Basic Elements of Neutron Inelastic Scattering

2012 NCNR Summer School on Fundamentals of Neutron Scattering

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

Φs

hω Outline 1. Introduction - Motivation - Types of Scattering

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

(( )) (( )) (( )) (( )) 4. Summary of Scattering Cross Sections ω0 ≠ 0 (( )) (( )) (( )) (( )) - 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

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

An ideal method of characterization would be one that can provide detailed information about both structure and dynamics. Types of Scattering

How do we “see”?

Thus scattering We see something when light scatters from it. conveys information!

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

However, the details of what we see are ultimately limited by the wavelength. Types of Scattering

The tracks of a compact disk act as a diffraction grating, producing a separation of the colors of white light.

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 à light is inadequate. λLight >> d ~ 4 A Moreover, most materials are opaque to light. Types of Scattering

To “see” atomic structure, we require a probe with a wavelength λ ~ length scale of interest.

X-rays EM - wave

Some candidates … Electrons Charged particle

Neutrons Neutral particle Types of Scattering

Which one should we choose?

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 à easy access

3. Fluxes are extremely high à can study small samples

However … Types of Scattering

Nucleus X-rays are electromagnetic radiation. Thus they scatter from the atomic electrons.

Electrons Consequences:

Low-Z elements Elements with similar X-rays are strongly attenuated are hard to see. atomic numbers have as they pass through the walls very little contrast. of furnaces, cryostats, etc.

Hydrogen Cobalt Nickel

(Z = 1) (Z = 27) (Z = 28) Types of Scattering

What about electrons?

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

1. Relatively cheap

2. Sources are not uncommon à good access

3. Fluxes are extremely high à can study tiny crystals

4. Very small wavelengths à more information However … Types of Scattering

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. Types of Scattering What about neutrons?

Advantages Disadvantages

Wavelengths easily varied Neutrons are expensive to match atomic spacings to produce à access is not as easy

Zero charge à not strongly Interact weakly with matter à attenuated by furnaces, etc. often require large samples

Magnetic dipole moment à Available fluxes are low can study magnetic structures compared to those for x-rays

Nuclear interaction à can see low-Z elements easily like H à good for the study of biomolecules and polymers.

Nuclear interaction is simple Low energies à Let’s consider à scattering is easy to model Non-destructive probe neutrons … The Neutron

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

Bertram Brockhouse 1994 Nobel Laureate in Physics -27 mn= 1.675x10 kg The Neutron Q = 0 S = ½ h

µn= -1.913 µN

1926: de Broglie Relation

λ = h/p = h/mnv

o o λ = 1 A λ = 9 A v = 4000 m/s v = 440 m/s E = 82 meV E = 1 meV Neutron Production

Neutrons not bound to a nucleus decay via - the weak force at a rate characterized by a n à p + e + νe lifetime of ~ 888 seconds (15 minutes).

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 Neutron Production Fission by Fission t

Nuclear fission is used in power 235 236 * and research reactors. 92U143 + n →[ 92U144 ] → X +Y + 2.44n

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

The fission process and moderator are confined by a large containment vessel. Neutron 2 3 (-m v /2kBT) Moderation Maxwellian Distribution Φ ~ v e

Liquid Hydrogen 25 K

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) Wave - Particle de Broglie Relation λ = h/mnv Duality

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 Basics of Neutron

Scattering

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). Q h = neutrons Φs( , ω) sec–cm2 Momentum Energy

2 2 ki kf hk = h(2π/λ) hωn = h kn /2m

hQ = hki - hkf hω = hωi - hωf 2θ

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

provides information = f{r (t), r (t), S (t), S (t)} Φs Φs i j i j about all of these quantities! Neutron Scattering These “cross sections” are what we measure Cross Sections experimentally.

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

d σ = Differential cross section d kf Ω

d2 Partial differential σ = dΩdEf cross section 2 Φi ki Neutron Scattering What are the physical meanings of these three cross sections? Cross Sections

σ Total # of neutrons scattered per second / Φi.

dσ Total # of neutrons scattered per second into dΩ / dΩ Φi. dΩ (Diffraction à 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. dΩ dEf (Inelastic scattering à dynamics. Small, but contains much info.) Neutron Scattering What are the relative sizes of the cross sections? Cross Sections

dσ d2σ Clearly: σ = dΩ = dΩ dEf dΩ dΩ dEf

dσ d2σ Thus: σ >> >> dΩ dΩ dEf

2 dσ . d σ σ d Ω dΩ dEf

d d2σ Typically, σ ~ 106 x dΩ dΩ dEf Elastic vs Inelastic Note that both of these d2σ cases are described by … Scattering dΩ dEf

d Elastic (k =k ) ≠ σ i f dΩ Elastic Scattering: • Change in neutron energy = 0 sinθ = (Q/2)/k

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

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

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

The scattered (final) neutron Ψ is a spherical wave: λ ∼ 2 Α ikr 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 1 fm = 10-15 m Ψi(r) ~ e (k = 2π/λ) Scattering From ANSWERS: a Single Nucleus

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 V(r) = 2πh2 b (r-r ) ( mn ) j δ j Σ j=1 ~ 10-15 m << 10-10 m

The details of V(r) are unimportant – V(r) can be described by a scalar parameter b that depends Fermi Pseudopotential only the choice of nucleus and isotope! Scattering From The Neutron Scattering Length - b a Single Nucleus

The parameter b is known as the X-rays neutron “scattering length” and has units of length ~ 10-12 cm.

Z It is independent of neutron energy.

Neutrons It is (usually) a real number; imaginary for absorption ...

It corresponds to the b varies randomly with Z and isotope à “form factor” for x- allows access to atoms that are usually ray scattering. unseen by x-rays. Scattering From We can easily calculate σ for a single, fixed nucleus: a Single Nucleus

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

dA v

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

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

dσ Since = v | |2 = v à = 2 Try calculating Φi Ψi σ 4πb dΩ Scattering From What if many atoms are present? Many Nuclei

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. Scattering From Two-particle or pair correlations Many Nuclei

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

φ2

(2) Superposition: 2 φ1 Amplitudes of scattered

neutrons φn add linearly.

1 Φs = φ1 + φ2 + …

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

After Andrew Boothroyd Depends on relative positions of 1 and 2 à pair correlations! PSI Summer School 2007 Pair 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: . d -i(ki-kj) (ri-rj) σ = b b e where the atoms are dΩ Σ i j and i,j what the atoms are. Pair Correlation These formulae are stated without proof … Functions

Partial Differential Cross-Section

Neutron Structure Factor

Pair Correlation Function Fourier Transform

Squires, Introduction to the theory of thermal neutron scattering (1996) Pair Correlation KEY IDEA – Neutron interactions are weak à 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.

2   ∂ σ i(Q⋅r −ωt)  3  Φs∝ ∝ e G(r,t)d rdt ∂Ω∂ω ∫∫

Real space Q-space Time space ω-space

Φs Φs

T ξ Γ ~1/ξ ~1/Γ Q ω 2π/d 2π/T d time Neutron Coherent and What happens when there are 2 or more different nuclei in the sample, each having a different value Incoherent Scattering of b, and these nuclei are randomly distributed?

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) dΩ dEf Σ i,j 2 bibj = b , for i=j = average Neutron Coherent and

Incoherent Scattering

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

2 d σ , where = σcSc(Q,ω) + σiSi(Q,ω) dΩ dEf

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

= +

The two isotopes are Deviations randomly distributed. ½(b1+b2) = b δb

= +

Q Q Q Total scattering Coherent scattering Incoherent scattering Neutron Coherent and What do these expressions mean physically? Incoherent Scattering

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) à interference effects. Gs(r,t) à no interference effects.

This cross section reflects This cross section reflects collective phenomena such as: single-particle scattering: Bragg Peaks Atomic Diffusion

Phonons Vibrational Density of States

Spin Waves Summary of (Q,ω) Space (r,t) Space Cross Sections

d σcoh σ = S(Q) dΩ coh 4π Q

σinc dσ = dΩ inc 4π Q

(( )) (( )) (( )) (( )) ω0 ≠ 0 (( )) (( )) (( )) (( )) 2 k σcoh (( )) (( )) (( )) (( )) d σ = f Scoh(Q,ω) (( )) (( )) (( )) (( )) dΩ dEf coh ki 4π ω

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

(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 Neutron Methods of specifying and measuring ki and kf Scattering Methods

1. Bragg Diffraction d nλ=2dSinθ

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

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

Thermal Neutrons

Cold Neutrons Different Types of Why so many different kinds? Neutron 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 à better resolution leads to lower count rates! Choose carefully …

∝ Courtesy of R. Pynn Instrumental Q-Resolution Matters! Q-Resolution

The “right” resolution depends on what you want to study.

X-ray Neutron

-0.050 -0.025 0.000 0.025 0.050 -0.050 -0.025 0.000 0.025 0.050 q (A-1) q (A-1) Instrumental hω-Resolution Matters! hω-Resolution

1200 Focusing Analyzer 800 Example – CuGeO3 400 Flat Intensity SPINS 0 8 1200 Focusing Analyzer 6 800 4 (meV) 400 5 Blades

ω 2 Intensity h Gap 0 0 0 1 2 3 1200 Q = (0,K,0.5) Focusing Analyzer 800

400 9 Blades Intensity 0 0 2 4 6 hω (meV) A 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 à 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) à interference effects. Gs(r,t)à no interference effects.

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

OK, after all of this, just exactly what can neutron scattering do for you?

Let’s look at a few examples … Neutron Elastic Neutrons can probe length scales o o ranging from ~0.1 A to ~1000 A Scattering

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

• Used to determine structure of

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

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

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) Do you see a Neutron Length pattern here? and Time Scales Larger “objects” tend to exhibit slower motions. Neutron Elastic Pop Quiz Scattering

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 … References for