Paul Scherrer Institute Villigen, Switzerland `
WWW.PSI.CH Neutron Diffraction
Jürg Schefer Laboratory for Neutron Scattering ETHZ&PSI Paul Scherrer 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 atoms in the presence of heavy ones such as Uranium • neighbored atoms in the periodic table may be distinguished • isotope 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 Neutron Diffraction, 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? Neutrons?
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(kk ')r i(kk ')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 atom j within the unit cell And Finally:
(r) ei(kk ')rd r
lattice atoms i(kk ')(Rh1h2h3r j ) bj e i(kk ')r j h1h2h3 j i(kk ')Rh1h2h3 bj e h1h2h3,ej F
maximal if n.2integer (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(kk ')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) eiqR j (t)d r (r) eiqR j eiqrj (t)d r
lattice atoms iq(Rh1h2h3r j ( t )) iqRh1h2h3 iqr j e e bj e ir j (t)q hkl j e Corrections: Example Debye Waller (step by step calculation)
i(kk ')R lattice atoms iq(Hhklr j ) iqHhkl iqr 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 : eiqr j (t) eiqr j eiqr j (t) 3 4 q sin iqr 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 : eW 1W ... 2 2 r j isotrop : x2 x2 x2 3
lattice atoms W i(kk ')R iq(Hhklr j ) iqHhkl iqr 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(kk ')rj Fhkl Fhkl e d r Rj F hkl F (r) ei(kk ')R j d r hkl isostropic : i(kk ')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(kk ')Rdr b eiq(Rh1h2h3r j ( t )) F eW 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 : speed of light 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 iqr 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 mj (qhkl ) : projection of magnetic moment m j to the z - axis (perpendicular to k,k') Magnetic Scattering (III)
2.5 iqr fMj (qhkl ) M j (r)e d r 2.0 unitcell
with the normlised magnetization density 1.5 3+ Nd
fMj (0) M j (r) d r 1. 1.0 unitcell 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 34 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 34 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