Materials for energy, heath, environment

Transcription

1 WHAT DO WE STUDY WITH NEUTRONS? Materials for energy, heath, environment archaeological artefacts, commercial products phase transitions magnetic orderings magnetic fluctuations exchanges bias soft matter polymers Page 1

2 WHY NEUTRONS? Why do we use neutrons? - Neutrons tell us about the positions and motions of atoms/magnetic moments in condensed matter - Neutrons interact with nuclei and magnetic moments the two interactions have similar strengths -Interaction with matter is gentle and simple: scattering data are easy to interpret - Neutrons are penetrating: bulk materials can be studied any sample can be contained in special environment - Experimental science: instrument design, data taking and data analysis Page 2

3 WHAT DO WE MEASURE? Scattering experiments: neutrons in and neutrons out! We measure the number of neutrons scattered by a sample against the number of incident neutrons (neutron flux) as a function of the change in direction and energy of the scattered neutrons as a function of polarisation or polarising magnetic fields Scattered intensities involve positions and motions of scattering centres: atoms/magnetic moments Scattered intensities are proportional to Fourier transforms (in space and time) pair correlation functions Page 3

4 HOW DO WE MEASURE? neutron diffraction source of neutron beams neutron spectroscopy Inelastic scattering source of neutron beams Page 4

5 OVERVIEW a bit history neutron properties interactions between neutrons and matter measured quantities scattering by atoms/nuclei scattering by magnetic moments key messages Page 5

6 NEUTRONS: INTRODUCTION A bit of history: W. Bothe & H. Decker discovered very penetrating radiation emitted when α particles hit light elements I. Curie & F. Juliot observed creation of p + in paraffin sheets & thought new radiation was γ-rays J. Chadwick a few months later discovers the neutron, a neutral but massive particle Nobel Prize in Physics 4 2He B 14 7 N n 4 2He Be 16 6 C n ( m He + m B )c 2 + T He = ( m N + m n )c 2 + T N + T n m n = ± a.m.u Page 6

7 NEUTRONS: INTRODUCTION A bit of history: E. Fermi showed that neutrons moderated by paraffin could be captured by various elements, producing artificial radioactive nuclei importance of neutron energy range D.P. Mitchell & N. Powers / H. v. Halban & P. Preiswerk showed that thermal neutrons can be diffracted by crystalline matter MgO crystals oriented (200) planes 22 corresponds to Bragg angle for peak of wavelength distribution of thermal neutrons ~0.16nm Page 7

8 NEUTRONS: INTRODUCTION A bit of history: O. Hahn, F. Strassmann & L. Meitner discovered the fission of 235 U nuclei through thermal neutron capture H. v. Halban, F. Joliot & L. Kowarski showed that 235 U nuclei fission produced 2.4 n 0 on average chain reaction E. Fermi & al first self-sustained chain reaction reactor CP1 Chicago C.G. Shull Proof of antiferromagnetic order in MnO C.G. Shull & B.N. Brockhouse Nobel Prize in Physics Page 8

9 Page 9 NEUTRONS: NEUTRON PROPERTIES free neutrons are unstable: β-decay proton, electron, anti-neutrino life time: 886 ± 1 sec wave-particle duality: neutrons have particle-like and wave-like properties mass: m n = x kg = u. (unified atomic mass unit) charge = 0 spin =1/2 magnetic dipole moment: μ n = μ N velocity (v) kinetic energy (E) temperature (T) wavevector (k) wavelength (λ) E = m n v 2 /2 = k B T = ( hk /2π) 2 /2m k = 2π / λ = m n n v /( h/2π) ( ) = 0.286/ ( E) 1/2 E in ev ( ) = k 2 (k in nm -1 ) λ (nm) = 395.6/ v m/s E mev T = L v = µsec λ Ao L m ( ) 1(A 0 ) 82 mev 124THz 950 K Example λ = 4 Å v=1000m/s E= 5 mev [ ] fortunately large value! monochromatisation: diffraction or time of flight

10 NEUTRONS: NEUTRON PROPERTIES Conversion chart P. A. Egelstaff ed. - Thermal Neutron Scattering Academic Press 1965 Page 10

11 NEUTRONS: NEUTRON PROPERTIES Neutron energy ranges Page 11

12 ULTRA-COLD NEUTRONS the very cold side v 20m/s E kin 2 µev T= K λ = 200 Å effect of gravity - neutrons are massive! mirror potential well for ultra-cold neutrons neutrons are stacked at distinct height levels (in the micrometer range!) note: cold neutron beams are bent by gravity ~ 1.2 cm at 100 m for 20 Å neutrons neutrons : objects to study fundamental interactions neutron β-decay free neutrons are not forever Page 12

13 KEY MESSSAGES neutrons are not elementary particles they are not for ever neutrons are not only powerful probe, they can be studied as objects Page 13

14 NEUTRON SCATTERING: INTERACTIONS Neutron scattering exploits cool neutrons: 0.05 mev< E n < 500 mev 0.4 Å < λ < 40 Å Neutrons X-rays Electrons Page 14

15 NEUTRON SCATTERING: INTERACTIONS Neutrons interact with nuclei: capture: absorption, emission of particles diffusion arising from very short range nuclear forces neutron wavelength much longer than nucleus size can t solve neutron structure Neutrons interact with electrons: magnetic interactions dipole-dipole interactions lead to scattering Strength of interactions? Measured through scattering cross sections In the case of neutrons, nuclear and magnetic scattering are equivalent Neutrons have no charge, but they interact (extremely weakly) with charges/electrical fields spin-orbit/schwinger scattering Page 15 see non-resonant magnetic X-ray scattering

16 NEUTRON SCATTERING: WHAT DO WE MEASURE? Scattering is measured in terms of cross-sections neutron counter set up (θ,φ) with area ds at a distance r from scattering system large compared compared to sample dimensions Φ = number of incident neutrons per area per second dσ dω d 2 σ dω de Page 16 σ = number of neutrons scattered per second / Φ an area (barns) number of neutrons scattered per second into dω = Φ dω number of neutrons scattered per second into dω & de = Φ dω de applies to all types of scattering events we have ignored the initial and final neutron spin states to be seen later..

17 HOW NEUTRONS INTERACT WITH MATTER NUCLEAR SCATTERING nuclear scattering from a single (fixed) nucleus range of nuclear forces << neutron wavelength point-like scattering s-wave scattering elastic scattering (fixed nuclei and no change in nucleus) same velocity before and after scattering no absorption (far away from resonance) incident plane wave h incident flux = neutron density x velocity = v = k neutron density = e i k i.x m n spherical scattered wave b : scattering length of the order of nucleus s size 2 number of neutrons per second into dω cross sections b expressed in fm m Page 17 scattering amplitude does not vary with scattering angle

18 HOW NEUTRONS INTERACT WITH MATTER NUCLEAR SCATTERING nuclear scattering from an assembly of nuclei: atom at incident wave: scattered wave at : cross section dσ dω = vdsψ scatt vdω wavevector transfer orders of magnitude: e i k i.r e i k i e ik i R f. r - R i i R R i i b i r -R i e i k Ψ scatt = e ik i R f.( r - R i ) i b i i r -R i nuclear scattering lengths, b s, depend on isotope, nuclear eigenstate, and nuclear spin orientation relative to neutron spin 2 = ds dω ei k f.r j b j e i ( k i k f ).R j r -R j i,j Κ is defined by Κ = k dσ i k f dω = beware! X-ray boys use different sign convention! 2 = b i * b j ( ) e -i ( k i k f ). R i R j ( ) b * i b i,j j e -iκ. R i R j Fourier transform ( ) Page 18

19 NUCLEAR SCATTERING COHERENT/INCOHERENT coherent and incoherent scattering consider an assembly of similar atoms/ions spins/isotopes are uncorrelated at different sites dσ dω = i,j averaged over all states ( ) b * i b j e -iκ. R i R j for a single nucleus b i = b + δb where δb i taken as real number i = 0 b i b j = b 2 + b ( δb i +δb j )+ δb i δb j with δb i δb j = 0 unless i=j = b i - b 2 = b 2 b 2 dσ dω = b 2 ( ) + b 2 b 2 e -iκ Ri Rj i,j δb i 2 ( )N σ coh = 4π b 2 σ incoh = 4π b 2 b 2 coherent scattering: correlations between different sites incoherent scattering: correlations on the same site (at different times) particular to neutron scattering ( ) sources of incoherent scattering: isotopic distribution and nuclear spin Page 19

20 NUCLEAR SCATTERING COHERENT/INCOHERENT If single isotope and zero nuclear spin, no incoherent scattering If single isotope and non-zero nuclear spin I nucleus+neutron spin: I+1/2 and I-1/2 scattering length b + and b - If neutrons and nuclei are un-polarised: probability plus [ ] b = 1 2I+1 ( I+1)b+ +Ib To reduce incoherent scattering (background): use isotope substitution use zero nuclear spin isotopes polarise nuclei and neutrons f + = I+1 probability minus 2I+1 f + = b 2 b 2 = I I+1 2I+1 ( ) b + b ( ) 2 ( ) 2 I 2I+1 Page 20

21 NUCLEAR SCATTERING COHERENT/INCOHERENT examples of scattering lengths: neighbouring elements can be easily identified Page 21

22 NUCLEAR SCATTERING SCATTERING LENGTHS most neutron scattering lengths are positive (same for X-rays) phase changes by after scattering positive b negative b e.g. hydrogen no change in phase at scattering point Page 22

23 NUCLEAR SCATTERING CONTRAST VARIATION Mixing 1 H and 2 H (H and D) allows contrast variation full contrast : external shapes intermediate contrast : details 100 % H 2 O 100 % D 2 O useful for imaging and scattering (small angle neutron scattering, liquids) Page 23

24 SCATTERING LENGTHS Orders of magnitude - numbers for neutron scattering cross sections are in ~ barns 1 barn = cm 2 area per atom ~ 10 Å 2 = barns 1 atom gives 10-9 probability scattering when beam hits it! to obtain 1% scattering (over 4π) requires 10 7 layers of atoms ~ 0.1 cm of sample! Take a single atom of C: b = 6.65 pm = m = cm σ = 5.56 barns The probability to observe a single scattered neutron per second requires a huge flux: I 0 =Φ σ Φ =I 0 /σ =1/ ( ) particules /cm 2 /sec actual neutron flux 10 7 n/cm 2 /sec, we must either wait for sec 570 million years or use ~ atoms ~ 0.4 µg only Page 24

25 Page 25 SCATTERING LENGTHS Numbers for neutron scattering typical neutron flux ~10 7 n/cm 2 /sec sample volumes in the fraction of cm 3 range counting time for incoherent scattering from Vanadium (σ ~ 5 barns) Questions about statistics: sample volume 1x1x0.1 cm 3 i.e. ~ atoms count rate ~ n/sec over 4π detector angular aperture ~ 1% leads to ~ n/sec experimental data are counts in the detector, independent events but with a fixed probability (scattering cross sections!): Poisson s like usual goal is to achieve 1% error per information unit: requires ~10,000 counts per bin dσ i.e. ~ minutes for typical elastic peak ( ) dω d 2 σ i.e. at least 10 times longer for inelastic studies ( ) dω de

26 SCATTERING LENGTHS - CONSEQUENCES Absorption essentially neutron capture random variation with atomic number and isotopes Applications for detection neutrons are captured by nuclei capture creates charged particles recoiling particles ionise gaseous materials 3 2He n 3 1 H +p MeV 10 5B n 7 3 Li He MeV Page 26

27 KEY MESSAGES neutrons interact with nuclei random variation of b s with atomic number isotropic scattering amplitude contrast and isotopic substitution low absorption coherent and incoherent scattering Page 27

28 SCATTERING LENGTHS - CONSEQUENCES Imaging with neutrons selectivity of neutrons selective imaging through absorption direct transmission through macroscopic a piece of rock from the Antartic the branch of an Arauc tree Page 28

29 SCATTERING LENGTHS - CONSEQUENCES Refractive index for neutrons For a single nucleus, Fermi pseudo-potential Inside matter, V = 2πh2 V( r) = 2πh2 m r b δ( r) scattering length density m ρ ρ = 1 volume i b i neutrons obey Schrödinger s equation 2 + 2m h 2 ( E V ) Ψ r ( ) = 0 In vacuo, V = 0 and E = E cin k i 2 = 2mE/h 2 In the medium, k 2 f = 2m( E - V )/h 2 = k 2 i 4πρ n = k f /k i 1 λ 2 ρ/2π With b>0, n<1 and neutrons are externally reflected by most materials Page 29

30 SCATTERING LENGTHS - CONSEQUENCES Applications neutron guides: critical angle γ Ni (Ni 58 c λ ρ/π ) γ c 0.1Þ 1 Neutron bottles: a bottle imposes n = o, k 2 i = 4πρ or λ = π ρ example SiO 2 density: 2.66 molecular weight: N =10 24 ( 2.66/60.08)N Avogadro = ρ = N( b Si + 2b O )= This works with very cold neutrons: λ > 864 Å v~ 4.6 m/s E~ 0.1µeV ~ 1mK!!! Page 30

31 HOW TO ACCUMULATE INTENSITIES? So far, we have added individual scattering intensities. How to combine them? neutron sources are chaotic: emission over 4π, at ill-defined times with wide distribution of energies, neutrons are moderated Do we have to take the neutron source into account? interference pattern from sample to detector L source x c x c x s interference is destroyed if >λ/2 max = λ/2π =1/k x i c / max = L/x s x c = ( λ/2π)l/x s = ( 1/k i )L/x s x c typical values: x c ~ y c ~10 nm (X-ray tomography with pin-hole source x c ~500µm) z c = λ 2 / λ 50 λ 10 nm Page 31

32 HOW TO ACCUMULATE INTENSITIES? particular case: small angle neutron scattering what is the largest object that can be measured? ILL D11 Page 32 size given by the lateral coherence length L~40 m, λ = 12Å, x s ~10-2 m (adjustable) x c ~ 1µm In general, ignore coherence of neutron sources and focus on constructive interference of scattered waves x c = ( λ/2π)l/x s Typically, objects smaller than 1µm are studied by scattering methods objects larger than 1µm are imaged

33 HOW TO ACCUMULATE INTENSITIES? Assume plane waves for incident neutrons far away from source interference pattern in front of detector interference pattern in front of detector spherical waves emitted by scattering centres plane waves in scattering system source How to combine scattered waves (amplitudes and phases)? Page 33

34 HOW TO ACCUMULATE INTENSITIES? Different ways to combine scattered waves Born approximation kinematic theory neutron wavefunction un-perturbed inside sample in general OK, away from Bragg reflections and total reflection Dynamical theory of scattering takes into account of the change in the neutron wave in the system see refractive index but generalised for all scattering vectors important near total reflection may be needed near Bragg reflections from perfect crystals with highly collimated beams In most cases, kinematic theory applies for neutrons Page 34

35 HOW TO ACCUMULATE INTENSITIES? Born approximation, i.e. first order perturbation method The incident wave travel unperturbed in scattering system scattering centres with potential V(r ) From the definition, dσ dω = k f in dω W k i,λ i k f,λ f Φ dω neutrons scattering system Use Fermi s Golden rule to calculate transition probabilities W k i,λ i k f,λ f k f in dω W k i,λ i k f,λ f = 2π k f h ρ 2 k f k f λ f V k i λ i in dω where is the density of k-states in dω per unit energy range ρ k f Page 35

36 HOW TO ACCUMULATE INTENSITIES? some algebra and manipulations as incident wavelength - plane wave in normalisation box neutron states are periodic in k-space, unit cell volume neutron flux Φ = h k m n ( ) 3 v k = 2π ρ k f de f = 1 v k k f 2 dk f dω is the number of n-states in dω between E f and E f +de f kinetic energy: de f = h2 k m f dk f n therefore: scattering cross section with k i, λ i and λ f fixed ρ k f = dσ dω 1 ( 2π) k 3 f λ i λ f = k f k i m n h 2 dω m n 2πh 2 2 k f λ f V k i λ i 2 energy of neutrons+ scattering system must be conserved E i +E λ i =E f +E λ fi d 2 σ = k 2 f m n 2 k de f dω k λ i λ f i 2πh 2 f λ f V k i λ i δ ( Ei E f +E λ i E λ f ) partial differential cross section for all scattering potentials Page 36

37 DIFFERENTIAL SCATTERING CROSS SECTION NUCLEAR SCATTERING more algebra and manipulations insert a potential V Fermi pseudo-potential - short range, scalar and central V has the form V = V j ( r R j )= V j ( x j ) with x j = r R j k f λ f V k i λ i j j * = χ λ f exp( ik f r)v ( x j )χ λ i exp( ik i r)dr 1 dr 2 dr 3...dR j...dr k f λ f V k i λ i = V j j j ( Κ) λ f exp iκ R j ( )λ i where V j * λ f exp( iκ R j )λ i = χ f Κ = k i k f ( ) ( Κ) = V j ( x j )exp iκ x j dx j exp( iκ R j )χ * f dr 1 dr 2 dr 3...dR j.. Fermi pseudo-potential V j ( x j )= 2πh2 m b δ x j ( j) V j ( Κ) = 2πh2 m b j d 2 σ de f dω λ i λ f = k f b k j λ f exp iκ R j i ( )λ i j 2 δ( E i E f +E λ i E λ f ) Page 37

38 DIFFERENTIAL SCATTERING CROSS SECTION NUCLEAR SCATTERING even more algebra and manipulations introduce time and energy to reach thermodynamics δ( E i E f +E λ i E λ f )= 1 + 2πh exp { i ( E λ f E λ i )t /h}exp -iωt - d 2 σ de f dω λ i λ f ( ) dt hω =E i E f = k f b k j b j' λ i exp -iκ R j ' i ( )λ f λ f exp( iκ R j )λ i 1 + 2πh exp i ( E λ f E λ i )t /h jj ' - d 2 σ de f dω λ i λ f = k f k i 1 2πh jj ' b j b j ' { } + λ i exp( -iκ R j' )λ f λ f exp( iht /h)exp( iκ R j )exp(-iht /h)λ i exp( -iωt) dt - where H is the Hamiltonian of the scattering system Introduce time-dependent operators R j ( ) dt exp -iωt ( t) = exp( iht /h)exp( iκ R j )exp(-iht /h) Page 38

39 DIFFERENTIAL SCATTERING CROSS SECTION NUCLEAR SCATTERING To get measured cross section, sum over all final λ f (with fixed λ i) and average over all λ i. δ( E i E f +E λ i E λ f )= 1 + 2πh exp { i ( E λ f E λ i )t /h}exp -iωt - Introduce partition function Z = d 2 σ d 2 σ = p de f dω λ i de f dω λ i λ f λ λ i λ f ( ) exp E λ /k B T ( ) dt, probability = k + f 1 b k i 2πh j b j' exp{ -iκ R j' (0)}exp iκ R j (t) jj ' - A compact form, but not easy to calculate: measure of pair correlation functions { } exp -iωt hω =E i E f p λ = 1 Z exp ( E λ /k B T) ( ) dt information on scattering system contained in time-dependent ops. and wavefunctions Page 39

40 DIFFERENTIAL SCATTERING CROSS SECTION NUCLEAR SCATTERING Consider a simple system with a single element but different b s d 2 σ = k + f 1 b dω de f k i 2πh jõ b j j', j exp( -iωt) dt b = f i b i b 2 2 = f i b i jj ' - i i no correlation between b s on different sites b jõ b j = b 2, j' j b jõ b j = b 2, j'= j Page 40 d 2 σ dω de f d 2 σ dω de f coh incoh = σ coh 4π = σ incoh 4π k f 1 k i 2πh jj ' + - exp{ -iκ R j ' (0)}exp{ iκ R j (t)} exp( -iωt) dt Coherent scattering: correlation between the position of the same nucleus at different times and correlation between the positions of different nuclei at different times interference effects k f 1 k i 2πh j + - exp{ -iκ R j (0)}exp{ iκ R j (t)} exp( -iωt) dt Incoherent scattering: only correlation between the position of the same nucleus at different times no interference effects

41 KEY MESSAGE neutron scattered intensities are proportional to space and time Fourier transforms of site correlation functions Page 41

42 KEY MESSAGE neutrons cover a wide range of length scales imaging/scattering BES DOE Page Octobre ans après le prix Nobel de Louis Néel C. Vettier 42

43 CRYSTALLINE MATERIALS Examples of objects on a lattice - crystalline/ordered materials Real space 1-d system: convolution of objects and lattice of Dirac functions N objects, N large Reciprocal space 1-d system: Fourier transforms For N large Similarly we define associated reciprocal spaces that reflect the symmetry and periodicities of real space lattices Page 43

44 NUCLEAR SCATTERING FROM CRYSTALLINE MATERIALS Crystalline materials: all atoms (nuclei) have an equilibrium position and they move about it atom in cell j: R j ( t) = j + u j ( t) { } can be generalised to non-bravais lattices exp{ -iκ R j' (0)}exp iκ R j (t) = N exp( iκ j) exp{ -iκ u 0 (0)}exp iκ u j (t) jj ' exp{ -iκ R j (0)}exp{ iκ R j (t)} = N exp{ -iκ u 0 (0)}exp iκ u 0 (t) j j { } { } displacements u(t) can be expressed in terms of normal modes or phonons q q q Page 44

45 NUCLEAR SCATTERING FROM CRYSTALLINE MATERIALS Coherent part exp U exp V exp{ -iκ R j' (0)}exp{ iκ R j (t)} = N exp( iκ j) exp{ -iκ u 0 (0)}exp iκ u j (t) jj ' it can be shown (Squires) expu expv = exp U 2 exp UV j { } d 2 σ dω de f d 2 σ dω de f coh coh = σ coh 4π = σ coh 4π k f k i k f k i N 2πh exp U2 exp j ( ) exp UV exp( -iωt) dt iκ j Debye-Waller factor 2W = U 2 = { K u} 2 N 2πh exp 2W + - ( ) exp( iκ j) 1+ UV + 1 2! UV j + - exp -iωt ( ) dt zero-th order: coherent elastic scattering - Bragg scattering 1 st order: coherent one-phonon scattering Page 45

46 NUCLEAR SCATTERING FROM CRYSTALLINE MATERIALS Coherent elastic scattering d 2 σ dω de f + coh = σ coh 4π k f k i N 2πh exp -2W diffraction ( ) exp( iκ j) exp( -iωt) dt exp( -iωt) dt = 2πh δ hω purely elastic scattering - d 2 σ = σ coh N exp( -2W) exp( iκ j)δ hω dω de f 4π coh el j dσ d 2 σ σ = de dω dω de f = coh N exp( -2W) exp( iκ j) f 4π dσ dω coh el coh el 0 ( )3 coh el = σ coh 4π N 2π exp -2W v 0 j ( ) k i = k f ( ) δ Κ τ τ + - ( ) ( ) j N: number of unit cells v 0 : volume of the unit cell non-bravais lattice with different atoms d in the unit cell dσ = N ( 2π)3 δ( Κ τ) F dω v N ( Κ) 2 F N ( Κ) = b d coh el 0 τ d structure factor exp( iκ d)exp( W d ) Page 46

47 NUCLEAR SCATTERING FROM CRYSTALLINE MATERIALS Bragg s law λ = 2d sinθ Practical application monochromators! Collecting intensities at Bragg peaks gives access to squared values of Fourier components of the structure Collect as many as possible to overcome the phase problem Page 47

48 NUCLEAR SCATTERING FROM CRYSTALLINE MATERIALS Coherent elastic scattering (diffraction) provides: periodicity in space, lattice symmetry and lattice constants positions of atoms in cells from powder and single crystals methods But requires inversion of intensities into phases/amplitudes F N ( Κ) 2 Self propagating synthesis High temperatures Phase transitions visible Page 48

49 CRYSTALLINE MATERIALS : DIFFRACTION INSTRUMENTS How does it work? monochromatisation crystal diffraction or time of flight neutron source Detector 2D detector Detector 1D Page 49

50 CRYSTALLINE MATERIALS : BEAM FILTERS Maximum wavelength for which no Bragg scattering can occur λ max = 2d max Bragg cut-off ( sinθ) max = 2d max d max is the maximum plane spacing Beyond λ max σ tot drops σ tot = σ scatt + σ absor Cut-off wavelengths for polycrystalline materials Transmission of pyrolytic graphite Be: 3.9 Å graphite: 6.7 Å Page 50

51 KEY MESSAGES neutron diffraction is an essential tool for structures determination it complements other diffraction methods neutrons probe bulk samples Page 51

52 NUCLEAR SCATTERING FROM CRYSTALLINE MATERIALS Inelastic coherent scattering (Bravais lattice) d 2 σ dω de f coh = σ coh 4π k f k i N 2πh ( ) exp( iκ j) 1+ UV + 1 2! UV exp 2W j + - exp -iωt ( ) dt Page 52 one-phonon coherent scattering (Bravais lattice) <UV> involves creation and annihilation of phonon modes UV = h 2MN s ( Κ e s ) exp{ -i( q j ω s t) }n s +1 +exp{ i( q j ω s t) }n s ω s d 2 σ = σ coh k f dω de f 4π k coh ±1 i δ( ω mω s )δ Κ mq τ ( ) conservation laws! Κ = τ ± q E i E f = hω = ± hω s [ ] ( 2π) 3 1 exp 2W v 0 2M ( ) s k f k i > k f τ energy loss k i ( Κ e s) n ω s +1/2 ±1/2 s Κ k i < k f k f energy gain k i Κ

53 NUCLEAR INELASTIC SCATTERING q Collective excitations phonons (magnons) q q How to measure? Go to a given Κ = τ ±q and energy transfer hω Orient sample: sample frame in coincidence with instrument frame Page 53

54 PHONON MODES classical instruments: triple-axis machines scan point by point either constant K or constant energy constant K applications: lattice interactions, soft mode phase transitions superconductivity, Page 54

55 SPECTROSCOPY Spectroscopy internal modes little dispersion of modes Excitations in hypothetical molecular crystal J.Eckert SpectroChim. Acta 48A, 271 (1992) Use incoherent scattering d 2 σ dω de f incoh ±1 = k f k i δ( ω mω s ) n s +1/2 ±1/2 s 2ω s r ( σ incoh ) r 4π 1 M r K e r 2 exp( 2W r ) Page 55

56 SPECTROSCOPY more global picture: time of flight and large detector coverage Page 56

57 TIME DOMAIN Time domain is reached through inelastic scattering the range of energy transfer that can be covered determines the accessible time domain (Fourier transform!) Depends on incident neutron energy and instrumental resolution Typical resolution in wavelength (and energy) are ~ a few percents except special techniques (backscattering, NSE) Page 57

58 TIME DOMAIN AND COMPLEMENTARITY complementarity with other methods Page 58

59 KEY MESSAGES accessible time and space domains cover a wide range of applications no single probe can cover the whole (K,ω) space that would allow an ideal Fourier transformation use complementary probes Page 59

60 DISORDERED MATERIALS liquids, glasses,..no equilibrium positions for atoms local order but no long range order dσ consider mono-atomic system dω = b * i b i,j j e -iκ. ( R i R j) for most liquids, glasses,... scattering depends on magnitude of averaging over polar angles exp -iκ ( R i R j ) ( ) { }= sin Kr ij replace sum over atoms by radial distribution function Kr ij g(r) for monoatomic liquid Page 60

61 DISORDERED MATERIALS - LIQUIDS mono-atomic liquid N dσ dω = N b2 + b i b j e -iκ.( Ri R j) = N b 2 +4πρ b 2 r 2 g r dσ( K 0) dω i j S(K): structure factor note that = N b 2 S( K) with = S( K) =1+ 4πρ K very different from F N (K) in crystalline materials relates to intensity (not amplitude) not well structured, only a few peaks ( ) ( ) 0 ( ) ( ) ( ) sin Kr Kr dr sin Kr r 2 dr = 0 unless K = 0 Kr 0 0 r [ g( r) 1] sin Kr ( ) dr Liquid Argon T= 85K Experimental data Inverted ρ(r) (Fourier transform) J.L. Yarnell et al. PRA 7, 2130 (1973) Page 61

62 DISORDERED MATERIALS - LIQUIDS In n-component systems, there are n(n+1)/2 site-site radial distributions To be measured, using isotopic substitution. In liquids, strictly speaking there is no elastic neutron scattering: nuclei recoil under neutron impact need to span all (K,ω) space or apply inelasticity corrections Placzek PRB 86, 377 (1956) Typical instruments ILL D4 Sandals ISIS Page 62

63 MAGNETISM! Neutrons have a spin ½ and therefore carry a magnetic moment neutron beams can be polarised µ n = γ µ N σ where µ N = eh γ = m p For comparison, electrons have µ e = 2µ B s where µ B = eh 2m e Neutron magnetic moments feel magnetic fields created in materials: - electrons: dipole moments and currents - nuclei: dipole moments (neglected here) The potential V in the scattering cross section should include these effects d 2 σ de f dω λ i λ f = k f k i m n 2πh k f λ f V k i λ i δ ( Ei E f +E λ i E λ f ) Page 63

64 MAGNETISM! Magnetic field created at distance R electron with momentum p Potential of a neutron in B 4π curl µ e R 2µ B R 2 h 4π γ µ 2 µ s R N B curl + 1 R 2 h Β = Β S + Β L = µ 0 V m = µ n B = µ 0 p R R 2 p R R 2 During scattering, neutron changes from state k i,σ i to k f,σ f d 2 σ de f dω σ i λ i σ f λ f = k f k i m n 2πh k f σ f λ f V m k i σ i λ i δ ( Ei E f +E λi E λf ) Complex evaluation: magnetic interaction is long range magnetic forces are not central Page 64

65 MAGNETISM! After long calculations d 2 σ = k f γ r de f dω k 0 i Q = The electrons x Q Q ( Κ) = ˆ σ i λ i σ f λ f ( ) exp iκ r x ( ) ( ) 2 σ f λ f σ Q σ i λ i 2 δ Ei E f +E λi E λ f ˆ Κ ( s x ˆ Κ )+ i hk p x ˆ ( Κ ) whereas for nuclear scattering ( ) where Q Κ K Q( Κ) K ˆ ( )= 1 M Κ 2µ B M(K) is the Fourier transform of M(r) ( ) ˆ K is a unit vector j ( ) b j exp iκ R j operator is related to the magnetisation of the scattering system separating spin and orbital contributions r 0 : classical radius of electron cm strength of scattering ~ (γr 0 ) 2 of the order of 0.3 barn Q is the vector projection of Q on to the plane perpendicular to K Page 65

66 KEY MESSAGES nuclear and magnetic interactions have similar strengths interactions with electrons connect to magnetisation densities magnetic scattered intensities are proportional to space and time Fourier transforms of site correlation functions for magnetic moments Page 66

67 MAGNETISM! Similarly to nuclear scattering, magnetic neutron scattering probes correlations ( ) 2 d 2 σ = k γr f 0 de f dω k i 2πh αβ geometrical factor δ αβ K ˆ ˆ ( K α β ) Q α ( Κ,0)Q β Κ,t Elastic scattering thermal average at infinite time M( Κ) or dσ dω el dσ dω el = γr 0 2µ B = γr 0 2 ( ) exp( iωt)dt Q β ( Κ,t) = exp( iht /h)q β ( Κ)exp( iht /h) ( ) 2 δ αβ ˆ equivalent to nuclear scattering K ˆ ( α K β ) Q α Κ αβ ( )Q β Κ ( ) Κ ˆ { M( Κ) ˆ Κ } 2 with Q( Κ)= 1 M Κ 2µ B contains all information on magnetic arrangements symmetry, periodicity, moments,.. ( ) Page 67

68 MAGNETIC DIFFRACTION Ferromagnets localised magnetic system has the same periodicity as lattice dσ dω el ( ) 2 N 2π = γr 0 ( )3 v 0 S η 2 1 τ 2 gf( τ) 2 ( ) 1 ˆ exp 2W τ η 2 { ( ˆ ) Aver }δ Κ τ ( ) - magnetic intensity on top of nuclear intensity - magnetic form factor not constant as b spatial distribution of magnetic electrons - Measurements of intensities give F(τ) which allow M(r) to be calculated Non-ferromagnets new periodicity in space leads to new Bragg peaks Page 68

69 MAGNETIC DIFFRACTION Non-ferromagnets neutrons allow to probe local magnetic order C. Shull et al powder samples or single crystals easy and routine experiments! One of the very strong points for neutrons More complex materials Important for new devices Page 69

70 KEY MESSAGE neutron diffraction (powder) is the the method of choice to determine magnetic structures (if not the only one ) Page 70

71 INELASTIC MAGNETIC SCATTERING OF NEUTRONS Inelastic magnetic neutron scattering probes magnetic correlations simple localised magnetic excitations (crystal field levels) spin waves fluctuations (spin density fluctuations) interactions between lattice and magnetism Page 71

72 INELASTIC MAGNETIC SCATTERING OF NEUTRONS very powerful experimental method investigating pairing mechanisms in superconductors phonon-like or magnetic? Sr 3 Ru 2 O 7 field-induced QCP phase diagram: low T and high H 8T 6T T=40 mk E=0.8 mev Page 72

73 NEUTRON POLARISATION scattering cross section involves neutron spin states another neutron degree of freedom use of polarised neutrons use of polarisation use of neutron spin precession in fields Larmor precession spin manipulation spin filters NSE methods Page 73

74 NEUTRON POLARISATION polarised neutron beams, up (u) and down (v) states P = n + n n + + n previous cross-sections gives rise to 4 cross-sections u u v v u v v u coherent nuclear scattering incoherent nuclear scattering u u v v u v v u b = ( I +1)b+ + I b 2I +1 b = 0 isotopes u u v v b 2 b 2 = ( I +1)b + + I b 2I +1 2 isotopes ( I +1)b + + I b 2I +1 2 isotopes b + b 2I +1 2 I( I +1) isotopes u v v u b 2 b 2 = 2 3 b + b 2I +1 2 I( I +1) isotopes particular cases: unpolarised neutrons Ni: all isotopes with I=0 Vanadium: only one isotope Page 74

75 NEUTRON POLARISATION Polarisation induces interference between nuclear and magnetic scattering ( )3 dσ dω = N 2π F v N Κ 0 { ( ) 2 + 2( P ˆ µ ˆ )F ( N Κ ) F ( Κ) + F ( Κ ) 2 } M M In ferromagnets, ( Κ) and F M( Κ) are non-zero for the same K vectors P ˆ ( µ ˆ )= ±1 F N application: polarisation devices for neutrons (anti-)parallel to B ( )3 dσ dω = N 2π F v N Κ 0 ( ) ± F M ( Κ) 2 polarising field B if matching F N and F M reflected beam is polarised Κ Similar effects/applications in reflectometry magnetic optical index polarising neutron guides guide fields Page 75

76 NEUTRON POLARISATION application: precise measurement of weak magnetic signals apply B perpendicular to K moments are aligned parallel to B ( )3 dσ dω = N 2π F v N Κ 0 ( ) ± F M ( Κ) 2 measure flipping ratio R R = dσ dω + / dσ dω = 1 γ 1+ γ 2 with γ = F M ( Κ)/F N Κ ( ) if γ is small, R~1-4γ allows to measure spin densities Iron C.G Shull et al. J.Phys.Soc.Japan 17,1 (1962) Page 76

77 SO MANY OTHER FIELDS neutron spin-echo: use of Larmor precession of neutron s spin time evolution of s=1/2 in magnetic field B ds r dt = γ s r B r ω L = γ B with γ = π Gauss -1 s 1 total precession angle φ = ω L t = γ B d/v depends neutron s velocity with B=10 Gauss ~29 turns/m for 4Å neutrons neutron spin-echo encodes neutron velocity quite high resolution without loss in intensity NSE breaks the awkward relationship between intensity and resolution: the better the resolution, the smaller the resolution volume and the lower the count rate! neutron reflectometry, SANS, Page 77

78 SUMMARY OF KEY MESSAGES neutrons have no charge low absorption nuclear and magnetic interactions have similar strengths interaction with nuclei very short range isotropy, isotope variation and contrast interactions with electrons lead to magnetisation densities neutron diffraction the method of choice to determine magnetic structures scattered intensities are proportional to space and time Fourier transforms of site correlation functions (positions and magnetic moments) accessible time and space domains cover a wide range of applications caveat: neutron sources are not very efficient.. Page 78

79 FURTHER READING Introduction to the Theory of Thermal Neutron Scattering G.L. Squires Reprint edition (1997) Dover publications ISBN Experimental Neutron Scattering B.T.M. Willis & C.J. Carlile (2009) Oxford University Press ISBN Neutron Applications in Earth, Energy and Environmental Sciences L. Liang, R. Rinaldi & H. Schober Eds Springer (2009) ISBN Methods in Molecular Biophysiscs I.N. Serdyuk, N. R. Zaccai & J. Zaccai Cambridge University Press (2007) ISBN Thermal Neutron Scattering P.A. Egelstaff ed. Academic Press (1965) Page 79