ACNS 2008 Tutorial Section SANS and Reflectometry for Soft Condensed Matter Research The Basic Theory for Small Angle Ne

1. ACNS 2008 Tutorial SectionSANS and Reflectometry for Soft Condensed Matter ResearchThe 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. v1’ particle 1 (projectile) v1 v2 particle 2 (target) v2’ Kinematics Aspect of Collision 12 variables: v1, v2, v1’ & v2’ • Conservation laws • energy (1) • mass(1) • momentum (3) • v1 and v2 are known (6) • 1+1+3+6 = 11

4. Is this reaction possible? • Does it violate any conservation law? closest approach Independent of the specific forces between the particles effective particle Possible existence of Neutron James Chadwick Nature, 129, 312, 1932 origin Kinematics Aspect of Collision 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. Schrödinger equation: Where is f(θ) in Schrödinger equation ? You put it in through boundary condition Phase Shift Analysis looking for far field solution (kr >> 1 , V(r) = 0) E > 0 LHS RHS expanded by partial wave d0is 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. d0→ 0 as k→ 0 Definition of Scattering Length a and

10. Accurate Measurements of the Scattering Length http://physics.nist.gov/MajResFac/InterFer/text.html

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

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 = 10-13 cm) Matching the wave functions and their flux for the exterior and interior regions, (E > 0) s = 2.3 barns

14. ~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/

15. triplet state (bound state) I = 1, parallel, EB = -2.23 MeV singlet state (virtual state) I = 0, antiparallel, E* = 70 keV Example: Neutron-Proton Scattering spin dependence interaction Eugene P. Wigner, Zeits. f. Physik83 253 1933 t 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. Real potential Requirement Fermi Approximation Step 2 – Fermi Pseudopotential Fictitious 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 Scientifica7 13 1936

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. For H Example: Neutron-Proton Scattering F = 10-13 cm t For D

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

27. For D2O For H2O can be adjusted to take on any value between these two extremes Basis of Contrast Variation For H For D For O t Scattering Length Density Calculator http://www.ncnr.nist.gov/resources/sldcalc.html

28. F = 10-13 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 (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 Review95 249 1954