Paul Scherrer Institute Villigen, `

WWW.PSI.CH Diffraction

Jürg Schefer Laboratory for ETHZ&PSI Institute, CH-5232 Villigen PSI

Copyright: [email protected] (1999) Outline

• goals • applications of diffraction • some basic tools for diffraction • diffraction instruments at PSI • examples • strengths and weaknesses • summary Goals of (Neutron) Diffraction

• structure (nuclear and magnetic) • thermal motions • occupation of interstitial sites • multiphase compositions Applications of Diffraction

• material specification: multiphase • structure/volume information as a function of temperature, pressure, light illumination, e.g • complementary information to X-Rays (nuclear density vs. electron-density) • localization of light in the presence of heavy ones such as Uranium • neighbored atoms in the periodic table may be distinguished • effects (e.g. H/D, B11,…) The Diffraction Method

• cross-section for diffraction • reciprocal lattice and Ewald sphere construction • structure factors and Bragg intensities Debye Waller factor as a correction example • magnetic diffraction

FOR MORE INFO... G.L. Squires, Thermal , Dover 1996 Diffraction Setup

sample detectors

sampl e

targe beam rductions are

HO-sca2 ter det

monochromats

C, 023eon levatG1 or SJ3// tanzboden (granite)

cradle d e polygon (1.5 - 380K) t e c monitor t o target r

secondary 1 cooled N2 -filter collimation neutron shutter primary collimation (fail safe)

beam reduction 3 horizontal collimation r to ec et high energy shutter d borpoly- ethylen

iron polyethylen

primary beam stop monochromator (Ge , C ) 311 002 mono- H O - scatterer 2 collimation chromator filtering X-Rays? ?

1.0 Neutrons 0.9

0.8

0.7

Neutrons 0.6 form factor [arb. units] [arb. factor form 0.5 X-Ray

0.4

0.0 0.1 0.2-1 0.3 0.4 sin()/ [Å ] Diffraction Method ...

We are using: • elastic scattering function S k k’ (momentum transfer q=k’-k, energy transfer =0) q • Born approximation: system undisturbed • sample size is indefinite • point symmetry of atoms yielding 3-dimensional symmetry (International tables)

periodicity in real space: basic vectors a1,a2,a3 • reciprocal space and Ewald sphere as a convenient definition Elastic Scattering Function

 scattering Density 2 2   V (r )    r  m

m 2  2 k ' V k    d r  exp  i k ' r  (r ) exp( i k r ) , 2  2   m

incoming neutron with direction k |k’| = |k| scattered neutron with direction k’ no energy transfer Scattering Density: continuos -> atoms -> lattice

2  2  2 V (r j, h1, h2, h3 )  b 2  j V (r )    ( r ) m m V ( r  r j, h1, h2, h3 )  0 Structure Factor: From the Continuum to Discrete Atoms

k'|V | k  ?  2 2 2  2   V ( R j)  b j V (r )   ( r ) m m V ( r  R j)  0 atoms i(kk ')r i(kk ')R j k'|V | k   (r)  e d r   bj  e j

bj: scattering “power”

Integral over (r) replaced by sum over all atoms j at position Rj in the substance R: Separation in Translation and Unit Cell Parts

Rj= Rhkl + rj

• translation symmetry

Rhkl = h1*a1 + h2*a2 + h3*a3

a1,a2,a3 lattice constants

h1,h2,h3 integers rj Rh1h2h3 • atoms within unit cell r =x *a + y *a + z *a j j 1 j 2 j 3 Rj (0  xj,yj,zj  1) position of j within the unit cell And Finally:

 (r)  ei(kk ')rd r 

lattice atoms i(kk ')(Rh1h2h3r j ) bj  e  i(kk ')r j h1h2h3  j  i(kk ')Rh1h2h3 bj e h1h2h3,ej  F

maximal if n.2integer (wave superposition) yields Laue conditions intensity Ewald sphere construction A Convenient Tool: Reciprocal Lattice and Ewald-Sphere

maximal if exponent is n.2 (wave superposition) i(kk ')Rh1h2h3 = e Laue conditions Ewald sphere construction use reciprocal space:   ai  a j*  2  ij  or

a2 xa3 a2 xa3 a1*  2   2  a1  (a2 xa3) Vunit

* * * Qhkl = h*a1 + k*a2 + l*a3 Symmetry: An addional help

Greek Geometry (Euklid) Existence of symmetry was known, but the importance not realized (example: Mirror plane between figures)

A-M. Legendre (Lectures 1794) Définition XVI: J`appelerai polyèdres symétriques deux polyèdre qui, ayant une base commune, sont construits semblablement, l’un au- dessus du plan, l’autre au-dessous avec cette condition que les sommets des angles solides homologues soient situés à égales distances du plan de la base, sure une même droite perpendiculaire à ce plan. Corrections: Debye Waller as an Example

Oscillator around rj(t) : rj(t)  rj  rj(t)

 to calculate :  (r) eiqR j (t)d r   (r) eiqR j eiqrj (t)d r

lattice atoms iq(Rh1h2h3r j ( t )) iqRh1h2h3 iqr j  e  e bj e ir j (t)q hkl  j  e Corrections: Example Debye Waller (step by step calculation)

i(kk ')R lattice atoms iq(Hhklr j ) iqHhkl iqr j (t) e d r  e  e bj e  hkl  j 2 Vibrating : r j(t)  r j  r j(t) 2 rj q r  q2  r2  cos 2 sin2  q2 to calculate : eiqr j (t)  eiqr j eiqr j (t)   3 4 q  sin  iqr j (t) 1 2 e  1 i q  rj   (q  rj)  ... 2 2 2 2  2 2 q r j 8 sin  r j 2 W   1 2 1 2 rj 2  (q  rj)  q 2 3  3 2 2 3 W 2 Example : eW  1W   ... 2 2 r j isotrop : x2  x2  x2  3

lattice atoms W i(kk ')R iq(Hhklr j ) iqHhkl iqr j e e dr  e  e bj e  hkl  j Easier: Folding Functions in Real Space Multiplying Functions in Reciprocal Space

rj(t)  rj  rj(t)

' i(kk ')rj Fhkl  Fhkl  e d r Rj F hkl  F  (r)  ei(kk ')R j d r hkl  isostropic : i(kk ')r W j 2 2 2  2 2 e  e d r q r j 8 sin  r j  W   2 3 2 3

lattice atoms F '  (r) ei(kk ')Rdr  b eiq(Rh1h2h3r j ( t ))  F eW hkl  h1h2h3  j j hkl Magnetic Scattering (I)

 ˆ ˆ ˆ  B (L  2S) ˆN where  L  r x pˆ : orbital angular momentum operator Sˆ : spin angular momentum operator

Field of a single electron with speed ve : γ : gyromagnetic ratio  Uˆ  ˆ H   ˆ  H c : m  N  xR e ve xR e : electron charge H  curl e  3 c 3 ˆ : Pauli spin operator R R  0 1 0 i 1 0 x  y   z  1 0 i 0 0 1  Magnetic Scattering (II)

lattice atoms iqr j FM (qhkl)   bMj(qhkl)  e Tj(qhkl) hkl  j e2 j bMj (qhkl )  2  fMj (qhkl )   m (qhkl ) mec

bMj : magnetic scattering amplitude fMj : magnetic form factor

Tj(qhkl) : Debye Waller

1 2  : spin of sampling neutron mj (qhkl ) : projection of magnetic moment m j to the z - axis (perpendicular to k,k') Magnetic Scattering (III)

2.5 iqr fMj (qhkl )  M j (r)e d r  2.0 unitcell

with the normlised magnetization density 1.5 3+ Nd

fMj (0)  M j (r)  d r  1.  1.0 unitcell magnetic form factors [form ] magnetic 3+ 0.5 Sm

0.0 02468101214-1 q [Å ] Magnetic Diffraction: Calculated Powder Patterns

calculated structures Nuclear nuclear: NaCl-type

220 AF 3+ Yb P Spiral

200 Ferro

111 calculated intensity

0 20406080 2 Nuclear and Magnetic: The Sum

unpolarised neutrons : incoherent superposition : 2 2 2 I(qhkl )  Fn  Fm  Fn  Fm  Inuclear  Imagnetic

polarized neutrons : coherent superposition :

incoming neutrons: spin up () or down (-)

 2 2 2 I (qhkl )  Fn  Fm  Fn  Fm  2 Fn  Fm flipping ratio can be changed Elastic Diffuse Magnetic Scattering from Paramagnets: Example Yb3+

Yb3+ : S=1/2, L=3, J=7/2 1.0

Dipole approximation: 0.9 3+ Yb 0.8

0.7

0.6 fdiffuse (q)  jo (q)  j1

0.5 j2 [arb. units] [arb. 0.4 Fdiffuse J (J 1)  L(L 1)  S(S 1) diffuse ,f 2

j (q)  ,j 0.3 2 1 3J (J 1)  L(L 1)  S(S 1) j 2 0.2 Idiffuse  fdiffuse 0.1

0.0 jv (q) : calculated by relativistic Dirac  Fock Approx. 0.0 0.1 0.2-1 0.3 0.4 0.5 cf. J. Brown, Int.Tables for Crystallography C (Kluver 1992) p.391 sin()/ [Å ]

7 9 1 3   34   2 2 2 2 Ref.: P.Fischer in: Magnetic Neutron Scattering, Ed. A. Furrer, fd  jo  j2  7 9 1 3 World Scientific (1995), Singapore, ISBN 981-02-2353-6 3   34   2 2 2 2 Refine!

Refined Experiment Model

Iobs (qhkl) Starting Model

2 Icalc (qhkl)  Fcal (qhkl i F(qhkl)  F(qhkl  e Diffraction Instruments at PSI

• HRPT high resolution powder diffraction • DMC medium resolution high intensity powder diffraction • TriCS single crystal diffraction Diffraction Instruments at PSI...

DMC HRPT TriCS Powder Diffractometer Powder Diffractometer Single Crystal Diffractometer Cold Neutrons Thermal Neutrons thermal neutrons Å Å Å  = 1.18 Å, 2.3 Å Ce3Cu3Sb4: Powder Diffraction at DMC

8000

7000 obs 6000 I(2K) - I(18K) calc 5000 =2.556Å difference DMC/SINQ 4000

3000

2000

1000 Intensity [counts]

0

-1000

-2000 0 20406080o 2 [ ]

T. Herrmannsdörfer et. al., Solid State Comm. (sub. 1999) Ce atoms plotted only! TriCS Layout