by

Roger Pynn

LECTURES 1 and 2: Theory of Nuclear Scattering of Neutrons

Overview

Introduction and theory of neutron scattering • Advantages/disadvantages of neutrons for scattering measurements

• Neutron properties

• Comparison with other structural probes

• Definition of scattering cross sections

• Fermi pseudopotential

• Refractive index for neutrons – two different calculations

• Elastic scattering and definition of the structure factor, S(Q)

• Coherent & incoherent scattering

• Inelastic scattering

• References

Why do Neutron Scattering?

• To determine the positions and motions of atoms in condensed matter– 1994 Nobel Prize to Shull and Brockhouse cited these areas

(see http://www.nobel.se/physics/educational/poster/1994/neutrons.html)

• Neutron advantages:– Wavelength comparable with interatomic spacings– Kinetic energy comparable with that of atoms in a solid– Penetrating => bulk properties are measured & sample can be contained– Weak interaction with matter aids interpretation of scattering data– Isotopic sensitivity allows contrast variation– Neutron magnetic moment couples to B => neutron “sees” unpaired electron spins

• Neutron Disadvantages– Neutron sources are weak => low signals, need for large samples etc– Some elements (e.g. Cd, B, Gd) absorb strongly– Kinematic restrictions (can’t access all energy & momentum transfers)

The 1994 Nobel Prize in Physics – Shull & Brockhouse

Neutrons show where the atoms are….

…and what the atoms do.

The Neutron has Both Particle-Like and Wave-Like Properties

• Mass: mn = 1.675 x 10-27 kg• Charge = 0; Spin = ½• Magnetic dipole moment: μn = - 1.913 μN

• Nuclear magneton: μN = eh/4πmp = 5.051 x 10-27 J T-1

• Velocity (v), kinetic energy (E), wavevector (k), wavelength (λ), temperature (T).

• E = mnv2/2 = kBT = (hk/2π)2/2mn; k = 2 π/λ = mnv/(h/2π)

Energy (meV) Temp (K) Wavelength (nm)Cold 0.1 – 10 1 – 120 0.4 – 3 Thermal 5 – 100 60 – 1000 0.1 – 0.4Hot 100 – 500 1000 – 6000 0.04 – 0.1

λ (nm) = 395.6 / v (m/s)E (meV) = 0.02072 k2 (k in nm-1)

Comparison of Structural ProbesNote that scattering methods provide statistically averaged information on structure rather than real-space pictures of particular instances

Macromolecules, 34, 4669 (2001)

Thermal Neutrons, 8 keV X-Rays & Low Energy Electrons:- Absorption by Matter

Note for neutrons:

• H/D difference

• Cd, B, Sm

• no systematic A

dependence

Interaction Mechanisms

• Neutrons interact with atomic nuclei via very short range (~fm) forces.• Neutrons also interact with unpaired electrons via a magnetic dipole interaction.

Brightness & Fluxes for Neutron & X-Ray Sources

Brightness(s-1 m-2 ster-1)

dE/E(%)

Divergence(mrad2)

Flux(s-1 m-2)

Neutrons 1015 2 10 x 10 1011

RotatingAnode

1016 3 0.5 x 10 5 x 1010

BendingMagnet

1024 0.01 0.1 x 5 5 x 1017

Wiggler 1026 0.01 0.1 x 1 1019

Undulator(APS)

1033 0.01 0.01 x 0.1 1024

Cross Sections

dEd

dE & d into secondper scattered neutrons ofnumber

d

d into secondper scattered neutrons ofnumber

/ secondper scattered neutrons ofnumber total

secondper cmper neutronsincident ofnumber

2

2

ΩΦΩ

=Ω

ΩΦΩ

=Ω

Φ==Φ

dEd

d

d

d

σ

σσ

σ measured in barns:1 barn = 10-24 cm2

Attenuation = exp(-Nσt)N = # of atoms/unit volumet = thickness

ikzinc e=ψIncident plane wave:

Scattered (circular) wave:

ikzuRik

ikzikrs

eeuR

b

eer

b

rr

rr−

−−=

−=ψ

Scattering by a single nucleus

Squires Eq. 1.18 generalized for the origin at any location.

λπ2

=kwhere

Scattering by a Single (fixed) Nucleus

units) (note 4 so bd

d b v

d

d

v v :is areasunit throughpassing neutronsincident ofnumber theSince

d b v /rb dS v dS v

:is scatteringafter secondper dSarea an throughpassing

neutrons ofnumber the),scatteringafter and before (same neutron theof velocity theis vIf

2total

22

2

incident

2222

scat

bπσσ

ψ

ψ

==ΩΦΩ

=Ω

==Φ

Ω==

• range of nuclear force (~ 1fm)is << neutron wavelength soscattering is “point-like”

• energy of neutron is too smallto change energy of nucleus &neutron cannot transfer KE to a fixed nucleus => scattering is elastic

• we consider only scattering farfrom nuclear resonances whereneutron absorption is negligible

Adding up Neutrons Scattered by Many Nuclei

0

)(

,

)()'(

,

2i

2

)'(

i

'.

2

)'.(

i

iscat

R.k i i

'by defined is Q r transfer wavevecto thewhere

get todS/r d use can weRr that soaway enoughfar measure weIf

R-r

1

R-r

b- is wavescattered theso

e is aveincident w theRat located nucleusa At

0

0

0

i0

kkQ

ebbebbd

d

eebd

dS

vd

vdS

d

d

ee

jiji

i

ii

RR.Qij

jii

RR.kkij

jii

R.kki rkii

scat

Rrki R.ki

rrr

rr

rr

r

rrrrrrr

rrrrr

rrrrr

rr

−=

==Ω

=Ω>>

Ω=

Ω=

Ω∴

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

−−−−

−

−

∑∑

∑

∑

σ

ψσ

ψ

The Fermi Pseudo-Potential

)(2

)( so )(2

'')2(

21 So,

1

x velocitydensity flux incident Further,

')2(dE'

statesr wavevectoofnumber i.e.

')2(

' volumein statesr wavevectoofnumber '

so ''

'2

'' isenergy neutron Final

box volume Y where/)2( is statek per volume the,ion"normalizatbox " standard Using

'k state in neutronsfor energy,unit per ,d in states momentum of # is where

)(12

'2

: RuleGolden sBy Fermi'

ns transitioall of iesprobabilitover is sum the where11

22).'(

2

2

2

23

23'

23

2'

222

3

'

2

).'(2'

2

'd in '

'

d in ''

rbm

rVrderVm

kVkdm

kY

d

m

k

Y

d

d

kmY

dm

kY

ddkk'Y

ddkk'dE

m

dkkdE

m

kE

Y

rderVY

kVkW

Wdd

d

rkki

k

k

k

Y

rkki

kkk

kk

kkk

vhvrv

h

rr

hhh

h

h

hh

v

rv

h

rr

h

vrv

r

r

vrv

rrr

rr

rrr

δπππ

πσ

πρ

πρ

π

ρ

ρπρπ

σ

=⎟⎠⎞

⎜⎝⎛=Ω

Ω=

Ω

===Φ

Ω==

Ω=Ω=

=⇒=

=

Ω

==

ΩΦ=

Ω

∫

∫∑

∑

−

−

Ω→

Ω→

Fermi pseudopotential

Use V(r) to Calculate the Refractive Index for Neutrons

materials.most from reflected externally are neutrons 1,ngenerally Since

2/1

:get we)A 10(~ small very is and ),definition(by index refractive Since

4or 2

222 givesenergy of onConservati

2 isenergy total themedium theInside

2

is vaccuumin neutron ofenergy total)(and kinetic The

tryreflectome & SANSfor used - (SLD) Density LengthScatteringnuclear thecalled is

1 where

2 :is medium theinside potential average theSo

nucleus. singlea for )(2

)( :by given is potential neutron-nucleus The

2

2-6-0

220

2222220

2

22

20

2

2

2

<=

==

=−+=+=

+

=

==

=

∑

πρλ

ρ

πρρπ

ρ

ρρπ

δπ

-n

nk/k

kkmm

kV

m

k

m

k

Vm

k

m

kE

bvolumem

V

rbm

rV

ii

hhhh

h

h

h

rhr

Now let’s do it the Hard Way – Calculate the Scattered Wavefunction….

( ) ⎥⎦⎤

⎢⎣

⎡

−+

+−≈

++−=

2

22

2222

2

11)(

)(

zd

yxzdr

yxzdr

for t & u << dViewgraph from M. R. Fitzsimmons

( )

( ) ( )

( )

ikds

tikd

zd

xik

zd

yikt

ikds

zd

yxik

ikd

ikzuRik

s

tei

k

zdi

zddze

dxedyezd

dze

dVeezd

bN

NdVeeuR

bd

ρλψ

πρ

ρψ

ψ

−=

−−

−=

−−=

−−≈

−−=

∫

∫∫∫ −−

−+

−

21

1

0

22

0

2

22

22

rr

rr A function with interesting consequences.

p.66 of Als-Nielsen & McMorrow

The scattered wave function at “d” from a plate of material illuminated by a plane wave.

Viewgraph from M. R. Fitzsimmons

Definition of the 1st Fresnel zone

The 1st Fresnel zone is the portion of the sample yielding phase from 0 to 180°, i.e., having the same sign of the imaginary component of exp(iku2/d).

m 10μλ ≈≤ du

d = 1 mλ = 1 nm

Viewgraph from M. R. Fitzsimmons

(1) The phase of the wave function in the forward direction ψt is related to the average (macroscopic) properties of the material, i.e., , and not its atomic structure, e.g., crystalline, liquid, etc.

(2)The phase can be retarded or advanced depending upon the sign of , since ρ = Nb.

( )( )

( )( ) ikdti

t

ikd

sinct

ikds

eed

eti

d

teid

ρλψ

ρλ

ψψψρλψ

−≈

−=

+=−=

1

t ρ

b

(scattering approach)

Viewgraph from M. R. Fitzsimmons

Let’s Calculate ψt(d) using a Macroscopic Approach

Relative to the front surface of Medium 1, the accumulated phase at d, φ(d) is:

( )tdktk −+′=+= 21 φφφ

Viewgraph from M. R. Fitzsimmons

( )( )

ikdktnit ee

kdktn

tdknkt

tdktk

)1(

)1(−=⇒

+−=−+=−+′=

ψ

φ

φ

From optics theory (Snell’s law), the perpendicular components of the wave vectors across an interface are related by the index of refraction n:

⊥⊥ =′ nkk

so

This is the result obtained from the macroscopic (optics)approach

Viewgraph from M. R. Fitzsimmons

(macroscopic approach)

Equating the Scattering & Macroscopic Approaches…ikdti

t ee ρλψ −=ikdktni

t ee )1( −=ψ( )

2

2

11

1

ρλπ

ρλ

−=⇒

=−

n

tktn

(scattering approach)

Index of refraction – same result as before

Phase advanced,

λ1<λ0

1+0.8x10-5-3.70.08Mn

Phase retarded,

λ1>λ0

1-2x10-510.30.09Ni

nb

(x10-5 Å)N (atoms/Å3)

Viewgraph from M. R. Fitzsimmons

Why do we Care about the Refractive Index?

• When the wavevector transfer Q is small, the phase factors in the cross section do not vary much from nucleus to nucleus & we can use a continuum approximation

• We can use all of the apparatus of optics to calculate effects such as:– External reflection from single surfaces (for example from guide surfaces)

– External reflection from multilayer stacks (including supermirrors)

– Focusing by (normally) concave lenses or Fresnel lenses

– The phase change of the neutron wave through a material for applications such as interferometry or phase radiography

– Fresnel edge enhancement in radiography

Coherent and Incoherent Scattering

The scattering length, bi , depends on the nuclear isotope, spin relative to the neutron & nuclear eigenstate. For a single nucleus:

)(

ji unless vanishes and 0but

)(

zero toaverages b where

22

,

).(2

2222

2

i

Nbbebd

d

bbbbb

bbb

bbbbbbbb

bbb

ji

RRQi

ii

ji

jijiji

ii

ji −+=Ω

∴

−=−=

==

+++=

+=

∑ −−rrrσ

δ

δδδ

δδδδ

δδ

Coherent Scattering(scattering depends on the direction & magnitude of Q)

Incoherent Scattering(scattering is uniform in all directions)

Note: N = number of atoms in scattering system

Nuclear Spin Incoherent Scattering

[ ] [ ]222 )())(1(12

1b and )1(

12

1 Thus

2)(2I)/(4I is state of occurence offrequency The

2)2)/(4I(2I is state of occurence offrequency The

y.probabilit same the

has state spin each d,unpolarize are spinsnuclear theand neutrons theIf

21)2/1(2 is )2/1( spin withstates ofnumber The

221)2/1(2 is )2/1( spin withstates ofnumber The

).2/1(or )2/1( be can system

neutron-nucleus theof spin The . spin withisotope singlea Consider

−+−+

−−

++

+++

=+++

=

+=

++=

=+−−+=+++

−+

bIbII

IbbII

b

fb

fb

a priori

III

III

II

I

Values of σcoh and σinc

0.024.936Ar0.01.5Al

0.57.5Cu0.04.2O

5.21.0Co0.05.6C

0.411.5Fe2.05.62H

5.00.02V80.21.81H

σincσcohNuclideσincσcohNuclide

• Difference between H and D used in experiments with soft matter (contrast variation)• Al used for windows• V used for sample containers in diffraction experiments and as calibration for energy

resolution• Fe and Co have nuclear cross sections similar to the values of their magnetic cross sections• Find scattering cross sections at the NIST web site at:

http://webster.ncnr.nist.gov/resources/n-lengths/

Coherent Elastic Scattering measures the Structure Factor S(Q) I.e. correlations of atomic positions

only. R of functiona is )( where

}.)(.{1)( ie

)()(1

)'()(.'1

or

)(.1

)( so

densitynumber nuclear theis where)(.)(.Now

1

)( whereatomssimilar ofassembly anfor )(.

00

.

.)'.(

2.

N ...

ensemble,

).(2

rrrrr

rrr

rrrrrrrrrr

rrr

rrrrr

rr

rr

rrrrr

rr

rrrrrr

rrr

∑

∫∫ ∫∫∫

∫

∫∑∫∑

∑

≠

−

−−−

−

−−−

−−

+−=

−+=

−==

=

=−=

==Ω

ii

RQi

NNRQi

NNrrQi

NrQi

NrQi

ii

rQi

i

RQi

ji

RRQi

RRR)Rg(

eRgRdQS

RrrerdRdN

rrerdrdN

)QS(

rerdN

QS

rerdRrerde

eN

QSQSNbd

d

i

ji

δ

ρ

ρρρρ

ρ

ρρδ

σ

g(R) is known as the static pair correlation function. It gives the probability that there is anatom, i, at distance R from the origin of a coordinate system, given that there is also a (different) atom at the origin of the coordinate system at the same instant in time.

S(Q) and g(r) for Simple Liquids

• Note that S(Q) and g(r)/ρ both tend to unity at large values of their arguments

• The peaks in g(r) represent atoms in “coordination shells”

• g(r) is expected to be zero for r < particle diameter – ripples are truncation errors from Fourier transform of S(Q)

Neutrons can also gain or lose energy in the scattering process: this is called inelastic scattering

General Expression for d2σ/dΩdE

• Squires (eqn 2.59) derives the following expression:

• Note that, because of the operators and the average over the states of the system, this expression is not easy to evaluate in the general case

• Note also that the exponential operators do not commute – each contains H and therefore p, and p and R do not commute.

λλ

λ

πσ

λλ

ω

Ap

Heeee

tR

dteeebbk

k

dEd

d

iHtRQiiHttRQi

i

ii

titRQiRQiii

ii

ii

∑

∑ ∫

=

=

=Ω

−−−

∞

∞−

−−

A operator,any for i.e. --scatterer

theof , states, possible over the average a thermal denotes and

scatterer theof an Hamiltoni theis where

i.e.operator rga Heisenbe is )( where

'2

1'

'

/./)(.

',

)(.)0(.'

2'

hrr

hrr

rrrr

rh

Correlation Functions

• Note again that the operators do not commute. If we ignore this fact, we can do the integration and obtain

∑

∑∫

∫

∑∫

+−=

−+−=

=⎟⎟⎠

⎞⎜⎜⎝

⎛Ω

=

=

−

−−

','

','

22

).(

',

)('.)0('.'.3

)}0()({1

),(

')}('{)}0('{1

),(

thatshows 4.17) to4.14 (eqn Squires

atom of typeoneonly is thereprovided ),('

'

find wen the.),(2

1),( and

'1

)2(

1),( :define weSuppose '

jjjjclassical

jjjj

coh

coh

trQi

jj

tRQiRQirQi

RtRrN

trG

rdtRrrRrN

trG

QNSk

kb

dEd

d

dtrdetrGQS

QdeeeN

trG jj

rrrr

rrrrrrr

r

rr

h

r

rr

rr

rrrrrr

δ

δδ

ωσπ

ω

π

ω

Correlation Functions (cont’d)

• We expressed the coherent scattering cross section in terms of G(r,t)

• If we use the classical variant given above, there is a clear physical meaning – G(r,t) is the probability that if particle j’ is at the origin at time zero, particle j will be at position r at time t.

• We can do the same thing with the incoherent scattering and express it in terms of a self-correlation function whose classical version is

• This says that the incoherent scattering is related to the probability that if a particle is at the origin at time zero, the same particle will be at position r at time t.

∑ +−=',

' )}0()({1

),(jj

jjclassical RtRrN

trGrrrr δ

)}0()({),( jjselfclassical RtRrtrG +−=

rr δ

Inelastic Neutron Scattering Measures Atomic Motions

In term of the pair correlation functions, one finds

dtrdetrGQSdtrdetrGQS

QNSk

kb

dEd

d

QNSk

kb

dEd

d

trQiss

trQi

sinc

inc

coh

coh

rr

h

rrr

h

r

r

r

rrrr

∫∫∫∫ −− ==

=⎟⎟⎠

⎞⎜⎜⎝

⎛Ω

=⎟⎟⎠

⎞⎜⎜⎝

⎛Ω

).().(

22

22

),(2

1),( and ),(

2

1),(

where

),('

.

),('

.

ωω

πω

πω

ωσ

ωσ

• Inelastic coherent scattering measures correlated motions of differentatoms

• Inelastic incoherent scattering measures self-correlations e.g. diffusion

(h/2π)Q & (h/2π)ω are the momentum & energy transferred to the neutron during thescattering process

Elastic Scattering as the tö¶ Limit of G(r,t)

• Elastic scattering occurs at ω = 0. Since it involves a δ(ω), only the part of G(r,t) which is constant contributes

• G(r,t) decays as t increases, so the constant part is G(r,¶)

• Since we only need the part of the correlation that is time-independent, we can write (noting that the correlation between the positions of j and j’are independent of t as t-> ¶)

any timeat operator density particle theis )'( where

')'()'(1

'}'{}'{1

),(

')}('{)}0('{1

),(

','

','

r

rdrrrN

rdRrrRrN

rG

rdtRrrRrN

trG

jjj

j

jjjj

v

vrrv

vrrrrrr

rrrrrrr

ρ

ρρ

δδ

δδ

+=

−+−=∞

−+−=

∫

∑∫

∑∫

G(r,¶) is called the Patterson function

Note – no trulyelastic scatteringfor a liquid

The Static Approximation

• In diffraction measurements, we measure scattered neutron intensity in a particular direction, independent of the change in neutron energy – i.e. we integrate the cross section over E = hω/2π. This is the Static Approximation.

the integral over ω picks out the t = 0 value of G to give

∑∫

∫−−

−

=

=⎟⎟⎠

⎞⎜⎜⎝

⎛Ω

',

)(.)0(..3

).(22

'1

)2(

1),( where

.),(2

1'

'

:had Earlier we

jj

tRQiRQirQi

trQicoh

coh

jj eeeN

trG

dtrdetrGNk

kb

dEd

d

rrrrrr

rr

r

rr

h

π

πσ ω

)(2

1)( Because ω

πδ ω hh det ti∫

∞

∞−

=

∫

∫=

=⎟⎠⎞

⎜⎝⎛Ω

−

rderGNb

ddtrdetrGNbd

d

rQicoh

trQicoh

static

coh

rr

rr

rr

rr

.2

).(2

)0,(

..),( ωσ ω

Comparison of Elastic Scattering and the Static Approximation

• These are not the same, except in an (unreal) system with no motion

• The elastic scattering cross section gives the true elastic scattering that results when the positions of different atoms are correlated for all times, as they are in a crystalline solid, even when phonons arepresent

• The static approximation, as its name suggests, gives the scattering for a system that is frozen in time

),(

)0,(

.2

.2

∫

∫

∞=⎟⎠⎞

⎜⎝⎛Ω

=⎟⎠⎞

⎜⎝⎛Ω

rderGNbd

d

rderGNbd

d

rQicoh

elastic

coh

rQicoh

static

coh

rr

rr

rr

rr

σ

σ

The Intermediate Scattering Function

• Another function that is often useful is the Intermediate Scattering Function defined as

This is the quantity measured with Neutron Spin Echo (NSE)

• It is not possible to derive exact expressions for I, G or S except for simple models. It is therefore useful to know the various analytical properties of these functions to ensure that models preserve them. Squires shows:

• There are also various sum & moment rules on these quantities that are sometimes useful (see Squires for details)

∫= rdetrGtQI rQi rrr rr.),(),(

),(),( );,(*),(

/

)/,( ),( );,(*),(

/ ωωωω ω −−==

+==+−−=−=

QSeQSQSQS

T)ki,-trG(-,t)rG(,-t);rG*(-,t)rG(

TkitQItQItQItQI

Tk

B

B

Brrrr

hrrrr

hrrrr

h

0 .001 0 .01 0 .1 1 10 100

Q (Å-1

)

En

erg

y T

ran

sfe

r (m

eV

)

Ne utro ns in Co nde ns e d Matte r Re s e arc h

S pa lla tio n - Ch opp er ILL - w ithou t s p in -ec ho ILL - w ith s p in -e ch o

Crystal Fields

Spin Waves

Neutron Induced

ExcitationsHydrogen Modes

Molecu lar Vibrations

Lattice Vibrations

and Anharmonicity

Aggregate Motion

Polymers and

Biological Systems

Critical Scattering

Diffusive Modes

Slower Motions

Resolved

Molecular Reorientation

Tunneling Spectroscopy

Surface Effects?

[Larger Objects Resolved]

Proteins in Solution Viruses

Micelles PolymersMetallurgical

SystemsColloids

Membranes Proteins

C rystal and Magnetic

Structures

Amorphous Systems

Accurate Liquid Structures

P recision Crystallography Anharmonicity

1

100

10000

0.01

10-4

10-6

Momentum DistributionsElastic Scattering

Electron- phonon

Interactions

Itinerant Magnets

Molecular Motions

Coherent Modes in Glasses

and Liquids

SPSS - Chopper Spectrometer

Neutron scattering experiments measure the number of neutrons scattered at different values of the wavevector and energy transfered to the neutron, denoted Q and E. The phenomena probed depend on the values of Q and E accessed.

References

• Introduction to the Theory of Thermal Neutron Scatteringby G. L. SquiresReprint edition (February 1997)Dover Publications ISBN 048669447

• Neutron Scattering: A Primerby Roger PynnLos Alamos Science (1990)(see www.mrl.ucsb.edu/~pynn)

• Elements of Modern X-Ray Physicsby Jens Als-Nielsen and Des McMorrowJohn Wiley and Sons

ISBN 0471498580