The Scattering Function: S(Q,ω)

Gail N. Iles

Bragg Institute

gail.iles@ansto.gov.au

• 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)