Neutron Diffraction. Juan Rodríguez-Carvajal Diffraction Group at the Institut Laue-Langevin

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$Neutron Diffraction Juan Rodríguez-Carvaal$

1 Neutron Diffraction Juan Rodríguez-Carvaal Diffraction Group at the Institut Laue-Langevin 1

2 Outline of the talk Characteristics of neutrons for diffraction Diffraction equations: Laue conditions Comparison neutrons synchrotron X-rays Magnetic neutron diffraction Examples of neutron diffraction studies 2

3 Neutrons for what? Neutrons tell you where the atoms are and what the atoms do (Nobel Prize citation for Brockhouse and Shull 1994) 3

4 Particle-wave properties energy-velocity-wavelength... kinetic energy (E) velocity (v) temperature (T). E= m n v 2 /2= k B T = p 2 /2m n = (ħk) 2 /2m n =(h/l) 2 /2m n momentum (p) ħ=h/2p p= m n v = ħ k wavevector (k) k= 2p/l = m n v/ħ wavelength (l) 4

Neutrons, a powerful probe Matter is made up atoms, aggregated together in organised structures The properties of matter and materials are largely determined by their structure and dynamics

5 Neutrons, a powerful probe Matter is made up atoms, aggregated together in organised structures The properties of matter and materials are largely determined by their structure and dynamics (behaviour) on the atomic scale distance between atoms ~ 1 Å = 1/ cm Atoms are too small to be seen with ordinary light (wavelength approx Å) The wavelength of the neutron is comparable to atomic sizes and the dimensions of atomic structures, which explains why neutrons can «see» atoms. The energy of thermal neutrons is similar to the thermal excitations in solids. Neutrons are zero-charge particles and have a magnetic moment that interacts with the magnetic dipoles in matter. Techniques using neutrons can produce a picture of atomic structures and their motion. 5

Particle-wave properties (Energy-Temperature-Wavelength) E = m n v 2 /2 = k B T = (ħk) 2 /2m n ; k = 2p/l = m n v/ħ Energy (mev) Temp (K) Wavelength (Å) Cold 0.

6 Particle-wave properties (Energy-Temperature-Wavelength) E = m n v 2 /2 = k B T = (ħk) 2 /2m n ; k = 2p/l = m n v/ħ Energy (mev) Temp (K) Wavelength (Å) Cold Thermal Hot Cold Sources intensity [a.u.] COLD THERMAL HOT Series1 Series2 Series velocity [km/s] 6

Scattering power of nuclei for neutrons Light atoms diffuse neutrons as strongly as heavy atoms The scattering power for X- rays is proportional to the atomic number (number of

7 Scattering power of nuclei for neutrons Light atoms diffuse neutrons as strongly as heavy atoms The scattering power for X- rays is proportional to the atomic number (number of electrons). The scattering power for electrons depends on the electrostatic potential The scattering power for neutrons is of the same order of magnitude similar for all atoms 7

Neutrons for magnetism studies Neutrons are strongly scattered by

magnetic moment of the neutron interacts strongly with the atomic

structures of materials and measure the magnetization with high

8 Neutrons for magnetism studies Neutrons are strongly scattered by magnetic materials Neutrons act as small magnets The dipolar magnetic moment of the neutron interacts strongly with the atomic magnetic moment Neutrons allow the determination of magnetic structures of materials and measure the magnetization with high precision. Magnetic Crystallography Ferromagnetic and antiferromagnetic oxides 8

9 Outline of the talk Characteristics of neutrons for diffraction Diffraction equations: Laue conditions Comparison neutrons synchrotron X-rays Magnetic neutron diffraction Examples of neutron diffraction studies 9

10 Interaction neutron-nucleus = number of incident neutrons /cm 2 / second = total number of neutrons scattered/second/ q k f Target d r ds q Direction q, f z Fermi s golden rule gives the neutron-scattering Cross-section number of neutrons of a given energy scattered per second in a given solid angle (the effective area presented by a nucleus to an incident neutron) 2 d d de' k' k m 2p 2 2 l, p l p l', ' k ' ' l' V kl 2 E E ) l l' 10

11 Interaction neutron-nucleus Weak interaction with matter aids interpretation of scattering data The range of nuclear force (~ 1fm) is much less than neutron wavelength so that scattering is point-like k Plane wave e ik r k Sample V(r) Plane wave e ik x r k 2q Detector k Spherical wave (b/r)e ik r Q Fermi Pseudo potential of a nucleus in r V 2p m 2 b ( rr ) Potential with only one parameter 11

12 Diffraction Equations For diffraction part of the scattering the Fermi s golden rule resumes to the statement: the diffracted intensity is the square of the Fourier transform of the interaction potential k' k 2 p / l 3 2 A( Q) V ( r)exp( iq r) d r I( Q) A( Q) A*( Q) A( Q) Q k ' k 2 p ( s - s ) / l 2p s 2p h 0 There are different conventions and notations for designing the scattering vector (we use here crystallographic conventions). A i d f i 3 X ( s) e ( r) ( r R )exp(2 p s r) r ( s)exp(2 p s R ) 3 f ( s) e ( r)exp(2 pis r) d r Atomic form factor. Scattering length 2p A b i d b i m 2 3 N ( s) ( r R )exp(2 p s r) r exp(2 p s R )

$Diffraction Equations for crystals In a crystal the atoms positions can be decomposed as the vector position of the origin of a unit cell plus the vector position with respect to the unit cell R R r$

13 Diffraction Equations for crystals In a crystal the atoms positions can be decomposed as the vector position of the origin of a unit cell plus the vector position with respect to the unit cell R R r l l A ( s) b exp(2 pis R ) exp(2 pis R ) b exp(2 pis r ) l l N l l l l 1, n exp(2 pis R ) 0 l for general exp(2 pis R ) N for s H HR L s H l l H Laue conditions: the scattering vector is a reciprocal lattice vector of the crystal I ( H) b exp(2 pih r ) F( H) N 1, n s 2 2 integer

14 Diffraction Equations for crystals The Laue conditions have as a consequence the Bragg Law Laue conditions: the scattering vector is a reciprocal lattice vector of the crystal s H s s 0 l L L s 2sinq 1 s H 2dhkl sinq l d hkl l s0 L l q 2q s L l d hkl

15 Ewald construction From Pecharsky and Zavali Detector s L /l s 0L /l

16 Ewald construction Ewald Sphere z L z D s L P s 0L 2q h O crystal x D D D O D y L x L Laboratory Frame and detector

$Diffraction patterns Single Xtal - 2D image +$ Int vs 2θ Single Crystal nλ=2d(sinθ) Powder

17 Diffraction patterns Single Xtal - 2D image + scan > 3D Int vs 2θ Powder - 2D image > 1D Int vs 2θ Single Crystal nλ=2d(sinθ) Powder or polycrystalline solid Courtesy of Jim Britten

18 Ewald construction Laue From Pecharsky and Zavali Detector s L /l s 0L /l 1/l max 1/l min

19 Laue image obtained in Cyclops

20 Single Crystal and Powder Diffraction Single Crystal diffraction allows to get with high precision subtle structural details: thermal parameters, anharmonic vibrations. Drawbacks: big crystals for neutrons, extinction, twinning Data reduction: Needs only the indexing and integration of Bragg reflection and obtain structure factors. List: h k l F 2 σ(f 2 ) Data Treatment: SHELX, FullProf, JANA, GSAS, Powder diffraction no problem with extinction or twinning. Data reduction: minimalistic, needs only the profile intensities and their standard deviations Data Treatment: FullProf, JANA, GSAS, TOPAS, y I ( T T ) b h h ci i i {h}

21 Outline of the talk Characteristics of neutrons for diffraction Diffraction equations: Laue conditions Comparison neutrons synchrotron X-rays Magnetic neutron diffraction Examples of neutron diffraction studies 21

22 Why neutrons? NEUTRON DIFFRACTION FOR FUNDAMENTAL AND APPLIED RESEARCH IN CONDENSED MATTER AND MATERIALS SCIENCE Location of light elements and distinction between adacent elements in the periodic table. Examples are: Oxygen positions in High-T C superconductors and manganites Structural determination of fullerenes an their derivatives, Hydrogen in metals and hydrides Lithium in battery materials Determination of atomic site distributions in solid solutions Systematic studies of hydrogen bonding Host-guest interactions in framework silicates Role of water in crystals Magnetic structures, magnetic phase diagrams and magnetisation densities Relation between static structure and dynamics (clathrates, plastic crystals). Aperiodic structures: incommensurate structures and quasicrystals 22

23 Why neutrons? The complementary use of X-ray Synchrotron radiation and neutrons (1) The advantages of thermal neutrons with respect to X-rays as far as diffraction is concerned are based on the following properties of thermal neutrons: constant scattering power (b is Q-independent) having a nonmonotonous dependence on the atomic number weak interaction (the first Born approximation holds) that implies simple theory can be used to interpret the experimental data the magnetic interaction is of the same order of magnitude as the nuclear interaction low absorption, making it possible to use complicated sample environments 23

24 Why neutrons? The complementary use of X-ray Synchrotron radiation and neutrons (2) Powder diffraction with SR can be used for ab initio structure determination and microstructural analysis due to the current extremely high Q-resolution. Structure refinement is better done with neutrons (or using simultaneously both techniques) because systematic errors in intensities (texture effects) are less important and because scattering lengths are Q- independent in the neutron case. 24

27 Why neutrons? The complementary use of X-ray Synchrotron radiation and neutrons (3) Magnetic X-ray scattering allows in principle the separation of orbital and spin components. However, SR cannot compete with neutrons in the field of magnetic structure determination from powders. The contribution of SR to that field is on details of magnetic structures (already known from neutrons) for selective elements using resonant magnetic scattering (rare earths, U,...) 27

28 Outline of the talk Characteristics of neutrons for diffraction Diffraction equations: Laue conditions Comparison neutrons synchrotron X-rays Magnetic neutron diffraction Examples of neutron diffraction studies 28

29 Magnetic scattering: magnetic fields The interaction potential to be evaluated in the FGR is: V μb Magnetic field due to spin and orbital moments of an electron: m s Electron p R R Neutron σ n Magnetic vector potential of a dipolar field due to electron spin moment O R+R 4p 0 B S L 2 2 B B B B μ n μ Rˆ 2 p Rˆ R R Biot-Savart law for a single electron with linear momentum p 29

30 Magnetic scattering: magnetic fields Evaluating the spatial part of the transition matrix element for electron : ' exp( ) ˆ ( ˆ i k k QR Q s Q) ( p Qˆ Vm i ) Q Where Q ( k k ') is the momentum transfer Summing for all unpaired electrons we obtain: k ' V k Qˆ ( M( Q) Qˆ ) M( Q) ( M( Q). Qˆ ). Qˆ M ( Q) m M (Q) is the perpendicular component of the Fourier transform of the magnetisation in the scattering obect to the scattering vector. It includes the orbital and spin contributions. 30

31 Scattering by a collection of magnetic atoms We will consider in the following only elastic scattering. We suppose the magnetic matter made of atoms with unpaired electrons that remain close to the nuclei. Vector position of electron e: R R r e l e The Fourier transform of the magnetization can be written in discrete form as M( Q) s exp( iq R ) exp( iq R ) exp( iq r ) s e e l e e e l e 3 F ( Q) s e exp( iq re) ρ ( r)exp( iq r) d r e F Q m r Q r r m 3 ( ) ( )exp( i ) d f ( Q) M( Q) ml fl ( Q)exp( iq Rl ) l 31

32 Scattering by a collection of magnetic atoms 3 F ( Q) s e exp( iq re) ρ ( r)exp( iq r) d r e F Q m r Q r r m 3 ( ) ( )exp( i ) d f ( Q) If we use the common variable s=sinq/l, then the expression of the form factor is the following: f ( s) W ( s) l0,2,4,6 ( s) U ( r) (4 psr)4pr dr l ) 0 l l 2 2 l s s A a s B b s C c s D for l l ( ) l exp{ l } l exp{ l } l exp{ l } l 2,4,6 ( s) A exp{ a s } B exp{ b s } C exp{ c s } D

33 M (Q) is the perpendicular component of the Fourier transform of the magnetisation in the sample to the scattering vector. 3 M( Q) M( r)exp iq r ) d r Q=Q e Magnetic scattering M M Magnetic interaction vector M e M e M e ( e M) d ( ) M M d 2 * r0 Magnetic structure factor: M( H) p m f ( H)exp(2 pih r ) Neutrons only see the components of the magnetisation that are perpendicular to the scattering vector N mag m1 m m m 33

34 Elastic Magnetic Scattering by a crystal (2) For a general magnetic structure that can be described as a Fourier series: M( h) S exp( 2 pikr ) f ( h)exp(2 pih R ) l k k l l l M( h) f ( h)exp(2 pih r ) S exp(2 pi( h k) R ) k l k l M( h) S f ( Q)exp(2 pi( H k) r ) m l k k The lattice sum is only different from zero when h-k is a reciprocal lattice vector H of the crystallographic lattice. The vector M is then proportional to the magnetic structure factor of the unit cell that now contains the Fourier coefficients S k instead of the magnetic moments m. S k exp 2p ikr l 34

35 Diff. Patterns of magnetic structures Portion of reciprocal space h = H+k Magnetic reflections Nuclear reflections Magnetic reflections: indexed by a set of propagation vectors {k} h is the scattering vector indexing a magnetic reflection H is a reciprocal vector of the crystallographic structure k is one of the propagation vectors of the magnetic structure ( k is reduced to the Brillouin zone) 35

$Diffraction patterns of magnetic structures Cu 2+ ordering Ho 3+$

36 Diffraction patterns of magnetic structures Cu 2+ ordering Ho 3+ ordering Notice the decrease of the paramagnetic background on Ho 3+ ordering 36

37 Magnetic refinement on D2B

38 Outline of the talk Characteristics of neutrons for diffraction Diffraction equations: Laue conditions Comparison neutrons synchrotron X-rays Magnetic neutron diffraction Examples of neutron diffraction studies 38

39 Two Axes Diffractometers: Powders and Liquids Low Resolution D20 High Resolution D2B 39

$Real time powder diffraction on D1B$ Boudada et al.; J. Solid State Chem.

$Diffraction pattern / 3 minutes.$

40 Real time powder diffraction on D1B Dehydration of MoO 3 2H 2 O [N. Boudada et al.; J. Solid State Chem. 105, 211 (1993)] MoO 3 2H 2 O RT RT to 400 ºC in vacuum. Diffraction pattern / 3 minutes. MoO 3 H 2 O T = 60 ºC MoO 3 T = 150 ºC 40

41 Some Applications: Liquid state D4 Earth Mars Io Ganymede What they have in common? Metallic cores Proximity in Solar System Fluid Outer Core: from 3000 to 5200 km Primarily Fe with some Ni 41

Hypothesis: M. Nasch, M. H. Manghnani, R. A. Secco, Science 277 (1997) 219.

42 Some Applications: Liquid state But is 5-10% less than pure Fe+Ni Volcanic Eruptions Fe Ni there is some light element, but which? S? O? H? Si? Hypothesis: M. Nasch, M. H. Manghnani, R. A. Secco, Science 277 (1997) 219. The light element helps aggregating clusters, which in turn are disaggregated by heating the system. 42

$Two-axis diffractometer Fe$

43 Some Applications: Liquid state FeNi and FeNiS alloys Liquid state High temperature Special furnace Two-axis diffractometer Fe Fe Fe Fe Fe 85 Ni 15 Fe 85 Ni 5 S 10 43

44 Levitation of Liquids Principle of the aerodynamic levitation and laser heating Pyrometer Mirror CO 2 Laser Video camera Sample Levitator Gas flow 44

45 Crystallisation Laser 1 st mirror CO 2 laser Neutron beam Laser Laser beam 2 nd mirror Pyrometer Neutron beam Sample Gas Cold Point 45

46 Again Neutron Crystallisation beam Laser Dust coming from the sample B 4 C Screen Laser beams Crystallisation due to contacts Gas Neutron beam Solutions: Improvement of the levitation Use of a screen in front of the nozzle B 4 C screens Gas 46

47 High Temperature Setup Video Camera Pyrometer Spherical mirror Video Camera B 4 C screens NaCl window Laser Sample 47

48 Three Lasers Setup Laser 1 Laser 2 Neutron beam B 4 C screens NaCl window Laser 3 Gas Detector 48

counts/s/0.1 counts/s/0.1 Self-propagating High-T Synthesis (SHS) D20 2.

K/min Acquisition time 500 ms (300 ms) Hot isostatic pressing expensive 2.3e4 0.

49 counts/s/0.1 counts/s/0.1 Self-propagating High-T Synthesis (SHS) D20 2.3e4 Titanium silicon carbide Ti 3 SiC 2 Self-propagating High-temperature Synthesis (SHS) Riley, Kisi et al.: 3 Ti : 1 Si : 2 C, 20 g pellet in furnace Heating from 850 C to 1050 C at 100 K/min Acquisition time 500 ms (300 ms) Hot isostatic pressing expensive 2.3e4 0.1e e4 30 time/s 35 time/s theta 0 D.P. Riley, E.H. Kisi, T.C. Hansen, A. Hewat, J. Am. Ceramic Soc. 85 (2002) theta

50 SHS: pre-ignition Ti a-b transition starting at 870 C Pre-ignition: TiC x growth during 1 min Melting (?) in 0.5 s temperature/c relative time/seconds theta/degrees relative time/seconds

51 SHS: intermediate product Intermediate phase TiC, Si substituted formed in 0.5 s, 2s delay Heating up to 2500 K afterwards decay in 5 s temperature/c relative time/seconds theta/degrees relative time/seconds

52 SHS: final product Product Ti 3 SiC 2 starts after 5 s incubation time constant about 5 s temperature/c relative time/seconds theta/degrees relative time/seconds

53 Diffraction patterns of LaMnO 3 vs Pressure (ISIS, POLARIS + Paris-Edinburgh cell) Stability of the Janh-Teller effect and magnetic study of LaMnO 3 under pressure L. Pinsard-Gaudart, J.Rodriguez-Carvaal, A. Daoud-Aladine, I.N.Goncharenko, M. Medarde, R.I. Smith and A. Revcolevschi. PRB 64, (2001)

54 Diffraction patterns of LaMnO 3 vs Pressure (ISIS, POLARIS + Paris-Edinburgh cell) Rietveld refinement of LaMnO 3 at RT and 14.6 kbar In this range of pressure the reflections in the diffraction pattern are still sharp enough for performing a proper refinement Stability of the Janh-Teller effect and magnetic study of LaMnO 3 under pressure L. Pinsard-Gaudart, J.Rodriguez-Carvaal, A. Daoud-Aladine, I.N.Goncharenko, M. Medarde, R.I. Smith and A. Revcolevschi. PRB 64, (2001)

55 LaMnO 3 vs Pressure (ISIS, POLARIS + Paris-Edinburgh cell) Behaviour of different structural parameters of LaMnO 3 up to 70kbar

56 Magnetic scattering of LaMnO 3 vs Pressure (LLB, G61 + Goncharenko cell) Rietveld refinement of the nuclear (fixed) and magnetic scattering of LaMnO 3 The analysis of the magnetic contribution to the diffraction patterns indicates that the mode A continues to be dominant for all pressure range studied. However the mode Ay is unable to explain the diffraction pattern at 67 kbar. The magnetic structure corresponds to the mixture of two representations Γ 3g (+-) and Γ 4g (--) of Pbnm for k=0 [Gx,Ay, Fz] ~ [0,Ay, 0] and [Cx,Fy,Az] ~ [0,0,Az] Magnetic structure at 67 kbar: [0,Ay,Az]

57 Magnetic scattering of LaMnO 3 vs Pressure (LLB, G61 + Goncharenko cell) Magnetic structure vs pressure

58 Conclusions LaMnO 3 Stability of the Janh-Teller effect and magnetic study of LaMnO 3 under pressure L. Pinsard-Gaudart, J.Rodriguez-Carvaal, A. Daoud-Aladine, I.N.Goncharenko, M. Medarde, R.I. Smith and A. Revcolevschi. PRB 64, (2001) 1: The main effect of pressure is to decrease the orthorhombic distortion by diminishing the tilt angle MnO 6 octahedra (the interatomic distances diminish near isotropically) 2: The Jahn-Teller effect is stable up to 70 kbar 3: The magnetic structure conserve the A-type of ordering but with a deviation of the magnetic moment from the b-axis towards the c-axis

59 Information on neutrons in the Web Thank you for your attention 59

Magnetic neutron diffraction. Rob McQueeney, Ames Laboratory and Iowa State University

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