single-atom cross-section. For a perfect quantum fluid,

151

where po is the momentum operator conjugate to R,. If a steady magnetic field B = curl A (where A is the vector potential) is applied to the electrons then p , = - iW~ + ( e / c ) A ,

(3.5)

where e is the electron charge and c is the velocity of light. The response function for magnetic neutron-electron scattering is

S(k,

oa) = ( 1 / 2 ~ r N ) f _ ~ d t exp( -i~0t)(Q+.

(3.6)

,90(k, oa) = ( o/2rrN ) f_2dt × e x p ( - i o a t ) Y',

Q(t))

Yoh(k, t)

a,b

= (og/N){l + n(~o)} Y'3{w

+

E(q)

For perfect quantum fluids, the response functions for neutron-nuclear and neutron-electron scattering (in zero magnetic field) are very similar. The response (3.6) for a perfect electron fluid is

q

S(k, ~o)= ( I / N ) E -E(k +q)}

(nq-

ha+q),

(f~-A+q).

(3.3)

Bose

-1

•

Fermi.

S( k ) =

The spin degeneracy factor g = (2i + 1), and o = % + o, is the total single-atom cross-section. The integration in (3.3) over the wave vector q can be made analytically [2]. The result (3.3) coincides with the result for a Boltzmann particle when we take the limits of high temperatures (T-+ m) or large scattering vectors. In this instance the static structure factor S( k ) = f ~ d ~

{c/,(k, ~0) / o } = 1:

Boltzmann,

whereas S(k) depends on k for perfect quantum fluids because of correlations induced by exchange forces. Consider now magnetic neutron scattering from electrons. The interaction operator is the sum of spin and momentum densities, [2]

Q = E exp(ik'R~){k×(i~xk)-ikxp~}/k2` a

(3.4)

(3.7)

The integration over the wave vector q can be performed analytically, and from the result we obtain the static structure factor for a perfect, degenerate electron fluid,

and

fq=(exp{(Eq-#)/T}+l)

q)2/k')}

xS{~o+ E ( q ) - E ( k +q)}(fq-fk+q).

Here, E ( q ) = q2/2M is the free-particle energy and the occupation functions for Bose and Fermi statistics are

nq=(exp{(Eq-bt)/T}-l)-):

{I + 2(k × q

3), [ 5 ~/-5+y 1 2 ~+5T2,

2 2

17y 2 120 ) '

0_-2,

where the reduced wave factor y = k/p r and the Fermi wave vector Pr is determined by the electron density p through pf = (3~'2p) W3. The structure factor approaches ½ for k >> pf, where correlations are negligible, and increase like 1/k in the limit k ~ 0. The latter effect arises from the momentum density in the interaction operator which is weighted by a factor 1/k.

3.1. 4He As the sample temperature is reduced to absolute zero, the detailed balance factor approximates to a step function. It is then convenient to express the response in terms of the dissipative part of the generalized susceptibility x(k, ~0), and

S. W. l.ocexev / 17woretical challenge.~ m m,utron .~catterm.e,

]52

at absolute zero

I'X"(k,~),

rrS( k. ~0 ) = '/0.

~,>0,

~ < 0.

(3.9)

Because the response vanishes for sufficiently large w. it follows that S(k. ~o) for a degenerate system possesses at least one m a x i m u m as a function of ~0, for fixed k. For particle-density fluctuations, observed in scattering off 4He, we have the fundamental sum rules d~0 ~ X " ( k . oa)= (~rl,e/M)

f

(3.10)

-L

and

f

3.11)

The mean-square energy width is the rauo of (3.10) and (3.11), 2

{ k : / M x ( k )}.

13.12)

In the long wavelength limit the compressibility sum rule for the isothermal susceptibility is lim x ( k ) = 1/Mi ,~, /,

S(k. ~)=

[(k/2.~t')8(~o-kc), 0.

¢ c > 0 . (3.141 0a < 0.

Because this result satisfies the f-sum rule (3.10) v,e are certain that the response at ~ = kc is the only contribution to S(k. ~e) in the limit "/'= 0 and k, ~o--+0. In view of this we might expect X"ik. ~) for larger k and o0. and small 7". to bc the sum of delta functions, to a good approximation. The expression

X"(k, ~ ) = { ~ , , x ( k ) / 2 }

d,~ {X"(k, ,,,)/,,,} =~×(k).

,,.,,, =

tion is consistent with an exact result for the coherent response of 4He in the limit 7 = 0 and small k and va,

(3.13)

,(1

where ~' is the isothermal sound velocity. The very different behaviour of dense 4 He con> pared with ~He (at low temperatures) is due largely to the difference in the q u a n t u m statistics for boson and fermion particles. For example, bosons tend to occupy the same single-particle states in marked contrast to particles subject to the Pauli exclusion principle. Whereas the low-lying states of a normal Fermi fluid can be expected to be similar to those of a perfect fluid, the low-lying excitations in an imperfect Bose fluid have a collective p h o n o n character. This latter feature of a Bose fluid is believed to reflect the existence of a condensate of z e r o - m o m e n t u m particles for temperatures below the ~.-transition. Numerous calculations of the condensate fraction n~ give no - 0.1 at absolute zero, whereas neutron scattering off aHe at T = 1.0 K is consistent with n . = 0.146 +_

0.035 [18,19]. The neutron data for 4He below the ?~-transition show that for small k the response is dominated by a sharp peak [16,17]. This observa-

x [at~

- ~,,)..

af~

+ <,)]

(3.15t

satisfies the sum rules (3.10) and (3.11), and it is consistent with (3.14) in the limit of small k. The frequency ~0o shows a marked k-dependence when the isothermal susceptibility reaches its maximum value. The dip in ~ is usually called the roton minimum for historic reasons. Such behaviour is in accord with the experimental finding but it lacks quantitative agreement, e.g. at k ~ 1.9 A ~. which is close to the minimum in ~,, the approximate dispersion w~,- 1.2 meV whereas the observed value is 0.75 meV. However the approximation is fair for smaller wave vectors, k <

0.SA '[16]. The experimental results contain a significant weight at high frequencies which is evidently not included in (3.15). For a wave vector k - 2.5 A the p h o n o n and high-frequency contributions to the response are, in fact, more or less equal, and increasing k further results in the total disappearance of the p h o n o n contribution. Although the interpretation of the neutron measurements has attracted much attention there is no really satisfactory theoretical picture yet. The p h o n o n mode persists above the k-transition at long wavelengths. We can attempt to interpret this using a Hamiltonian for the imperfect Bose fluid that is the sum of the kinetic energy' of particles with an effective mass M*, and a simple pair-wise interaction. Within the random phase approximation, discussed in more detail in the

S. IV. Lovesey / Theoretical challenges in neutron scattering

next subsection,

x ( k , to)= K(k, t o ) / ( 1 - u ( k ) K ( k ,

to)}, (3.16)

where u(k) is the spatial Fourier transform of the pair-potential, and K(k, to) is the susceptibility of a perfect Bose fluid, in which the chemical potential is determined from (3.17)

N=Y'.nq. q

To incorporate the measured values of the particle density as a function of temperature we make the replacement

( E( k ) - ~ , ) / r = ( M / M * ) E ( k ) +O, where O is determined by (3.17), and O = 0 at the h-transition. The phonon dispersion is obtained from the equation

1 - u(k ) K ( k , to)=O. For large to,

K( k, to)-* (pk2/M*to2), and setting to = sk we conclude that the phonon velocity

s = { pu(O)/M*

},/2

(3.18)

The value of u(0) > 0 can be determined from the phonon energy observed at 2.3 K. Using the saturated vapour pressure value of the particle density p at the h-transition, M * = 1.4M, and 8 increases from zero at the transition to 0.48 at T = 4.8 K. The theory then predicts that the phonon velocity should increase as the temperature is increased from 2.2 to 4.2 K, whereas the experimental results show the opposite trend. Hence, a seemingly reasonable theory is in conflict with observation, and no satisfactory theory exists at present. The random phase approximation (3.16) is also inadequate for Fermi fluids, as discussed in the following subsection.

3.2. ~He Notwithstanding the immense technical difficulties in neutron scattering experiments with 3He, caused by large absorption, several studies of 3He have been reported [20-22]. To date there is no wholly

153

adequate theory for the interpretation of the data. Similar shortcomings of the theory of density fluctuations in Fermi fluids are found in the interpretation of hard X-ray scattering from simple metals [23]. The exclusion principle tends to keep particles apart in an imperfect Fermi fluid, and the independent-particle approximation is quite successful in computing properties of normal fluids. By the same token, the Pauli principle limits fermion collisions at low temperatures, to such an extent that the time between collisions r increases at 1 / T 2 as T --, 0. This has a profound effect on the propogation of ordinary sound because the kinematic viscosity, and hence the damping of ordinary sound, is proportional to r. At T = 0, ordinary sound no longer propagates at any frequency, since local thermal equilibrium, which exists for tot << 1, is never attained. Density fluctuations in an imperfect Fermi fluid participate in a collective motion that is sustained by a self-consistent interaction between the particles. Called zero-sound by Landau, this collective mode is a distortion of the Fermi surface and it occurs only in the coilisionless regions where tot >> 1. Hence, if we keep the frequency fixed and decrease the temperature, ordinary sound will eventually be completely damped and zero sound will emerge at a sufficiently low temperature. Collisionless sound propogation in an imperfect Fermi fluid can be demonstrated within the random phase approximation which is a simple theory of self-consistent interactions in many-particle systems. The approximate susceptibility is given by (3.16). Given that the Fourier transform of the pair-potential is positive at long wavelengths, the sound velocity, in the strong coupling limit (u(0) much larger than the Fermi energy) is given by (3.18). Hence the imperfect Fermi fluid, with pairwise interactions, supports a collective oscillation of the particle density with an infinite lifetime and a dispersion of the same form as obtained for ordinary sound. The dissipative part of x(k, to) as a function of to for a fixed, small value of k consists of a delta-function peak

( ~rk/2Ms )B( to - ks),

(3.19)

due to zero sound, and a broad component at low

154

S. 14/. Love~'ev / T h e o r e t w a l c'hallenge~ m neutron ~'cattcring

frequencies. The latter arises from the continuum of particle--hole states, and it extends through the range 0 < ~ < { E r ( y 2 + 2 y ) } . where the Fermi energy E r = p f / 2 M and v = k / p r . The contribution (3.19) satisfies the f-sum rule (3.10), independent of the precise value of the zero-sound velocity. However, (3.11) and the compressibility sum rule imply .s-~= t "2, whereas it is generally believed that s > r. The theory described is not in accord with the experimental findings. For wave vectors k ~ 2p~, the width of the observed response spectrum is smaller than the theoretical prediction. However. theory and experiment are more or less in line if the 3He mass is increased by a factor three, i.e. the effective mass M * + 3M. The conclusion is consistent with the results of Landau's theory valid for small k and o:. Hence, the neutron experiments imply that the particle interactions in 3He persist at wave vectors as large as k - 2pr, and free-particle scattering, centred about the recoil energy (k2/2M), emerges at wave vectors very much greater than this. Even with an effective mass M * - 3 M , the thedry is an inadequate representation of the data, particularly in the region in which the zero-sound mcxte merges with the continuum. The differences between theory and experiment are greatly reduced by the introduction of a frequency-dependent potential that is chosen to match with Landau theory in the appropriate limit [24---26]. Such a theory is a good starting point from which to explore the pressure dependence of the response because the Landau parameters as a function of pressure are well established. ]'he zero-sound mode energy shows a marked increase with applied pressure, and the intensity of the density response decreases which exacerbates further the immense technical difficulties of the experiments [25]. Thus far we have discussed only the density response. However, the measured response is the weighted sum of the density and paramagnetic scattering, of. (3.1). The latter is predicted to be localized about a frequency which is well below the zero-sound contribution for k - P f , and to possess an intensity that is comparable with that from zero sound. This rather substantial pardmagnetic response is yet another complicating is-

sue in performing an interpretation of the particle density response [22]. 3.3. M a g n e t i c neutron

electron scattering

The copious supply of energetic neutrons from advanced spallation neutron sources can be used for inelastic scattering off electrons in metals, where the characteristic energy is the Fermi energy that is typically a few electron volts. These new experiments have far reaching ramifications in materials science. The qeutron-electron interaction (3.4) contains the electron spin and rnometaturn densities, whereas electron loss experiments and inelastic X-tax, scattering probe the electron particle density. In consequence, neutron electron scattering provides fundamentally different information from that derived from these other, established, experimental techniques. The difference is strikingly apparent in the f-sum rules discussed below. The feasibility and usefulness of magnetic neutron electron scattering can be assessed b~, calculating the appropriate response function (3.6) for models of electrons in materials. For example. effects due to band structure can be explored in numerical calculations of the response using oneelectron wave functions and energies generated with a realistic ion potential [27]. Effects due to Coulomb, electron phonon and magnetic field interactions, say. are difficult to include in band structure calculations. However, we can assess their significance by combining the (exact) f-sum rule for magnetic scattering, and approximate calculations of S ( k , ce) based on the one-component plasma model, for example [28]. The f-sum rule for particles described b5 a Hamitonian t t is [29], ~2

2f

-f_

d,+ ,uS{k, , u ) = = E(k.

>([[O+,HI'O] B).

(3.20)

where the second equality defines E ( k , B), and B is the external magnetic field. Note that (3.20) is exact: the proof relies on the identity S ( k , t o ) = S( - k, o:) which is valid even in the presence of an external field. The Hamihonian in (3.20) is sum of the free-

s. w. Lovesey/ Theoreticalchallengesin neutronscattering electron Hamiltonian

155

in which vq, Uq and #q are, respectively, the matrix element, p h o n o n displacement operator, and the q t h Fourier c o m p o n e n t of the electron particle density. For zero magnetic field,

trons interaction (3.4) and hence increases in value at long wavlengths. In the presence of a steady magnetic field, free electrons occupy Landau states in which there is free particle motion parallel to the field and harmonic oscillator states for motion perpendicular to the field. A calculation of S(k, ~o), treating the C o u l o m b interaction with the r a n d o m phase approximation, predicts a field induced coupling of neutrons to the lowest energy L a n d a u level and the collective density oscillation (hybrid mode) [31]. This result is consistent with the magnetic field contributions to the sum rule (k perpendicular to B)

E( k, O)= ( kZ/2M *) + ( 4 / 3 ) Eki .

E(k, B) = E(k, O ) - ( 1 / M * ) ( V ~

(1/2M*)(i~7-(e/c)A)2+ gl~Bi.B,

(3.21)

where B = curl A, the C o u l o m b interaction, and an e l e c t r o n - p h o n o n interaction with the generic form

Hcp= E v,Uqoq,

(3.22)

q

- ( 4 / 3 ) Eti, + ( g * J J k 2 ) - "r~%(i: ).

- ( l / k 2 N ) E q 2 { 1 - ( k . 0) 2 } v,(u,o,)

(3.25)

q

+(o/k2) fd,

+ V/)

{g(r)-l}

× ( c o s ( k . r ) - 1 ) ( k . ~7)2(e2/r). (3.23) Here, Ekm is the exact kinetic energy per particle, and g(r) is the pair distribution function, related to the static structure factor

S(k)=l+pfdrexp(ik.r){g(r)-I

(3.24)

It is evident that the sum rule depends on electron correlation s , in contrast to the f-sum rule for particle density fluctuations (3.10). We can gauge the sensitivity of the n e u t r o n - e l e c t r o n cross-section to electron correlations by c o m p a r i n g values of E(k, 0) for a perfect electron fluid with those obtained with Monte Carlo data for Ek~. and g ( r ) [30]. There is negligible difference between the two sets of results for high electron densities (r.~ = 2), which might be expected since r~ < 1 is the weak coupling limit [28]. However, there are significant (35%) differences for q = 5 and intermediate wave vectors (k = Pr ). Free particle behaviour is achieved in the limit k ---, oc, as is evident in (3.23). The effect of the e l e c t r o n - p h o n o n interaction is significant at long wavelengths. This stems from a p h o n o n induced, diffuse contribution to S(k, col spread over a wide range of ~0, which is engaged by the m o m e n t u m density in the neutron-elec-

Here, ~, = 2 + gM*/2 Me, the magnetic field defines the z-axis and the cyclotron frequency ~0~ = (eB/M*c). The magnetic field contribution to the cross-section is enhanced at small wave vectors. Moreover, it is readily discriminated from the increasing e l e c t r o n - p h o n o n contribution by its sharp ~0 dependence (the Landau and collective m o d e contributions are delta-functions in the limit of small k) and its field dependence. The latter affords the means of a large signal-to-background observation of Landau and collective m o d e states, using synchronously pulsed neutrons and magnetic field.

References

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156

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