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

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

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) 1+1+3+6 = 11

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

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

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

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

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

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

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

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

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

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

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

Example: Neutron-Proton Scattering spin dependence interaction Eugene P. Wigner, Zeits. f. Physik 83 253 1933 t triplet state (bound state) I = 1, parallel, EB = -2.23 MeV singlet state (virtual state) I = 0, antiparallel, E* = 70 keV s = 20 barns

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

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

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

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 7 13 1936

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

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

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

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

Expression of s(q): Coherent & Incoherent Contribution

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

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

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 http://www.ncnr.nist.gov/resources/sldcalc.html

F = 10-13 cm Lecture 1 Overview of Neutron Scattering & Applications to BMSE – Neutron Scattering for Biomolecular Science Roger Pynn, UCSB, 2004

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