1. Neutron Scattering Sciences Division Spallation Neutron Source
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

2. 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 s(q)
Coherent and Incoherent Scattering
Contrast variation
References

3. Kinematics Aspect of Collision
particle 1 (projectile)
particle 2 (target) v1 v2
v1’
v2’
12 variables: v1, v2, v1’ & v2’
Conservation laws
energy (1)
mass(1)
momentum (3)
v1 and v2 are known (6)
= 11

4. Kinematics Aspect of Collision
origin
effective particle
closest approach
Possible existence of Neutron
James Chadwick
Nature, 129, 312, 1932
Is this reaction possible?
Does it violate any conservation law?
Independent of the specific forces
between the particles
What is the possibility that the projectile will scatter off the target at that specific angle?
Interaction: hidden in Cross Section
scattering ≡ (initial constellation = final one), elastic scattering ≡ conservation of kinetic energy
A + B → A + B

5. Dynamics Aspect of Collision: Concept of Cross Section
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 Q (T-1)
Reaction probability ≡
s : a proportionality constant of reaction probability with dimension of L2
To calculate s one must be to be able to calculate reaction probability

6. Scattering Experiment
angular differential cross section
Given the interaction potential V(r), how can one calculate σ(θ)?

7. Phase Shift Analysis Schrödinger equation:
Where is f(θ) in Schrödinger equation ?
You put it in through boundary condition
looking for far field solution (kr >> 1 , V(r) = 0) E > 0
LHS
RHS
expanded by
partial wave
d0 is introduced as one of
the integration constants
matching the coefficients of
exp(ikr) and exp(-ikr) from RHS and LHS
and

8. Reasoning of S-wave Scattering for Low Energy Scattering (kr0 << 1)
Classically
Quantum Mechanically
Only neutrons with l = 0 will be scattered

9. Definition of Scattering Length a
and
d0 → 0 as k → 0

10. Accurate Measurements of the Scattering Length

11. Physical Significance of Sign of Scattering Lengthuo r0
a < 0
a > 0 r

12. Example: Neutron-Proton Scattering
Lecture 2 Basic Theory - Neutron Scattering for Biomolecular Science
Roger Pynn, UCSB, 2004

13. Example: Neutron-Proton Scattering
From the capture of a low-energy neutron by hydrogen
n + H1 → H2 + g (2.23 MeV)
Solving the Schrödinger equation with this binding energy, (E < 0)
V0 = -36 MeV and r0 = 2 F (F = cm)
Matching the wave functions and their flux for the exterior and interior regions, (E > 0)
s = 2.3 barns

14. Example: Neutron-Proton Scattering
The “Barn Book”
Brookhaven National Laboratory Report
BNL-325, 1955
~20 barns
2.3 barns
Experimental Nuclear Reaction Data (EXFOR / CSISRS)
National Nuclear Data Center

15. Example: Neutron-Proton Scattering
spin dependence interaction
Eugene P. Wigner, Zeits. f. Physik t
triplet state (bound state)
I = 1, parallel, EB = MeV
singlet state (virtual state)
I = 0, antiparallel, E* = 70 keV
s = 20 barns

16. Fermi Approximation Step 1 – Born Approximation
Why we need Born Approximation?
The many-body problem of thermal neutron scattering
What is Born Approximation?
Another way to solve the Schrödinger Equation
Compare with
Born approximation eliminates the need of solving Schrödinger equation

17. Can Born Approximation be Applied to Neutron Scattering?
If we use the potential parameters for n-p scattering
No with real potential, too large for Born Approximation to be applicable

18. Fermi Approximation Step 2 – Fermi Pseudopotential
Fictitious potential
Requirement
Real potential
With this fictitious potential, Born Approximation is valid

19. Fermi Approximation Step 2 – Fermi Pseudopotential
V0* ~ 10-6V0
r0* ~ 102r0
Why delta function?
What is b ?
actual neutron-nucleus
interaction potential
Fermi pseudopotential
Enrico Fermi, Ricerca Scientifica

20. Neutron Scattering Data for Elements and Isotopes
Neutron Diffraction George E. Bacon

21. Chemical Binding Effect
s ~ m2
low energy (0.025 eV)
high energy (~10 eV)

22. Lecture 2 Basic Theory - Neutron Scattering for Biomolecular Science
Roger Pynn, UCSB, 2004

23. A Typical Reactor-based SANS Diffractometer
angular differential cross section
Lecture 5 Small Angle Scattering - Neutron Scattering for Biomolecular Science
Roger Pynn, UCSB, 2004

24. Expression of s(q): Coherent & Incoherent Contribution

25. Example: Neutron-Proton Scattering
F = cm t
For H
For D

26. Contrast Variation 10-12 F = 10-13 cm
Neutron Diffraction George E. Bacon

27. Basis of Contrast Variation
For H
For D
For O
For H2O
For D2O t
can be adjusted to take on any value between these two extremes
Scattering Length Density Calculator

28. F = cm
Lecture 1 Overview of Neutron Scattering & Applications to BMSE –
Neutron Scattering for Biomolecular Science
Roger Pynn, UCSB, 2004

29. References and Further Reading
Roger Pynn - An Introduction to Neutron Scattering (
- Neutron Physics and Scattering (
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