Inelastic (neutron) Scattering: a brief introduction

John A. Stride School of Chemistry, UNSW

"for pioneering contributions to the development of neutron scattering techniques for studies of condensed matter"

"for the development of neutron spectroscopy"

"for the development of the neutron diffraction technique"

Bertram N. Brockhouse (1918-2003)

Clifford G. Shull (1915-2001)

McMaster University Hamilton, Ontario,

Canada

Massachusetts Institute of Technology (MIT) Cambridge, MA, USA

The Nobel Prize in Physics 1994

2-axis: diffraction

3-axis: dynamics

2

ki

ko Q

2

ki

ko

2

Q

ki

ko

2 x y

h

 

Q2 = y 2 + h2 = ki - ko cos 2q( )( )2

+ ko sin 2q( )( )2

Q2 = ki

2 + ko

2cos2 2q( ) - 2ki ko cos 2q( ) + ko

2sin2 2q( )

= ki

2 + ko

2cos2 2q( ) + sin2 2q( )[ ] - 2ki ko cos 2q( )

cos2 a + sin2 a =1:

Q2 = ki

2 + ko

2- 2ki ko cos 2q( )

Consider the properties of a photon

If we take X-rays, then ≈ 1 Å; what is the associated photon energy?

• If we take a typical vibration of ~1500 cm-1:

 

l =hc

E=

hc

h ¢ c u =

1

100 ´1500= 6.667 ´10-6 m

that diffracts << associated with atomic motions

Now consider the properties of a neutron

For neutrons of ≈ 1 Å, what is the associated energy?

Vibrations and lattice modes lie up to ~3500 cm-1

l =h

mnv(de Broglie's equation); E =

1

2mnv

2

\v =h

mnl® E =

1

2mn

h

mnl

æ

èç

ö

ø÷

2

=h2

2mnl2

=6.626´10-34( )

2

2 1.675´10-27( ) 1´10-10( )=1.311´10-20 J

E=1.311´10-20

1.602´10-19eV = 81.8meV» 660cm-1

that diffracts = associated with atomic motions

Neutron spectroscopy - why?….

Neutron spectroscopy - why?….

So, how do we do spectroscopy with neutrons..?

• Monochromators can be used to define ki

• However, need to determine ko to obtain E

1. use a second monochromator (analyzer) to set ko or use a filter to set ko

scan ki at the monochromator (and ko at analyzer)

2. time-of-flight (TOF) analysis with fixed ki (direct geometry) time-of-flight with fixed ko (indirect geometry)

need a precise knowledge of the instrument flight paths

Triple axis spectroscopy….. (Taipan & Sika)

IN2

0, I

LL Ideally suited to probing small regions of (Q,)

Very high flux, but low (Q,) coverage

Usually requires large single crystal samples

Time-of-flight spectroscopy….. (Pelican)

… for a given packet of neutrons, that start at some moment:

Time-of-flight spectroscopy….. (Pelican)

… for a given packet of neutrons, that start at some moment:

Time-of-flight spectroscopy….. (Pelican)

… for a given packet of monochromatic neutrons, that start at some moment and then scatter from a sample:

Whilst all neutrons arrive at the sample at the same time

- some will gain energy & others lose energy at the

sample

TOF basics….

De Broglie's equation: l =h

mnv=

ht

mns

But E =1

2mnv

2 ® v =2E

mn

DE = Ei - E f and tTOF = ti + t f

If L1 = clock start - to - sample distance

and L2 = sample - to - detector distance:

tTOF =L1

vi

+L2

v f

=L1

2Ei

mn

+L2

2E f

mn

ü

ý

ïïïïïïïïï

þ

ïïïïïïïïï

If distances are known:

either Ei can be fixed & tTOF measured (direct geometry)

OR

Ef can be fixed & tTOF gives Ei

(indirect geometry)

In BOTH cases, we then have E

TOF instruments Pelican Layout

Ideal instrument to start INS on:

can run many sample types single sample position set Ei & measure…. full spectra immediately (do not sequentially scan) large coverage of Q- space – ideal for survey scans

Correlation functions

Most of the analysis of scattering data is actually in the form of so-called pair correlation functions

Think of Bragg’s Law:

What we are measuring is the correlations between regions (planes) of

scattering density

- this may be electron density for X-rays (photons) & electrons - or nuclei (predominantly) for neutrons

What about inelastic scattering?

We measure the change in some parameter (polarizability, amplitude of

oscillation, etc.)

- if the wave vectors are similar in length to the distance between points

in the reciprocal lattice, then we can have structural and temporal

information!

- alternatively, mis-match gives either structural or temporal information…

Correlation functions

The intensity of elastic, coherent neutron scattering is proportional to the

spatial Fourier Transform of the Pair Correlation Function, G(r); the probability

of finding a particle at position r if there is simultaneously a particle at r = 0.

The intensity of inelastic coherent neutron scattering is proportional to the

space and time Fourier Transforms of the time-dependent pair correlation

function function, G(r,t) = probability of finding a particle at position r at time t

when there is a particle at r = 0 and t = 0.

For inelastic incoherent scattering, the intensity is proportional to the space

and time Fourier Transforms of the self-correlation function, Gs(r,t); the

probability of finding a particle at position r at time t when the same particle

was at r = 0 at t = 0.

Correlation functions

The intensity of elastic, coherent neutron scattering is proportional to the

spatial Fourier Transform of the Pair Correlation Function, G(r); the probability

of finding a particle at position r if there is simultaneously a particle at r = 0.

The intensity of inelastic coherent neutron scattering is proportional to the

space and time Fourier Transforms of the time-dependent pair correlation

function function, G(r,t) = probability of finding a particle at position r at time t

when there is a particle at r = 0 and t = 0.

For inelastic incoherent scattering, the intensity is proportional to the space

and time Fourier Transforms of the self-correlation function, Gs(r,t); the

probability of finding a particle at position r at time t when the same particle

was at r = 0 at t = 0.

Correlation functions

The intensity of elastic, coherent neutron scattering is proportional to the

spatial Fourier Transform of the Pair Correlation Function, G(r); the probability

of finding a particle at position r if there is simultaneously a particle at r = 0.

The intensity of inelastic coherent neutron scattering is proportional to the

space and time Fourier Transforms of the time-dependent pair correlation

function function, G(r,t) = probability of finding a particle at position r at time t

when there is a particle at r = 0 and t = 0.

For inelastic incoherent scattering, the intensity is proportional to the space

and time Fourier Transforms of the self-correlation function, Gs(r,t); the

probability of finding a particle at position r at time t when the same particle

was at r = 0 at t = 0.

Applications suited to TOF….

A typical TOF experiment gives all correlation functions:

G(r) along the elastic line

G(r,t) + Gs(r,t) outside of the elastic region

The given coverage of S(Q,) - kinematic limits - makes TOF ideal for dynamics that

are not restricted to regions or points of reciprocal space

There is no need for single crystal samples - although these can be accommodated

- polycrystalline, powder, amorphous, liquid,……

Molecular spectroscopy & lattice dynamics Guest:host interactions (confinement)

Dynamics of inclusions Magnetism (low dimensional & frustrated)

Single molecule magnets Amorphous materials & liquids (local dynamic/ structure relation)

Recent (unpublished) data…. PMB

Localised modes dominated by methyl group dynamics.

• Absence of dispersion • Increasing intensity as a f(Q) • Agreement between TAS & TOF data (a good thing)

Samples ca. 100 mg of powdered C6(CH3)5H Data collected < 10 K.

M. Mudge, B.K. Ng, M. Bhadbhade, K.C. Rule, R.A. Mole & J.A. Stride; unpublished

IN4, ILL

Taipan, ANSTO

kx

E=÷÷

ø

öççè

æ2

2

d

d

Numerical modelling

measure the

sum over all

wave-vectors

in S()

J.A. Stride, Acta Cryst. B, 61 (2005) 61 200

comparison of DFT to expt:

0.0 10.0 20.0 30.0 40.0

0

5

10

15

0 81 161 242 323

neutron energy loss / cm-1

S(w

) /

ar

bit

rary

un

its

neutron energy loss / meV

J.A. Stride, J.M. Adams & M.R. Johnson, Chem. Phys., 317 (2005) 143

IN4

, ILL

d-HMB: isotopic substitution

1.34

1.32

 

2 =1.41

HMB

d-HMB

IN4

, ILL

6

C 12.01

14

Si 28.09

32

Ge 72.59

50

Sn 118.71

82

Pb 207.19

Methyl tunnelling • Very low energy transitions (much lower than the torsional motions) can result in tunnelling

spectra – Very sensitive probe of the potential energy surface that the rotor sits in

– CH3, NH3, NH4+, etc.,

Stride et al., Chemistry, Eur. J., 2009, 15, 6569.

0 2! /3 4! /3

rotational angle

barr

ier

heig

ht

!

Tunnel

splitting

Methyl tunnelling • Very low energy transitions (much lower than the torsional motions) can result in tunnelling

spectra – Very sensitive probe of the potential energy surface that the rotor sits in

– CH3, NH3, NH4+, etc.,

Stride et al., Chemistry, Eur. J., 2009, 15, 6569.

0 2! /3 4! /3

rotational angle

ba

rrie

r he

igh

t

!

Tunnel

splitting

X-(C6H4-CH3)4

M.C.C. Ng, D.J. Craig, J.B. Harper, L. van-Eijck & J.A. Stride, Chemistry, Eur. J., 15 (2009) 6569 M.C.C. Ng, J.B. Harper, A.P.J. Stampfl, S. Rols, G.J. Kearley & J.A. Stride, Chem., Eur. J., 18 (2012) 13018

IN4, ILL Ta

ipan

, AN

STO

IN1

6, I

LL

magnetic sponges: M3(HCO2)6

M = Mn2+, Fe2+, Co2+, Ni2+, Zn2+

bulk ferro-, ferrri- & non-magnetic examples

porous structure

magneto-structural correlations?

Z. Wang, B. Zang, H. Fujiwara, H. Kobayashi & M. Kurmoo, Chem. Commun., (2004) 416

adsorbed species….

Currently >40 guests characterised

Z. Wang, B. Zang, H. Fujiwara, H. Kobayashi & M. Kurmoo, Chem. Commun., (2004) 416

[Mn3(HCO2)6] @ i = 3.6 Å; Ei = 6.3 meV

Tc = 8.0 K

T = 6 K

T=1.5 K T = 4 K

T = 15 K

J.A. Stride, M. Kurmoo and Z. Wang, Physica B, 385-386 (2006) 465

IN4

, ILL

[Mn3(HCO2)6] @ i = 4.1 Å; Ei = 4.9 meV

Pronounced degree of dispersion Magnetic Excitations (Q-dependence)

Samples ca. 500 mg of perdeuterated powders.

Strong long-range magnetic excitations

Chain-like coupling of spins

(unusually well-defined dispersion curves from a powder).

IN5

, ILL

adsorbed species…?

Unloaded, T = 1.5 K Loaded with C6H6, T = 1.5 K

Mn3(HCO2)6 i = 1.8 Å; Ei = 25.2 meV:

IN4

, ILL

adsorbed species…?

Mn3(HCO2)6 i = 1.8 Å; Ei = 25.2 meV:

• Assumes that the framework phonons are not very different upon loading

• all modes show Q2 phonon-like behaviour

• no indication of magnetic scattering in difference plot

IN4

, ILL

phonon density of states of confined

species:

2 4 6 8 10 12 14 16 18

16.1 32.3 48.4 64.5 80.6 96.8 112.9 129.0 145.2

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

neutron energy loss / cm-1

GD

OS

/

arb

itra

ry u

nit

s

neutron energy transfer / meV

IN4

, ILL

Conclusions (take home messages)

Inelastic neutron scattering can provide complementary – often unique - information to

other scattering techniques

The whole gamut of condensed matter dynamics is accessible to neutrons:

• from meV through to near eV

• largely motions of atoms & nuclei (coherent, incoherent, diffusive)

• sensitive to fluctuations in magnetic moments (magnons, spin waves)

Isotopic variation can ‘silence’ certain modes • incoherent scattering of 1H (when present) dominates spectra

• most molecular species have 1H – great sensitivity….

• dynamics on confinement

Absence of selection rules for vibrational modes • particle-particle interactions – ease of numerical modelling

World class facilities available at ANSTO – complementary techniques in Japan,

Europe & USA.

thank you