Atomic Motion via Inelastic X-Ray Scattering
Cheiron School Beamline Practical - Monday ONLY at BL35
Alfred Q.R. Baron & Satoshi Tsutsui

We will introduce students to the use of inelastic x-ray scattering, IXS, to investigate atomicmotions. A short discussion of the technique and tour of the instrumentation will be followed bymeasurements of some simple materials. If student interest is strong, we may also attempt toinvestigate a more complicated (structural phase change) material. The notes below are meant toprovide the students with a bit of guidance and background, and the "exercises" to provide somepoints for discussion. The Cheiron school audience is very broad, and the discussion below is
probably too simple for some and may be too advanced for others. We will try to fill in the gaps
during discussions.
1. Introduction

to the simple macroscopic picture (sound velocity anddamping) mentioned above.

Atoms in materials are always in motion – be it longrange diffusive motion, as in liquids, or oscillatory,phonon, motions about equilibrium positions in a solid.
Historically, the importance of this motion was notedbecause the thermodynamic behavior of materials (e.g.
specific heat) made no sense until one postulated
models of atomic motion – simple models include
those suggested by Einstein and Debye.
More
recently, it has been understood that atomic motions
are intimately connected to many facts of material
behavior, including many structural phase transitions,and some important electronic ones. Perhaps most
famous among these is the superconducting transition:
superconductivity in most materials is driven byinteraction of the atomic motions (phonons) and theelectronic system. There are then strong reasons, bothfrom the point of view of fundamental science, and for
applied reasons (how does a new type ofsuperconductor work?) to try to investigate dynamics.

2. The Method: Inelastic X-ray Scattering
X-rays have long been known to be sensitive tothermal motion via thermal diffuse scattering or
Debye-Waller factors. The difficulty is that to measureatomic length scale correlations you should use x-rays
that have a wavelength similar to the spacing between
atoms (~Å) and this then gives the x-ray energy in thekeV region, with 1 Å x-ray having an energy of 12.4
keV. Meanwhile, the energy scale of the atomic
motions is typically ~0.001 eV. Thus, to measureatomic motions with x-rays, one requires energy
resolution ΔE/E ~ 10-7, which is a severe requirement.
Furthermore, as the cross section for inelastic
scattering is small, one must be vary careful to preserve
the maximum possible flux.
[Exercise. Neutrons vs X-Rays. How muchenergy does a neutron with a 1Å deBroglie
wavelength have?
What is the requiredenergy resolution (ΔE/E) to measure ~meV
excitations with 1 Å neutrons? Note that theanswer to this question helps explain the factthat the field of inelastic neutron scatteringdeveloped some 40 years earlier thaninelastic x-ray scattering.]

In this practical we will investigate relatively simple
atomic motion, longitudinal acoustic modes. These
modes are the direct extrapolation of conventionalsound – if you tap a crystal, sound will propagate
through it, and, depending on the material properties,
the sound will propagate with different speeds, andwith different damping or attenuation.
Considerstriking a hard material, for example: naively (andcorrectly) one can expect the sound will propagate along way, and relatively quickly. Meanwhile, a noise
made in the water, will propagate much more slowly,
and generally not so far.

Beamline 35 of SPring-8 (BL35XU) [1] is designed formeasuring dynamics of atoms using x-rays.
The
beamline layout is shown in figure 1. Possibly themost notable fact at first glance is that is very large fora synchrotron radiation instrument. This in fact
follows rather directly from Bragg’s law, which also
underlies many x-ray optics, thus we spend some time
to discuss it here.

We will look at microscopic sound in a hard material,
diamond, and a liquid (an extreme version of a softmaterial), and see how these are different and how
these differences reveal interesting information abotu
the materials and may be related behave, and how our1

BL35XU Hutch Layout
Optics Hutch

High Heat Load
Monochromator

NRS-2

NRS-1

Analyzer Hutch

IXS
SamplePosition55°

High Resolution
Optics Table

Backscattering HutchNRS
Sample Table

Beam Path for Inelastic X-Ray Scattering
AnalyzerDetector

MirrorBackscattering
Monochromator

Optical Table
Vertical beam offset

MirrorSampleMonochromator

Figure 1. Schematic of BL35XU. The top figure shows the hutch outlines, as viewed from above, while thelower part shows the beam path through the beamline, The part shown is about 50 m long.
Bragg’s law governing
crystalline materials states

x-ray

diffraction

achieve meV resolution at 21.7 keV, howmany planes are needed? In backscattering(δ = 0), how thick (mm) does the crystal haveto be? How many materials do you thinkmight have the required level of perfection?]fromλ = 2d sin(Θ B )where d is the spacing between the planes of the crystal
lattice, and ΘB the Bragg angle the incident radiation
makes with those planes. Differentiation of Bragg’s
law immediately gives[Exercise. Temperature scans. Assume whiteradiation is incident on a silicon Braggreflection optic. What is the fractional energychange (ΔE/E) of the reflected beam if thetemperature of the optic is changed by 1
degree C? Note the thermal expansion ofsilicon is isotropic, with an expansion
coefficient of α = 2.6 x10−6/K near roomtemperature (ie: a change of 1K intemperature leads to a fractional expansionof 2.6 ppm.) How much do you think one
can practically change the reflected energy
using temperature in this way?]Δλ = 2Δd sin(Θ B ) + 2d cos(Θ B ) ΔΘ
which relates small changes in the various quantities.
Assuming the lattice constant of the material is now
fixed (Δd=0) then this may be re-written asΔΘ = tan(Θ B )ΔλλThis makes it immediately clear that in order toachieve a maximum angular acceptance for our optics,large ΔΘ, for a fixed energy resolution, it is desirable
to operate at Bragg angles near to 90 degrees (ΘB=π/2δ, δ<<1).
In fact we typically operate with δ ~ 0.3
mrad. The scattered angle is then twice delta, so it isclear one needs very long path lengths (~10m) toseparate the incident and reflected beams from anoptic. This explains the large size of the beamline.

3. The Dynamic Structure Factor, S(Q,ω)
Naïvely, one might think that to understand thedynamics of a material, one should measure thepositions, as a function of time, of all the atoms in thesample of interest. Certainly this information is
sufficient to “know” the dynamics of a material.
However, this may not help you to understand thedynamics, as it is a huge amount of information, andone would need a fast computer to somehow process it
and answer meaningful questions. Also, while thisinformation can be measured in some special cases forslow motion, it is, now, nearly impossible to measurefor the fast motions we discuss here. Thus, most often
people measure correlations of motions in atoms, andnot the detailed motion of atoms. It turns out, in fact,that these correlations are the natural direct product ofa standard scattering experiment.

[Exercise. Acceptable angular divergence.
Assume you use a Bragg reflection nearbackscattering, δ = 0.3 mrad at an x-ray
energy of 21.747 keV.
What is thecontribution to the energy resolution if the xray beam divergence is 80 micro-radians.
How about if δ = 1 mrad][Exercise. Perfect silicon. The intrinsic energy
resolution of a Bragg reflection, ΔE/E, isapproximately given by 1/N, where N is thenumber of planes reflecting radiation.
ToScattering theory in physics is highly developed. The
reader is directed to such classic texts as Squires [2] for2

probed at a momentum transfer of Q. Whatis correlation length probed for |Q| = 2 nm—
1? 20 nm-1?]

a treatment, as well as the original literature (theseminal paper by van Hove [3], is in fact very readableand clear).
In short, within some very simple
approximations - that the scattering is weak, that thescatterer is far from the detector and far from thesource, etc - the scattered intensity is directly related tothe density-density correlation of a material. Thus
scattering experiments naturally answer the question:“If I slightly perturb one atom, how does theperturbation propagate to other atoms in the material”.

4. DiamondDiamond is well known as a jewel whose beauty of thecrystal attracts people as shown, figure 2, but given its
hardness, especially, it is also a useful material forindustrial applications. This is related to the strong
covalent bond based on the carbon sp3 hybridized
configuration. On the other hand, diamond is wellknown to be is easy to cleave in the {1 1 1} direction,
leading to beautiful gemstones, as seen in figure 2.

We now introduce some standard terminology. The
scattered intensity at some particular angle and energy
transfer in our measurements is proportional to thedouble differential cross section,⎛ d 2σ ⎞= re 2 εα * • εβ S(Q,ω)⎜⎟⎝ dΩdE ⎠ kα−>k'β()Q ≡ k' − kE = ω = E − E'
This introduces the dynamical structure factor of amaterial, S(Q,ω), which is related to the cross sectionthrough a scale factor describing the interaction ofelectrons and light: the product of the classical radiusof the electron, re, and the usual polarization factor forThomson (dipole) scattering*.
We also define themomentum transfer, Q, as the difference between theincident and scattered wave vectors. The dynamicstructure factor (sometimes also called the scatteringlaw), is then related to the equilibrium density-density
correlations of the material by+∞
1 +∞
−iωt
S (q,ω ) =dt ed 3 r d 3 r' e iq•( r−r')∫∫
2π −∞−∞Figure 2. Photos of diamond crystals.
The physical properties of diamond are very useful for
research of solid state physics. The hardness is useful,e.g., for making high pressure cells using "diamondanvils" (DAC) which are the best way of approaching(statically) the pressures in the center of the earth.
Diamond also has excellent thermal conductivity(much better than copper) which makes it useful forsome x-ray optical devices such high heat-load

ρ(r,0) ρ (r',t )+which explicitly shows the dependence on the densitydensity operator.
Notably, the expression in terms momentum/frequency
space, (q,ω), makes this quantity response to thenormal modes in periodic materials: i.e. measuring
S(Q,ω) is exactly the appropriate way to investigate
phonons in crystals.
[Exercise. Correlation lengths. The definitionof is Q=K’-K where K and K’ are the incident
and scattered wave vectors, respectively,
with their magnitude given by |K|=2π/λ. By
analogy to Bragg’s law, write down anexpression for the correlation length, d,*We have also dropped a phase space factor that is
unity for x-ray, though not for neutrons.

3Figure 3. Diamond lattice cubic unit cell. Thetwo atoms in a primitive unit cell are circled.

beamline monochromators.

2The unit cell of diamond is shown in figure 3 and has
cubic symmetry, Fd3m, with 8 atoms/unit cell. The
lattice constant is a = 3.566 Å. However, the primitive
cell has only two atoms. The number of phonons
possible at any momentum transfer is just 3 times thenumber of atoms per primitive cell, giving 6 phononmodes in diamond. The elastic properties are alreadyknown well. The longitudinal sound velocity is very
fast, 18 km / sec, as compared to several km/s typicalof a metal. The Debye temperature of diamond is very
high, 2,200 K, several times higher than in usual metal.
The high sound velocity and high Debye temperaturemean that diamond is an extremely hard material.

ZQj =Qj(⎡⎤
1 ⎢ nω qj + 1 δ ω − ω qj ⎥)= ⎢⎥ω qj + n
ω qj δ ω + ω qj ⎥⎢⎣⎦andFqjHarmonic ω(())

but we do not discuss them in detail here. We only
note that the second term, F, essentially gives peaks atthe phonon frequencies, while the first gives theintensity at different momentum transfers, which
depends sensitive on the directions of atomic motions
given by the e qjd . Note the over-all scale factor for
Z is as Q2, so higher momentum transfers, larger Q,
give more intensity.
With the help of the instructors you will measure theacoustic mode dispersion of diamond.
[Exercise.
Consider how to measure thesound velocity of diamond. Does the value
measured using IXS agree with the known
sound velocity?][Exercise. Check the line width of each peakin the spectra you measure.
Are they
resolution limited? Why or why not?]The expression for the scattered intensity, S(Q,ω) , fora crystalline material can be written as:∑ Z

fd (Q) −WdeQ • e qjd eiQ•d
2M dd
Atoms/CellThe phonon dispersion was reported some years ago byinelastic neutron scattering, and is shown in figure 4 asa plot of the phonon frequencies at different
momentum transfers - these are essentially the peak
positions in measurements of S(Q,ω) . Modes that goto zero frequency at zero momentum transfer areacoustic, corresponding to the usual longitudinal andshear sound modes. Generally, for acoustic modes, thetwo atoms in a primitive cell will move together, inphase with eachother, for acoustic modes. The highenergy modes are optical modes, where the two atoms
tend to move opposite eachother.
Recently,superconductivity was discovered in boron-doped (Bdoped) diamond at low temperature [5]. Correlation
between its superconductivity and phonon response has
been demonstrated.

S ( Q, ω )1p =∑[Exercise. We will measure the longitudinal
acoustic modes. How might one measure
the transverse modes?

Fqj (ω )j3rModeswhere ZQj is the “structure factor” and F is thefrequency dependence. For completeness, these terms
are given by

Figure 4. Phonon dispersion in diamond at room temperature investigated by inelastic neutron scattering. Markers
show the experimental data points, and solid lines show the calculated results based on a shell model [4].

4data to estimate the sound velocity? How
does it compare with the low-frequency limitof about 1500 m/s in liquid water?]

4. Liquid
The second measurement will be the dynamic structure
factor of a liquid. Below we will discuss the generalliquid properties in terms of liquid water, however, themeasurements, will be of polydimethylsiloxane(PDMS) CH3(C2H6OSi)nSi(CH3)3, which is a silicone
oil, in this case one that is used as a low-temperature
cooling bath fluid. The dynamics of this material, tothe best of our knowledge, have not yet been explored,
so this measurement will provide new information,however, we expect many of the features (broadphonons, central peak, some diffusion) will be, broadl
speaking, similar to those in water.

[Exercise. Diffusion and the Einstein relation.
Estimate the diffusion constant from thelinewidth of the central peak (do correct forthe resolution!). Does it approximately obey
the Einstein relation for Brownian motion?
What is the diffusion coefficient?]

5. Conclusions
We have investigated the dynamics of two samples
exemplifying crystalline material, diamond, and adisordered liquid. Both materials show acoustic modesat low momentum transfers, corresponding toapproximate expectations from the fact that bothmaterials propagate sound waves. However, while thewater diamond has very well defined modes with ahigh sound velocity, characteristic of a hard solid, thewater has highly damped modes characteristic of a softmaterial. The liquid also has a strong central peakowing to its disorder.

Figure 5, below, shows data from water at several
momentum transfers. There is a strong central peak,with wings on the side that are the phonons, analogous
to the longitudinal acoustic modes in diamond. Herehowever, one can note several differences:
1.
2.

3.

The central (elastic) peak is strong compared
to diamond – this results from the disorder ofthe material leading to enhanced scattering.
The central peak is NOT resolution limited.
This is due to the diffusion of the molecules in
the liquid (a glassy material, which is also
disordered, would also have a strong central
peak, but it would be resolution limited in thissample.
The phonon peaks are very broad. This
indicates that the phonon modes have a short
lifetime, again due primarily to the disorder.

6. References.
[1] A. Q. R. Baron, Y. Tanaka, S. Goto, K. Takeshita,T. Matsushita, and T. Ishikawa, The Journal of Physics
and Chemistry of Solids 61, (2000) 461-465.
See also arXiv 0910.5764.
[2] G. L. Squires, Introduction to the theory of
Thermal Neutron Scattering (Cambridge UniversityPress, New York, 1978).
[3] L. van Hove, Phys. Rev. 95, (1954) 249-262.
[4] J. L. Warren, J. L. Yarnell, G. Dolling and R. A.
Cowley, Phys. Rev. 158, 805 (1967).
[5] E. A. Ekimov et al., Nature 428, 542 (2004).

[Exercise. Sound Velocity. Use the measured

1.6 nm-1x5

3.2 nm-1x54.6 nm-1x5-20-1001020Energy Transfer (meV)

Figure 5. Inelastic scattering from water at theindicated momentum transfers. The red lines arefits to the data5