Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter
Inelastic neutron scattering techniques are introduced here as one of the most important experimental techniques in the investigation of collective excitations in fluids (liquids and compressed gases) and amorphous solids. The correlation functions involved, the spectra of which are determined in...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter / J.-B. Suck // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 7-18. — Бібліогр.: 26 назв. — англ. |
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Suck, J.-B. 2017-06-03T03:08:01Z 2017-06-03T03:08:01Z 2008 Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter / J.-B. Suck // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 7-18. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 61.12.Ex, 63.20.Dj, 63.50.+x DOI:10.5488/CMP.11.1.7 https://nasplib.isofts.kiev.ua/handle/123456789/118995 Inelastic neutron scattering techniques are introduced here as one of the most important experimental techniques in the investigation of collective excitations in fluids (liquids and compressed gases) and amorphous solids. The correlation functions involved, the spectra of which are determined in inelastic neutron scattering experiments, the dispersion relations of the collective excitations and how they are obtained from the measured spectra and finally two of the most often used instrumental techniques are briefly discussed. Методику непружнього розсiяння нейтронiв представлено тут як одну з найважливiших експериментальних методик для дослiдження колективних збуджень у плинах (рiдинах та стиснутих газах) та аморфних твердих тiлах. Коротко обговорюються вiдповiднi кореляцiйнi функцiї, спектри яких визначаються в експериментах по непружньому розсiянню нейтронiв, дисперсiйнi спiввiдношення для колективних збуджень та як вони можуть бути отриманi з вимiрюваних спектрiв, а також – двi з найбiльш часто використовуваних методик. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter Непружнє розсiяння нейтронiв в застосуваннi до дослiджень колективних збуджень у топологiчно невпорядкованих системах Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter |
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Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter Suck, J.-B. |
| title_short |
Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter |
| title_full |
Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter |
| title_fullStr |
Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter |
| title_full_unstemmed |
Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter |
| title_sort |
inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter |
| author |
Suck, J.-B. |
| author_facet |
Suck, J.-B. |
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2008 |
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English |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
| title_alt |
Непружнє розсiяння нейтронiв в застосуваннi до дослiджень колективних збуджень у топологiчно невпорядкованих системах |
| description |
Inelastic neutron scattering techniques are introduced here as one of the most important experimental techniques
in the investigation of collective excitations in fluids (liquids and compressed gases) and amorphous
solids. The correlation functions involved, the spectra of which are determined in inelastic neutron scattering
experiments, the dispersion relations of the collective excitations and how they are obtained from the measured
spectra and finally two of the most often used instrumental techniques are briefly discussed.
Методику непружнього розсiяння нейтронiв представлено тут як одну з найважливiших експериментальних методик для дослiдження колективних збуджень у плинах (рiдинах та стиснутих газах) та аморфних твердих тiлах. Коротко обговорюються вiдповiднi кореляцiйнi функцiї, спектри яких визначаються в експериментах по непружньому розсiянню нейтронiв, дисперсiйнi спiввiдношення для колективних збуджень та як вони можуть бути отриманi з вимiрюваних спектрiв, а також – двi з найбiльш часто використовуваних методик.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118995 |
| citation_txt |
Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter / J.-B. Suck // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 7-18. — Бібліогр.: 26 назв. — англ. |
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| first_indexed |
2025-11-24T18:41:11Z |
| last_indexed |
2025-11-24T18:41:11Z |
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| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 1(53), pp. 7–18
Inelastic neutron scattering applied to the investigation
of collective excitations in topologically disordered
matter
J.-B.Suck
Institute of Physics, University of Technology Chemnitz, D–09107 Chemnitz, Germany
Received November 22, 2007, in final form December 17, 2007
Inelastic neutron scattering techniques are introduced here as one of the most important experimental tech-
niques in the investigation of collective excitations in fluids (liquids and compressed gases) and amorphous
solids. The correlation functions involved, the spectra of which are determined in inelastic neutron scattering
experiments, the dispersion relations of the collective excitations and how they are obtained from the mea-
sured spectra and finally two of the most often used instrumental techniques are briefly discussed.
Key words: inelastic neutron scattering, correlation functions, dispersion relations, collective excitations,
topologically disordered matter
PACS: 61.12.Ex, 63.20.Dj, 63.50.+x
1. Introduction
In a similar manner as inelastic neutron scattering (INS) has played a decisive role in the
experimental investigation of phonon dispersions in crystal lattices in the second half of the last
century, it also has played the main role in the experimental studies of the collective excitations
in topologically disordered matter such as fluids and amorphous solids in the same period. Light
scattering has for a long time been used in order to investigate the atomic dynamics in disordered
matter next to ~q = 0, where ~q is the wave-vector of the excitation. However, the application of highly
resolving Brillouin scattering spectrometers [1], which makes it possible to determine the dispersion
and the damping of the modes with high accuracy [2], has also shown quite a durable period of their
development. After the development of inelastic X-ray scattering (IXS) spectrometers by the end
of the last century, this technique was very successfully applied to the investigation of collective
excitations in topologically disordered matter as well [3,4]. Inelastic light and X-ray scattering
techniques are described in detail in separate articles in this volume. Here the focus is on inelastic
neutron scattering techniques in its application to the investigation of the collective excitations in
disordered matter. Further articles in this volume will concentrate on the results obtained using
this method. A more extensive discussion including results is given elsewhere [5].
2. Neutron scattering
What advantage does scattering of cold, thermal and epithermal neutrons have in this context?
The neutron is neutral and therefore its interaction with condensed matter is weaker in a scattering
experiment than that of charged particles such as electrons. This permits to use solid sample
containers and even solid pressure cells, if the material is appropriately chosen. On the other hand,
the reduced interaction also imposes the use of larger samples, often several g of a substance, and
the small absorption frequently leads to a non-negligible amount of multiply scattered neutrons.
Neutrons are scattered either from the nucleus of an atom (nuclear scattering) or from the
electron shell of the atom (magnetic scattering). In spite of the fact that the magnetic scattering
of thermal neutrons is of primary importance in the studies of the magnetic structure and the spin
c© J.-B.Suck 7
J.-B.Suck
dynamics, there have been relatively few investigations of collective magnetic excitations (spin
waves) in disordered matter. Therefore, we shall concentrate on nuclear scattering.
In this case one does not have to deal with form factors as one does in magnetic scattering and
in case of all the other probes, which are scattered from the electron shell of the atom. Thus, the
interaction of neutrons with a wavelength of the order of 0.1 nm with the sample nuclei (with a
diameter of some fm) reduces to the scattering constant, b (scattering cross-section σsc = 4π b2),
which varies in a non-systematic way from element to element and even from isotope to isotope of
the same element. In addition to this, the scattering is also spin dependent. While nuclear scattering
is coherent, if all scatterers are identical, isotopic mixtures and spin dependent scattering can lead
to incoherent scattering. If the scattering lengths, bi, bj , of the atoms at the positions ri, rj in the
sample are uncorrelated, which is most often the case, one can separate the information on the
structure and dynamics of the sample from its interaction with the probe [6]. Due to this simple
separation and the very good knowledge of the scattering lengths it is also possible to normalize the
data very reliably and to produce results on absolute scale. This is one of the greatest advantages
in the present context, as the experimental results normally turn out to be rather smooth functions
and absolute values are an advantage as a further important parameter in comparison with theories.
Figure 1. The adiabatic dispersions for liquid
Rb from experiment and model calculations
(lower dispersion). The high frequency disper-
sion from experiment, model calculations [9] and
computer simulations [10] (upper dispersion).
The crosses between these two dispersions were
obtained from the maxima of Jl(Q, ω).
The non-systematic variation of the scat-
tering length from element to element and from
isotope to isotope also allows us to measure
light elements aside from heavy elements and
to mark elements by using different isotopes
(isotopic substitution) of the same element in
different measurements. This possibility is used
to determine the partial dynamic structure fac-
tors of multi-component systems.
A further advantage of NIS comes from the
fact that the energy of cold, thermal and epi-
thermal neutrons, depending onto which neu-
tron moderator the beam tube used is directed,
just spans the range of the energy of the exci-
tations in fluids and amorphous solids. Thus,
the incident energy, E0, of the neutrons can
be optimized to the excitation under investiga-
tion and the energy transfer, ~ω is then of the
same order as E0 and can therefore be highly
resolved. Likewise the de Broglie wavelength is
of the same order as are the atomic distances
in condensed matter (0.3 nm). Thus, the wave-
length dependence of the atomic dynamics can
be measured with good resolution as well.
Due to its large mass, even small changes
of the neutron velocity, ~v, can be connected
with large momentum transfers, ~ ~Q. Therefore,
INS is most favorably used for investigating the
atomic structure and dynamics far out into re-
ciprocal space within the higher order Brillouin zones (BZ) of the reciprocal lattice of single crystals,
i.e., using large wave-vector transfers ~Q = 2 π ~τ ±~q. Here 2πτ is the vector to the reciprocal lattice
point, from which the dispersion is measured. Thus, the large momentum transfer connected even
with a small energy transfer gives plenty of freedom in order to optimally chose the BZ, in which
one wants to measure the phonon dispersion in single crystals, – normally a further advantage of
inelastic neutron scattering.
Indications for the existence of ‘smeared’ out reciprocal lattice ‘points’ at the maxima of the
continuous static structure factor, S(Q) and low order analogues of BZ, called pseudo Brillouin
8
Experimental techniques: INS
Figure 2. Dispersion of collective excitations in liquid Rb measured near its triple point: due
to the shift of the maxima of the excitations to higher energies on multiplication with ω2 the
dispersion obtained from Jl(Q, ω) has the same shape as the dispersion obtained (by a fit of a
model) from S(Q, ω) but is shifted to higher energies.
zones (PBZ), exist also in disordered systems as it is suggested experimentally by the shape of the
observed dispersion relations (see e.g. figures 1, 2). However, these are of little help (compared to
the case of single crystals mentioned above) as the dispersion of collective excitations in disordered
systems is measured as a function of the wave-vector transfer, ~Q, and not of the wave-vector of the
excitation, ~q. Only at smallest Q this can be identified with q. In addition, collective excitations
in topologically disordered condensed matter are best investigated at small ~Q, i.e., within the
first PBZ by Neutron Brillouin Scattering (NBS) (see chapter on NBS in this volume), because
there, as all computer simulations and experiments show, the collective excitations are best defined
and therefore most easily separated from the foot of the elastic (amorphous samples) or quasi-
elastic (fluids) line. For this reason, the usual advantage of a large neutron mass becomes to some
extent a disadvantage in the investigation of collective excitations in disordered matter with severe
consequences for the experiments (see section 5).
As in other scattering experiments, in INS one measures the intensity of the scattered neutrons,
I( ~k0
~k), as a function of the wave vector of the incident, ~k0, and of the scattered neutron, ~k, and
of the incident neutron intensity on the sample, I0. From these quantities the double differential
scattering cross-section, d2σ/dΩdE is calculated, which still contains the information on the sample
properties and of the neutron-sample interaction. Here σ,Ω,dE are the scattering cross-section,
the solid angle under which the detector surface appears at the centre of the sample and dE is
the energy bin, where the scattered neutron is recorded. To obtain absolute intensities, numerous
corrections have to be performed, partly from measurements done under conditions identical to the
sample measurement (like Cadmium-, empty can- and Vanadium calibration measurements), partly
from analytical calculations (like energy and scattering angle dependent absorption corrections and
the energy dependence of the detector efficiency) and partly from e.g. a Monte Carlo computer
simulation (for multiple scattering corrections).
The contribution Ic from the neutrons scattered from the empty sample container to the scat-
tered intensity Isc in the sample measurement is obtained in the empty can measurement. Assuming
absorption of all neutrons falling on the Cd-sample with the same geometry as the sample, in the
Cd-measurement, the background ICd passing aside the sample and therefore not attenuated by
the sample and its container is measured. Having normalized all the measurements to the same
incident intensity (the same monitor count rate), this intensity is directly subtracted from Isc and
Ic, while in the subtraction of the empty can intensity the absorption of neutrons in the sample has
to be taken into account by the absorption factor Aec(θ,E). Here θ is the scattering angle (angle
between ~k0 and ~k).
After calculating the sample absorption, As(θ,E), that of the empty can, Ac(θ,E), and the
detector efficiency, ε(θ,E), the intensity MS(θ,E) of the multiply scattered neutrons has still to
be calculated (analytically or (better) in a computer simulation [7] of the experiment), because
9
J.-B.Suck
only for neutrons scattered once in the sample, ~k0 and ~k can be determined and hereby the energy
– and momentum of the excitation in the sample
~ω = E0 − E = ~
2/(2m)(k0
2 − k2), (1)
~ ~Q = ~(~k0 − ~k), (2)
Q =
√
k0
2 + k2 − 2k0 k cos θ. (3)
As disordered systems are normally isotropic, in what follows the modulus of ~Q will be used in
place of the corresponding vector for the wave-vector transfer. After the rather involved multiple
scattering correction, which one tries to keep small by using sample geometries, which do not
scatter more than 10 to 15 % of the neutron beam incident on the sample, the fully corrected
intensity from the sample, Icorr can be calculated
Subtraction of Cd − intensity : Isc − ICd ,
Ic − ICd ,
Subtraction of empty can scattering : Is = Isc − Aec(θ,E) Ic ,
Icorr =
Is − MS(θ,E)
Ns ε(θ,E) As(θ,E) Ac(θ,E)
, (4)
where Ns is the number of scattering units calculated from the number density, n, of the sample
and its volume.
The Vanadium calibration run and the V-holder measurements are done under identical condi-
tions as the sample measurements (the same sample geometry, the same experimental conditions)
and the same type of corrections are applied to the measured intensity as in the case of the sample
experiment. As V scatters neutrons almost exclusively incoherently (spin incoherence), Icorr V is
used for the calibration of the relative efficiencies, if several different detectors are used in the
spectrometer, and for the normalization of the sample intensity to the known cross-section of V in
the calculation of the absolute values of the double differential scattering cross-section
d2σ
dΩdE
=
Icorr
Nsu∆Ω∆EIcorr V
. (5)
Hence, the desired information on the dynamics of the sample, the dynamic structure factor S(Q,ω)
can be separated from the neutron sample interaction under the conditions mentioned above [6]:
Sexp(Q,ω) =
4π
σsc
k0
k
d2σ
dΩdE
, (6)
where σsc is the total scattering cross-section of the defined scattering unit. What really Sexp(Q,ω)
in equation 6 represents, depends on the correlations covered in the experiment. If only hetero-
correlations contribute to Sexp(Q,ω), i.e., purely coherent scattering contributes to d2σ/dΩdE,
then σsc = σcoh and Sexp(Q,ω) = S(Q,ω), the dynamic structure factor. That is the information
one needs to determine the dispersion of collective excitations. Therefore one tries, if possible, to
maximize the coherent part of the scattered intensity by choosing appropriate elements. If only
auto-correlations contribute, i.e., purely incoherent scattering contributes to d2σ/dΩdE, then σsc =
σincoh and Sexp(Q,ω) = Ss(Q,ω), the self-part of the dynamic structure factor. In most cases both
correlations will contribute and it then depends on the amount of incoherent scattering, whether one
has to calculate the incoherent contribution from an appropriate model and subtract this (correctly
normalized) from Sexp(Q,ω) to obtain the desired contribution from hetero-correlations only (see
below) or one has to interpret the coherent scattering ‘on top’ of the incoherent ‘background’.
In addition, what kind of dynamic structure factor one has determined depends on the number
of different elements in the sample. If this contains one element only, the dynamic structure factor
is determined (taking a weighted mean value over the scattering length of the isotopes of the
element). If several different elements are present, of which each couples with a different strength
10
Experimental techniques: INS
(scattering length) to the incident neutron, the total dynamic structure factor is determined, which
is the weighted sum over the partial dynamic structure factors, Si,j(Q,ω)
σS(Q,ω) = 4π
∑∑
bibj
√
cicjSi,j(Q,ω) +
∑
σinc
i ciS
s
i (Q,ω), (i, j = 1 . . . n), (7)
where ci, σinc
i and Ss
i (Q,ω) are the relative concentration, the incoherent scattering cross-section
and the self-part of the dynamic structure factor (auto-correlations) of the element i. One has
to realize that the dispersions obtained from the total dynamic structure factor (as was done in
nearly all cases up till now) and those determined the same way from the partial dynamic structure
factors (after separating them out via isotopic substitution) will not be the same in most cases.
S(Q,ω), a real function, is not symmetric on the energy scale, since the excitation of a mode is
always more likely than its absorption. This detailed balance condition S(Q,ω) = eβ S(−Q,−ω),
with β = ~ ω/ kB T and kB the Boltzmann constant, is often used to symmetrize (artificially) the
dynamic structure factor
S̃(Q,ω) = e
−β
2 S(Q,ω). (8)
This symmetrized representation is sometimes very helpful if one wants to obtain the dispersion
of the collective excitations from the neutron energy gain and loss side of S(Q,ω) under most
similar conditions, (see figure 3). However, one has to take care to include the symmetrization
factor explicitly in all further calculations when using S̃(Q,ω).
Figure 3. left from below: S(Q, ω), S̃(Q, ω), right from above: Jl(Q, ω) and F (Q, ω) for the same
cut at Q = 21.5nm−1 through the spectra of the metallic glass Mg70Zn30. F (Q, ω) is strongly
approximated as it still contains contributions from incoherent scattering and multi-excitations.
3. Correlation functions
S(Q,ω) is the spectrum of the van Hove correlation function G(r, t) [6,8], the time dependent
correlation function of a pair of local densities
n(r, t) =
1√
N
N
∑
1
δ(~r − ~Rl(t)), (9)
n(Q, t) =
1√
N
N
∑
1
ei(~Q·~Rl(t)), (10)
where the second line is the local density after Fourier transformation.
G(r, t) =
1
n
〈
∑
l,l′
δ
(
~r′ − ~Rl′(0)
)
δ
(
~r − ~Rl(t)
)
〉
, (11)
11
J.-B.Suck
where n is the number density (n = N/V ) and r =| ~r − ~r′ |. Thus, in the van Hove correlation
function, two particles or local densities at Rl and Rl′ in a distance r are correlated at two different
times, herein 0 and t. If the correlated entities are different particles, then G(r, t) = Gd(r, t), (d =
distinct correlation) describing the collective dynamics, which is reflected in the coherent neutron
scattering. If the same particle is correlated at two different times, the single particle dynamics is
reflected in Gs(r, t) (s = self-correlation) and the incoherent scattering.
S(Q,ω) and G(r, t) are related to each other by double Fourier transforms
S(Q,ω) =
∫
∞
−∞
dt e−iω t
∫
dr ei(~Q·~r) G(r, t),
S(Q,ω) =
∫
dt e−iωt F (Q, t),
F (Q, t) =
∫
d3rei(~Q·~r) G(r, t), (12)
where the Fourier transformation of G(r, t) into Q-space (as is the Fourier transform of S(Q,ω)
from energy into time space) is called the intermediate scattering function, F (Q, t). In most cases,
the modelling of the dynamics does not start from G(r, t), the correlation function calculated in
general from the results of molecular dynamics calculations (MD), nor from the experimentally
determined spectrum of it, S(Q,ω), but from F (Q, t), which is the correlation function of the
Fourier transformed dynamical variables (see equation 10) and can be obtained from either of the
functions by one Fourier transformation
F (Q, t) =
〈
n∗( ~Q, 0) n( ~Q, t)
〉
(−n(2π)3δ( ~Q)). (13)
In this equation, the second term, the forward scattering term, is normally neglected. Thus F (Q, t)
is a central function at which the results from theory, computer simulations and experiments, done
in a sufficiently large energy range and with good statistical accuracy, best meet. In the classical
approximation it describes the time decay of the correlations represented by the static structure
factor of the system at t=0, i.e., F (Q, 0) = S(Q).
As the scattering units are moving, in place of the local densities one can also use the time
dependent local currents ~j(~r, t) as dynamical variable
~j(~r, t) =
1√
N
N
∑
1
~vl(t) δ(~r − ~Rl(t)),
jα(Q, t) =
1√
N
N
∑
l=1
vl,α ei(~Q·~Rl(t)), (α = x, y, z). (14)
Hence, the current correlation function, J(r, t), the corresponding intermediate scattering function,
J(Q, t), and its spectrum, J(Q,ω), are deduced just the same way as it was done for the density as
dynamical variable, except that now velocity vectors are dealt with, which have different directions
in space, and which have to be correlated as well. Consequently, the corresponding correlation
function now splits up additionally into longitudinal and transverse correlations.
Jαβ(r, t) = V 〈jα(~r′, 0)jβ(~r, t)〉 , (α, β = x, y, z, and V the volume). (15)
As in the case of the dynamic structure factor, the spectrum of the current correlation function is
obtained by Fourier transformation of the corresponding intermediate scattering function J(Q, t)
Jαβ(Q, t) =
〈
j∗α( ~Q, 0) jβ( ~Q, t)
〉
, (16)
Jαβ(Q, t) =
QαQβ
Q2
Jl(Q, t) +
(
δαβ − QαQβ
Q2
)
Jt(Q, t), (17)
12
Experimental techniques: INS
where δαβ is the Kronecker δ. Neutrons (and X-rays) couple directly only to longitudinal motions
(parallel to ~Q), to transverse motions only via (for topologically disordered matter: diffuse) Umk-
lapp scattering. Therefore for INS (and IXS) experimentally accessible is only the longitudinal part
of the spectrum, Jl(Q,ω), of the current correlation function
Jl(Q,ω) =
∫
∞
−∞
dt e−iω t Jl(Q, t), (18)
while in computer simulations, the transverse part of it can be determined as well. As only one
scattered intensity is recorded, from which S(Q,ω) and Jl(Q,ω) are calculated, the spectrum of
the current correlation function has to be connected to the spectrum of the density correlation
function [6,8]
Jl(Q,ω) =
ω2
Q2
S(Q,ω). (19)
In spite of this fact, Jl(Q,ω) plays a central role in the experimental determination of dispersion
relations in disordered matter. Due to the multiplication of S(Q,ω) with ω2 and due to the fact
that S(Q,ω) falls to zero with ω faster than any power of ω, Jl(Q,ω) has always one or more
maxima, which emphasize the weak and broad maxima in the inelastic part of S(Q,ω). However,
one has to interpret these maxima with some caution, as the function is forced to show at least
one maximum on the neutron energy gain and loss side of the spectrum, – even for a Lorentzian.
4. Dispersion relations
How can the dispersion of collective excitations in disordered systems be obtained from the
measured spectra of the density or the current correlation functions? In principle, most methods
mentioned herein below can be applied to fluids and amorphous solids. However, in practice some
of them have been applied with preference to fluids, – for which presently a larger amount of
different results have been obtained – the others with preference to amorphous solids.
In models, describing the collective dynamics of disordered systems such as fluids, dispersion
relations defined by the ratio of subsequent lowest even frequency moments of the dynamic structure
factor, 〈ωn〉, are most frequently used.
〈ωn〉 =
∫
∞
−∞
ωn S(Q,ω) dω. (20)
The adiabatic ( = isothermal for γ = Cp/CV ≈ 1) dispersion ω0(Q) is the ratio of the second
to the zeroth frequency moment, kBTQ2/M and S(Q), where M is the mass of the scattering unit.
The high frequency dispersion, ωl(Q), is the ratio of the fourth to the second moment [6].
ω0 =
kBTQ2
M S(Q)
, (21)
ω2
l = 3
kBTQ2
M
+ Ω2(0) − Ω2(Q), (22)
Ω2(Q) =
n
M
∫
d3r g(r) cos (Q z)
d2φ(r)
dz2
, (23)
where φ is the inter-atomic potential. While the interaction potential enters the adiabatic dispersion
only indirectly via S(Q), the high frequency dispersion also contains, besides the free particle
dispersion (first term in the high Q limit) in its second and third terms, a potential part (similar to
the dynamical matrix for harmonic system, but for the topologically disordered system weighted
with the particle distribution according to the radial distribution function g(r)). Here the effect of
the potential on the dynamics comes in explicitly. The two parts of the high frequency dispersion
react very differently on temperature and pressure changes. While the free particle dispersion reacts
13
J.-B.Suck
most strongly on the changes in temperature, the changes in pressure (and therefore in density)
have much greater effect on the potential part of the dispersion [12].
One can determine these dispersions either from the ratio of experimentally determined fre-
quency moments, as it was first done for liquid Rb (see figure 1) and later for fluid Ar [11] or from
a fit of a model, where these dispersions are treated as parameters [12], or from models for the
frequency moments. The inclusion of these dispersions in the (visco-elastic) models is very impor-
tant, because they guarantee that the model fulfils the condition of the existence of all frequency
moments up to the fourth.
A more straightforward approach is to fit the models for the collective atomic dynamics to
the dynamic structure factor and determine the position (energy) and the width (damping) of the
maxima or shoulders corresponding to the excitations. If the data were taken in an Q − ω range
corresponding to the hydrodynamic limit, hydrodynamic equations (essentially three Lorentzians
with two S-shaped additions) or kinetic or linear response theory, taken in the hydrodynamic limit
(all parameters are constants, for an explicit comparison see [13]), can be used.
If the data were taken outside the hydrodynamic limit, one can fit generalized hydrodynamic
models (the same equations but parameters become functions of Q (and ω)) e.g. kinetic or linear
response theory or mean-field (visco-elastic) models to the dynamic structure factor. Models for bi-
nary systems do exist [14], but they have got quite a few (not necessarily independent) parameters.
Starting from the local current as dynamical variable, one can fit convex functions to the maxima
of Jl(Q,ω) and determine the dispersion from these peak positions. Only if the excitations can be
represented by δ-functions, as would be the case for ideal phonons with an infinite lifetime, the
dispersion obtained this way will be identical to the dispersion determined by fitting the inelastic
maxima in S(Q,ω). As shown in figure 2, for excitations in topologically disordered systems the
dispersion relations have approximately the same shape (as a function of Q), but the dispersion
obtained from Jl(Q,ω) will always have higher energies, because the excitation peaks (or shoulders)
are broad and are shifted to higher energies by multiplication with ω2. It should be noted that the
dispersion obtained by fitting the frequently used model of the damped harmonic oscillator (DHO)
to the inelastic part of S(Q,ω), corresponds to the dispersion obtained from Jl(Q,ω). Likewise,
the dispersion obtained from fitting Bloch’s spectral function, F (Q,ω),
F (Q,ω) =
〈
∑
j
(
~̂
Q · ~ej( ~Q))2 δ(ω − ωj( ~Q)
〉
(24)
(where ~e is the polarization vector of the excitation,
~̂
Q is the unit vector in direction of ~Q, ωj(Q)
is the dispersion branch j and 〈. . .〉 indicating the ensemble average) corresponds to the dispersion
determined from Jl(Q,ω) (see below). As equation 24 demonstrates, F (Q,ω) is the obvious choice in
order to define from its maxima the dispersions of collective excitations in topologically disordered
solids [15]. In figure 3 the spectra of the (symmetrized) density- and of the current correlation are
compared with the spectral function for identical data.
In computer simulation it is not difficult to determine F (Q,ω) and even fij(Q,ω), the partial
spectral functions for a multi-component system and in fact it is highly desirable while investi-
gating the collective excitations in disordered solids that simulation and experiment should meet
at the spectral function. However, the experimental determination of F (Q,ω) (related to the one
-excitation part of the recorded intensity only) from even a well corrected intensity involves differ-
ent laborious subsequent corrections (even in a most simple approach discussed below), which all
can only be performed within approximations:
I( ~k0,~k) ∼ [Sexp(Q,ω)] ∗ R(Q,ω),
Sexp(Q,ω) = S(Q,ω) + (Ss(Q,ω)), (25)
S(Q,ω) =
4π
σsc
∑
bibj
√
cicj Sij(Q,ω),
S(Q,ω) = S1phon(Q,ω) + Smult.phon(Q,ω),
14
Experimental techniques: INS
S1phon(Q,ω) =
1
~ω (1 − e−β)
~
2 Q2
M
e−2W F (Q,ω),
F (Q,ω) =
∑
ij
eWi+Wj
√
cicj
MiMj
fij(Q,ω)
/
∑
ij
eWi+Wj
√
cicj
MiMj
, (26)
where W (Q) are mean Debye-Waller factors. Of these corrections, the de-convolution of the resolu-
tion function, R(Q,ω), is usually not performed. Instead the model used to fit the final data in order
to obtain the energy and width of the collective excitations, is convoluted with R(Q,ω). To sub-
tract the self-part of the dynamic structure factor (incoherent scattering) one has to use a model,
e.g. the multi-phonon expansion of Ss(Q,ω) [16], based on the known vibrational density-of-states,
g(ω), and calculate Ss(Q,ω) in the harmonic approximation
Ss(Q,ω) = A δ(ω) + e−2W (Q)
∞
∑
n=1
fn(ω)
n!
(
~Q2
2M
)n
, (27)
fn(ω) =
∫
∞
−∞
fn−1(ω
′) f1(ω − ω′)dω′, (28)
f1(ω) =
g(ω)
~ω (1 − e−β)
. (29)
Here A δ(ω) represents the elastic scattering. The multi-phonon part of these equations (n > 2)
can finally be used in order to calculate (in the incoherent and harmonic approximation) and
subtract the multi-excitation part, Smult.phon(Q,ω) of the coherent scattering. The remaining one-
excitation-part, S1phon(Q,ω) is then directly proportional to F (Q,ω), which still is a weighted sum
over the partial spectral functions, fij(Q,ω), in case of a multi-component system.
Inverting the relation between S1phon(Q,ω) and F (Q,ω) (see equation 26) and taking the Bose-
Einstein occupation factor in its high temperature approximation leads to a relation completely
the same as that of Jl(Q,ω) and S(Q,ω), except that the equation still contains the Debye-Waller
factors and the mass of the scattering unit and the one-excitation part of S(Q,ω) in place of
S(Q,ω) [17]
F (Q,ω) =
2M
~2Q2
~ω(1 − e−β)e2W (Q) S1phon(Q,ω), (30)
F (Q,ω) ≈
(
2M
~Q2
e2W (Q)
) (
ω
Q
)2
S1phon(Q,ω). (31)
5. Instruments
Dispersions of collective excitations in fluids and later topologically disordered solids have been
determined using INS by triple-axis and by time-of-flight spectrometry since the second half of the
1960s. Both methods have remained the main sources of information on the collective dynamics in
disordered systems using INS (see e.g. [17]). Even though occasionally other types of spectrometers
have been used such as a Neutron Spin-Echo (NSE) spectrometer, we will focus here on the two
most frequently used spectroscopic methods.
In general, the attempts to measure the dispersion of collective modes at small wave-vector
transfers Q turn out to be extremely difficult. To measure the linear part of a dispersion, the
velocity, v0 of the incident neutrons has to exceed the sound velocity, vs (optimally by a factor of√
2 [18]) in order to ‘cover’ the dispersion in the Q−ω-space of the experiment. This high incident
energy considerably decreases the good energy resolution available in INS experiments at smaller
E0. The smallest Q at high incident k0 = m v0/~ are only reached at the smallest scattering angles,
as shown in equation 3. Under these conditions one has to measure an exceptionally small signal
(because the static structure factor in the low Q range is of the order of 1% of its maximal value)
aside from the high background next to the incident beam (due to the small scattering angles)
15
J.-B.Suck
Figure 4. Q−ω-space of Neutron Brillouin Scattering experiments with incident energies of 2.5
(dark grey), 25 (light grey) and 250 meV for scattering angles between 1 and 6 degrees. The
dispersion of collective excitations in a solid, e.g., with a sound velocity of 4000 m/s can only
be measured with the highest incident energies. Insert: enlarged small Q section.
– and this is at a considerably larger distance from the sample (to separate the scattered from
the incident beam) than one would do in normal INS experiments. At the same time, the usual
multiple scattering intensity, which normally is relative structureless in this Q-range, makes up
about 40% of the recorded intensity leading to large multiple scattering corrections. To meet all
these contradicting conditions, the spectrometers and the sample containers (insertions of absorbing
grids to reduce the multiple scattering) have to get a special set-up for this kind of experiments.
Figure 5. The principle of the IN1 triple.axis spec-
trometer at the High Flux reactor of the Institut
Laue-Langevin in Grenoble. In an NBS-experiment,
a large cylindrical vacuum container would be
mounted on the sample table and the analyzer
would be behind the sample, very few degrees devi-
ating from the direction of the incident neutrons.
In triple-axis-spectrometer (TAS) ex-
periments, ~k0 and ~k are determined via
Bragg reflections from two single crystals,
the monochromator, which selects the de-
sired ~k0 from the incident ‘white’ neutron
beam in the direction of the sample outside
the monochromator shielding, and the ana-
lyzer, which in the same way determines the
~k of the scattered beams (see figure 5). The
transmitted bandwidths ∆k0 and ∆k, and
hence the Q − ω resolution ellipsoid of the
spectrometer, are determined essentially by
the mosaic spread of the single crystals
and the collimation chosen along the flight
path. For NBS experiments, the paths of
the neutrons next to the sample are evac-
uated or flushed with He, which scatters
and absorbs neutrons less than air. The ad-
vantage of the TAS is its large flexibility
(within the accessible kinematic range (see
figure 4)) and relative low background due to the numerous collimators and good detector shielding.
The flexibility permits to take data (within the kinematically accessible region) at fixed energy- or
fixed wave-vector transfer, which frequently makes it possible to immediately interpret the data.
However, each point of these cuts at ω or Q = constant has to be measured with a different configu-
ration of the spectrometer, thus making both quantitative corrections and absolute measurements
more difficult. In addition, while the high flexibility is a great advantage, if only certain regions of
16
Experimental techniques: INS
S(Q,ω) have to be measured, as is the case in phonon dispersion measurements based on a good
model for the dynamics of the system under investigation, for topologically disordered systems,
due to the weak and broad excitations maxima, S(Q = const, ω) has to be scanned in a complete
energy region, requesting very long measuring times for these NBS experiments on TAS, especially
because most of these instruments today still work with a small detector surface [19].
Figure 6. Principle of the new IN5B time-of-flight
spectrometer at the High Flux reactor of the Institut
Laue-Langevin in Grenoble. For NBS-experiments
normally only the two dimensional position sensitive
(pixel) detector will be used, permitting to collect
the complete intensity of a small angle scattering
cone. The flight path of nearly 4 m is evacuated or
flushed with He.
This is different in time-of-flight
(TOF) spectroscopy, because there all ac-
cessible points of S(Q,ω) are measured at
the same time and under the same experi-
mental conditions without any change of
the initially chosen parameters of the ex-
periment. This makes quantitative correc-
tions and the measurement of S(Q,ω) on
absolute scale easier. In TOF spectrom-
etry, ~k0 is determined either by several
choppers (see figure 6), which finally let
only a small energy band ∆E0 pass onto
the sample or (in hybrid TOF spectrom-
eters) by one (or several) monochromator
crystal such as in a TAS and one or several
choppers, producing neutron pulses with
the desired mean ~k0 incident on the sam-
ple. The pulses are required, because the
modulus of ~k is determined from the time-
of-flight of the scattered neutrons between
the sample and the large detector area while its direction is determined from the (in NBS experi-
ments: small) scattering angle. The advantage of simultaneous measurements of the scattered
intensity at all accessible Q and ω values is counterbalanced by the fact that the pulsing reduces,
in the mean, the incident intensity by a factor of 100 to 200 in comparison with the continuous
flux at a TAS. In spite of this, the complete coverage of the interesting part of S(Q,ω), which per-
mits to ‘follow’ the dispersion even if the maxima are getting broad at lager Q, the determination
of absolute values for the comparison with theories and especially the collection of the complete
intensity of the scattering cone at smaller scattering angles on the large position sensitive detector
in modern TOF and especially NBS spectrometers, makes the TOF method very attractive for the
investigations of collective excitations in disordered matter.
The energy resolution of the elastically scatter neutrons dE0 is of the order of 2 to 3 % on
TOF [20,21] and TAS [22,23], – except in the very rare cases e.g. when a TAS is used with a NSE-
equipment, achieving then an energy resolution down to 1 µ eV [24,25]. The energy resolution in
the neutron energy gain spectra (ω < 0) normally deteriorates with the increasing energy transfer,
which has to be taken into account when fitting the peaks caused by collective excitations. For
the neutron energy loss spectra, the resolution varies little in TOF spectrometry, improving slowly
toward larger energy transfers, being a further reason – besides the intensity argument discussed
above – for the preference for these spectra.
IXS experiments, due to the very high incident energy of up to 21 keV, do not suffer from
the kinematic restrictions discussed above for INS. In addition, except for dispersions situated
in the range of a few meV such as for compressed gases [13], the resolution down to 1.6 meV
(using the Si(11,11,11) reflection) is very good compared to (non-NSE) INS experiments with
high incident energies, though the shape of the resolution function of the IXS backscattering
spectrometer crystals (Lorentzian) is not very favorable. Due to the extremely long flight-path of
the incident photons, forced by the backscattering geometry, the experiments need about the same
time with IXS and INS. Even though it is more difficult to determine S(Q,ω) on absolute scale,
very good results have been obtained for disordered systems using IXS [3,4,26]. In spite of all theses
advantages, IXS works well only on well selected samples, and therefore one has to expect that
17
J.-B.Suck
INS and IXS will remain complimentary methods in the investigation of collective excitations in
disordered systems for some time in future.
References
1. Vacher R., Courtens E. Brillouin Scattering. – In: International Tables of Crystallography. Vol. D
’Physical Properties of Crystals‘. Ed. Authier A., Kluver, Dordrecht 2003, p. 329 – 335
2. Rat E., Foret M., Massiera G., Vialla R., Arai M., Vacher R., Courtens E., Phys. Rev. B, 2005,
72, 214204.
3. Ruocco G., Sette F., J. Phys.: Condens. Matter, 2001, 13, 9141
4. Scopigno T., Ruocco G., Rev. Mod. Phys., 2005, 77, 881
5. Suck J.-B., Experimental Investigations of Collective Excitations in Disordered Matter. –In: Collective
Dynamics of Nonlinear and Disordered Systems Ed. Radons G., Häussler P., Just W., Springer Berlin,
Heidelberg 2005 p. 147–170
6. S.W. Lovesey: Theory of Neutron Scattering from Condensed Matter, (Clarendon Press, Oxford 1984)
7. Copley J.R.D., Verkerk P., van Well A.A., Fredrikze H., Comp.Phys.Commun., 1986, 40, 337
8. J.-P. Boon, S. Yip: Molecular Hydrodynamics, (McGraw-Hill New York 1980)
9. Suck J.-B., Report KFK 1975 2231
10. Rahman A., Phys.Rev.Lett. 1974 32, 52
11. van Well A.A., Verkerk P., de Graaff L.A., Suck J.-B., Copley J.R.D., Phys.Rev. A, 1985, 31, 3391
12. Pratesi G., Suck J.-B., Egelstaff P. A. J. Non-Cryst. Solids 1999 250-252, 91
13. Suck J.-B., J. Phys.: Condens. Matter, 1991, 3, F73
14. Chushak Ya., Bryk T., Baumketner A., Kahl G., Hafner J., Phys.Chem.Liq. 1996 32, 87
15. Hafner J.: From Hamiltonian to Phase Diagrams, (Springer, Berlin, Heidelberg, New York, Lon-
don,Paris, Tokyo 1987)
16. Sjölander A., Ark.Fys. 1958 14, 315
17. Suck J.-B., Int. J. Mod. Phys. B, 1993, 7, 3003
18. Robinson R.A., Physica B 1989 156 – 157, 557
19. Kempa M., Janousofa B., Saroun P., Flores P., Boehm M., Demmel F., Kulda J., Physica B 2006
385–386, 1080
20. Copley J.R. D., Neut.Instr.Meth.Phys.Res. A 1990 291, 519
21. Jahn S., Suck J.-B., Appl. Phys. A 2002 74, S1465
22. Popovici M., Stoica A.D., Ionita I., J.Appl. Cryst. 1987 20, 90
23. Saroun J., Kulda J., Neutr.News 2002 13, 1573
24. Zeyen C.M.E.., J.Phys.Chem.Solids 1999 60, 1573
25. Kulda J., Farhi E., Zeyen C.M.E., , Physica B 2002 316–317, 383
26. Scopigno T., Suck J.-B., Angelini R., Albergamo F., Ruocco G.,, Phys.Rev.Lett. 2006 96, 135501
Непружнє розсiяння нейтронiв в застосуваннi до дослiджень
колективних збуджень у топологiчно невпорядкованих
системах
Й.-Б.Сук
Iнститут фiзики, Технологiчний унiверситет Хемнiтца, Хемнiтц, Нiмеччина
Отримано 22 листопада 2007 р., в остаточному виглядi – 17 грудня 2007 р.
Методику непружнього розсiяння нейтронiв представлено тут як одну з найважливiших експери-
ментальних методик для дослiдження колективних збуджень у плинах (рiдинах та стиснутих газах)
та аморфних твердих тiлах. Коротко обговорюються вiдповiднi кореляцiйнi функцiї, спектри яких
визначаються в експериментах по непружньому розсiянню нейтронiв, дисперсiйнi спiввiдношення
для колективних збуджень та як вони можуть бути отриманi з вимiрюваних спектрiв, а також – двi з
найбiльш часто використовуваних методик.
Ключовi слова: непружнє розсiяння нейтронiв, кореляцiйнi функцiї, дисперсiйнi спiввiдношення,
колективнi збудження, топологiчно невпорядкованi системи
PACS: 61.12.Ex, 63.20.Dj, 63.50.+x
18
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