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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.
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