Neutron & X-ray Scattering in Materials Research
Introduction
Scattering of neutrons and X-rays allows a wide variety of materials to be probed:
magnetic or nonmagnetic; crystalline or amorphous; organic or inorganic.
Additionally, the wavelength of the incident radiation, λ, may be selected such that
scattering from a particular size of entity is optimised. Consequently, neutrons and Xrays are able to probe entities of characteristic dimension between that of typical
interatomic spacing (~ 1Ǻ) to, for instance, polymers of dimension ~ 100nm [1]. Xrays correspond to electromagnetic radiation of wavelength between 10-8 and 10-12 m.
A neutron’s wavelength and thermal energy are directly related to each other: by
somehow changing the effective temperature of neutrons, an appropriate wavelength
may be selected. Such temperature control allows neutrons of comparable wavelength
to X-rays to be produced.
X-rays interact with atomic electrons; X-ray scattering cross-section is sensitive to
variations in electron density. In general, atomic electron density increases with
atomic number: X-ray scattering therefore allows an electron density profile – a map
of the position of atomic electron clouds – to be established [2]. On the other hand,
neutrons interact, via short-range (~femtometer) forces with atomic nuclei. The crosssection for neutron scattering also varies with atomic number although this variation
is not regular across the Periodic Table. Neutron scattering can also be used to
construct a structural map: position of atomic nuclei - rather than the electron clouds –
may be determined using neutron scattering. Additionally, neutrons also have a
magnetic dipole moment and may therefore interact with magnetic electrons – those
electrons in atoms which are unpaired. Neutron scattering allows magnetic order – or
the lack of magnetic order – to be probed.
Sources and Scattering
Neutrons may be produced by the fission of Uranium nuclei in a nuclear reactor or,
alternatively, by bombarding a metal target, such as Tungsten or Tantalum, with high
energy protons [1]. This latter method, known as spallation, allows a pulsed neutron
beam to be generated. In contrast, a reactor provides a continuous neutron flux.
Generally, compared to reactor sources, spallation sources provide a greater flux of
neutrons. Additionally, the between-pulse background flux of spallation neutrons is
likely to be very low.
The thermal energy of a neutron of effective temperature T is 3kBT/2; kB is
Boltzmann’s constant. This energy is equal to the neutron’s kinetic energy, mv2/2 (m
and v are the neutron’s mass and velocity respectively). The de Broglie relationship
between wavelength and velocity – λ = h/mv; h is Planck’s constant – allows the
temperature and wavelength to be related. Appropriate wavelengths may be attained
by slowing down the neutrons as they emerge from their source. Such slowing down –
also known as moderation – is achieved by directing the ‘hot’ neutrons towards, for
instance, liquid methane or liquid deuterium. The cross-section for inelastic collisions
of neutrons in such materials is sufficient such that the neutron velocity can be
reduced significantly. Despite moderation, a spread in neutron velocity – and hence
wavelength – about a mean value is observed. Monochromation of the neutron beam
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may be achieved by directing the flux towards a single crystal [3]. If a flux of
radiation is incident at an angle θ to a single crystal in which the parallel atomic
planes are separated by a distance d, the Bragg condition for diffraction – nλ = 2dsinθ;
n = 1, 2, 3… - implies that radiation of particular wavelength may be achieved in
crystals with different d or by rotating a particular crystal. Alternatively,
monochromation may be achieved by directing the neutron flux through a mechanical
velocity selector known as a chopper.
X-rays may be generated by accelerating electrons into a metal target – Tungsten,
for instance [2]. In addition to a broad (‘white’) spectrum of radiation, well-defined
X-rays – of wavelength corresponding to the emission lines of the metal target – will
also be produced. Alternatively, X-ray sources of greater brilliance and smaller beam
cross-section may be generated in a synchrotron source. In such synchrotrons,
electrons are accelerated along a curved trajectory and will, ultimately, lose energy in
the form of electromagnetic radiation [4]. This radiation may be monochromated by a
single crystal such that an appropriate X-ray wavelength is selected.
kf
Q
2θ
kf
kf
Q
Q
2θ
2θ
ki
ki
(a)
(b)
ki
(c)
FIG 1. Wavevector diagrams for (a) elastic scattering and inelastic scattering in which the radiation (b)
loses energy and (c) gains energy. ki and kf refer to the wavevector before and after scattering; 2θ is the
scattering angle. Redrawn from [1].
Upon interacting with the material sample being studied, the incident radiation
may be scattered elastically or inelastically. In all scattering processes, momentum,
and hence wavevector, must be conserved. In elastic scattering, shown in Figure 1(a),
the direction of the wavevector of the emergent radiation, kf, will, in general, be
different to that of the incident radiation, ki; the magnitude of this wavevector will not
change. In inelastic scattering processes, the magnitude and direction of the incident
wavevector will change: kf will be greater or lesser than ki if the radiation increases or
decreases in energy respectively. The wavevector of emergent radiation relates to the
angle through which the neutrons have been scattered; a change in wavevector, Q = kf
– ki, also known as the scattering factor [3], may be measured by varying the angular
position – relative to the sample – of the radiation detector.
Structure Determination
Elastic scattering of neutrons or X-rays may be used to determine the structure of
crystalline materials [5]. Figure 2 shows the neutron and X-ray diffraction patterns of
Iron. Both patterns show peaks which correspond to diffraction from different planes
of the crystal; 110, 200 and 211 are the appropriate Miller indices. The difference in θ
between the X-ray and neutron patterns is a consequence of the difference in
wavelength of the two sources.
2
I
(a)
I
(b)
FIG 2. Diffraction patterns of Iron obtained using (a) X-rays and (b) neutrons. I is the scattering
intensity. From [5].
Scattering of X-rays from the 6 electrons of a Carbon atom is more significant
compared to the scattering from the single electron in a Hydrogen atom [3].
Compared to X-rays, neutron scattering from Hydrogen atoms is significant: neutron
scattering is therefore an excellent probe of organic materials. Figure 3 shows the
atomic structure of anthracene and benzene determined from X-ray and neutron
scattering respectively. In anthracene, the Hydrogen atoms appear insignificant
compared to the Carbon atoms. In contrast, the Carbon and Hydrogen atoms are
distinct in benzene. Such sensitivity to Hydrogen atoms allows bond lengths to be
determined accurately.
(a)
(b)
FIG 3. The atomic structure of (a) anthracene and (b) benzene determined from X-ray and neutron
scattering respectively. From [5].
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Neutron scattering cross-section varies with atomic mass number and, therefore, is
sensitive to isotopic variation. Such isotopic dependence may be exploited in the
study of organic condensed matter: 1H atoms may, for example, be replaced by 2H
(Deuterium) atoms. Such deuteration may be utilised in the study of polymers;
neutron scattering can probe interactions between polymer chains. In a single, isolated
polymer, the constituent monomers will connect together in a random manner such
that the mean end-to-end distance of the polymer chain will correspond to an ideal
random-walk [6]. In a dilute polymer solution, repulsive interactions between polymer
chains may lead to a ‘self-avoiding random-walk’ and consequently, the mean length
of the chain cannot be described by ideal random-walk statistics. Such dilute solutions
could be probed by neutrons, X-rays or, if sufficiently dilute, visible light. In a
concentrated polymer solution, despite many interchain interactions, theoretical
predictions suggest that, due to smaller concentration gradients, the random-walk
result should be observed. Scattering of light or X-rays is insufficient for imaging
such concentrated solutions. However, by utilising selective (rather than complete)
deuteration, individual polymer chains, as a consequence of a sufficient deuteriumhydrogen contrast, may be imaged. Neutron scattering does indeed prove that
monomeric random-walking is observed in concentrated solutions.
Magnetic Scattering
Unlike X-rays, neutrons may interact with electronic spin and therefore can probe
magnetic ordering in solids: if long-range magnetic ordering exists, additional
scattering of neutrons, relative to the nuclear contributions, will be observed [3].
Figure 4 (a) and (b) show the neutron diffraction patterns of Iron held above and
below the Curie temperature, TC, respectively. Below TC, the nuclear diffraction peaks
are enhanced by a magnetic contribution. This additional magnetic contribution is
superimposed onto the nuclear contribution because, as shown in Figure 4 (c), the
magnetic and chemical unit cells of Iron are coincident. In contrast, in an
antiferromagnetic material – in which the magnetic and chemical unit cells are
different – any directional magnetic scattering will be observed as new, additional
peaks in the neutron diffraction pattern. Figure 5 shows the neutron diffraction
patterns of a Gold-Manganese alloy at temperatures greater and lesser than the Néel
temperature, TN (the antiferromagnetic transition temperature). For T < TN, additional
magnetic contributions to neutron diffraction are observed. Relative to Figure 5(a), in
Figure 5(b), the additional peaks exist midway between the nuclear peaks: this
suggests that the periodicity of the magnetic order is exactly twice the periodicity of
the magnetic ions, namely the Manganese. The structure of AuMn, shown in Figure
5(c), confirms such an antiferromagnetic arrangement. In ferromagnetic and
antiferromagentic materials, as the material is heated above the magnetic transition
temperature, magnetic scattering of neutrons does occur but, because the direction of
the magnetic moments is random, this scattering will not be directional (as it is below
the transition temperature) but, rather, will contribute to the scattering background.
Such an increase in background scattering is confirmed by comparing Figures 4(a) &
(b) and 5(a) & (b).
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I
(a)
I
(b)
(c)
FIG 4. The neutron diffraction patterns of iron at temperature (a) greater than and (b) lesser than the
Curie temperature. The magnetic contribution to scattering is shaded. The arrangement of magnetic
spins (arrows) and atoms (circles) in Iron are illustrated in (c). From [5].
I
(a)
I
(b)
(c)
FIG 5. The neutron diffraction patterns of AuMn at temperature (a) greater than and (b) lesser than the
Néel temperature. The magnetic scattering is shaded. The atomic and magnetic structure is shown in (c).
From [5].
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Small Angle Scattering
The Bragg condition implies that, generally, large objects scatter incident radiation
through small angles. If the wavelength of incident radiation is smaller than the
dimension of the scattering entity, low-order diffraction peaks will not be observed
although scattering will occur. Large entities (~ 100 nm maximum [1]) may be probed
by small angle neutron or X-ray Scattering (SANS and SAXS respectively); the
small-angle scattering intensity will be maximised for a particular cluster size.
SANS has been used to probe strong coupling between electrons and phonons in
Manganese oxides known as manganites. In solids, phonon-phonon interactions are
responsible for any thermal expansion of the solid [7]: if phonons do not interact with
each other, thermal expansion is prohibited. Strong electron-electron interactions can,
for example, influence the magnetic exchange of electrons. For instance, if a
ferromagnetic material is heated above its Curie temperature, if neighbouring
magnetic spins do not interact with each other, the inverse magnetic susceptibility will
increase linearly with temperature; this is the Curie-Weiss law. Any deviation from
such behaviour indicates that neighbouring electrons are interacting with each other
[8].
FIG 6. The SANS intensity (points) and anomalous volume expansion (line) near the Curie temperature
in La1-xCaxMnO3. The SANS wavelength was 4.5 Ǻ; the volume expansion was measured using a
strain-gauge technique. From [8].
Figure 6 shows the temperature dependence of the anomalous volume expansion
(due to phonon-phonon interactions) and SANS intensity near the Curie temperature
in La1-xCaxMnO3; the SANS intensity is proportional to the number of distinct
ferromagnetic scattering centres. For T < TC, all valence electrons are delocalised:
there are very few distinct ferromagnetic scattering centres and, consequently, the
SANS intensity is low. At T > TC, sufficient thermal energy exists such that the spin
associated with the scattering centres is randomized: the material exhibits
paramagnetism such that few distinct ferromagnetic clusters are present.
Consequently, the SANS intensity is suppressed. At T = TC, the bandwidth for
electron-electron interactions is sufficient such that some, but not all, electrons can
couple feromagnetically with their neighbours: distinct ferromagnetic clusters form
and, consequently, the SANS intensity is maximised. SANS therefore provides a
measurement of electron-electron interactions. The results in Figure 6 show that the
electron-electron and phonon-phonon interactions have identical temperature
dependence. The combination of an electron and the strain field associated with the
nearest lattice phonon is known as a polaron: it is therefore appropriate to consider
the results in Figure 6 a consequence of polaron-polaron interactions. SANS has
therefore shown that the paramagnetic phase of manganites can be successfully
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described as a polaronic liquid rather than, as was originally believed, an ideal
polaronic gas.
Small angle scattering is also ideal for studying biological polymers such as
proteins. Gliadins are important proteins found in wheat; interactions between these
gliadins are believed to influence the mechanical response of wheat dough [9].
Solutions of these gliadins have been studied by SAXS. Figure 7 shows relationships
between the scattering intensity and the scattering factor associated with a particular
gliadin. These quantities may be plotted relative to each other is such a way that the
radius of gyration of the protein, Rg, and the radius of gyration of the protein crosssection, Rc, may be determined. Rg and Rc may then be mathematically related to the
length and diameter of a cylindrical rod or, alternatively, to the axes of an ellipsoid.
(a)
(b)
FIG 7. Relationship between (a) loge(I) and (b) loge(IQ) as a function of Q2 in α-gliadin. The different
symbols correspond to various concentrations of solvent. Rg and Rc may be determined by linear
extrapolation of data for which QRg and QRc are less than 1 in (a) and (b) respectively. From [9].
Appropriate analysis of the scattering data (Q and I) therefore allows the shape of the
gliadins to be determined. The data shown in Figure 7 was used to establish that the
gliadins may be modelled as prolate ellipsoids. Aggregation of such ellipsoids, which
is different to the aggregation of cylindrical rods, is believed to contribute to the
viscoelasticity (viscous or elastic mechanical response, depending on the period of
deformation) of wheat dough.
Inelastic Scattering
In solids, excitations, such as phonons, are likely to have energy ~ kBT such that, for
typical temperatures (up to 500 K, for instance) these excitations are in the meV
energy range. Typical X-rays and neutrons ~ 1 – 10 Ǻ have energy ~ keV and ~meV
respectively. Upon inelastically scattering from the aforementioned excitations, the
percentage change in neutron energy will be much greater than the corresponding
percentage change in X-ray energy and will, therefore, be much easier to measure.
Inelastic neutron scattering is, therefore, frequently used to study these excitations.
Upon scattering, the wavevector of the excitation is equal to the change in neutron
wavevector, Q; the momentum of the excitation is ħQ. By comparing the energy and
wavevector of the incident and emergent neutrons, the energy and momentum of the
excitations may be determined.
Phonon dispersion in solids can be studied using inelastic neutron scattering.
Figure 8(a) shows the dispersion of the 4 phonon modes in Silicon determined by
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inelastic neutron scattering; the phonon frequency is proportional to phonon energy.
For a centrosymmetric covalent solid such as Silicon, at small k, the transverse- and
longitudinal-optical phonon modes are expected to be degenerate: neutron scattering
confirms this prediction. Neutron scattering also shows that, in contrast, in a polar
solid such as KBr, the optical phonon modes are not degenerate. This observation also
agrees with theoretical predictions.
(a)
(b)
FIG 8. The variation of the phonon frequency with normalised wavevector, K/Kmax, in (a) Silicon and
(b) KBr determined by inelastic neutron scattering. ‘A’ and ‘O’ refer to acoustic and optical phonons
respectively; ‘L’ and ‘T’ correspond to the longitudinal and transverse modes. From [7].
(a)
(b)
FIG 9. The variation of magnon frequency in (a) MnPt3 and (b) RbMnF3 with k2 and k respectively
determined by inelastic neutron scattering. From [7].
Phonons are elementary lattice excitations; the elementary magnetic excitations in
solids are known as magnons [7] and, as for phonons, the dispersion of these magnons
may be measured using inelastic neutron scattering. In an antiferromagnet, the
magnon energy is expected to vary linearly with k while, in contrast, in a ferromagnet,
the magnon energy should be proportional to k2. Figure 9 shows the variation of
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magnon energy as a function of wavevector in MnPt3, which is ferromagnetic, and
RbMnF3, which is an antiferromagnet, as determined by neutron scattering. The
respective square and linear dependences of magnon energy on wavevector are
confirmed by such scattering.
Conclusions
Scattering of neutrons and X-rays allows many condensed matter systems to be
studied. Diffraction of neutrons or x-rays allows crystal structure to be successfully
determined. Due to their sensitivity to Hydrogen atoms, neutrons are particularly
useful for studying organic matter; deuteration may also assist the study of such
materials. One major advantage associated with neutrons is their ability to interact
with atomic magnetic moments; neutron scattering may determine the presence or
absence of magnetic ordering in materials. To probe larger entities, small angle
scattering may be employed. In the examples given, SANS was used to probe strong
coupling between electrons and phonons in a magnetic material while SAXS provided
data which allowed interactions between proteins to be determined. Both neutrons and
X-rays may scatter inelastically; ~ meV neutrons are often used to study phonons or
magnons in solids. Inelastic neutron scattering has allowed theoretical dispersion
relationships to be confirmed experimentally.
References
[1] Pym R., Neutron Scattering Lecture Notes, http://www.mrl.ucsb.edu/~pynn
[2] Bacon G. E., X-ray and Neutron Diffraction, Pergamon Press (1966)
[3] Bacon G. E., Neutron Diffraction, 3rd Edition, Clarendon Press (1975)
[4] http://www.kent.ac.uk/physical-sciences/cmg/xray-web/
[5] Bacon G. E., Neutron Physics, Wykeham Publications (1969)
[6] Jones R. A. L., Soft Condensed Matter, Oxford University Press (2002)
[7] Kittel C., Introduction to Solid State Physics, 6th Edition, Wiley (1986)
[8] De Teresa J. M. et al., Nature, 386 (1997) 256
[9] Thomson N. H. et al., Biochim. et Biophys. Acta, 1430 (1999) 359
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