Atomic motion in materials using X-ray
Satoshi Tsutsui and Hiroshi Uchiyama (JASRI/SPring-8)
We will introduce investigation of atomic motion using high resolution inelastic x-ray
scattering (IXS) as beamline practical of Cheiron School.
more complicated than the other beamlines.
also too broad to show in this textbook.

The beamline optics for IXS is

Scientific field covered with IXS beamline is

We will try to show simpler guidance including

scientific background through the beamline practical to understand IXS experiment.
Introduction
Atoms in materials do not stay even in solids.

Atoms in solids are vibrating at a

certain atomic position with keeping structural symmetry.

Atomic vibrations in solids arecalled phonon, which is frequently affected by electronic properties as well as connectedwith elastic properties in solid state materials.

Meanwhile, such atomic vibrations

connected with physical properties exist in liquid and gas. One of the contributions ofatomic vibration is propagation of sound.
Several experimental techniques have been used in laboratory apparatuses to detect
excitations from atomic vibrations: specific heat experiment is applicable to investigation ofphonon density of states when electronic contribution is excluded; sonic wave
measurements and Brillouin scattering are useful to investigate acoustic phonons throughsound velocity; Raman scattering and infrared scattering experiments can detect some
optical phonons.

Although phonon energy depends on phases of vibrations, theseapparatuses are difficult to apply to investigation of phase dependence of phonon energy,
phonon dispersion relations.
Phononinvestigateddispersion

relationsonlyinelasticbyscattering (INS) until about 1990.

wereneutronSoon after

inelastic X-ray scattering (IXS) for phonon
studies were established [1], third generationsynchrotron radiation facilities of ESRF inEurope [2], APS in United States [3] andSPring-8 in Japan [4] were built.

One of thedifficulties in phonon measurements using
X-ray is energy resolution for experiments.
This is connected with why INS for phonon
1Fig. 1. crystal structure of diamond.

Fig. 2. Phonon dispersion relation in diamond. [5]studies was established by about 40 years earlier than IXS.
of order of meV or THz.

Typical energy of phonons isFigure 1 and 2 show crystal structure of diamond and its

dispersion relations obtained by INS, respectively [5].

In addition, wave length required

to investigate phonon dispersion relations is comparable to diffraction experiments.
In this beamline practical, we prepared a single crystal sample of a simple metal. We
can know how to measure IXS spectra and polarization effect of IXS spectra, which areuseful to carry out IXS to know phonon dispersion in your interesting compounds.
Q1. Why was INS for phonon studies established earlier than IXS?

Discuss differences

between IXS and INS on viewpoints of correlation between energy and wave length.
Brief introduction of inelastic X-ray scattering
As mentioned above, meV energy resolution is required in IXS.

To achieve such highenergy resolution, higher order reflections from nearly perfect crystals are used as a highresolution monochromator (HRM).

Reflectivity of high order reflections in X-ray is much

weaker than that of low order one because of form factor of X-ray scattering.
signal intensity obtained in most of inelastic experiments is weak.

In addition,

This means that highbrilliant X-ray source is crucial to carry out inelastic scattering experiments.
reduction of photon flux at HRM should be as small as possible in IXS.

In addition,

Although several

types of HRM’s exist to achieve meV resolution, BL35XU chooses a backscattering setup,
whose acceptance is large in HRM.
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IXS requires changing energy as well as high energy resolution. The way of changing
energy in a backscattering setup is different from that in conventional setups such as abeamline monochromator (e. g. double crystal monochromator).
relation of 2 sin

Bragg law gives a, where d denotes space of atomic planes,  denotes scattering angleand  denotes wave length of X-ray.

Since  ~ 90 degrees in the backscattering setup,changing scattering angle is not allowed to changing energy unlike a beamline
monochromator.

Instead of changing scattering angle, space of atomic plane is changed

with thermal expansion.
Employment of backscattering optics enlarges beamline size.
BL35XU [6].

There are two reasons why BL35XU is large.

beamline optics.

The other is caused by analyzer optics.

Figure 3 is layout ofOne is caused by main

The main beamline optics islocated in the backscattering hutch. This consists of offset optics and HRM optics.

Theformer one works for changing beam height to make space around sample area as well as
reducing heat load of backscattering monochromator crystal. The latter one works only
for reducing the band width of X-ray.

To avoid interference of the offset optics and HRM

one, a certain space between X-ray beams towards and away of backscattering
monochromator is needed.

Meanwhile, analyzer optics requires analyzer acceptance aswell as backscattering geometry among sample, analyzer crystals and detector elements.
As mentioned above, backscattering geometry in X-ray optics is connected with energy
resolution. Keeping the backscattering geometry is crucial for high resolution inelastic
X-ray scattering.

Fig. 3.

Concerning the analyzer acceptance, BL35XU employs a multi analyzer

Layout of inelastic x-ray scattering beamline of BL35XU [6].
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system [7].

This system is useful to save time for investigation of phonon dispersion

relations because signal intensity of IXS measurement is weak.

Details of the multianalyzer system will be explained later.
Q2. Si(n n n) backscattering setup is employed in BL35XU.

The energy resolutions of Si(88 8), Si(9 9 9) and Si(11 11 11) setups are 5.5, 3 and 1.5 meV. How precise temperature
control and readout is required in such IXS setups?

The lattice constant of silicon is 0.5431

nm and thermal expansion of silicon is 2.6 x 10-6 K-1.
IXS and INS as a tool for investigation of dynamical structure factor
IXS

enables

ustoinvestigate

dynamical structure factor of materials
using

X-ray.

Since(a)disorderedmaterials such as glasses and liquids
have

nosymmetries,

momentumtransfer in these materials is scalar.
In other words, momentum transfer isQ, q

determined only by scattering angle
formeasurements.

Dynamical

structure factor depends only on
scattering angle.
interestedin(b)Then, if you are

dynamicalstructure

factor of disordered materials, youwill

investigate

scatteringangle

dependence of IXS spectra to obtain
dynamical structure factor.

On theQ

other hand, since crystalline materials
have a certain symmetry based onstructure,Fig. 4. Schematic drawing of the relationshipmomentum transfer in these materials

between reduced momentum transfer andis a three-dimensional vector.

analyzer location in BL35XU.

theirown

crystalThis(a) Horizontalmeans that dynamical structure factor

analyzers can be aligned approximately along

depends on sample orientation as well

the longitudinal direction, and (b) Verticalas scattering angle.

ones can be aligned approximately along theCrystalline

materials

translational symmetry.

haveaFor example,

transverse

direction

mounted properly.
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ifyoursampleiswhen you are interested in dispersion relations of acoustic modes, you have in principle
several choices in reciprocal space, one of which is of course the best position to investigate.
Therefore, a standard position to investigate phonon dispersion relation is a Bragg peak,which corresponds to a center of Brillouin zone.

To consider momentum transfer Q,

existence of translational symmetry is important.

Let us consider acoustic modes forsimplicity. Now, momentum transfer Q is represented as, where Q0 denotesa center of Brillouin zone and q denotes a reduced momentum transfer in a Brillouin zone.
When you choose q parallel to Q0, you can observe a purely longitudinal acoustic mode;
when you choose q perpendicular to Q0, you observe a purely transverse acoustic mode.
If you choose q which is neither parallel nor perpendicular to Q0, you can observe mixed
spectra of longitudinal and transverse modes. This is polarization effect of phonons inIXS and INS.
BL35XU has a 4 x 3 multi-analyzer system [7].
the scattering plane of the spectrometer.
scattering plane.

Four horizontal analyzes are located on

The other eight ones are shifted from theThis multi-analyzer system is useful especially for investigation ofphonon dispersion relations in crystalline samples.

Figure 4 shows schematic drawings

for relationships between analyzer arrays and reduced momentum transfer in crystalline
samples.

If you mount your sample appropriately, you can measure IXS spectra

approximately along longitudinal or transverse direction you are interested in.

Deviationfrom ideal reciprocal positions in longitudinal or transverse direction occurs due to lattice
constants and structural symmetries of you sample even if you mount and align yoursample perfectly.
Comparison with INS
IXS is a complementary technique to INS in investigation of dynamical structure factor
based on atomic motions in materials.

Since difference between incident energy andscattered one is much larger in IXS than in INS, IXS allows investigation of dynamicalstructure factor free from kinematic limitation.

Then, IXS is a useful technique toinvestigate atomic dynamics in liquid and glass, which require measurements at highenergy and low momentum transfers.

In addition, since the beam size at the sample

position in IXS is small, say 100 x 100 m2 or smaller, IXS is applicable to small samples.
Then, if you are interested in a sample under high pressure, using diamond anvil cell for
example, you can investigate phonon dispersion relation of the materials in high pressure.
IXS differs from INS in viewpoint of scattering process. Basic scattering process of IXSis Thomson scattering proportionate to number of electrons per each atom.
scattering intensity in atoms with larger atomic number is stronger.
5This means

In addition, crosssection of magnetic and/or electronic excitations is too small to detect by IXS. In INS,meanwhile, scattering intensity of atoms is independent of atomic number but depends on
nuclear isotope.

In addition, cross section of magnetic excitations is comparable to that ofphonon excitations in INS.

Such differences produce advantages and/or disadvantage ofIXS to INS other than sample size and range of momentum transfer.

For example, IXS iseasy to observe atomic motion of heavy elements and to avoid observation of magnetic
excitations.
Q3. What is difference of relationships between energy and momentum transfers in X-ray
from that of neutron?
Phonon studies in solid state physics
Phonon dispersion relations reflect elastic properties in material.

For example, soundvelocities and/or elastic constants in some directions are given by acoustic modes.
Instability of crystal structure in materials is also given by phonon dispersion relations.
Such instabilities are frequently caused by electronic properties.

In other words, phonon

dispersion relations are sometimes affected by electronic states through electron-phonon(el-ph) interaction.

This reflects one atom binding another neighbor atom with electrons.

Kohn anomaly is a phenomenon reflecting el-ph interaction.

In materials containing

strong electron-phonon coupling, discontinuous phonon dispersion relation caused by
Kohn anomaly can be observed at 2 kF, where kF is Fermi wave number. Investigation ofel-ph coupling in phonon dispersion relations is important to know electronic properties.
References[1] E. Burkel: Springer Tracts in Modern Physics, 125, Springer, Berlin, 1991.
[2] http://www.esrf.eu[3] http://www.aps.anl.gov[4] http://www.spring8.or.jp[5] J. L. Warren, J. L. Yarnell, G. Dolling, and R. A. Cowley: Phys. Rev. 158, 805 (1967).
[6] A. Q. R. Baron, Y. Tanaka, S. Goto, K. Takeshita, T. Matsushita, and T. Ishikawa: J. Phys.
Chem. Solids: 61, 461 (2000).
[7] A. Q. R. Baron, J. P. Sutter, S. Tsutsui, H. Uchiyama, M. Masui, S. Tajima, R. Heid, andK.-P. Bohnen: J. Phys. Chem. Solids 69, 3100 (2008).

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