2019 NCNR Summer School on Methods and Applications of Neutron Spectroscopy

Basic Elements of Neutron Inelastic Scattering

Peter M. Gehring National Institute of Standards and Technology NIST Center for Neutron Research Gaithersburg, MD USA

Φs

hω Outline 1. Introduction - Motivation - Scattering Probes

2. The Neutron - Production and Moderation - Wave/Particle Duality

3. Basic Elements of Neutron Scattering - The Scattering Length b - Scattering Cross Sections - Pair Correlation Functions - Coherent and Incoherent Scattering - Neutron Scattering Methods

(( )) (( )) (( )) (( )) ω0 ≠ 0 4. Summary of Scattering Cross Sections (( )) (( )) (( )) (( )) - Elastic (Bragg versus Diffuse) (( )) (( )) (( )) (( )) - Quasielastic (Diffusion) - Inelastic (Phonons) ω (( )) (( )) (( )) (( )) Motivation Structure and Dynamics

The most important property of any material is its underlying atomic / molecular structure (structure dictates function).

Bi2Sr2CaCu2O8+δ

The motions of the atoms (dynamics) are extremely important because they provide information about the interatomic potentials.

An ideal method of characterization would provide detailed information about both structure and dynamics. Motivation Why Should We Scatter?

Lyft

Conveys information: We see something when light scatters from it. location, speed, shape

Light is composed of electromagnetic waves. λ ~ 4000 Å – 7000 Å

However, the details of what we can see are ultimately limited by the wavelength. Motivation

The tracks of a compact disk act as a diffraction grating, producing a separation of the colors of white light when it scatters from the surface.

From this one can determine the nominal distance between tracks on a CD, which is 1.6 x 10-6 meters = 16,000 Angstroms.

To characterize materials we must determine the underlying structure. We do this by using d the material as a diffraction grating.

Problem: Distances between atoms in materials o are of order Angstroms  light is inadequate. λLight >> d ~ 4 A Moreover, most materials are opaque to light. Scattering To measure atomic structure requires a Probes probe with a λ ~ length scale of interest.

X rays EM - wave

Some candidates … Electrons Charged particle

Neutrons Neutral particle 1929 Nobel Laureate in Physics Scattering Probes

X rays: Pros and Cons

If we wish only to determine relative atomic positions, then we should choose x rays almost every time.

1. Relatively cheap

2. Sources are ubiquitous  easy access

3. High flux  can study small samples

4. Extremely good resolution However … Scattering Probes

X rays are electromagnetic radiation. Nucleus Thus they scatter from the charge density.

Consequences: Electron orbitals

Low-Z elements Elements with similar X rays are strongly attenuated are hard to see. atomic numbers have when passing through furnaces, very little contrast. cryostats, and samples too.

Hydrogen Cobalt Nickel

??

(Z = 1) (Z = 27) (Z = 28) Scattering Probes

Electrons: Pros and Cons

Electrons are charged particles  they see both the atomic electrons and nuclear protons at the same time.

1. Relatively cheap

2. Sources are not uncommon  easy access

3. Fluxes are extremely high  can study tiny crystals

4. Very small wavelengths  more information

However … Scattering Probes

Electrons have some deficiencies too ...

Requires very Radiation damage Magnetic structures are hard to thin samples. is a concern. determine because electrons are deflected by the internal magnetic fields.

M Scattering Probes

Neutrons: Pros and Cons

Zero charge  not strongly Neutrons expensive to produce attenuated by furnaces, etc.  access not as easy

Magnetic dipole moment  Interact weakly with matter can study magnetic structures  often require large samples

Nuclear interaction  can see low-Z Available fluxes are low elements easily like H  good for the compared to other methods study of biomolecules and polymers.

Nuclear interaction is simple Low energies   scattering is easy to model Non-destructive probe Scattering To characterize atomic dynamics requires Probes a probe with hω ~ energy scale of interest.

Neutrons: X rays: E = 81.81/λ² E = 12,398,000/λ Scattering To characterize atomic dynamics requires Probes a probe with hω ~ energy scale of interest.

Phonons

Neutrons: E = 81.81/λ² Scattering Neutron: A Lucky Coincidence Probes

E = p²/2m = h²k²/2m = 81.81/λ²

The mass of the neutron is such that one can simultaneously study structure and dynamics.

Thus neutron scattering methods can directly measure the geometry of dynamic processes. The Neutron

“If the neutron did not exist, it would need to be invented.”

Bertram Brockhouse 1994 Nobel Laureate in Physics The Neutron

“… for the discovery of the neutron.”

Sir James Chadwick 1935 Nobel Laureate in Physics The Neutron -27 mn= 1.675x10 kg Q = 0 S = ½ h

μn= -1.913 μN

Interactions: Strong, Electro-weak, Gravity

o o λ = 1 A λ = 9 A de Broglie Relation: v = 4000 m/s v = 440 m/s λ = h/p = h/m v E = 82 meV n E = 1 meV Production

Free neutrons decay via the weak force. - Lifetime ~ 888 seconds (15 minutes). n  p + e + νe

A useful source of neutrons requires a nuclear process by which bound neutrons can be freed from n the nuclei of atoms and that is easily sustainable.

There are two such processes, spallation and fission … Spallation Fission Fission t * Nuclear fission is used in power 235U + n → 236U → X +Y + 2.44n and research reactors. 92 143 [ 92 144 ]

A liquid medium (D2O, or heavy water) is used to moderate the fast fission neutrons to room temperature (2 MeV  50 meV).

The fission process and moderator are confined by a large containment vessel. (-mv2/2k T) Moderation Maxwellian Distribution Φ ~ v3e B

Liquid Hydrogen 25 K

6

5 Heavy Water (D2O) 4

3 Hot Graphite 300 K 2

1 2000 K

0 0 5 10 15 “Fast” neutrons: v = 20,000 km/sec Neutron velocity v (km/sec) Moderation de Broglie Relation λ = h/mnv

Fast Neutron, ~ 0.00002 nm V ~ 20,000,000 m/sec

Thermal Neutron, ~ 0.2 nm V ~ 2,000 m/sec

Cold Neutron, V ~ 200 m/sec ~ 2 nm Scattering Basics

Neutron scattering experiments measure the flux Φs of neutrons scattered by a sample into a detector as a function of the change in neutron wave vector (Q) and energy (hω). Φ (Q,hω) = neutrons s sec–cm2 Momentum Energy 2 2 ki kf hk = h(2π/λ) hωn = h kn /2m 2θ hQ = hki - hkf hω = hωi - hωf

The expressions for the scattered neutron flux Φs involve the positions and motions of atomic nuclei or unpaired electron spins.

Contains information about Φs structure and dynamics! Scattering These “cross sections” are what Basics we measure experimentally.

Consider an incident neutron beam 6 9 2 with flux Φi (~10 to 10 n/sec/cm ) We define three cross sections: and wave vector ki on a non-absorbing sample. σ = Total cross section

dσ Ω = Differential cross section kf d

d2σ Partial differential = dΩdEf cross section 2 Φi ki Cross What are the physical meanings Sections of these three cross sections?

σ Total # of neutrons scattered per second / Φi. (Typically of order 1 barn = 10-24 cm2.)

dσ Total # of neutrons scattered per second into dΩ / dΩ Φi. dΩ (Diffraction  structure. Signal is summed over all energies.)

d2σ Total # of neutrons scattered per second into dΩ with a final energy between Ef and dEf / dΩ dEf Φi. dΩdEf (Inelastic scattering  dynamics. Small, but contains much info.) Cross What are the relative Sections sizes of the cross sections?

dσ d2σ Clearly: σ = dΩ = dΩdEf dΩ dΩdEf

dσ d2σ Thus: σ >> >> dΩ dΩdEf

2 dσ . d σ σ dΩ dΩdEf

dσ d2σ Typically, ~ 106 x dΩ dΩdEf Elastic vs Note that both of these d2σ Inelastic cases are described by … dΩdEf

dσ Elastic (ki=kf) ≠ dΩ Elastic Scattering: • Change in neutron energy = 0 sinθ = (Q/2)/k

• Probes changes in momentum kf θ Q = 2ksinθ = 4πsinθ/λ only θ

ki

Inelastic (ki≠kf) Inelastic Scattering: • Change in neutron energy ≠ 0 • Probes changes in both k k f Q f Q momentum and energy 2θ 2θ

ki ki Energy loss (hω>0) Energy gain (hω<0) Nuclear Consider the simplest case: Scattering A fixed, isolated nucleus.

The scattered (final) neutron

Ψf is a spherical wave: λ ∼ 2 Å ikr-ωt Ψf(r) ~ (-b/r)e

QUESTIONS:

1. The scattering is

elastic (ki = kf = k). Why?

x 2. The scattering is isotropic. Why? The incident neutron Ψi is a plane wave: Nucleus ikx-ωt 1 fm = 10-15 m Ψi(r) ~ e (k = 2π/λ) Nuclear ANSWERS: Scattering

1. The scattering is elastic because the nucleus is fixed, so no energy can be transferred to it from the neutron (ignoring any excitations of the nucleus itself).

2. A basic result of diffraction theory states: if waves of any kind scatter from an object of a size << λ, then the scattered waves are spherically symmetric. (This is also known as s-wave scattering.)

N << λ 2 V(r) = 2πh b δ(r-r ) ( mn )Σ j j j=1 ~ 10-15 m << 10-10 m

Details of V(r) are unimportant! V(r) can be parametrized by a scalar b that Fermi Pseudopotential depends only the nucleus and isotope! Nuclear Calculate σ for a Scattering single, fixed nucleus:

. Def. σ Φi = Total number of neutrons scattered per second by the nucleus. v = Velocity of neutrons (elastic  same before and after scattering).

dA v

2 Total number of neutrons scattered per second through . v dA |Ψf| = dA

2 2 2 2 Ω 2 . v dA |Ψf| = v(r dΩ)(b/r) = v b d = v 4πb = σ Φi

σ Since Φ Ψ 2  σ π 2 d i = v | i| = v = 4 b Try calculating dΩ Nuclear The Neutron Scattering Length - b Scattering

Units of length: Analogous to f(Q), the Varies randomly with Z and isotope b ~ 10-12 cm. x-ray form factor.  Neutrons “see” atoms x rays can’t.

Total Scattering Cross Section: σ = 4πb2

X rays (Q = 0)

H D C N O Al Si Z

Neutrons Nuclear What if many atoms are present? Scattering

Scattering from Scattering from one nucleus many nuclei

Scattered neutron Ψf is a spherical wave:

Constructive interference

The incident neutron

Ψi is a plane wave: Get strong scattering in some directions, but not in others. Angular dependence yields information about how the nuclei are arranged or correlated. Magnetic Magnetic vs Nuclear Scattering Scattering

Nuclear Potential Magnetic Potential

2πh2 VN(r) = bδ(r) VM(r) = -µn B(r) mn

Scalar interaction  Vector interaction  Isotropic scattering Anisotropic scattering

Very short range Longer range

Depends on nucleus, Depends on neutron isotope, and nuclear spin spin orientation.

Polarized neutrons can measure the different components of M. Magnetic Magnetic Form Factor Scattering

Nuclear Potential Magnetic Potential

2πh2 VN(r) = bδ(r) VM(r) = -µn B(r) mn

Large angle

Small angle Magnetic Neutrons Scatter from Scattering M Perpendicular to Q

Magnetic scattering depends on Fourier

transform of interaction potential VM(r):

VM(Q) = -µn B(Q)

F.T.

James Clerk Maxwell (1831 – 1879) Correlation Two-particle or Functions pair correlations

(1) Born Approximation: Assumes neutrons scatter only once (single scattering event).

φ2

(2) Superposition: 2 φ1 Amplitudes of scattered

neutrons φn add linearly.

Φs = φ1 + φ2 + … 1

2 2 2 2 Intensity = |Φ| = |φ1 + φ2 + …| = |φ1| + |φ2| + … + φ1*φ2 * + φ2*φ1 + ...

After Andrew Boothroyd  PSI Summer School 2007 Depends on relative positions of 1 and 2 pair correlations! Correlation From Van Hove (1954) … Functions

The measured quantity Φs depends only on time-dependent correlations between the positions of pairs of atoms.

This is true because neutrons interact only weakly with matter. Thus only the lowest order term in the perturbation expansion contributes.

Differential Cross-Section Depends only on: . dσ -i(ki-kj) (ri-rj) where the atoms are dΩ = Σ bibj e and i,j what the atoms are. Correlation From Squires (1996): Functions Introduction to the theory of thermal neutron scattering

Partial Differential Cross-Section

Neutron Structure Factor

Pair Correlation Function Fourier Transform Correlation KEY IDEA – Neutron interactions are weak  Scattering only probes two-particle correlations Functions in space and time, but does so simultaneously!

The scattered neutron flux Φs(Q,hω) is proportional to the space (r) and time (t) Fourier transform of the probability G(r,t) of finding an atom at (r,t) given that there is another atom at r = 0 at time t = 0.

∂ 2σ     Φ ∝ ∝ ei(Q⋅r −ωt)G(r,t)d 3rdt s ∂Ω∂ω ∫∫

Real space Q-space Time space ω-space

Φs Φs

ξ T Γ ~1/ξ ~1/Γ Q ω 2π/d 2π/T d time Correlation Functions

There are two main ways of measuring the neutron scattering cross section S(Q,ω).

Φ TO Constant-E scans: CE vary Q at fixed hω.

Q

Φ TA Constant-Q scans: vary hω at fixed Q. CQ

hω Classic BCS Superconductivity Example Coherent vs What happens when two different nuclei are Incoherent randomly distributed throughout the crystal?

This situation could arise for two reasons. 1. Isotopic incoherence 2. Nuclear spin incoherence

Both reasons can occur because the scattering interaction is nuclear.

Recall that

Then the above equation must be generalized:

2 2 d σ bibj = (b) , for i=j = bibj Sij(Q,ω) dΩdEf Σ i,j 2 bibj = b , for i=j = average Coherent vs What happens when two different nuclei are Incoherent randomly distributed throughout the crystal?

Our partial differential cross section can then be recast into the form:

d2σ = σcSc(Q,ω) + σiSi(Q,ω) , where dΩdEf

2 σc = 4π(b) c = coherent 2 2 σi = 4π{b - (b) } i = incoherent Coherent vs Consider a system composed of two b1 = different scattering lengths, b and b . Incoherent 1 2 b2 =

= +

The two isotopes are Deviations δ randomly distributed. ½(b1+b2) = b b PSI Summer School 2007 School Summer PSI Boothroyd Andrew After

= +

Q Q Q Total scattering Coherent scattering Incoherent scattering Coherent vs What do these expressions Incoherent mean physically?

Coherent Scattering Incoherent Scattering

Measures the Fourier transform Measures the Fourier transform of the *pair* correlation function of the *self* correlation function

G(r,t)  interference effects. Gs(r,t)  no interference effects.

This cross section reflects This cross section reflects collective phenomena such as: single-particle scattering:

Phonons Atomic Diffusion

Spin Waves Vibrational Density of States Brief (Q,ω) Space (r,t) Space Summary

dσ σcoh = S(Q) dΩ coh 4π Q

σ σinc d = dΩ inc 4π Q

(( )) (( )) (( )) (( )) ω0 ≠ 0 (( )) (( )) (( )) (( )) d2σ k σcoh (( )) (( )) (( )) (( )) = f ω Scoh(Q, ) (( )) (( )) (( )) (( )) dΩdEf coh ki 4π ω

Quasielastic 2σ k σ d = f inc Sinc(Q,ω) dΩdEf inc ki 4π ω ω = 0 Basics of Elements of all scattering experiments Scattering

(Ei,ki) Detector(s) A source

A method to specify the (Ef,kf) incident wave vector ki

A well chosen sample + good idea

A method to specify the final wave vector kf Basics of Methods of specifying and measuring k and k Scattering i f

1. Bragg Diffraction d nλ=2dSinθ

L 2. Time-of-Flight (TOF) v = L/t

3. Larmor Precession B Neutron You will see many of these on the tour … Instruments

Thermal Neutrons Cold Neutrons Neutron Why so many different kinds? Instruments

Because neutron scattering is an intensity-limited technique. Thus detector coverage and resolution MUST be tailored to the science.

Uncertainties in the neutron wavelength and direction imply Q and ω can only be defined with a finite precision.

The total signal in a scattering experiment is proportional to the resolution volume  better resolution leads to lower count rates! Choose carefully … ∝ Courtesy of R. Pynn Quick Please try to remember these things … Review

Neutrons scattering probes two-particle correlations 1 in both space and time (simultaneously!).

The neutron scattering length, b, varies randomly with Z  2 allows access to atoms that are usually unseen by x-rays.

3 Coherent Scattering 4 Incoherent Scattering

Measures the Fourier transform Measures the Fourier transform of the pair correlation function of the self correlation function

G(r,t)  interference effects. Gs(r,t) no interference effects.

This cross section reflects This cross section reflects collective phenomena. single-particle scattering. More Examples

OK, after all of this, just exactly what can neutron scattering do for you? More After Bruce Gaulin Phonons NXS 2016 Examples

 Normal modes in periodic crystal  wavevector 1 ˆ u(l,t) = ∑ε j (q)exp(iq⋅ l)B( qj,t) NM jq  Energy of phonon depends on q and polarization

FCC structure

LongitudinalTransverse mode mode

Lynn, et al., Phys. Rev. B 8, 3493 (1973). FCC Brillouin zone More One-Phonon Scattering Cross Section Examples

Structure (polarization) factor 2 1 −Q 2 u2 Q ⋅ ε j (q) S1+ (Q,ω)= e ∑ (1+ n(ω))δ(Q − q − τ)δ(ω −ω j (q)) 2NM jq ω j (q)

Long

Trans Phonon Energy Phonon

0 q, [ξ00] 1/2

After Bruce Gaulin NXS 2016 More Diffusion Scattering Cross Section Examples

For times that are long compared to the collision time, atom diffuses

Liquid Na

r2 (t) ≈ 6Dt

Auto-correlation function   −3 / 2 r2 G (r,t) = 6π r2(t) exp −  s { }  2   6 r (t)  Cocking, J. Phys. C 2, 2047 (1969)..

1  hω  DQ2 ω = S(Q, ) exp  2 2 2 After Bruce Gaulin πh  2k T  ω + (DQ )  B NXS 2016

Do you see a Length and pattern here? Time Scales Larger “objects” tend to exhibit slower motions. Elastic Neutrons can probeo length scaleso Scattering ranging from ~0.1 A to ~1000 A

Mitchell et. al, Vibrational Spectroscopy with Neutrons (2005)

Neutrons needed to determine

structure of 123 high-Tc cuprates because x rays weren’t sufficiently sensitive to the oxygen atoms.

Pynn, Neutron Scattering: A Primer (1989) Inelastic Neutrons can probe time scales Scattering ranging from ~10-14 s to ~10-8s.

hω = 0

Probes the vibrational, magnetic, and lattice excitations (dynamics) of materials by measuring changes in the neutron momentum and energy simultaneously.

Mitchell et. al, Vibrational Spectroscopy with Neutrons (2005) Elastic Scattering Pop Quiz

Can one measure elastic scattering from a liquid?

If Yes, explain why? If No, explain why not?

Hint: What is the correlation of one atom in a liquid with another after a time t? Nobel Prize The Fathers of Neutron Scattering in Physics “For pioneering contributions to the development of neutron 1994 scattering techniques for studies of condensed matter"

“For the development of the “For the development of neutron diffraction technique" neutron spectroscopy"

Clifford G Shull Ernest O Wollan Bertram N Brockhouse MIT, USA ORNL, USA McMaster University, Canada (1915 – 2001) (1910 – 1984) (1918 – 2003)

Showed us where Did first neutron Showed us how the atoms are … diffraction expts … the atoms move … Useful References

http://www.mrl.ucsb.edu/~pynn/primer.pdf

“Introduction to the Theory of Thermal Neutron Scattering” - G. L. Squires, Cambridge University Press

“Theory of Neutron Scattering from Condensed Matter” - S. W. Lovesey, Oxford University Press

“Neutron Diffraction” (Out of print) - G. E. Bacon, Clarendon Press, Oxford

“Structure and Dynamics” - M. T. Dove, Oxford University Press

“Elementary Scattering Theory” - D. S. Sivia, Oxford University Press