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• By the end of this lecture you will be able to:
– Recognise the energy ranges of neutron
spectrometers
– Recognise the shape of a characteristic S(Q,ω) curve
depending on scientific problem
– State the relevance of each component of the
scattering function
Outline
Neutrons in Condensed Matter Research
The ESS Project, Vol II, ed. By D. Richter (FZ Jülich, 2002), p.5-4
Probes of Condensed Matter
• Dynamical ranges
– Real space
• (r , t)
– Reciprocal space
• (Q,ω)
• Diffraction
– Ewald sphere of reflection
– Incoming neutron is a plane wave, k
– Elastic scattering; only direction changes
– Magnitude of k remains the same
– Momentum transfer between neutron and sample
• Measurements
– Neutron intensity vs. dQ i.e. spanning reciprocal space
– More ‘stuff’ in real space, more intensity in reciprocal
space
A word on Reciprocal Space
Neutron Scattering - Elastic
• Bragg’s Law
wavelength
wavevector
• The neutron is either a wave or a particle
– In diffraction we talk about interference patterns, as
though the neutron were a wave
– When collecting data, we talk about He atoms in
detector tubes capturing neutrons, as particles
Wave-Particle Duality
• In neutron scattering you will often need to convert
your measurements
– Energy, E
– Wavelength, λ
– Optical frequency, f
– Velocity, v
– Wave vector, k
– Temperature, T
– Angular frequency, ω
Units
Using:
Thermal neutrons @ 293K = 25meV
Cold neutrons @ 20K = 2meV
Basic properties of the neutron
• Kinetic energy of slow neutrons with velocity v
• E = mv2 / 2
• de Broglie wavelength of the neutron
• λ = h / mv
• Wavevector k of the neutron has magnitude
• k = 2π / λ (direction being that of its velocity)
• I.e. Momentum of neutron: p=ћk
• A neutron with incident wave vector ki, interacts with a
sample
• The neutron’s outgoing wave vector is kf
Wave Vector, k
ki
kf
Q
2θ
REAL Space
• In reciprocal space, we create the scattering triangle
• Scattering vector, Q = ki – kf
• Q denotes the momentum transfer
Momentum Transfer, Q
ki
kf Q
2θ RECIPROCAL
Space
• Reciprocal space scattering diagram
• Elastic case: Qi = Qf
• Inelastic case: Qi ≠ Qf
Momentum Transfer, Q
ki kf
Elastic
case
Q
Qi Qf
2θ
Inelastic
case
ki kf
Q
Qi Qf
φ
E.g. Positive
momentum transfer
Energy Transfer – ћω
• In terms of energy:
• Ei = ћ2ki2 / 2mn and Ef = ћ2kf
2 / 2mn
(where ћ = h / 2π)
• Energy transfer:
• ћω = Ei – Ef = ћ2 / 2mn (ki2 - kf
2) (where ω = 2πf)
• Combining equations for energy and momentum transfer:
• Q2 = ki2 + kf
2 – 2ki kf cos φ
• Scattering occurs in an elementary cone of solid
angle dΩ
Total Cross-Section
x
dA
dΩ Plane wave
Spherical
wave
ki
Incident
beam
kf
Scattered
beam
φ
• Total cross-section is defined by:
• Incident plane wave of neutrons:
• The probability of finding a neutron in a volume dV is: however,
– refers to density of one neutron per unit volume in all space
• The flux of neutrons incident normally on unit area per second is:
Total Cross-Section
k = wavenumber ψ𝑖 = 𝑒−𝑖𝑘𝑥
ψ𝑖 = 𝑒−𝑖𝑘𝑥
• Wave scattered by an isolated nucleus:
– The minus sign makes most b values positive
• The scattered flux If is and the number of neutrons
scattered per second is flux x area:
• This is the effective area of the nucleus viewed by the neutron
– Units of cross-section = barns [1 barn = 10-28m2]
– Units of scattering lengths = fermis [1 fermi = 10-15m]
Total Cross-Section
r = distance from scattering nucleus
b= scattering length of nucleus ψ𝑓 = −𝑏
𝑒−𝑖𝑘𝑟
𝑟
ψ𝑓2× 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =
𝑏2
𝑟2𝑣
b = scattering length (fermi)
Neutron Data Tables
Z = atomic number
A = mass number
p = abundance (in %)
T = half-life T1/2
I = nuclear spin
σ = cross sections (barns)
• The differential cross-section is defined as:
• This is measured using a diffractometer
– Measures scatters intensity as a function of wave vector transfer
– Does not involve any information about changes in energy
• No. of neutrons scattered per second into this angle is flux x area
• The solid angle dΩ subtended by detector at the sample is dA/r2
• The units of the differential cross-section are barns/steradian
Differential cross-section
𝑑𝜎
𝑑Ω=𝑛𝑜. 𝑜𝑓 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑠𝑐𝑎𝑡𝑡𝑒𝑟𝑒𝑑 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 𝑖𝑛𝑡𝑜 𝑎 𝑠𝑜𝑙𝑖𝑑 𝑎𝑛𝑔𝑙𝑒 𝑑Ω
𝐼0𝑑Ω
𝑑𝜎
𝑑Ω= ψ𝑓
2𝑟2
• The double differential cross-section is defined as:
• This is measured using a spectrometer
– Measures the inelastic scattering cross-section which involves a change in energy
– Dimensions are area per unit energy
• The quantity S(Q,ω) is the scattering function of the system
– Probability that energy of system changes by Ei - Ef = ћω and momentum of system
changes by ћki – ћkf = ћQ
Double differential cross-section
𝑑2𝜎
𝑑Ω𝑑𝐸𝑓=
𝑛𝑜. 𝑜𝑓 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑠𝑐𝑎𝑡𝑡𝑒𝑟𝑒𝑑 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 𝑖𝑛𝑡𝑜 𝑎 𝑠𝑜𝑙𝑖𝑑 𝑎𝑛𝑔𝑙𝑒 𝑑Ω𝑤𝑖𝑡ℎ 𝑓𝑖𝑛𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐸𝑓 𝑎𝑛𝑑 𝐸𝑓 + 𝑑𝐸𝑓
𝐼0𝑑Ω𝑑𝐸𝑓
𝑑2𝜎
𝑑Ω𝑑𝐸𝑓= 𝑏2𝑘𝑓
𝑘𝑖𝑁𝑆 𝑸,𝜔
Scattering Function
Neutron and Synchrotron Radiation for Condensed Matter Studies, J. Baruchel, 1993
• Scattering per atom is given by a
double differential cross section
– Scattering cross section σs
– Scattering function S(Q,ω)
• Elastic scattering: ω=0
• Inelastic scattering at any Q
– Localised motion
• Q-dependent frequencies in
S(Q,ω)
– Propagating motions in r(t)
Inelastic Data Traces
Maths Karlsson. Phys. Chem. Chem. Phys., 2015,17, 26-38
T Chatterji, N Jalarvo, C M N
Kumar, Y Xiao, T Bruckel
J. Phys.: Condens. Matter 25
(2013) 286003
Neutron Scattering
Q ω r t Physics sample Instrument
Elastic Q 0
r S(Q,0) Atomic and magnetic structure
Single crystal 4-circle diffr.
Q 0 r S(Q,0) Atomic and magnetic structure
Powder 2-axis diffr.
0 S(Q,0) Large structures Polymers, clusters, biolog. Molec.
Small-angle scattering
Total Q r 0 S(Q) Structures Liquids, amorphous
Liquid/Am. Diffr.
Inelastic Q ω r t S(Q, ω)
Propagating modes Single crystal TAS
Q ω r t S(Q, ω) Dynamics Liquid TOF
Q ω r t S(Q, ω) Spectrum Powder TOF
ω r=0 t S(ω) Localised motion Molecules TOF
Quasi-elastic
S(ω) Slow motion TOF, BS Spin-echo
• The scattering function S(Q, ω):
• The dynamic structure factor S(Q, ω) is a measurable
quantity, so it has to be a real function
• Just for fun, let’s look at the double Fourier transform
of S(Q, ω):
S(Q,ω)
S 𝑄, 𝜔 =1
2𝜋ћ 𝐼 𝑄, 𝑡 𝑒−𝑖𝜔𝑡𝑑𝑡
• This is the time-dependent pair-correlation function
• It describes the probability to find an atom at a position r, at a time, t, provided an
atom (the same or another) was found at r=0 and t=0.
• This also applies for T>>ћω/kB (the classical limit) where G(r,t) becomes a purely
real function
• Why do we need this? We can’t see a direct picture of atoms. To do this we would
need to detect the phase and amplitude of their wave function.
• We can only measure the intensity of scattered neutrons
Correlation Function
G 𝑟, 𝑡 =1
2𝜋 3 𝑑𝑸𝑒−𝑖𝑸.𝑟 𝑑𝜔
∞
−∞
𝑒𝑖𝜔𝑡𝑆 𝑸,𝜔
• Different isotopes of the same element have characteristic scattering lengths
• If the nucleus has non-zero spin, each isotope has two different scattering lengths b+ and b-
– These represent the two states of the compound nucleus • nucleus ↑ + neutron ↓ = b- (antiparallel) => I – 1/2
• nucleus ↑ + neutron ↑ = b+ (parallel) => I+1/2
• Existence of isotopes and spin effects gives rise to cross-sections with components of BOTH coherent and incoherent scattering
Coherent/Incoherent Nuclear Scattering
• Coherent scattering
– Interference effects
– Space and time relationships between different atoms
Coherent Nuclear Scattering
Neutron diffraction pattern and Rietveld refinement for La0.5Pb0.5FeO3 (Budapest Neutron Centre)
• Incoherent scattering – No interference effects
– Arises from deviations of the scattering lengths from their mean values
– Gives information about properties of individual atoms
Incoherent Nuclear Scattering
G.N. Iles, F. Devred, P.F. Henry, G. Reinhart, T.C. Hansen. J Chem Phys. 41(3):034201 (2014)
• The differential cross-section for the elastic scattering from an assembly of fixed nuclei in the sample is:
• The summation is over all atoms
• This can be written as a double summation:
• Assuming the isotopes and spin states are distributed randomly and are uncorrelated, the quantity bjbj’ can be replaced with its mean value
Coherence
𝑑𝜎
𝑑Ω= 𝑏𝑗𝑒𝑥𝑝 𝑖𝑸. 𝒓𝑗𝑗
2
𝑑𝜎
𝑑Ω= 𝑏𝑗𝑏𝑗′𝑒𝑥𝑝 𝑖𝑸. 𝑟𝑗 − 𝑟𝑗′
𝑗′𝑗
• If j and j’ refer to different sites i.e. j ≠ j’ (or different particles) and there is no correlation between the values of bj and bj’:
• However, if it is the same site, or one particle i.e. j = j’:
• Our differential cross-section then becomes:
Coherence
𝑏𝑗𝑏𝑗′ = 𝑏𝑗 𝑏𝑗′ = 𝑏2
𝑏𝑗𝑏𝑗′ = 𝑏𝑗2 = 𝑏2
𝑑𝜎
𝑑Ω= 𝑏 2𝑒𝑥𝑝 𝑖𝑄. 𝑟𝑗 − 𝑟𝑗′ + 𝑏2 − 𝑏 2
𝑗𝑗,𝑗′
• Differential cross-section:
• For an individual atom:
Incoherent Nuclear Scattering
𝑑𝜎
𝑑Ω= 𝑏 2𝑒𝑥𝑝 𝑖𝑄. 𝑟𝑗 − 𝑟𝑗′ + 𝑏2 − 𝑏 2
𝑗𝑗,𝑗′
Coherent scattering:
Each nucleus has the
average scattering length
<b>
Incoherent scattering:
Magnitude is mean-square deviation
of the scattering length from average
value
𝜎𝑐𝑜ℎ = 4𝜋 𝑏2 = 4𝜋𝑏𝑐𝑜ℎ
2
Total coherent scattering
cross-section
𝜎𝑖𝑛𝑐𝑜ℎ = 4𝜋 𝑏2 − 𝑏 2
Total incoherent scattering
cross-section
b = scattering length (fermi)
Neutron Data Tables
σ = cross sections (barns)
• Measuring the double differential cross-section gives
information about both space and time correlations:
Inelastic scattering
𝑑2𝜎
𝑑Ω𝑑𝐸𝑓=𝑑2𝜎
𝑑Ω𝑑𝐸𝑓 𝑐𝑜ℎ+𝑑2𝜎
𝑑Ω𝑑𝐸𝑓 𝑖𝑛𝑐𝑜ℎ
Dependent on correlation at
different times between
positions of different nuclei
Dependent only on correlation
between positions of the
same nucleus at different
times
Absent if no isotope or spin
incoherence in sample!
Example:
Phonon dispersion relations
Example:
Molecular spectroscopy or diffusion
• In terms of the double differential – The coherent inelastic cross-section:
– Incoherent inelastic cross-section:
• Where
– S(Q,w) is incoherent scattering function
– G(r,t) is the time-dependent self pair-correlation function
Inelastic scattering
𝑑2𝜎
𝑑Ω𝑑𝐸𝑓 𝑐𝑜ℎ= 𝑏 2
𝑘𝑓
𝑘𝑖𝑁𝑆 𝑸,𝜔 <b> mean scattering length
𝑑2𝜎
𝑑Ω𝑑𝐸𝑓 𝑖𝑛𝑐𝑜ℎ= 𝑏2 − 𝑏 2
𝑘𝑓
𝑘𝑖𝑁𝑆𝑖𝑛𝑐 𝑸,𝜔
S 𝑸,𝜔 =1
2𝜋ћ 𝐺𝑠 𝒓, 𝑡 𝑒𝑥𝑝 𝑖 𝑸. 𝒓 − 𝜔𝑡 𝑑𝒓𝑑𝑡
• Remember this equation?
• Written in terms of energy:
Kinematics of inelastic scattering
𝑄2 = 𝑘𝑖2 + 𝑘𝑓
2 − 2𝑘𝑖𝑘𝑓 cos 2𝜃
ћ2𝑄2
2𝑚𝑛= 𝐸𝑖 + 𝐸𝑓 − 2 𝐸𝑖𝐸𝑓 cos 2𝜃
= 2𝐸𝑖 − ћ𝜔 − 2 𝐸𝑖 𝐸𝑖 − ћ𝜔 cos 2𝜃 For direct geometry
spectrometer
= 2𝐸𝑓 + ћ𝜔 − 2 𝐸𝑖 𝐸𝑓 + ћ𝜔 cos 2𝜃 For indirect geometry
spectrometer
• Nuclear and magnetic interactions ~ similar magnitude
• Differences: – Cross-section for nuclear scattering is a scalar quantity
– Magnetic cross-section has vector component due to applied field
– Form factor for scattering:
• Combined amplitudes in forward direction from all unpaired electrons are in phase
• As 2θ increases, phase differences occur – Electronic cloud is comparable with neutron wavelength
• Marked effect for neutrons where scattering is from the outermost electronic orbitals
Magnetic scattering
Summary
• Properties of the neutron
• Compared elastic with inelastic neutron scattering
• Momentum and energy transfer
• Cross-sections
• Scattering lengths
• Correlation function
• Coherent/incoherent scattering
• Inelastic neutron scattering in terms of S(Q,w)