Phonons and Inelastic neutron and Xray Scattering

1
Phonons and Inelastic neutron and X-ray Scattering

• Paolo Ghigna, Dipartimento di Chimica Fisica M.
  Rolla, Università di Pavia

2
Summary

• The study of atomic dynamics in condensed matter
  at momentum transfers, Q, and energies, E,
  characteristic of collective motions is,
  traditionally, the domain of neutron
  spectroscopies.
• The experimental observable is the dynamic
  structure factor S(Q,E), which is the space and
  time Fourier transform of the density-density
  correlation function.
• Neutrons as probing particle are particularly
  suitable, since
• the neutron-nucleus scattering cross-section is
  sufficiently weak to allow for a large
  penetration depth,
• the energy of neutrons with wavelengths of the
  order of inter-particle distances is about 100
  meV, and therefore comparable to the energies of
  collective excitations associated to density
  fluctuations such as phonons, and
• the momentum of the neutron allows to probe the
  whole dispersion scheme out to several Å-1, in
  contrast to inelastic light scattering techniques
  such as Brillouin and Raman scattering which can
  only determine acoustic and optic modes,
  respectively, at very small momentum transfers.

3
What is a phonon?

• In physics, a phonon is a quantized mode of
  vibration occurring in a rigid crystal lattice,
  such as the atomic lattice of a solid. Phonon can
  also be used to describe an exitation of such a
  mode. The study of phonons is an important part
  of solid state physics, because phonons play an
  important role in many of the physical properties
  of solids, such as the thermal conductivity and
  the electrical conductivity. In particular, the
  properties of long-wavelength phonons gives rise
  to sound in solids -- hence the name phonon. In
  insulating solids, phonons are also the primary
  mechanism by which heat conduction takes place.
• Phonons are a quantum mechanical version of a
  special type of vibrational motion, known as
  normal modes in classical mechanics, in which
  each part of a lattice oscillates with the same
  frequency. These normal modes are important
  because, according to a well-known result in
  classical mechanics, any arbitrary vibrational
  motion of a lattice can be considered as a
  superposition of normal modes with various
  frequencies in this sense, the normal modes are
  the elementary vibrations of the lattice.
  Although normal modes are wave-like phenomena in
  classical mechanics, they acquire certain
  particle-like properties when the lattice is
  analysed using quantum mechanics (see
  wave-particle duality.) They are then known as
  phonons. Phonons are bosons possessing zero spin.

4
X-rays as a probe of phonons

• While it has been pointed out that X-rays can in
  principle as well be utilised to determine the
  S(Q,E), it was stressed that this would represent
  a formidable experimental challenge, mainly due
  to the fact that an X-ray instrument would have
  to provide an extremely high energy resolution.
  This is understood considering that photons with
  a wavelength of l0.1 nm have an energy of about
  12 keV. Therefore, the study of phonon
  excitations in condensed matter, which are in the
  meV region, requires a relative energy resolution
  of at least DE/E10-7.
• On the other hand, there are situations where the
  use of photons has important advantages over
  neutrons. A specific case is based on the general
  consideration that it is not possible to study
  acoustic excitations propagating with a speed of
  sound vs using a probe particle with a speed v
  smaller than vs.
• This limitation is not particularly relevant in
  neutron spectroscopy studies of crystalline
  samples. Here, the translation invariance allows
  to study the acoustic excitations in high order
  Brillouin zones, thus overcoming the above
  mentioned kinematic limit on phonon branches with
  steep dispersions.
• On the contrary, the situation is very different
  for topologically disordered systems such as
  liquid, glasses and gases. In these systems, in
  fact, the absence of periodicity imposes that the
  acoustic excitations must be measured at small
  momentum transfers. Thermal neutrons have a
  velocity in the range of 1000 m/s, and only in
  disordered materials with a speed of sound
  smaller than this value (mainly fluids of heavy
  atoms and low density gases) the acoustic
  dynamics can be effectively investigated.
• Another advantage of the inelastic X-ray
  technique arises from the fact that very small
  beam sizes of the order of a few tens of
  micrometers can be presently obtained at third
  generation synchrotron sources. This allows to
  study systems available only in small quantities
  down to a few 10-6 mm3 and/or their investigation
  in extreme thermodynamic conditions, such as very
  high pressure. These differences with respect to
  inelastic neutron scattering motivated the
  development of the very high resolution inelastic
  x-ray scattering (IXS) technique, and following
  the pioneering experiments in 1986, the IXS
  technique rapidly evolved. To date there are four
  instruments operational at the ESRF (2), APS (1)
  and Spring-8 (1), and several more under
  construction.

5
What is a phonon?

• Due to the connections between atoms, the
  displacement of one or more atoms from their
  equilibrium positions will give rise to a set of
  vibration waves propagating through the lattice.
  One such wave is shown in the figure below. The
  amplitude of the wave is given by the
  displacements of the atoms from their equilibrium
  positions. The wavelength ? is marked.
• It should be noted that there is a minimum
  possible wavelength, given by the equilibrium
  separation a between atoms. As we shall see in
  the following sections, any wavelength shorter
  than this can be mapped onto a wavelength longer
  than a.
• Not every possible lattice vibration has a
  well-defined wavelength and frequency. However,
  the normal modes (which, as we mentioned in the
  introduction, are the elementary building-blocks
  of lattice vibrations) do possess well-defined
  wavelengths and frequencies. We will now examine
  these normal modes in some detail.

6
Inelastic Scattering

• We will consider inelastic scattering where there
  is a change in the energy of the scattered beam
  with respect to the incident beam due to
  interactions of the incident wave with the
  sample.
• This has proved to be a fruitful area of
  investigation, particularly with neutrons where
  the energy of thermalised neutrons is comparable
  to that of phonons. This was recently recognised
  in the award of the 1995 Nobel Prize in Physics.

7
Optical, Acoustic, Transverse, Longitudinal Phonon

• In real solids, there are two types of phonons
  "acoustic" phonons and "optical" phonons.
  "Acoustic phonons", which are the phonons
  described above, have frequencies that become
  small at the long wavelengths, and correspond to
  sound waves in the lattice. Longitudinal and
  transverse acoustic phonons are often abbreviated
  as LA and TA phonons, respectively.
• "Optical phonons," which arise in crystals that
  have more than one atom in the unit cell, always
  have some minimum frequency of vibration, even
  when their wavelength is large. They are called
  "optical" because in ionic crystals (like sodium
  chloride) they are excited very easily by light
  (in fact, infrared radiation). This is because
  they correspond to a mode of vibration where
  positive and negative ions at adjacent lattice
  sites swing against each other, creating a
  time-varying electrical dipole moment. Optical
  phonons that interact in this way with light are
  called infrared active. Optical phonons which are
  Raman active can also interact indirectly with
  light, through Raman scattering. Optical phonons
  are often abbreviated as LO and TO phonons, for
  the longitudinal and transverse varieties
  respectively.

8
High Energy X-ray Inelastic Scattering

• Ever since the epoch-making DuMond experiments on
  beryllium, which provided the first evidence for
  the validity of Fermi-Dirac as opposed to
  Maxwell-Boltzmann electron momentum
  distributions, inelastic x-ray scattering has
  been established as a probe of the ground state
  properties of electrons in solids.
• Inelastic scattering refers to a number of
  interactions between x-rays and atoms in which
  the energy of the scattered photon is less than
  that of the incident one. The term high energy
  indicates the relative magnitude of the incident
  photon energy in comparison to the electron
  binding energy.
• Amongst the effects are the Compton effect,
  plasmon scattering (or collective excitation),
  x-ray Raman, x-ray resonant Raman and phonon
  scattering. In a typical scattering experiment an
  incident beam of photons of energy hn1 and
  wavevector k1 is scattered by the sample into a
  beam of photons of average energy hn2 and
  wavevector k2. The spectral distribution then
  provides information about the electronic
  structure of the sample.
• One advantage of this type of experiment is that
  the scattering probe is a quasi-particle of
  energy hn h(n1-n2 ) and momentum (h/2p)k
  (h/2p)(k1-k2) and the functional dependence
  between k and w is multi-valued and can be
  tuned appropriately to study a desired
  interaction. Large momentum transfers correspond
  to Compton scattering providing information on
  the ground state momentum of the individual
  electrons.

9
High Energy X-ray Inelastic Scattering

• Intermediate and low momentum transfer are
  governed by the differential scattering cross
  section
• where e (w,k) is the dynamic dielectric function
  and S(w,k), the dynamic structure factor provides
  information about the correlation of the valence
  electrons in space and time. For the core
  electrons x-ray Raman spectra provide information
  about the density of states near the Fermi energy
  and many body effects while x-ray resonant Raman
  and phonon inelastic scattering can provide
  details of the spin dependent momentum
  distribution

10
X-ray Inelastic Scattering

• DuMond (in 1929!) developed from first
  principles, a relation between the electron
  momentum distribution I(p) of an isotropic
  ensemble of electrons and the spectrum of a
  monochromatic x-ray beam inelastically scattered
  by them. This relation
• which leads to
• DuMonds experimental spectra of Be were clearly
  most similar to (i) which was the first direct
  evidence for the validity of the Fermi-Dirac
  electron momentum distribution.

11
Inelastic Neutron Scattering

• This is the principle technique for determining
  phonon dispersion curves and is of major
  importance in the study of phonons generally.
• Thermal neutrons (i.e. those in the meV energy
  range) have both the right energy and wavevector
  to interact with phonons. This is in marked
  contrast to other probes such as infrared
  radiation (where the energy is similar but
  (h/2p)k is only a very small fraction of the
  Brillouin zone boundary, and hence only phonons
  with k?0 can be measured) or x-rays ( where
  (h/2p)k can be of the correct order but the
  energy is many orders of magnitude larger than
  that of phonons).
• Note that a lattice vibration of angular
  vibration w connotes the movement of phonons,
  each with energy hn and crystal momentum (h/2p)k
  . Use of the term crystal momentum does not mean
  that ordinary momentum is transferred through the
  crystal at the corresponding rate, but crystal
  momentum is a conserved quantity in interactions
  involving the creation or annihilation of crystal
  phonons.

12
Inelastic Neutron Scattering

• A neutron of velocity v has a wavevector and
  kinetic energy
• Suppose that in either absorbing or creating a
  phonon, the energy and wavevector of the neutron
  are changed to E and Kn respectively. Then the
  angular frequency w and wavevector k of the
  phonon involved will be related by the
  conservation equations
• The vector G is either zero or a reciprocal
  lattice vector and can be included since a phonon
  of wavevector k is identical to a phonon of any
  (kG). The positive signs in are for the creation
  of a phonon (Stokes process) and the negative
  signs for the annihilation (anti-Stokes) of a
  phonon.

13
Inelastic Neutron Scattering
14
Triple-axis spectrometer

• The beam of neutrons is monochromated to the
  desired wavelength via Bragg reflection of a
  crystal monochromator. After the beam is
  scattered by the sample an analyser crystal is
  adjusted so that it Bragg reflects a defined
  energy of the scattered neutrons towards the
  detector.
• In the triple-axis spectrometer the incident
  energy can be kept constant by keeping qM
  constant. The scattered energy is adjusted by
  changing qA, but the consequent change in ½k½
  can be compensated by simultaneously altering the
  scattering angle f to keep ½K½½k-k½ constant
  also the angle y of the sample can be varied so
  that K maintains its orientation with respect to
  the crystal axes. Thus by simultaneously changing
  the three angles qA, f and y, it is possible to
  investigate the variation of neutron count rate
  with the energy E of the scattered neutrons at
  fixed values of the incident neutron energy E and
  scattering vector K.

15
Phonon dispersion curves

• The technique can thus be applied to measuring
  the phonon frequencies as a function of
  wavevector throughout the Brillouin zone of the
  material. The figures display the results for two
  f.c.c metals and for f.c.c silicon which has two
  atoms in the basis and thus displays both
  acoustic and optic branches in the Brillouin zone

16
Comparison IXS versus INS

• Similarities and differences between the IXS and
  INS cross section are summarised below
• Q - E limitation for INS
• Small beam size for IXS
• X-rays couple to the electrons of the system with
  a cross-section proportional to the square of the
  classical electron radius, ro2.82x10-13 cm, i.e.
  with a strength comparable to the neutron-nucleus
  scattering cross-section b.
• The IXS cross section is proportional to fj(Q)2.
  In the limit Q?0, the form factor is equal to the
  number of electrons in the scattering atom, Z
  for increasing values of Q, the form factor
  decays with decay constants of the order of the
  inverse of the atomic wavefunction dimensions of
  the electrons in the atom.
• The total absorption cross-section of X-rays
  above 10 keV energy is limited in almost all
  cases (Zgt4) by the photoelectric absorption
  process, and not by the Thomson scattering
  process. The photoelectric absorption, whose
  cross-section is roughly proportional to Z4,
  determines therefore the actual sample size along
  the scattering path. Consequently the Thomson
  scattering channel is not very efficient for
  system with high Z in spite of the Z2 dependence
  of its cross-section.
• As a consequence, multiple scattering processes
  can in general be neglected
• The magnetic cross section is negligible for IXS,
  whereas it is comparable to the nuclear cross
  section for neutrons.
• The IXS cross section is highly coherent
• The shape of the IXS instrumental energy
  resolution is not Gaussian as it is for a neutron
  triple-axis spectrometer, but Lorentzian.

17
General optical lay-out

• The optical lay-out is based on the triple-axis
  principle, composed of the very high energy
  resolution monochromator (first axis), the sample
  goniometry (second axis) and the crystal analyser
  (third axis). The figure below shows a schematic
  view of the optics and the distances involved.
  Due to the backscattering geometry the beamline
  is fairly long in order to acquire a sufficient
  beam offset between the incident photon beam from
  the X-ray source and the focused very high-energy
  resolution beam at the sample position

18
Backscattering monochromator

• The main monochromator consists of a flat perfect
  single crystal, operating at a Bragg angle of
  89.98º and utilising the silicon (n,n,n,)
  reflection orders. This close-to-exact-backscatter
  ing configuration insures that the spectral
  angular acceptance, the so-called Darwin width,
  is larger than the x-ray beam divergence, and
  therefore that all the photons within the desired
  energy bandwidth are transmitted. The crystal is
  temperature controlled in the mK region by a high
  precision platinum 100W (Pt100) thermometer
  bridge in closed-loop operation with a PID
  controlled heater unit. The complete electronic
  unit was purchased from Automatic Systems
  Laboratory (Milton Keynes, England).
• The energy scans are performed by varying the
  temperature of the monochromator and keeping the
  temperature of the analyzer fixed.

19
Very high energy resolution spherical crystal
analyzer

• Although the required energy resolution is the
  same for the monochromator and the analyser,
  there is a big difference concerning their
  angular acceptance. The spatial angular
  acceptance of the monochromator should be
  compatible with the divergence of the synchrotron
  beam, whereas the analyser must have a much
  larger angular acceptance, which is only
  restricted by the required Q resolution. An
  angular acceptance up to 4x10mrad2 is an adequate
  compromise of Q-resolution and signal
  maximisation. These constraints necessitate
  focusing optics. Since it is not possible to
  elastically bend a crystal without introducing
  important elastic deformations, which in turn
  deteriorate the intrinsic energy resolution, one
  has to position small, unperturbed crystals onto
  a spherical substrate with a radius, fulfilling
  the Rowland condition. This polygonal
  approximation to the spherical shape yields the
  intrinsic energy resolution, provided the
  individual crystals are unperturbed and the
  geometrical contribution of the cube size to the
  energy resolution is negligible. The solution
  realised at the ESRF consists of gluing 12000
  small crystals of 0.6x0.6x3 mm3 size onto a
  spherical silicon substrate. This procedure,
  yield very good results, and provided the record
  energy resolution of 0.9 meV, utilising the
  silicon (13,13,13) reflection order at 25704 eV.