ACNS 2008 Tutorial Section

SANS and Reflectometry for Soft Condensed Matter Research

The Basic Theory for Small Angle Neutron Scattering

Wei-Ren Chen

Neutron Scattering Sciences Division Spallation Neutron Source

Oak Ridge National Laboratory

May 11th 2008

Outline

Two Aspects of Collision: Kinematics vs. Dynamics

Cross Section Calculation I: Method of Phase Shift

Cross Section Calculation II: Fermi Approximation

Expression of Scattering Cross Section

Coherent and Incoherent Scattering

Contrast variation

References

Kinematics Aspect of Collision

particle 1 (projectile)

particle 2 (target)

v1 v2

v1’

v2’

Conservation laws

• energy (1)• mass(1)

• momentum (3)• v1 and v2 are known (6)

1+1+3+6 = 11

12 variables: v1, v2, v1’ & v2’

Kinematics Aspect of Collision

What is the possibility that the projectile will scatter off the target at that specific angle?

Possible existence of NeutronJames ChadwickNature, 129, 312, 1932

Interaction: hidden in Cross Section

111

21 mm

origin

effective particle

closest approach

Is this reaction possible? Does it violate any conservation law?

Independent of the specific forces between the particles

scattering ≡ (initial constellation = final one), elastic scattering ≡ conservation of kinetic energy

A + B → A + B

Dynamics Aspect of Collision: Concept of Cross Section

d

azimuthal axis

polar axis

x

area A

density N

d

d

Intensity I

Beam size A (L2)Intensity of beam I (T-1)Thin sample thickness Δx (L)Number density of sample N (L-3)no. of reaction occurring per second (T-1)

Reaction probability ≡

To calculate one must be to be able to calculate reaction probability

: a proportionality constant of reaction probability with dimension of L2

A

xNA

I

A

xNA

I

Scattering Experiment

ikzrki ee in

r

ef

ikr

sc

vvJ in*inin

dRvdN 2sc

*sc

Given the interaction potential V(r), how can one calculate σ(θ)?

2

in

/ fJ

ddN

d

d

2

2

sc*scsc R

fvvJ

angular differential cross section

dd

d

Phase Shift Analysis

looking for far field solution (kr >> 1 , V(r) = 0) E > 0

Where is f(θ) in Schrödinger equation ? You put it in through boundary condition

ErV2

2

2

Schrödinger equation:

is introduced as one of the integration constants

LHS r

efe

r

efe

ikrikr

ikrikz cos RHS

expanded by partial wave

matching the coefficients ofexp(ikr) and exp(-ikr) from RHS and LHS

llk

2

02

sin124

2

02

cossin121

l

lli Pel

kl and

ll lkrAru 2/sin0

Reasoning of S-wave Scattering for Low Energy Scattering (kr0 << 1)

b

r0

v

z

Classically

Quantum Mechanically

sec1021 27 ergll

Only neutrons with l = 0 will be scattered

sec10

sec/101051067.130

61324

erg

cmcmgmbvL

u0

0

sin(kr)sin(kr+)

Definition of Scattering Length a

1 when sin1

cossin121

002

2

2

02

krk

Pelk l

lli l

and 1 when sin4

002

2 kr

k

0 → 0 as k → 0

k

fak

0

0lim

2a24 a

Accurate Measurements of the Scattering Length

http://physics.nist.gov/MajResFac/InterFer/text.html

Physical Significance of Sign of Scattering Length

arAkkakrAkrAu sinsin 00

a < 0 a > 0

r0

uo

r0

Example: Neutron-Proton Scattering

Lecture 2 Basic Theory - Neutron Scattering for Biomolecular ScienceRoger Pynn, UCSB, 2004

36 MeV-Vo

-EB 2.23 MeV

r0=2F

V(r)

r

Example: Neutron-Proton Scattering

From the capture of a low-energy neutron by hydrogen

Solving the Schrödinger equation with this binding energy, (E < 0)

V0 = -36 MeV and r0 = 2 F (F = 10-13 cm)

Matching the wave functions and their flux for the exterior and interior regions, (E > 0)

n + H1 → H2 + (2.23 MeV)

= 2.3 barns

~20 barns

2.3 barns

Example: Neutron-Proton Scattering

The “Barn Book”Brookhaven National Laboratory Report

BNL-325, 1955

Experimental Nuclear Reaction Data (EXFOR / CSISRS)National Nuclear Data Center

http://www.nndc.bnl.gov/

Example: Neutron-Proton Scattering

Eugene P. Wigner, Zeits. f. Physik 83 253 1933

spin dependence interaction

t

= 20 barns

S0

2T0

22

sin4

1sin

4

31

k

triplet state (bound state)I = 1, parallel, EB = -2.23 MeV

singlet state (virtual state)I = 0, antiparallel, E* = 70 keV

Fermi Approximation Step 1 – Born Approximation

Another way to solve the Schrödinger Equation

rVrirdf exp

4

2 32

''exp'exp'2

32

1 rkirVrkirdr

eer

ikrikz

Why we need Born Approximation?

The many-body problem of thermal neutron scattering

What is Born Approximation?

Born approximation eliminates the need of solving Schrödinger equation

Compare with r

efe

ikrikz

Can Born Approximation be Applied to Neutron Scattering?

12

200

rV

7.310

104106.11036106.154

2612624

2

200

rV

If we use the potential parameters for n-p scattering

No with real potential, too large for Born Approximation to be applicable

Fermi Approximation Step 2 – Fermi Pseudopotential

Real potential

constant

7.3

10~

300

2

200

40

rV

rV

kr

02*

0

06*

0

10~

10~

rr

VV

Fictitious potential

300

320

~20

rVrrVdm

fakr

300

3*0

*0

22

200

2*0

1103

10~

rVrV

rV

kr

With this fictitious potential, Born Approximation is valid

Requirement

constant

1

1

300

2

200

0

rV

rV

kr

V(r)

r0

-V0

V(r)

r0

-V0*

*

actural neotron-nucleusinteraction potential

Fermi pseupotential

Fermi Approximation Step 2 – Fermi Pseudopotential

actual neutron-nucleusinteraction potential Fermi pseudopotential

300

320

~20

rVrrVdm

fakr

V0* ~ 10-6V0

r0* ~ 102r0

Enrico Fermi, Ricerca Scientifica 7 13 1936

N

iii rrb

mrV

1

2* 2

Why delta function?What is b ?

cmr

cmr

cm

11*0

130

8

10~

10~

10

Neutron Scattering Data for Elements and Isotopes

Neutron Diffraction George E. Bacon

Chemical Binding Effect

~ 2

high energy (~10 eV)

5.0

21

1

1

11

barns 80420

111

Tmmn

low energy (0.025 eV)

1~

1~)water(18

1

1

11

barns 20

Lecture 2 Basic Theory - Neutron Scattering for Biomolecular Science

Roger Pynn, UCSB, 2004

A Typical Reactor-based SANS Diffractometer

Lecture 5 Small Angle Scattering - Neutron Scattering for Biomolecular ScienceRoger Pynn, UCSB, 2004

angular differential cross section

d

d

Expression of (): Coherent & Incoherent Contribution

N

iii rrb

mrV

1

2* 2 rVrirdf

exp

4

2 32

N

i

N

jjiji rrkibbf

d

d

1 1

2exp

2inc

222coh

2

222

bbbbb

bbbbbbb jijii

kSNbNb

rkibbbNN

ii

2coh

2inc

2

1

222

exp

Example: Neutron-Proton Scattering

t

kSNbNb

rkibbbNN

ii

2coh

2inc

2

1

222

exp

12 :number quantum magnatic , 2

1or

2

1 sIsIs

F

bbb 8.3

4

27.2324.53

4

3

222

2 6494

3 F

bbb

barns 8.18.3 cohcoh Fbb

barns 2.805.25 inc

22inc Fbbb

For D

F

bbb 7.6

4

10.0295.04

6

24

22 5.60 Fb

barns 6.5coh

barns 0.2inc

For H 0 1 ss

2

1

2

3 ss

F = 10-13 cm

Contrast Variation

Neutron Diffraction George E. BaconF = 10-13 cm

10-12

Basis of Contrast Variation

t

For H For D For O

Fb 8.5 Fb 8.3 H Fb 7.6

D

For H2O

Fb 8.1 OH2

For D2O

Fb 2.19 OD2

can be adjusted to take on any value between these two extremessolvent b

Scattering Length Density Calculatorhttp://www.ncnr.nist.gov/resources/sldcalc.html

Lecture 1 Overview of Neutron Scattering & Applications to BMSE –

Neutron Scattering for Biomolecular ScienceRoger Pynn, UCSB, 2004

F = 10-13 cm

References and Further Reading

Roger Pynn - An Introduction to Neutron Scattering (http://www.mrl.ucsb.edu/~pynn/)

- Neutron Physics and Scattering (http://www.iub.edu/~neutron/)

Sidney Yip et al. - Molecular Hydrodynamics

Sow-Hsin Chen et al. - Interaction of Photons and Neutrons With Matter

Peter A. Egelstaff - An Introduction to the Liquid State

M. S. Nelkin et al. - Slow Neutron Scattering and Thermalization

Anthony Foderaro - The Element of Neutron Interaction Theory

Paul Roman - Advanced Quantum Theory

Jean-Pierre Hansen et al. - The Theory of Simple Liquids

Stephen W. Lovesey - Condensed Matter Physics: Dynamic Correlations

Peter Lindner and Thomas Zemb – Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter

Ferenc Mezei in Liquids, Crystallisaton et Transition Vitreuse, Les Houches 1989 Session LI

Léon Van Hove Physical Review 95 249 1954