Neutron Scattering John R.D. Copley Summer School on Methods and Applications of High Resolution Neutron Spectroscopy and Small Angle Neutron Scattering NIST Center for NeutronNCNR Summer Research, School 2011 June 12-16, 2011 Acknowledgements National Science Foundation NIST Center for Neutron (grant # DMR-0944772) Research (NCNR) Center for High Resolution Neutron Scattering (CHRNS) 2 NCNR Summer School 2011 (Slow) neutron interactions Scattering plus Absorption Total =+Elastic Inelastic scattering scattering scattering (diffraction) (spectroscopy) includes “Quasielastic neutron Structure Dynamics scattering” (QENS) NCNR Summer School 2011 3 Total, elastic, and inelastic scattering Incident energy Ei E= Ei -Ef Scattered energy Ef (“energy transfer”) Total scattering: Elastic scattering: E = E (i.e., E = 0) all Ef (i.e., all E) f i Inelastic scattering: E ≠ E (i.e., E ≠ 0) D f i A M M Ef Ei S Ei D S Diffractometer Spectrometer (Some write E = Ef –Ei) NCNR Summer School 2011 4 Kinematics mv= = k 1 222 Emvk2m==2 = Scattered wave (m is neutron’s mass) vector k , energy E 2θ f f Incident wave vector ki, energy Ei N.B. The symbol for scattering angle in Q is wave vector transfer, SANS experiments is or “scattering vector” Q =Q is momentum transfer θ, not 2θ. Q = ki - kf (For x-rays, Eck = = ) (“wave vector transfer”) NCNR Summer School 2011 5 Elastic scattering Q = ki - kf G In real space k In reciprocal space f G G G kf Q ki 2θ 2θ G G k Q i E== 0 kif k Q =θ 2k i sin NCNR Summer School 2011 6 Total scattering, inelastic scattering Q = ki - kf G kf G Q Qkk2kkcos2222= +− θ 2θ G if if ki At fixed scattering angle 2θ, the magnitude (and the direction) of Q varies with the energy transfer E. NCNR Summer School 2011 7 Structure and Neutron Diffraction Neutron diffraction measures structure. The scattered intensity, I(2θ), is measured – hence S(Q). The structure of a solid is determined from elastic scattering (Bragg reflection) intensities which are generally measured using a diffractometer (no energy analysis). The structure of a fluid is related to the total scattering and is measured using a diffractometer; there is no elastic scattering from a fluid. NCNR Summer School 2011 8 Dynamics and Neutron Spectroscopy Neutron spectroscopy (inelastic scattering) measures dynamics. The measured intensity yields S(Q,E). Scattering with nonzero E, e.g. to study vibrational excitations, is called inelastic scattering. Inelastic scattering that is centered at E = 0 and associated with diffusional behavior, is called quasielastic neutron scattering (QENS). N.B. In an alternative notation the energy transfer is written = ω = E and the scattering function SQ,( ω=) = SQ,E( ) NCNR Summer School 2011 9 Elastic, inelastic, quasielastic, … The schematic spectrum shown below is resolution-broadened. neutron gain neutron loss ω < 0 ω > 0 ∫∫SQ,( ωω=) d SQ,EdE( ) = SQ( ) NCNR Summer School 2011 Diffraction and Spectroscopy The Nobel Prize in Physics 1994 “to Professor Clifford G. Shull .. for the development of the neutron diffraction technique” “to Professor Bertram N. Brockhouse .. for the development of neutron spectroscopy” Both methods are based on the use of neutrons flowing out from a nuclear reactor. When the neutrons bounce against (are scattered by) atoms in the sample being investigated, their directions change, depending on the atoms' relative positions. This shows how the atoms are arranged in relation to each other, that is, the structure of the sample. Changes in the neutrons' velocity, however, give information on the atoms' movements, e.g. their individual and collective oscillations, that is their dynamics. ... Clifford G. Shull has helped answer the question of where the atoms "are" ... Bertram N. Brockhouse [has helped with] the question of what the atoms "do" http://nobelprize.org/nobel_prizes/physics/laureates/1994/press.html NCNR Summer School 2011 11 Advantages and disadvantages ¾ Neutrons have wavelengths comparable with interatomic spacings and energies comparable with material energies; the geometry of motions can be studied ¾ Little absorption → bulk probe: containment is simplified ¾ Weak interaction simplifies interpretation of data ¾ Sensitivity to isotope (esp. H/D) may be used to advantage ¾ Magnetic interaction enables studies of magnetic materials BUT… ¾ Neutron sources are weak, intensities are low ¾ Some elements absorb strongly ¾ Kinematics restricts available (Q,E) space NCNR Summer School 2011 12 Neutron scattering intensities ⎛⎞kf ⎛⎞σS Intensity∝ ⎜⎟⎜⎟ S() Q,E ⎝⎠k4i ⎝⎠π The measured intensity is proportional to the product of quantities that depend … • on the method of measurement, e.g. the choice of Ei, • on the strength of the interaction between neutrons and the sample (i.e. on the “scattering cross section”), and • on the sample itself, through the scattering function S(Q,E). NCNR Summer School 2011 13 Cross sections Consider a “thin” sample placed in a neutron beam. The neutrons are transmitted, absorbed, or scattered, with probabilities pT, pA and pS respectively. NCNR Summer School 2011 14 The scattering probability pS N atoms; area of sample = A NNtσ σ pt= SS==Σ SSAV where σS is the microscopic scattering cross section (barn/atom) (1 barn = 10-24cm2) t is the sample thickness, V is its volume, ρ = N/V is the number density, -1 ΣS=ρσS is the macroscopic scattering cross section (cm ) NCNR Summer School 2011 15 Event rates The sample is placed in a beam whose current density (or “flux”) is Φ (n/cm2/s). The current, i.e. the number of neutrons hitting the sample, is IA 0 =Φ n/s. The scattering rate is: IIpS0S==ΦΣ=ΦΣ=Φσ( A)( S t) V S N S The absorption rate is: IIpAA0A==ΦΣ=ΦΣ=Φσ( )( A t) V A N A Hence the transmission rate is IIpIIIT0T0AS==−−=Φ−Σ( A1t)( T) where Σ=Σ+ΣTAS is the total removal cross section. NCNR Summer School 2011 16 “Not necessarily thin” samples The scattering, absorption and transmission probabilities for a sample that is “not necessarily thin” are as follows: −ΣTt Transmission: peT = (Beer’s law) Absorption −ΣTt p1eAAT= ( −ΣΣ)( ) Scattering (N.B. Scattering may be p1e=−−ΣTt ΣΣ followed by absorption, or by SST( )( ) additional scattering) NCNR Summer School 2011 Absorption cross sections 1 barn (b) = 10-24cm2 N.B. log scale NCNR Summer School 2011 18 Absorption ¾ As compared with x-ray absorption cross sections, neutron absorption cross sections are generally small. ¾ Exceptions include 3He, 6Li, 10B, 113Cd, 135Xe*, 157Gd. *(2.6 × 106 barns) Rhodes, Richard, 1986, “The Making of the Atomic Bomb”, Simon and Schuster, ISBN 0-671-44133-7. ¾ For most elements and isotopes the “1/v” law applies: σ∝abs 1v ∝λ ¾ Important exceptions to the “1/v” law include Cd and Gd. NCNR Summer School 2011 19 Scattering cross sections linear 1 barn (b) = 10-24cm2 scale H 82b Yb Hg Cl Ni Nd N Scattering cross section (b/atom) Atomic number Z NCNR Summer School 2011 20 Comparison with x-rays As compared with x-ray (Q=0) cross sections, which vary as Z2: HDCN OAlSi Fe neutron scattering cross sections show little systematic variation with atomic number: HDCN OAlSi Fe X-ray cross sections vary with Q; neutron cross sections do not. NCNR Summer School 2011 21 Cross section examples 1 mm of aluminum has 99% transmission 0.020” of cadmium has 0.3% transmission 1 m of dry air scatters 4.8%, absorbs 0.7% 0.1 mm of water scatters 5.5% (These results were obtained using thermal neutron (2200 m/s, 1.8 Å) absorption cross sections.) NCNR Summer School 2011 The single differential cross section For a “thin” sample, the total When there is one type of atom we integrated scattering is: obtain, in the static approximation, IE=φ N σ E. Si( ) Si( ) dσ σ ()E,2θ= S S() Q The measured intensity in a d4Ω i π diffraction experiment (on a “thin” sample) is related to (Q= 4πθλ sin / ) the single differential ONLY DEPENDS scattering cross section: ON THE SAMPLE ⎛⎞dσ Thus the single differential cross section is IE,2Si()θ=φ N⎜⎟ ΔΩ proportional to the “structure factor” S(Q). ⎝⎠dΩ The measured intensity is solid angle proportional to S(Q). NCNR Summer School 2011 Correlation functions – S(Q) G Neutron diffractometers measure S(Q). G G S(Q) is the Fourier transform of the pair distribution function gr:( ) GGG GG S(Q)=+ρ 1∫[ g(r) − 1] exp( iQ.r) dr GG1 G GG g(r)=+ 1⎡⎤ S(Q) − 1 exp − iQ.r dQ 3 ∫ ⎣⎦() ρπ()2 Averaging over directions within the sample we obtain: 4πρ S(Q)=+ 1 r[] g(r) − 1 sin Qr dr Q ∫ 1 sin() Qr g(r)=+ 1 Q2 [] S(Q) − 1 dQ 2Qrπρ2 ∫ Pair distribution functions contain detailed information about structure. NCNR Summer School 2011 The double differential cross section The double differential cross section is The measured intensity in a related to the “scattering function” spectroscopy experiment is related to the double SQ,E( ) differential scattering cross When there is one type of atom, we section: obtain 2 dkσ σ f ()E,2,Eifθ= S() Q,E, ddEΩπfiG 4= k IE,2,ESi()θ= f G kf ⎛⎞d2σ E ONLY DEPENDS ki 2θ f φN⎜⎟ΔΩΔ Ef . ON THE SAMPLE ⎝⎠ddEΩ f Ei The measured intensity is proportional to energy window (kkSQ,E.fi) ( ) NCNR Summer School 2011 Correlation functions – S(Q,ω) Most neutron spectrometers measure S(Q,ω). The quantity I(Q,t) is known as the “intermediate scattering function”. GG I(Q,t)= =∫ S(Q,ωωω )exp(i t)d GG1 S(Q,ω= ) I(Q,t)exp( −ω i t)dt 2π= ∫ Neutron spin echo measures I(Q,t) directly. The quantity G(r,t) is the “time-dependent pair correlation function”: GG1 G GG G(r,t)=− I(Q,t)exp iQ.r dQ 3 ∫ ( ) ()2π GGGGG I(Q,t)= ∫ G(r,t)exp() iQ.r dr The functions I and G contain detailed information about the collective (pair) dynamics of materials.