Neutron Sciences | Neutron Science at ORNL

1 By Roger Pynn Los Alamos National Laboratory LECTURE 1: Introduction & Neutron Scattering Theory . Overview 1. Introduction and theory of Neutron scattering 1. Advantages/disadvantages of neutrons 2. Comparison with other structural probes 3. Elastic scattering and definition of the structure factor, S(Q). 4. Coherent & incoherent scattering 5. Inelastic scattering 6. Magnetic scattering 7. Overview of Science studied by Neutron scattering 8. References 2. Neutron scattering facilities and instrumentation 3.

2 Diffraction 4. Reflectometry 5. Small angle Neutron scattering 6. Inelastic scattering 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 ). 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 ( Cd, B, Gd)

3 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 = x 10-27 kg Charge = 0; Spin = . Magnetic dipole moment: n = - N. Nuclear magneton: N = eh/4 m p = x 10-27 J T-1. Velocity (v), kinetic energy (E), wavevector (k), wavelength ( ), temperature (T). E = mnv2/2 = kBT = (hk/2 )2/2mn; k = 2 / = mnv/(h/2 ).

4 Energy (meV) Temp (K) Wavelength (nm). Cold 10 1 120 3. Thermal 5 100 60 1000 Hot 100 500 1000 6000 (nm) = / v (m/s). E (meV) = k2 (k in nm-1). Comparison of Structural Probes Note 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.

5 Neutrons also interact with unpaired electrons via a magnetic dipole interaction. Brightness & Fluxes for Neutron &. X-Ray Sources Brightness dE/E Divergence Flux (s-1 m-2 ster-1) (%) (mrad2) (s-1 m-2). 15 11. Neutrons 10 2 10 x 10 10. 16 10. Rotating 10 3 x 10 5 x 10. Anode 24 17. Bending 10 x 5 5 x 10. Magnet 26 19. Wiggler 10 x 1 10. Undulator 1033 x 1024. (APS). Cross Sections = number of incident neutrons per cm2 per second = total number of neutrons scattered per second / . measured in barns: d number of neutrons scattered per second into d.

6 = 1 barn = 10-24 cm2. d d . d 2 number of neutrons scattered per second into d & dE. = Attenuation = exp(-N t). d dE d dE N = # of atoms/unit volume t = thickness Scattering by a Single (fixed) Nucleus range of nuclear force (~ 1fm). is << Neutron wavelength so scattering is point-like . energy of Neutron is too small to change energy of nucleus &. Neutron cannot transfer KE to a fixed nucleus => scattering is elastic we consider only scattering far from nuclear resonances where Neutron absorption is negligible If v is the velocity of the Neutron (same before and after scattering), the number of neutrons passing through an area dS per second after scattering is : 2.

7 V dS scat = v dS b 2 /r 2 = v b 2 d . Since the number of incident neutrons passing through unit areas is : = v incident = v 2. d v b 2 d . = = b2 so total = 4 b 2. d d . Adding up Neutrons Scattered by Many Nuclei r r r i k 0 .R i At a nucleus located at R i the incident wave is e r r -b r r r . so the scattered wave is scat = e i k0 .Ri r ri e ik '.(r Ri ) . r - Ri .. 2. d vdS scat 2. r r r r r dS 1. = = bi e ik '.r r r e 0 k ') .Ri i ( k d vd d r - Ri If we measure far enough away so that r >> R i we can use d = dS/r 2 to get d r r r r r r r = bi b j e = bi b j e i ( k 0 k ').

8 ( Ri R j ) iQ.( Ri R j ). d i , j i, j r r r where the wavevector transfer Q is defined by Q = k ' k0. Coherent and Incoherent Scattering The scattering length, bi , depends on the nuclear isotope, spin relative to the Neutron & nuclear eigenstate. For a single nucleus: bi = b + bi where b averages to zero bi b j = b + b ( bi + b j ) + bi b j 2. but b = 0 and bi b j vanishes unless i = j 2. b = bi b = b b 2 2 2. i d r r r e iQ .( Ri R j ). = b + ( b2 b ) N. 2 2. d i, j Coherent Scattering Incoherent Scattering (scattering depends on the (scattering is uniform in all directions).)

9 Direction of Q). Note: N = number of atoms in scattering system Values of coh and inc Nuclide coh inc Nuclide coh inc 1H V 2H Fe C Co O Cu Al 36Ar 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: Coherent Elastic Scattering measures the Structure Factor S(Q) correlations of atomic positions d r r 1 r r r e 2 iQ.

10 ( Ri R j ). = b (Q) for an assemblyof similar atoms where S (Q) =. d N i, j ensemble r r r iQr .rr r r r iQr .rr r Now e = dr .e (r Ri ) = dr .e N (r ) where N is the nuclear number density i i r 1 r iQr .rr r 2. so S (Q) =. N dr .e N (r ). r 1 r r iQr .(rr rr ') r r 1 r r iQr .Rr r r r or S(Q) = dr ' dr .e N (r ) N (r ' ) = dR dr e N (r ) N (r R). N N. r r r iQr .Rr ie S (Q) = 1 + (R).e r r r r r where g(R) = (R Ri + R0 ) is a function of R only. i 0. g(R) is known as the static pair correlation function.