Neutron Scattering

Neutron Scattering Basic properties of neutron and electron neutron electron −27 −31 mass mkn =×1.675 10 g mke =×9.109 10 g charge 0 e spin s = ½ s = ½ −e= −e= magnetic dipole moment µnn= gs with gn = 3.826 µee= gs with ge = 2.0 2mn 2me =22k 2π =22k Ek== E = 2m λ 2m energy n e 81.81 150.26 Em[]eV = 2 Ee[]V = 2 λ ⎣⎡Å⎦⎤ λ ⎣⎦⎡⎤Å interaction with matter: Coulomb interaction — 9 strong-force interaction 9 — magnetic dipole-dipole 9 9 interaction Several salient features are apparent from this table: – electrons are charged and experience strong, long-range Coulomb interactions in a solid. They therefore typically only penetrate a few atomic layers into the solid. Electron scattering is therefore a surface-sensitive probe. Neutrons are uncharged and do not experience Coulomb interaction. The strong-force interaction is naturally strong but very short-range, and the magnetic interaction is long-range but weak. Neutrons therefore penetrate deeply into most materials, so that neutron scattering is a bulk probe. – Electrons with wavelengths comparable to interatomic distances (λ ~2Å ) have energies of several tens of electron volts, comparable to energies of plasmons and interband transitions in solids. Electron scattering is therefore well suited as a probe of these high-energy excitations. Neutrons with λ ~2Å have energies of several tens of meV , comparable to the thermal energies kTB at room temperature. These so-called “thermal neutrons” are excellent probes of low-energy excitations such as lattice vibrations and spin waves with energies in the meV range. Basic idea of neutron scattering experiment: K K K K neutron Qk=−fik kii,ω incoming neutron source K = 22 q , ω excitation ωω=−fiω= ()kkf−i 2mn K neutron k ,ω scattered neutron ω = 0 elastic scattering detector ff ω ≠ 0 inelastic scattering – 1 – Neutron sources Research reactor 235Un+→A+B+2.3n (A, B: fission fragments) ⇒ chain reaction, keeps going by itself until "fuel" (uranium enriched by 235U ) is exhausted ⇒ source of both energy (nuclear power reactors) and neutrons (research reactors) research reactors optimized for neutron flux ⇒ low power fission reaction most favorable for thermal neutrons ⇒ fast neutrons slowed down by "moderator" ( HO22, DO) neutron Maxwellian profile flux ~30meV energy Research reactor FRM-II in Munich, Germany Outside view of the building (left) and layout of the experimental hall (right). The neutron beam tubes (blue) tap into the flux emitted from the reactor core (center) and guide the neutrons to various neutron scattering instruments. http:///www.frm2.tu-muenchen.de – 2 – Spallation source p + Hg → A + B + xn Fragmentation of target atoms induced by high energy proton beam from accelerator. Typical energy of spallation neutrons >> thermal energy ⇒ need to use moderator for scattering experiments. Magnetism and Crystal Analyzer Chopper Spectrometer Microvolt Liquids Spectrometer Reflectometers Disordered Materials Instrument 100-Micr ovolt Spectrometer SANS Instrument Engineering Diffractometer Single Crystal Powder Diffractometer Diffractometer Layout of the Spallation Neutron Source (SNS) at Oak Ridge National Lab (USA) Protons are accelerated in the linear accelerator, and various neutron scattering instruments are grouped around the target, where the neutrons are produced in a spallation reaction. http://www.sns.gov – 3 – Neutron detectors Since neutrons are electrically neutral, they are difficult to detect directly. One therefore converts them into charged particles via a nuclear reaction such as: nH+→33e H+p The protons are collected by a high electric field and converted into electric current. 10 10 Another type of neutron detector is based on a gas of BF3. B has a high neutron capture cross section via the nuclear reaction nB+→10 7 Li+4He. The energetic nuclei produced in this reaction ionize gas molecules, which are again collected by a high electric field. Interaction of neutrons with matter elastic scattering inelastic scattering strong-force interaction position of nuclei in solid lattice vibrations (“nuclear scattering”) (lattice structure) (phonons) magnetic interaction position and orientation of spin excitations electronic magnetic moments (magnons, spin waves) in solids (ferromagnetism, antiferromagnetism) Elastic nuclear scattering dσ basic quantity: differential cross-section dΩ detector ∆Ω solid angle element covered by detector # of neutrons incident neutron beam, flux φ = area ⋅ time dσ = # of neutrons scattered into solid angle element dΩ per unit time, dΩ normalized to incident flux. Material-specific quantity, experimental conditions (size of detector, incident flux) normalized out. ⎡⎤dσ 1 dimensions: == area ⎣⎦⎢⎥dtΩ∆[]Ω[][φ] dimensionless – 4 – dσ calculation of through Fermi's Golden Rule: dΩ 2π KK2 transition rate ()# of transitions per unit time : Wk= Vkρ ()E = fi f Density of final states K K 1 ik ⋅r ⎫ | kei i = ⎪ L3 ⎪ plane waves, normalized to sample size L K K ⎬ 1 ik ⋅r f ⎪ | kef = L3 ⎭⎪ 3 K ⎛⎞L dk f ρ f ()E = ⎜⎟ ⎝⎠N2π dE density of states in k-space K 2 dk ff=Ωk dk fd 33 2 ⎛⎞LL2 dk fn⎛⎞m k fdE= k f ρ ff()Ek=Ω⎜⎟ d=⎜⎟ 2 dΩwith = ⎝⎠22ππdE ⎝⎠= dk fnm velocity =k incident neutron flux: = i 33 LmnL kkif= for elastic scattering 2 KK 2 K dWσ ⎛⎞mn ik()if−⋅k r K ⇒= =⎜⎟Ve dr dΩ incident flux ⎝⎠2π =2 ∫ 2 K K 2 ⎛⎞mn KK−⋅iQ r = ⎜⎟Vr()e dr "Born approximation" ⎝⎠2π =2 ∫ For short range strong force, use approximate interaction potential KK2π =2 K Vr()=−bδ ()r R m n position of nucleus "scattering length" b depends on the details of the nuclear structure and varies greatly (and almost randomly if you don’t know a lot about nuclear physics) among elements/isotopes. A table of scattering lengths can be found on the web under http://www.ncnr.nist.gov/resources/n-lengths/ dσ 2 for single nucleus: = b dΩ dσ total cross section: σ =Ωdb=4π 2 ∫ dΩ – 5 – KK2π =2 K G lattice of nuclei: Vr=−bK δ r R b K : scattering length of nucleus at lattice site R () ∑K R ()R mn R 2 K K dσ KKK iQ⋅r =−∫ dr ∑bR δ ()r R e dΩ R K K 2 iQ⋅R = ∑beR R K For most Q , matrix elements are very small because phase factors contributed K K by different nuclei cancel out. Matrix element is large only for Q= K (reciprocal KK lattice vector, defined as eiK⋅R =1). If this condition is satisfied, the constructive interference leads to a huge enhancement (“Bragg peak”) of the scattering rate. K K 2 3 iQ⋅k ()2π K K Use relation eN=δ Q−K derived on problem set. ∑K∑K() kkv0 reciprocal lattice vector volume of unit cell total # of unit cells If all nuclei are identical: 3 dσ N ()2π K K =−bQ2 δ K ∑K () dvΩ o K K for unit cell with several atoms, basis vector d 3 dσ ()2π K KK2 =−NQδ ()KFN ()K dvΩ ∑K 0 K K K K FK= eiQ⋅dbK “nuclear structure factor” N ()∑K d d Relationship between reciprocal lattice vector and scattering angle Θ at Bragg peak: K K Kn12π λ sin Θ= = ⇒ 2dnsin Θ= λ “Bragg's law” 22kd2π K K k Θ k i f n = integer d = lattice constant Collect intensities of all Bragg reflections ⇒ determine lattice structure (crystallography). – 6 – This calculation is strictly valid only for a single isotope with nuclear spin I = 0 . However, almost all elements are found naturally with a distribution of isotopes, most of which have I ≠ 0 . The scattering length b depends on details of the nuclear structure and can vary strongly from one isotope to the other. It also depends on the combined spin of the nucleus and the neutron which can take the values I +½ or I −½ . The nuclear spins are randomly oriented in a solid, except at extremely low temperatures. For a given isotope, one therefore has to average over the scattering lengths b+ for I +½ and b− for I −½ ; e.g. for hydrogen ( I = ½ ) bf+ =10.85 m and b− = −47.50 fm, and for deuterium ( I =1) bf+ = 9.53 m and bf− = 0.98 m. The number of states for these two situations is: 2½()II+=2+2 for I+½ 2½()II−=2 for I−½ Further, the different naturally occurring isotopes of each element are incorporated randomly in a solid, with no correlations between different lattice positions. Taking these complications into account, the differential cross section for elastic nuclear neutron scattering from a solid containing a single element per unit cell becomes K K 2 K K JJK dσ iQ⋅R iQ⋅−()R R′ ==beK bbKJJK e ∑K r ∑KJJK R R′ dΩ R RR′ where ... denotes the average over isotopes and nuclear spin states. Since there are no correlations between b-values for different nuclei, we have 2 ⎡⎤22II+ 2 K JJK KJJK ξξ+− ′ bbR R′ ==b ∑cξξ⎢⎥b+ bξ for R=R ξ ⎣⎦⎢⎥42IIξξ++42 ⎡⎤22II+ 222 K JJK KJJK 2 ξξ+− ′ bbR R′ ==b ∑cξξ⎢⎥()b + ()bξ for R≠R ξ ⎣⎦⎢⎥42IIξξ++42 where ... denotes the different naturally occurring isotopes and cξ their relative abundance. Splitting the sum into two terms, we have K K JJK dσ 2 iQ⋅−()R R′ 2 =+be b K∑∑JJK K dΩ RRK, JJ′K R RR≠ ′ K K JJK 22iQ⋅−()R R′ 2 =+be ⎡⎤b−b K∑∑JJK K RR, ′ R ⎣⎦ where b 2 has been added to the first term and subtracted from the second ∑K R K JJK term in the sum. With the restriction R ≠ R′ thus removed, the first term is identical to the result obtained above for the case of a single element per unit cell, except that b2 was replaced by b 2 . – 7 – 3 dσ ()2π 22K K Therefore =−Nb∑δ ()QK+N⎡ b2 −b⎤ dVΩ K ⎣ ⎦ 0 K "coherent" scattering "incoherent" scattering Again, the chemical properties of the elements are dictated entirely by the number of protons and electrons and completely independent of the number of neutrons per nucleus (isotope) or the nuclear spin.

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