* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Momentum distribution and final state effects in liquid neon
Survey
Document related concepts
Cross section (physics) wikipedia , lookup
Tight binding wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Quantum state wikipedia , lookup
Wave–particle duality wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Canonical quantization wikipedia , lookup
Coherent states wikipedia , lookup
Atomic theory wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Renormalization group wikipedia , lookup
Hydrogen atom wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Particle in a box wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Transcript
Jl~urJtltlo/Low Tt'nlpcrature P/ll's'icL I/ol. 109. NoN I/'2 1997 Momentum Distribution and Final State Effects in Liquid Neon R. T. Azuah, ~'* W. G. Stirling, ~'* H. R. Glyde, 2 and M. Boninsegni 2 ~Department of Physics University ~/" Keele, Keele ST5 5BG, UK 2Depurtment of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA [Received December 23, 1996; revised June 13, 1997) We report high precision inelastic neutron scattering measurements in liquid Neon at a temperature of" 25.8 K and saturated vapour pressure ~ The data covers a wide range o f energy and momentum transfer (2 A I <~Q <~ I3 ~I i). The atomie momentum distribution, n(p), and final state effects ( F S E ) can be readily extracted from this intermediate wavevector transJer data provided a suitable method of" analysis is used. We find that the momentum distribution in liquid Neon is marginally sharper than a Gaussian and that final state effects contribute predominantly an anti~ymmetric component to the the dynamic structure factor. The width o f n(p) and the kinetic energy are increased by 3 7 % above the classical values due to quantum effects. The experimental results are in agreement with theoretical valaes obtained by a Path Integral Monte Carlo simulation. 1. I N T R O D U C T I O N As one moves down the periodic table for the group eight elements there is a smooth progression in the behaviour and properties of the condensed phases of these elements. Liquid Helium requires a fully quantum treatment while a classical description is sufficient for Xenon. Not surprisingly, this character can be attributed chiefly to the larger atomic mass and partly to the increasing interatomic interaction for a greater atomic number. For these "inert" gases, as they are commonly known because of their closed outer electronic shells, the interatomic forces are characterized *Current address: Hahn-Meitner-lnstitut, Glienicker Strasse 100, 14109 Berlin, Germany. *Current address: Physics Department, University of Liverpool Oliver Lodge Laboratory, Oxford Street, Liverpool L69 3BX, UK. 287 0022-229U97/1000-0287512.50/0J') 1997PlenumPublishingCorporation 288 R . T . Azuah et aL by a weak van der Waals attraction and a strong repulsion at short distance. The zero-point energy, which is inversely proportional to the atomic mass, is a consequence of the uncertainty principle and is a purely quantum effect. For Helium the zero-point motion is large, due to the small atomic mass, and hence quantum effects are important. For the heavier inert gases, on the other hand, zero-point energy is much smaller and does not appear to play any rote in shaping their properties. Liquid Neon sits between these two extremes as a semi-classical system where quantum corrections are sufficiently large to be observed. Its phase diagram is relatively simple, featuring a very narrow liquid phase with a melting temperature of 24.5 K and boiling temperature of 27.5 K at saturated vapor pressure (SVP). One physical property which can be directly measured and which depends on the microscopic structure and dynamics of a system is the atomic momentum distribution, n(p). For a classical system n(p) is a Maxwell-Boltzmann distribution with a root-mean-square momentum, Po, determined by the equilibrium temperature T, i.e. hpo=(Mkl~T) 1'2 where M is the atomic mass and k , is Boltzmann's constant. Any deviation from this result will therefore imply non-classical behaviour, i.e., the presence of quantum eft~cts. Inelastic neutron scattering is the most effective means of measuring n(p). The momentum and energy of thermal neutrons match closely the momentum and energy of atomic excitations in condensed matter. In a neutron scattering measurement the observed intensity contains all the information required to specify the structure and dynamics of a system; for example, the atomic momentum distribution can be deduced from it. Neutron scattering measurements of liquid Neon have been performed in the past ~4 to obtain information on collective properties and on the atomic momentum distribution. For wavevectors in the range 0.8 A ~~< Q~<4 A -~, the observed dynamic structure factor, S(Q, co) is dominated by a "central" diffusive peak centered at co = 0. This peak has wings on the c o > 0 side characteristic of weak collective behaviour. These wings disappear from S(Q, co) at Q ~ 4 A I, and the central peak moves away from co = 0 to positive co, characteristic of scattering from a single atom interacting with its neighbours. Between 4 A-~ ~< Q ~<8 A-~ coherent effects in S(Q, co) are important. This is revealed by oscillations of the peak position and the width of S(Q, co) with Q. At Q~>8 A ~, the oscillations cease and S(Q, co) appears to be well approximated by the incoherent Si(Q, co). The moments of Si(Q, co) have a simple dependence on Q and its width and peak position cannot oscillate. Sears 2 first presented and analysed liquid Neon data to extract n(p). The data covered the intermediate wavevector range between 5 and I0 A - ~ and he was able to determine that n(p) was a Gaussian but with a large quantum enhancement of Momentum Distribution and Final State Effects in I,iquid Neon 289 its width above the classical value. Later measurements by Peek et al. 3 at much larger momentum transfers (20 to 28 A-~) confirmed the Gaussian character of n(p), although they observed a larger quantum correction to its width than observed by Sears. In this context, we present measurements of S(Q, co) in the range 2 ~ ~ < Q ~ < 1 3 A ~. We focus on the incoherent range 6 A ~<Q~< 13 A ~ where S(Q, co) is dominated by the momentum distribution, n(p), but final state (FS) contributions (discussed below) are not negligible. Our goal is to determine both n(p) and final state contributions making use of a method of analysis proposed by Glyde. 5 The data indicate a nearly Gaussian n(p) and confirm the large quantum correction found in the previous high Q measurements. 3 In addition, we are able able to determine final state effects from which quantities such as the Laplacian of the interatomic potential and the force experienced by the scattered atom are extracted. 2. T H E O R E T I C A L B A C K G R O U N D We now consider briefly some neutron scattering principles. The intensity measured in a neutron inelastic scattering experiment is the double differential cross section which may be expressed as ~' d2a df~ dE/ a k~ 4~h ~/S( Q, co) (1-) Here, ki, k/and Ei, El are the wave vectors and energies of the neutron before and after scattering from the sample and hQ and hco are the momentum and energy transferred to the sample. S(Q, co), the dynamic structure factor, contains all the information necessary to specify the structure and dynamics of the sample. S(Q, co) the Fourier transform of the density autocorrelation function of a system. This is dominated by collective massdensity excitations at low momentum transfers. As Q increases, the contribution from the collective excitations is less important as the interactions between atoms becomes relatively smaller. At sufficiently large Q values, S(Q, co) simplifies considerably and arises mainly from excitations of single particles; this is known as the incoherent limit. In the incoherent approximation, S(Q, co) has well defined frequency moments; for example the second moment is directly related to the single particle kinetic energy. As Q increases further, the scattering time becomes so short that the scattered atom does not have time to interact with the rest of the system and, in the limit of infinite Q, independent single-particle properties of the system are measured. This limit is called the impulse approximation (IA). 290 R . T . Azuah et aL In the IA, the dynamic structure factor can be obtained directly from the atomic momentum distribution, n(p), through (2) where heoaR=h2Q2/(2M) is the free atom recoil energy, M is the atomic mass and the 0-function connects the momentum to the energy transferred during the scattering process. The struck atom is assumed to recoil freely in its final state. As (2) only holds in the Q--* oo limit, the measured scattering function at finite Q will generally differ from S~A. This difference between the observed S(Q, (n) and S~A(Q, co) is usually referred to as Final State Effects (FSE), accounting for the interactions that occur at finite Q between the struck atom and its neighbours. The observed S(Q, o~) can be formally expressed in terms of S~A(Q, ~o) as -f. S(Q, co) = f dco'R(Q, co-co')Sia(Q,r (3) where R(Q, o)) accounts for FSE and is denoted the FS convolution function. v When SEA(Q, Co) is broad with respect to R(Q, co), it is difficult to determine the function R(Q, co) from experiment over all co. In this case, it is useful to expand R(Q, on) about its value at co=coR ~ and obtain FSE corrections as additive corrections to the IA. Various theories and models 8-~~ for FSE have been put forward and have been considered in detail in the recent article by Sosnick et al.Jt Here we consider a method of analysis proposed by Glyde, 5 which extracts both FSE and n(p) from experimental data. In this method, the intermediate scattering function S(Q,t), i.e., the Fourier transform of S(Q, co), is expanded in cumulants of powers of the scattering time, giving S(Q, t)= S,A(Q, t) R(Q, t) = S(Q) e -i~''R' exp n~__~ I~,,( - i O n (4) Here, S~A(Q, t) and R(Q, t) are the Fourier transforms of S~A(Q, o)) and R(Q, co) respectively and the multiplicative relationship in time implies a convolution in energy consistent with (3). S(Q) is the static structure factor and the time can be related to the displacement s of the struck atom parallel to Q by t = (M/hQ)s. Large values of Q and co correspond to a short scattering time limit. The parameters/~, depend on the atomic interactions and single particle momentum distribution in the sample and are Momentum Distribution and Final State Effects in Liquid Neon 291 therefore calculable from first principles. Furthermore, they have a well defined dependence on Q, which can be used to identify their contribution to S(Q, co). The/2,, provide a basis for the interpretation of the data and their expected Q-dependence is crucial indistinguishing between n(p) and FSE contributions to S(Q, co). Thus it is essential to have experimental data covering a wide Q range. In this article we will concentrate on the additive approach (AA) variant of the method s in which the cumulants in the exponentials of (4) are expanded and terms with n > 4 are ignored. The resulting expression is then Fourier transformed analytically to give S( Q, co) = S(i( Q, co) + St(Q, co) + S2( Q, co) (5) where Sa(Q, co)= S(Q) ~/2 exp [-L- (co-- co-)-/'~ (2gN2) - S~(Q, co)= ~3 (co-re'R) 8it; 1 - 2co2, + 2,t.t2 1- - J Sa(Q, co) (6) So;( Q, co) with os and c o 5 = ( c o - coR) , 2/la2. The parameters /z 2, /.z3 and P4 are related to the second, third and fourth moments of S(Q, co) by ,ttz=M2, /z3=M 3 and / z 4 = M a - 3 M ~ where M,,=~dco(co-coR)" S(Q, co)/S(Q). In the incoherent limit the parameters have the following dependence on Q,5. h2~ ", = (20) 2 ~2 h3/.t3 = (j.Q)2 a3 (7) h4~4 = (j.Q)2 d4 + (2Q)4 o~4 where 2 = h2/M = 0.2076 meV A 2 for Neon. The ~, and 8, are independent of Q. ~2 ( = ( p ~ > ) and ~4 ( = < P ~ > - 3 ( p ~ > ) are the second and fourth moments of n(p) along the wave vector Q and obviously depend on n(p). d 3 and d4 depend on the pair potential energy, V(r), as follows (see Eq. (4.15) of Ref. 8 and Eq. (35) of Ref. 5) a3= ~<V2V(r)> ~i4 = <F~> = <VQ V(r)2> (8) 292 R . T . Azuah et al. where Ve V(r) is the gradient of V(r) and (F~)> = (F2>/3 is the average square force in the direction of the scattering vector. The d3 and ~ are final state contributions to S(Q, co). In liquid Neon it is expected 2 that n(p), and there/ore StA(Q, col will be nearly Gaussian despite the presence of quantum effects. However, the root-mean-square momentum, Po, will no longer be given by the classical equipartition value, (hpo)2=h2(p~)> =Mk/~T. Quantum effects can be approximately accounted for using the equipartition result if T is replaced by an effective temperature, T~n.. This is discussed further in Sec. 5.2. 3. E X P E R I M E N T A L DETAILS The measurements were carried out using the MARl time-of-flight (TOF) spectrometer at the ISIS facility at the Rutherford Appleton Laboratory (RAL). ISIS is a spallation neutron source which generates a high flux of epithermal neutrons, allowing measurements to be carried out at high momentum transfers. MARI is a direct geometry chopper spectrometer with a range of incident energies available from approximately 10 meV to in excess of 1000 meV. There are more than 540 3He gas detectors providing an almost continuous angular coverage of scattering angle, ~b, of 3 <~b < 135 ~ in steps of 0.43 ~ This enables a huge range in (Q, ~)) space to be covered in a single experimental run and thus makes it the ideal choice for measuring the Q dependence of the dynamic structure factor. The spallation source generates short pulses of neutrons with a broad distribution of energies. The incident energy is selected by a curvedslit mechanical chopper situated at a distance of about 10 m from the moderator. At that distance, the pulse is spread in time according to the neutron velocity (energy) and emission time at the moderator. A variable chopper time delay phase-locked to the main pulse allows selection of the incident energy. Choosing from a set of 5 choppers, it is possible to obtain excellent energy resolution (AE/Ei~ 1-2%) combined with high neutron fluxes at all energies 20 < Ei < 1000 meV. The sample position is at 11.4 m from the moderator. After scattering from the sample, the neutrons travel for another 4 m before reaching the gas detectors which lie in a vertical scattering plane. The sample container was a cylinder made from aluminium of length 5 cm and diameter 3 cm. To minimize multiple scattering the sample length was split into 5 mm thick disc segments by 1 mm thick cadmium discs which have a high neutron absorption cross section. The container was then mounted inside a standard 4He "orange" cryostat and the Neon sample (99.994% Neon-20 from BOC gases, CP Grade) condensed with the aid of a dedicated gas handling rig. The sample temperature was Momentum Distribution and Final State Effects in I,iquid Neon 293 maintained at 25.8 _+0.05 K using a Lakeshore temperature controller and a Rh/Fe resistance sensor. As an independent check, the resistance thermometry readings were compared to the vapour pressure readings of the sample. The measurements reported here were made at saturated vapour pressure (SVP). To cover the intermediate momentum transfer region of interest, an incident neutron energy of 120 meV was employed. Two scans were made, the first with the sample cell empty and the second with the liquid Neon in place. 4. DATA R E D U C T I O N The data, collected in neutron TOF, is shown in Fig. l(a) at large scattering angles (130-135 degrees) where the elastic line is more easily discernible. The solid line in the figure shows the empty cell scattering which was directly subtracted from the total scattering to obtain the Ne scattering indicated by the trangles. The data was converted to energy transfer (~o) at I 03 c I I (a) --o-- N e o n + Cell - - E m p t y Cell - - ~ - - Neon 0.2 c -~ 0.1 o 0 0.0 3200 I I 3400 3600 3800 time-0f-flight ( g s e c s ) Fig. 1. (a) The "raw" data in neutron time-of-flight showing sample cell contribution to the scattering at high scattering angles (130-135 degrees). A direct subtraction was used to obtain the Ne contribution. (b) The observed dynamic structure factor, S(O, ~o), tbr liquid Ne. The lines are a fit of two hall:Gaussians plus exponential tails as described in the text to obtain peak positions and widths. 294 R.T. Azuah et al. constant scattering angle S(~b, 03) using standard procedures. As a result of the continuous angular coverage of detectors it was straightforward to obtain constant-Q slices ( S ( Q , 03)) from S(~b, co) at Q bins of 0.2 A -~ A sample of the Ne data displayed in Fig. l(b) shows that the statistical accuracy of the data is far superior to what has been obtained to date. For a meaningful interpretation of the data, an accurate knowledge of the instrumental broadening contribution to the observed scattering is essential. A Monte Carlo method t2' 13 was used to calculate the instrument resolution function by simulating the neutron scattering experiment. We have used the known instrument parameters and sample cell geometry. The incident beam characteristics were modeled using the Ikeda-Carpenter ~4 speed distribution function with the three adjustable parameters refined by fitting to the experimental monitor peaks situated before and alter the sample. A model impulse approximation scatterer was used to represent the sample and the simulation results were obtained at each detector in TOF. The simulation was then transformed to energy transfer following the same procedure as for the real experimental data. By treating both the simulated and experimental data in the same way, we ensure that any broadening (b) 0.2 ) = 2.0 A-I '> eo E ~0.1 c) 6.0 c.~ 8.5 12.0 0.0 -20 20 ho~ ( m e V ) Fig. 1. (Continued) 40 60 295 Momentum Distribution and Final State Effects in Liquid Neon effects inherent in the data reduction procedure, such as summing detectors together, would be equally represented in both. We then determine the model scattering profiles with no instrumental broadening. The resolution function, I(Q, co) is then obtained as the difference, in convolution, between the ideal and the instrument broadened simulation. The calculated resolution function for Q = 7.0 A - ~ is shown as a dotted line in Fig. 2. It is clear from this that the resolution function is 0.2 : 13.2 ,~-1 12.6 "0.1 11.0 9.4 8.6 0.0 7.0 i -20 -10 l 0 10 20 30 Y (i-I) Fig. 2. The longitudinal momentum distribution, J(Q, Y)=(hQ/M)S(Q,e)), Y= (oJ-oJR)(M/hQ) for liquid Neon at several Q values. Note the asymmetry in the data with more intensity in the positive Y side of the peaks. The lines are fits of the AA expression in Eq. (5) to J(Q, Y). 296 R.T. Azuah et aL much narrower than the Neon scattering peak; the function narrows even further as Q increases. No corrections were made for multiple scattering since it is estimated that the ratio of double to single scattering is less than 3 % for our experimental configuration, spread over a broader range in energy than the Neon recoil peak. To improve the statistical precision of the data, it is useful to consider the Y-scaling aspects of S~A(Q, co). In the IA the scattering function can be portrayed in terms of a longitudinal momentum distribution which depends on a single scaling variable Y=(CO--COR)M/(hQ), and not on Q and co separately. This breaks down at moderate Q values where FSE are important in the data. However, Y-scaling is still approximately observed experimentally and hence a generalised longitudinal momentum distribution can be used and is defined as hQ J(Q, Y ) = ~ - S ( Q , co) (9) which is weakly Q dependent as a result of FSE. The interest in considering this function instead of S(Q, co) is that it is possible to sum data over a wider Q range with only negligible broadening effects. Much of the data analysis reported here was carried out in J(Q, Y). 5. ANALYSIS AND DISCUSSION 5.1. Approach to the Incoherent Approximation In the Q range where coherent effects are important, we have analysed the data by fitting a generalised peak shape function consisting of two half Gaussians plus exponential tails to the data. ~5 It is expressed as follows exp(alx2+blx+cl) if x<~al Hexp if a~<x~<O Hexp if O<<.x<<.a2 f k exp(a2x- + b , x + c2) if x >~a 2 where x = (co-cop) and hcop is the energy at which the S(Q, co) is a maximum. H, a~, a2, COp, a~ and a 2 are fitted while the remaining parameters are determined by enforcing a continuous function. This function was chosen as it is sufficiently flexible to capture the asymmetry of the data and Momentum Distribution and Final State Effects in Liquid Neon 297 provide reliable information on the Q-dependence of the peak position and of the full width at half maximum. The individual parameters have no physical significance. A sample of these fits are displayed in Fig. 1 and it is clear that the function represents the data well. The extracted parameters from the fits are plotted in Fig. 3. The peak position, h%,, is higher at low Q values than the Neon recoil energy represented by the dashed line in Fig. 3(a). he% then drops rapidly with Q and goes below hcoR by 4 A -~ where it oscillates with Q and asymptotically approaches coR from below. The large deviation observed at Q values below 3 A-~ does indicate a major divergence from the expectations of the incoherent approximation and probably originates from some persistent collective response in the liquid. However, it is also possible that there might be a small contamination of the data by the empty container elastic scattering at these low Q's where a complete subtraction is more difficult. The rest of the results are consistent with general observation of neutron scattering data in the incoherent regime. The negative h(co,-co~) is a signature of the presence of final state effects whose influence diminish slowly as Q increases. The extracted widths, Wo, defined as the average of a~ and a, above, of the data are shown in Fig. 3(b). We observe that there are some oscillations with Q which are most significant at lower Q values. Close examination of the results show that the oscillations are in antiphase with those of the structure factor, S(Q), which is shown as the dotted curve in the figure. This result was not observed in pre'vious studies. In this coherent range, Eqs. (5) and (6) apply with parameters/,t 2, P3 and P4 that depend on and oscillate" with S(Q). With the equations and the accompanying moments, it appears quite possible to relate the oscillations in co, and Wc) to those in S(Q). The incoherent approximation, which is central to the analysis in the following section, applies to the data tbr wavevectors above about 6A-I 5.2. Momentum Distribution and Final State Effects In order to obtain information on the momentum distribution and Final State effects, we have chosen to analyse the data in the incoherent limit using Eqs. (5) and (6) expressed in terms of J(Q, y ) This is J(Q, Y ) = 1 - ~ zp 2 x8p 2 1-2x-+ Yc,(X) (10) where x = y/~/2, y= (co- coo)/(hQ/M), ~2 = (P~), Jc,(x) = (2rr~2)-'/2 e x p ( - x 2 / 2 ) is the gaussian component of the longitudinal momentum distribution and/~2, ll3 and/~4 are defined in (7). The lower limit of the R. T. Azuah et aL 298 (a) 0 ,.< > Data Neon recoil o .... O cO E 2 O Cq o{ . . . . . . . . {.~ - - ~ 3 0 -o~;%;a~%~,,o~ - 9 5 10 15 (b) .< > s CY 1 o 3: o zx . . , . o o 0 5 . . . Data Guide to eye S(Q) 10 15 Q (,~,-1) Fig. 3. (a) Extracted peak positions, cop, and (b) widths, WQ, of S(Q, ~o) for liquid Ne. We believe that the oscillations in WQ arise purely from variations of the structure factor, S(Q), through the normalisation condition. Momentum Distribution and Final State Effects in Liquid Neon 299 incoherent region was set to 6 A-~ on the basis of the results above. The available data therefore spans the range 6 A - l to 13.2 A-~ in steps of 0.2 A-~ and we note that even with such fine Q binning, the data are of sufficiently good statistical precision to allow the extraction of information with good accuracy. Are presentative sample of the data is presented in Fig. 2 and a note worthy feature of the line shapes is their asymmetry with more intensity in the positive Y side of the peaks at all Q values. J(Q, Y) was convoluted with the calculated instrumental resolution function and fitted to the data at all wave vectors with the kt,, as free parameters. The fitted AA function is shown as the solid line in Fig. 2 and reproduces the experimental line shapes well. The fine details of the fits can be understood in terms of the components of the AA fits as indicated in Fig. 4 for one low and one high momentum transfer. The following observations can be made about the components of the fits: is a Gaussian centered at Y= 0 with a width which is independent of Q. This component is defined by the parameter i~2 = (t~Q/M)< p~>. 9 Ja(Y) 9 Jl(Q, Y), and therefore/~3, shifts the peak intensity to Y < 0 and is responsible for the observed asymmetry in J(Q, Y). It is now clear why h(ogp- (.OR)is negative in the incoherent approximation. 9 J2(Q, Y), defined b y / & , redistributes the intensity in J(Q, Y) such that it becomes narrower than a Gaussian. The Final State contribution from this term reduces rapidly with Q and is almost negligible by 13.2 ~ - t To decompose the data into its n(p) and FSE parts, it is necessary to plot the Q dependence of the extracted p,, in order to determine the quantities ~, and &,. We recall that ~,, give rise to n(p) while FSE depend on d,. To achieve this separation we have fitted the/4, given in Eq. (7) to the extracted parameters. These fits are displayed in Fig. 5. We see that both P2 a n d / ~ follow exactly their expected Q dependence in (7) and it follows that values for ~2 and d3 are given directly from the slopes of the fits, i.e., ~2= < p ~ ) = 14.6 _0.7A -2 and ~ 3 = < V Z v ) / 6 = 6 2 + 7 m e V A -2. A fit of h4fl4=(~Q)2a4-[-(~Q)4(x 4 t o the observed values of h4/~4(Q) is also shown in Fig. 5; it yields d4=<F2Q>/3=2OO+_2OO(meV/A) 2 and 0~4 ~---< p ~ > - - 3<p~> = 20 + 15 A-4, i.e., they are less precisely determined. The parameter 0~4 is clearly very small, and could be zero. These values are the essential results of this paper. They are displayed in Table I and are used below to calculate some relevant quantities for liquid Neon at T -- 25.8 K. R. T. Azuah et al. 300 Data, Q = 7.0 ,~-1 AA Fit IA Comp., JG - Asym. Comp., J~ Comp., J2 ~ , ~_ ~, "~ 0.10 >-. c9 0.05 0.00 -20 -10 0 0.10 .~ 10 20 30 Data, Q = 13.2 ~-i AA Fit IA Comp., Jo i'~ -asym. Comp., Jt ! ~al . . . . . Symm" C~ J2 d 0.05 0.00 T -20 -10 0 10 20 30 Y (,~k-1) Fig. 4. The fitted components of the AA expression (Eq. (6)) at low and high momentum transfers. The scattering is dominated by Jr = (hQ/M)Se;( Q, u~) which is the first approximation to S~a( Q, to). The contribution from the antisymmetric component, Jt(Q, Y} = (hQ/M)S~(Q, eJ), the first approximation to FSE decays as Q-~ while the symmetric component J2(Q, Y ) = ( h Q / M ) S 2 ( Q , w ) decays as Q--" and appears insignificant at the highest Q's studied. Momentum Distribution and Final State Effects in Liquid Neon 120 > 301 g2 80 O E cq 40 450 g3 =a3 Q2 ,, ~ " > O E 300 =7. 150 0 40 g 4 = O-4Q 4 + a 4 Q 2 ~>0 20 E 0 -20 0 50 100 150 200 250 Q2 (~-2) Fig. 5. The observed Q-dependence of the extracted best-fit parameters deduced from the AA fits to the data. R . T . Azuah et aL 302 TABLE I Best Fit Parameters for the AA Model Scattering Function [or Liquid Neon Data, Compared to Theoretical P I M C Values. Note that 2 = h 2 / M = 0.2076 meV A 2 Obtained from Data ~ A-z 63 meV A -2 c~4 A -4 d4 ( meV A - ~)2 ( V : V ( r ) ) meV A - 2 ( F ~ > (meV A - I ) 2 9~2 (J3 350 + 60 372 + 43 d4 Theory {P I M C ) 200 + 200 300 + 20 420 + 50 14.6 _+ 0.7 62 + 7 20 _+ 15 200 + 200 We begin by determining the longitudinal momentum distribution, Equation(10) reduces t o J J A ( Y ) when / t 3 = 0 and #4/(/z2)2= 0~4/(0~2)2= 6, where 6 = 0.10 _+0.07 is the excess of the distribution. Since 0Z4 is small, JIA(Y) is nearly a gaussian, Jbx ~ Ja. The 3D momentum distribution corresponding to J I A ( Y ) is JIA(Y). 3/2 ~4<5 10p2 + 1P4~] (11) With ~2 determined, the average single particle kinetic energy, {E,v > = 3(h20~2)/(2M) can be calculated. We obtained { E x ) = 52.9 + 2.5 K (i.e., 4.56+0.22 meV). This agrees with 52.8 + 3.8 K obtained by Peek et al 3 at larger momentum transfers (20-28 A-~) where FSE are expected to be less important. However, it is marginally larger than Sear's value of 48.8 K determined from data covering a similar Q range to ours. The data at each Q has been reduced to J I A ( Y ) by subtracting off the FS contributions, determined below, and the results summed to produce a c o m p o s i t e J I A ( Y ) data as shown in Fig. 6. We return now to the effective temperature of the liquid referred to in Sec. 2. To first order, quantum effects will increase the kinetic energy but leave the momentum distribution unchanged, iv For small quantum effects, the increase in { E x ) in a liquid can be expressed in terms of an effective temperature {Ex)=3(ksT~tr)/2 where Tefr= T [ I + i 2 < T ) 1 -2-~T) +2-~T) (12) M o m e n t u m l)istribution and Final State Effects in I,iquid Neon 303 0.10 0 0 0 0 0 0 0 0.08 0 0 0 0 0 0 0 0 0 0 0 0 0.06 0 0 0 0.04 0 0 0 0 0 0 0 ~ O O 0 0 0 0.02 0 0.00 i -20 L -10 0 y 10 20 (s Fig. 6. The experimental h m g i t u d i n a l m o m e n t u n l distribution for liquid N e o n at at t e m p e r a t u r e of 25.8 K. T h e d i s t r i b u t i o n is G a u s s i a n with a n a v e r a g e r o o t - m e a n - s q u a r e m o m e n t u m o f Po = 3.82 _+ 0.09 A - ~ [ 1- d i m e n s i o n value). | being related to the pair potential by ~2 ~ tl2 "~ - ~ (V-V(r) ) (13) The leading correction is exact and the higher terms are obtained assuming the velocity distribution is a Gaussian. We have included many terms since this series is found to be poorly convergent for the observed ( E x ) of liquid Ne. The T~tr of 35.3 K is 36.7% above the real temperature ( T = 2 5 . 8 K). The | value required in (12) is about 60+_~5 K although the series does not converge satisfactorily for the upper limit of ( E K ) . Equation (12) is therefore unreliable and an alternative must be considered. The kinetic energy of a solid and liquid at equal densities are observed to be the same. If we 304 R.T. Azuah et al. treat the liquid as a harmonic solid, the quantum corrections to ( E x ) imply an effective temperature of ~ __ 6__0 .8 (0)4 + 1 Again, the leading correction is exact and a Debye model has been used in the rest of the terms to relate the higher moments to | ( 0 ] ~ = ( 5 / 3 ) O 2 gives the correct second moment). This series converges well for T~r= 35.3+ 1.7 K and 0 = 5 7 + 5 K . From (13) and (14) we find (V2V(r)) = 350 + 60 meV A 2 This value, obtained from ~,, agrees well with the value (V2V) = 6~ 3 = 372 + 43 meV A -2 obtained from ~3. In the AA method, the final state function is not recovered as a single function but as additive components. The first and most dominant component is the antisymmetric J~(Q, Y) determined by the parameter /~3. Inspection of the fitted results shows that Jj constitutes a significant part of the observed scattering at all Q values and decreases as 1/Q. The second component of FSE in J2(0-, Y) decreases with Q as 1/0.2. In general, J2 contributes both to n(p) and FSE but in liquid Neon the n(p) part is found to be small. This is established by the (2 dependence of/.t 4 in Fig. 5 where 0~4 is observed to be essentially zero. In contrast to J~, J2 is symmetric and is barely discernible at the largest 0- values studied as indicated in Fig. 4. The extracted FS parameters ri 3 and d4 also provide information on the potential and forces which are experienced by the struck atom during the neutron scattering process. The exact relations for these were laid out in Eq. (8). We note that d3 and d4 cannot be negative, since both depend on squared quantities and this is indeed confirmed by the results in Table I. From a visual perspective, this is dramatically confirmed experimentally, in the case of~3, by the fact that the asymmetry in the data is such that there is more intensity on the positive Y side of the peak. This situation would be reversed if d 3 w e r e negative. The average Laplacian of the potential along 0- of the struck atom is (V2 V(r)) = 372 + 4 3 meV A -2 (15) There is very good agreement between this (V2V(r)) and that calculated from the effective temperature, T~jr, of the liquid. The average squared force along Q on the struck atom, ( F ~ ) , on the other hand, is not easily determined by fitting to the extracted P4 displayed in Fig. 5. A reasonable estimate for ( F ~ ) is 200 + 200 (meV/~) 2. The large uncertainty quoted is due to the large scatter in/z4. In order to obtain independent and reliable quantitative theoretical estimates of ( V 2 V ( r ) ) and ( F ~ ) , as well as of the kinetic energy per Momentum l)is|ribution and Final State Effects in Liquid Neon 305 particle (E~.), we carried out a Path Integral Monte Carlo (PIMC) simulation of liquid Neon. The PIMC method is a well-established tool that enables the computation of thermodynamic properties of quantum many-body systems at finite temperature directly from the microscopic Hamiltonian ceperley. Here, we utilized it to calculate the kinetic energy per particle as well as <V2V(r)) and <F~)) in liquid Neon at T = 2 5 . 8 K and at a density p=0.0363 A -3. PIMC simulations of 54 and 108 Neon atoms interacting via an accepted pairwise potential, 2~ with periodic boundary conditions were performed; this type of simulation has been shown to afford a quantitatively accurate microscopic description of solid Neon. 2' The calculated kinetic energy per particle, ( E ~ - ) = 5 3 . 1 +0.1 K is in excellent agreement with the experimentally obtained value of 52.9 + 2.5 K. Calculated values of <V2V(r)) and (F~2) are <V2V(r)) = 300_+ 20 meV A 2 (16) (F~)) = 420 + 50 ( meV/A )2 which are also consistent with the observed values, within experimental uncertainties. In order to test the sensitivity of the results for (E~,.), <V2V(r)) and (F~?) on the potential used, we performed a separate PIMC simulation using a Lennard-Jones potential brown. The results obtained were consistent with those reported above, indicating that none of the sequantities is particularly sensitive to the detailed features of the hardcore of the interaction potential. It is interesting to compare the above results to those obtained for the same quantities in liquid Argon at SVP. Quantum effects in Argon are much smaller than in Neon, due to the larger atomic mass. We performed a PIMC simulation of liquid Ar at T = 8 4 K and at a density of 0.009287 A-3, based on a Lennard-Jones potential. 23 Quantum corrections to the kinetic energy per particle can be estimated to be less than 4 % of the classical value 3kBT/2. We obtained <V2lZ(r)) = 2 3 0 + 2 0 meV A 2 and ( F ~ ) = 590 _+50 (meV/A) 2. For a classical system, it is 17 (F~) = T(V2V(r))/3 (17) The failure of (17) to hold for a particular system is a sensitive indicator of the importance of quantum effects; for example, our results for liquid Ar satisfy (17) only to within 10%, although quantum corrections to the kinetic energy per particle are considerably smaller than that. Our results for liquid Ne show a large deviation from the prediction of (17), indicating a significant quantum contributions. 306 R . T . Azuah et aL We may compare our/13 with the Final State effects determined by Fradkin etal. 25 They confined themselves to the incoherent limit and analysed their data using (10), with o~4 set to zero, i.e., they assumed a gaussian IA, JIa =JG" We have found ~4 small but nonzero (see Table I). F r o m their data, Fradkin etal. determine the coefficient /t 3 in the form A 3 = 3! ,u3/(hQ/M) 3= M(V2V)/36h2Q. They find a value ( V 2 V ) = 342600K/nm2=295meV/A 2, consistent with their observed A 3 values. This is also consistent with the value ( V 2V) = 372 + 43 meV/A 2 which we obtain here from/z 3. Fradkin et al. found the Final State contribution to/z4 too small to be determined from their data. Thus, their results are consistent with ours up to the coefficient/z3. 6. C O N C L U S I O N S Liquid Neon is a semi-classical liquid in which quantum corrections are clearly observable. We find that the width of the m o m e n t u m distribution at T = 2 5 . 8 K and SVP is increased by 37% above the classical, Maxwell-Boltzmann width, ((hpo) 2) = M k ~ T , by quantum effects. The observed kinetic energy ( E x ) = 52.9 +2.5 K agrees with that found by Peek et al., 3 as well as with our P I M C calculated value of 53.1 + 0.1 K, and is larger than earlier estimates. In addition, we find that n(p) is slightly sharper than a Gaussian with a small positive excess of 6 = c ~ 4 / ~ = 0.10_+ 0.07. Q u a n t u m corrections, to order h 2, are expected to increase the width of n(p) but leave it Gaussian in shape. Departure from a Gaussian is a signature that corrections of order h 4 or higher are significant in Ne. In normal liquid 4He at T = 2 . 3 K, we found 2~' we tbund a much larger deviation from a Gaussian, with n(p) having an excess 6 =0.57. Peek et al. found that their calculations of (E~.) based on calculations of (V2V(r)) and the first two terms of the series in (12) in powers of h 2 lies 10% below their observed value of 52.8_+3.8 K. This difference may arise because quantum corrections are sufficiently large that the series in powers of h 2 is poorly convergent for Ne. In the incoherent regime, Q > 6 A-~, we also determined the leading two terms of the final state broadening function R(Q, s) defined by J(Q, s) = JiA(S) R(Q, s) where s = (hQ/M) t is the length conjugate of the Y-scaling variable. This is R ( Q , s ) = I + - - ia3 3! (2Q) s3 + . a4 . . S 4. _ _ 4! (2Q) z where a 3 / ) . = 300 + 30 A -4 and d4/,~ 2 = 4000 + 4000 A 6. The additive form of R(Q, s) is useful in Neon because JIA(Y) is much broader than R( Q, Y). Momentum Distribution and Final State Effects in Liquid Neon 307 At lower Q values, namely Q ~<8 A-~, we found that the width of S(Q, co) oscillates with Q in agreement with the data of Buyers et al.l These oscillations arise from coherent contributions to S(Q, co) and are similar to the oscillations observed in liquid helium. They can be related to the oscillations in S(Q) through equations (5) and (6) and moments ~,, that oscillate with S(Q) when coherent effects are important. 5 ACKNOWLEDGMENTS This work was supported by the U.K. Engineering and Physical Science Research Council and in part by the National Science Foundation through grants NSF INT 93-14661 and DMR 96-23961. We are grateful to Dr. S. M. Bennington and other staff of the ISIS Facility, Rutherford Appleton Laboratory for their assistance with the measurements. REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. W. J. L. Buyers, V. V. Sears, P. A. Lonngi, and D. A. Lonngi, Phys. Rev. A II, 697 (1975). V. F. Sears, Can. J. Phys. 22, 555 (19811. D. A. Peek, I. Fujita, M. C. Schmidt, and R. O. Simmons, Phys. Rev. B 45, 9671 (1992). A. A. van Well and L. A. de Graaf, Phys. Rev. A 32, 2396 ( 1985i. H. R. Glyde, Phys. Rev. B 50, 6726 (I994). S. W. Lovesey, Theory of Neutron Scatterhlg J?om C~mdensed Matter, Volume I. Oxford University Press, Oxibrd (1984). H. A. Gersch and L. J. Rodriguez, Phys. Rev. A 8, 905 (1973). V. F. Sears, Phys. Rev. B 30, 44 (1984). R. N. Silver, Phys. Rev. B 39, 4022 (1989). C. Carraro and S. E. Koonin, Phys. Rev. Lett. 65, 2792 (1990). T. R. Sosnick, W. M. Snow, R. N. Silver, and P. E. Sokol, Phys. Rev. B 43, 216 (1991). C. K. Loong, S. lkeda, and J. M. Carpenter, Nucl. hlstr, and Math. A 260, 381 (1987). P. E. Sokol, private communication. S. lkeda and J. M. Carpenter, Nucl. Instr, and Meth. A 239, 536 (1985). K. H. Andersen, A Neutron Seattering Study of Liquid Helium-4. PhD thesis, Keele University ( 1991 ). See equations (14) and (25) in Ref. 5. V. F. Sears, Phys. Rev. A 1, 1699 (1970); V. F. Sears, Phys. Rev. A 5, 452 (1972); V. F. Sears, Can. J. Phys. 63, 68 (1985). A. A. Maradudin, E. W. Montroll, and G. H. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation, Solid State Physics Supplement 3, Academic Press, New York (1963), p. 117. For a thorough description of the PIMC methodology, see D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995). R. A. Aziz, in Inert Gases: Potentials, Dynamics and Energy Transfer in Doped Crystals, M. L. Klein (Eds.), Springer-Verlag, Heidelberg (1984). D. N. Timms, A. C. Evans, M. Boninsegni, D. M. Ceperley, J. Mayer, and R. O. Simmons, J. Phys. CM 8, 6665 (1996). S. Brown, Proe. Phys. Soe. 891 897 (1966). 308 R. I". Azuah et aL 23. A. Cuccoli, A. Macchi, M. Neumann, V. Tognetti, and R. Vaia, Phys. Ree. B 45, 2088 (1992). 24. E. M. Lit~hitz and L. P. Pitaevskii, Statistical Physics, 3rd Edition, Part l, Pergamon Press, New York (1980), p. 98. 25. M. A. Fradkin, S.-X. Zeng, and R. O. Simmons, Phys. Rev. B 49, 15563 (1994). 26. R. T. Azuah, W. G. Stifling, H. R. Glyde, P. E. Soko[, and S. M. Bennington, Phys. Rel'. B 51, 605 (1995).