Neutron scattering. Max Wolff. Division for Material Physics Department of Physics and Astronomy Uppsala University Uppsala Sweden

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1 Neutron scattering Max Wolff Division for Material Physics Department of Physics and Astronomy Uppsala University Uppsala Sweden

2 Outline Why is neutron scattering important? Peculiarities of neutrons How does it work? What do we measure? How do we measure? What is it good for? Applications

3 Neutron scattering 3

4 Time and length scales 4

5 Why neutrons? N S Neutrons are neutral: high penetation power, nondestructive Neutrons have a magnetic moment: magnetic structures and excitations Neutrons have a spin: polarized neutrons, coherent and incoherent scattering, nuclear magnetism Neutrons have thermal energies: excitation of elemenatry modes, phonons, magnons, librons, rotons, tunneling, etc. Neutrons have wavelengths similar to atomic spacings: structural information, short and long range order, pore and grain sizes, cavities, etc. Neutrons see nuclei: sensitive to light atoms, exploiting isotopic substitution, contrast variation with isotopes

6 The neutron Properties: Life time: T 1/2 = 890 s Mass: m = * kg Charge: Q = 0 Spin: S = /2 Mag. Moment: µ/µ n = Dispersion: E = 1 2 mv = 2 2 k 2m 2 (Photon: E = ck λ = 1.8 Å E 7 kev) Thermal neutron: T = 293 K v = 2200 m/s λ = 2π/k = 1.8 Å E = 25 mev (atomic distances) (molecular/lattice excitations) 8

7 Energy regimes Energy momentum relation : E = λ -2 = 2.07 k 2 = 5.23 v 2 λ [Å] E [mev] T [K] v [m/s] Cold neutrons : mev D 2 at 25 K Thermal neutrons : mev D 2 O at room T Hot neutrons : mev graphite at 2400 K

8 Comparison Energy momentum relation : E = λ -2 = 2.07 k 2 = 5.23 v 2 λ [Å] E [mev] T [K] v [m/s] X ray properties : E = ħck, c = m/s Inelastic x-ray scattering : λ = 1 Å E = 12.4 kev Raman, infra-red,... : Å E = ev

9 Heisenberg principle Position Momentum

10 Heisenberg principle Heisenberg s heritage : nothing is certain! ΔE Δt ħπ Δν Δt ½ Δω Δt π Δp Δx ħπ Δλ -1 Δx ½ Δk Δx π Let s get real : Spectrometer with ΔE/E of 5% (0.05%) utilized with neutrons of E = 2 mev - what can we measure? ΔE = 0.1 mev Δt s = 20 ps ΔE = 1 μev Δt s = 2 ns

11 Interaction 11

12 Absorption 12

13 What can we measure? Φ σ dσ/dω d 2 σ/dωde = number of incident particles per area and time = total number of particles scattered per time and Φ = number of particles scattered per time into a certain direction per area and Φ = number of particles scattered per time into a certain direction per area with a certain energy per energy interval and Φ

14 Definitions k i Q k f Q = k f ki = k2 f 2m k 2 i 2m

15 Mathematical description Schrödinger equation: t =(ˆp2 2m + ˆV ) i( r, ) = e i( k x t) f (r, ) = e i(kx t) + f( ) e i(kr t) r

16 Kinematic approximation Kinematic approximation means: Combination of s-wave scattering (for neutrons) with the Born approximation (single event scattering) The following effects are neglected (dynamic theory): - Attenuation of the beam. - Absorption. - Primary extinction (attenuation due to scattering). - Secondary extinction (attenuation due to multiple scattering). We will also not consider the finite size of the sample (lattice sum). All these effects are important for perfect crystals that hardly exist is disordered materials.

17 Scattering function Scattering potential for neutrons: V (r, t) (b n (r R)+b m (r R)) The scattering function: S( Q, )= 4 s k i k f 2 S( Q, ) = f, i (E f ) k f, f, ˆ f ˆV ˆ i, i,k i 2 ( f, i) This equation is also known as scattering law or dynamical structure factor. It completely describes the structural and dynamic properties of the sample. s = 2 d d =4 b2

18 Potential For neutrons (scattering at the core) wave length (10-10 m) is much larger than the size of the scattering particle (10-15 m). This implies that no details of the core can be seen and the scattering potential can be described by a single constant b, the scattering length: V ( r) = 2 h m b ( r R) This is called the Fermi pseudo potential. B is a phenomenological constant and has to be determined experimentally.

19 Cross section The Fermi pseudo potential is true neutron-nucleus potential, but it is constructed to provide isotropic s-wave scattering in the Born approximation. Inserting this potential into the scattering cross section results in: The total scattering cross section is then:

20 Ensemble The potential for an ensemble of scatterers can be described by the sum of the scattering length: V ( r) = 2 h m i b i ( r Ri ) The sum is taken over all nuclei at positions R i. For the cross section this results in: = i b i e i Q r 2 = i,j b i b j e i Q( R i Rj) = ij b i b j e i Q R ij R ij is the distance between the scatterers or the lattice parameter for a crystal.

21 Average In a scattering experiment the ensemble average or time average is measured: = ij b i b j e i Q r Assuming that the scatterers are fixed at their positions the average has to be taken only over the scattering length: = ij b i b j e i Q R ij Formally we can divide up the double sum into to sums: = i=j b i b j e i Q R ij + i=j b i b j e i Q R ij

22 Averaging As δb i averages to 0. =0 =0 For no correlations i=j. Inserting into the cross section gives: b coh =<b> and b inc =(<b 2 >-<b> 2 ) ½ is then called coherent and incoherent scattering length, respectively.

23 Spin incoherent scattering For a one atom with nuclear angular moment J the total moment of the neutron nucleus pair has to be considered. The neutron is a Spin ½ particle. In an unpolarised neutron beam S= ½ and S=- ½ neutrons are present. For the total moment of the system we get: M=J±S. Accordingly two scattering length have to be considered: b + for M=J+ ½ and b - for M=J- ½ Generally, 2J+1 different values are possible for the angular moment J. This results in a total number of 2(2J+1) eigenstates (2(J+1) for b + and 2J for b - ).

24 Spin incoherent scattering The probability for measuring b + is then: And for b - : For J=0 we get 1 for b + and 0 for b -. The coherent and incoherent scattering length can then be calculated:

25 Example 1 H This results in: p + =3/4 and p - =1/4. Hydrogen scatters mainly incoherent!

26 Example 2 H This results in: p + =2/3 and p - =1/3. Deuterium scatters mainly coherent!

27 Correlation functions Coherent scattering: Bragg reflections, liquid structure factor, phonon scattering etc. Example (Bragg law): Incoherent scattering: Scatterers exist, self motion etc. Example (Random jump diffusion):

28 Real - reciprocal space Observable in INS and Diff. Observable in NSE, simple models Simple models, computer simulations

29 Scattering cross section Coherent scattering cross section H D C O Al Si Incoherent scattering cross section

30 Scattering length Scattering length b [fm] Atomic number 83

31 Summary Weakly interacting Spin Core interaction Low energy Neutrons: High penetration Magnetic sensitivity Isotope sensitivity Coherent/Incoherent scattering Good energy resolution Scattering function describes a sample completely Instrumentation: Time of flight Angle dispersive

32 Nuclear reactor Reaction products neutron Access neutrons Lise Meitner und Otto Hahn Target... discovered 1938 together with Fritz Strassmann the nuclear fission reaction of Uranium after irradiation with neutrons n U! U! Ba Kr +3n + 177MeV

33 Floor plan

34 Spallation

35 ESS - How it will look like MAX IV Target station: SP: 50 Hz LP: 16.6 Hz Pulse length 2 ms Instrumentation Linear accelerator: NC: SC: Energy: Current: Power: 769 m 432 m GeV 3.75 ma 5 MW

36 Angle dispersive neutron scattering Source Incident beam Spin flipper Monochromator/Polarizer Detector Collimator k i Q Exiting beam k f Sample Collimator Spin flipper Analyzer

37 TOF neutron scattering Polarizer Spin flipper Chopper Incident beam Source Collimator Detector Exiting beam k i Q Sample Analyzer Collimator Spin flipper k f

38 Scattering geometry Elastic scattering Rayleigh process Neutron energy gain Anti-Stokes process Neutron energy loss Stokes process Constant directions of k i and k f but changing Q! Constant directions of k i and Q but changing k f!

39 Kinematics Kinematics of inelastically scattered neutrons : energy gain energy loss E i = 100 mev ħq [Å -1 ] ħω [mev] B.T.M. Willis & C.J. Carlile, Experimental Neutron Scattering, Oxford University Press: For 2θ = 0, 180 Q = Δk, E ~ Δk 2

40 Time of flight Inelastic scattering instruments : Time-of-Flight Spectrometer energy gain energy loss E i = 100 mev ħq [Å -1 ] ħω [mev] Equipped with large detector arrays to cover a wide 2θ range with typically 1 detector tube per degree of 2θ. Data are to be interpolated to constant Q.

ħω [mev] Typically equipped with a single detector but with

41 Three axis Inelastic scattering instruments : Three-Axis Spectrometer energy gain energy loss E i = 100 mev ħq [Å -1 ] ħω [mev] Typically equipped with a single detector but with highly flexible positioning capabilities to map out, e.g., constant Q slices.

[mev] Equipped with large detector arrays to cover a wide 2θ range.

42 Backscattering Inelastic scattering instruments : Back-Scattering Spectrometer energy gain energy loss E i = 100 mev ħq [Å -1 ] ħω [mev] Equipped with large detector arrays to cover a wide 2θ range. Dynamic range is limited and data are recorded approximately at constant Q.

43 Kinematics Kinematics of inelastically scattered x rays and visible light: energy gain energy loss Energy of interest : ħω = 10 mev ħq [Å -1 ] X rays : E i ~ 10 kev k i = k f ΔE/E ~ 10-7 Q = 2k i sin(θ) (~ 2θk i ) Raman : k i = k f ~ 10-3 Å -1 Q = 0 ħω [mev]

44 Neutron Spin Echo Velocity Spin /2 B 0 B 1 /2 Spin Detector Selector Polarizer Analyzer Sample n beam magnetic field profile spin precession spin defocusing

45 NSE Velocity selector Axial Precession field, B o sample Axial Precession field, B 1 Detector Supermirror polarizer /2 /2 Mezei flipper coils Supermirror analyser A B C So far we have assumed that the neutrons enter and leave the spectrometer with the same distribution of velocities (i.e. any scattering by the sample is elastic) Quasi-elastic scattering process will change the energy and therefore the velocity of the neutrons. The accumulated precession angle is now L = n [(L o B o )/v o - (L 1 B 1 )/v 1 ] Thus the accumulated precession angle L can be used as a measure of the energy transfer associated with the scattering process

46 Elastic/inelastic scattering Inelastic discrete lines Elastic line Quasielastic signal Sample Resolution

47 Example - clathrate CH 4 in H 2 O-clathrate: Enormous amounts trapped in perma-frost soils and deep-sea sedimants! Environmental risk through global warming! CO 2 in H 2 O-clathrate : Opportunity for dumping our combustion junk! Composition decomposition mechanism? Stability regime and conditions?

48 Example - clathrate Tetrahydrofuran (THF) in H 2 O hydrate-clathrate :

49 Example - clathrate Text Phonon Density of States : Sum up all detectors! M. Koza et al.

50 Example - clathrate Elastic Structure Factor: Sum up intensity in the energy channels around the elastic line only! Static Structure Factor: Sum up intensity in all energy channels! DIFFRACTION!!!

51 Example - detailed balance Neutron energy gain Anti-Stokes line Neutron energy loss Stokes line Clathrate : Ba 8 Zn x Ge 46-x

52 Example - ballistic motion Particular interest : Is there a change in diffusion dynamics from ballistic at short distances (time scales) to Brownian like motion at long distances (time scales)? Brownian Lorentz line Ballistic Gaussian line

53 Example - balistic motion Instrument : IN13 with cooled/heated monochromator λ = 2.2 Å ΔE = 8 μev E range = μev At r <2.4 Å indication of ballistic motion?

54 Example - confined water

55 Example - confined water Size effect of micelles on diffusion properties of confined water Instruments : IN5 and IN16 at ILL BASIS at SNS, USA

56 Rouse model The current description of the Rouse dynamics is based on the bead and spring model for polymer chains. I pair Rouse I self Rouse 2 Wl ( ) = 2 t 4 4 Q, t exp Q t exp Wl Q 3N Winter School on Neutron 2 Scattering in Soft Matter R π Björkliden, x 3/25-30/ 12 9π ( Q, t) = du exp{ u Ω t h [ u Ω t ]} h ( y) = 12 2 Q l dx cos R ( ) 4 4 xy [ ( )] 2 Q l 1 exp x ; Ω = W R 36

57 NSE - rouse - self NSE results validate the theory based on Rouse dynamics: PDMS, T=373 K Winter School on Neutron Scattering in Soft Matter Björkliden, 3/25-30/ 12 D. Richter, et al., Phys. Rev. Lett., 62, 2140 (1989).

58 NSE - rouse - pair NSE results validate the theory based on Rouse dynamics: Winter School on Neutron Scattering in Soft Matter Björkliden, 3/25-30/ 12 H. Montes, et al., J. Chem. Phys., 110, (1999).

Example - Reptation Reptation in polyethylene The dynamics of dense polymeric systems are dominated by entanglement effects which reduce the

lateral diffusion - although several other models have also been proposed The measurements on IN15 are in agreement with the reptation model.

59 Example - Reptation Reptation in polyethylene The dynamics of dense polymeric systems are dominated by entanglement effects which reduce the degrees of freedom of each chain de Gennes formulated the reptation hypothesis in which a chain is confined within a tube constraining lateral diffusion - although several other models have also been proposed The measurements on IN15 are in agreement with the reptation model. Fits to the model can be made with one free parameter, the tube diameter, which is estimated to be 45Å Schleger et al, Phys Rev Lett 81, 124 (1998)

60 Example - glass Glassy Dynamics in Polybutadiene The first peak in the structure factor corresponds to interchain correlations (weak Van der Waals forces), the second peak is dominated by intrachain correlations (covalent bonding) Each NSE spectrum was rescaled by the experimentally determined characteristic time, t h, for bulk viscous relaxation for the corresponding temperature. S(Q) S(Q) ( (a) 4K 160K 270K Scattering vector, Q, (nm -1 ) at the first peak the spectra follow a single, stretched exponential ( =0.4) universal curve, indicating that the interchain dynamics very closely follow the -relaxational behaviour associated with macroscopic flow of polybutadiene. S(Q,t)/S(Q,0) (b) Q=14.8nm K 230K 240K 260K 280K t f /

61 Example - glass Glassy Dynamics in Polybutadiene Those correlation functions measured at the second peak of the structure factor do not rescale. They are characterised by a simple exponential function, with an associated temperature dependent relaxation rate which follows an Arrhenius dependence with the same activation energy as the dielectric -process in polybutadiene, and which is unaffected by the glass transition. A. Arbe et al Phys. Rev. E 54, 3853 (1996) S(Q) S(Q,t)/S(Q,0) (a) 4K 160K 270K Scattering vector, Q, (nm -1 ) (c) Q=27.1nm K 180K 190K 205K 220K 240K 260K 280K 300K t f /

62 Summary Why is neutron scattering important? Peculiarities of neutrons How does it work? What do we measure? How do we measure? What is it good for? Applications

Neutron Instruments I & II. Ken Andersen ESS Instruments Division

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