Chapter 4: Diffraction Georg Roth Institute of Crystallography RWTH Aachen University

http://www.ifk.rwth-aachen.de http://www.frm2.tum.de

The „crystal palace“ in Aachen-Burtscheid Outstation of the IfK at FRM II in Garching Outline Diffraction part 1: Geometric crystallography in a nutshell

• Crystal lattices the lattice concept, unit cells, lattice points, lattice directions, lattice planes • Crystallographic coordinates systems The 7 crystal systems, the 14 Bravais lattices • Crystallographic symmetry-operations and -elements Rotation, roto-inversion, glide planes, screw axes • Crystallographic point groups and space groups The 32 crystallographic point groups (crystal classes), symmetry directions, Hermann-Mauguin symbols, the 230 space groups Matter exists in (at least) 3 states: gas – liquid - solid  Crystals are representatives of the solid state Definition of a crystal: Thermodynamically stable solid state with regular 3dim. periodic arrangement of atoms, ions or molecules in space in contrast to other states of condensed matter: liquids, glasses and amorphous solids

Halite - NaCl + rock salt Na - and Cl–- c ions

b 0 a How to choose a unit cell? (2D-example)

hexagonal symmetry 60° 60° 60° 60° 60° 60°

b4 As small as possible unit cell b2 As much symmetry as possible a4

a b a a b 2 b3

a3 choice of origin: point of highest symmetry (6-fold rotation point) crystal = lattice  motive

motive basis vectors

a b 3a 2b How to adress a lattice point? Linear combination of a and b

 = 3a + 2b (+ 0c)

General translation vector  : Motive here: A 2D-snowflake  = ua + vb + wc; u, v, w  Z In a crystal: Set of atoms in In 3 dimensions the unit cell How to address atoms in the unit cell?

a b r y b 1 1 x1a

r1 = x1a + y1b (0  x, y < 1)

Positions of atoms in the unit cell:

Position vector rj = xja + yjb + zjc (0  x, y, z < 1) (3dim) There is more than just lattice & motive: Symmetry

Rotational symmetry

60° a b Translational symmetry

120° 180° 3 dim. periodicity of crystals  3D crystal lattice

3 non-colinear basis vectors a, b and c (in the script a1, a2, and a3) define a parallelepiped, called unit cell of the crystal lattice and the crystallographic coordination system with its origin.

angles between basis vectors:

angle (a1,a2) = 

angle (a2,a3) = 

angle (a3,a1) =  faces of unit cell:

face (a1,a2) = C

face (a2,a3) = A

face (a3,a1) = B

any lattice point (point lattice) is given by a vector

 = ua1 + va2 + wa3 (u, v, w  )  is also known as translation vector lattice points and lattice directions according to the translation vector  = ua1 + va2 + wa3 (u, v, w  ) • lattice points are indicated by the corresponding integers  uvw • lattice directions or lattice rows by  [uvw] [001]

[-2-3-1] or [ 2 3 1 ]

[231]

[010]

[100] lattice planes (crystal faces are special lattice planes) 3 non-collinear lattice points define a lattice plane: • interceptions of a lattice plane with the axes

X (a1), Y (a2) and Z (a3): ma1, na2, oa3 • reciprocal values: 1/m, 1/n, 1/o (with smallest common denominator no/mno, mo/mno, mn/mno) • Miller indices: h = no, k = mo, ℓ = mn  Miller indices (hkℓ) describe a set of equally spaced lattice planes.

I: 1 1 1 1/1 1/1 1/1 h = 1 k = 1 ℓ = 1  (111) II: 1 2 2 1/1 1/2 1/2 h = 2 k = 1 ℓ = 1  (211) Lattice spacing d(hkl): Distance between lattice planes (hk0): Solely depends on the lattice parameters

Y (100)

X d(100) Diffraction of X-rays, neutrons, electrons, ... allows us to measure lattice spacings directly

The observable is the scattering angle  which may be converted into lattice spacings d(hkℓ)  Bragg equation for the reflection condition 2d(hkℓ)·sin(hkℓ) =  d(hkℓ): interplanar distance of a set of lattice planes (hkℓ) (hkℓ): scattering angle, angle between the incident beam and the lattice plane (hkℓ) : wavelength of the radiation

[hkℓ]

(hkℓ) (hkℓ)

(hkℓ) d(hkℓ) The lattice spacing d(hkℓ) can be calculated from the lattice parameters for each crystal system, e.g.:

• cubic crystal system (a = b = c,  =  =  = 90°), ex. NaCl a d(hkl)  h2  k 2  l 2 • hexagonal crystal system (a = b, c,  =  = 90°,  = 120°), ex. Graphite 1 d(hkl)  4 h2 k 2 hk  l 2 3 a2 c2 • orthorhombic crystal system (a, b, c,  =  =  = 90°) 1 d(hkl)  h2  k 2  l 2 a2 b2 c2 • tetragonal crystal system (a = b, c,  =  =  = 90°) 1 d(hkl)  h2 k 2  l 2 a2 c2 7 different coordinate systems in 3D  7 crystal systems in 3D name of system minimum symmetry conventional unit cell

triclinic 1 or -1 a  b  c;     

a  b  c;  =  = 90°, monoclinic one diad – 2 or m (‖Y)  > 90° three mutually perpendicular orthorhombic a  b  c;  =  =  = 90° diads – 2 or m (‖X, Y and Z)

tetragonal one tetrad – 4 (‖Z) a = b  c;  =  =  = 90°

trigonal a = b  c;  =  = 90°, one triad – 3 or -3 (‖Z) (hexagonal cell)  = 120° a = b  c;  =  = 90°, hexagonal one hexad – 6 or -6 (‖Z)  = 120° four triads – 3 or -3 cubic a = b = c;  =  =  = 90° (‖space diagonals of cube) 7 primitive + 7 centred unit cells 14 Bravais lattices (represented by their unit cells)

triclinic P monoclinic P monoclinic A orthorhombic P monoclinic axis‖c (0,0,0 + 0, ½, ½)

• orthorhombic I orthorhombic C orthorhombic F tetragonal P (0,0,0 + ½, ½, ½) (0,0,0 + ½, ½,0) (0,0,0 + ½, ½,0 ½,0, ½ + 0, ½, ½) 14 Bravais lattices (cont.)

tetragonal I hexagonal P hexagonal/ cubic P rhombohedral 14 Bravais lattices = 7 primitive lattices P for the 7 crystal systems with only one lattice point per unit cell + 7 centred (multiple) lattices cubic I cubic F A, B, C, I, R and F with 2, 3 and 4 lattice points per unit cell Definition of symmetry: Symmetry operations are isometric transformations (or movements) where one object is transformed into another in a congruent or enantiomorphous manner, i.e. all distances, angles and volumes remain invariant That means: an object and its transformed object superimpose in a perfect manner  they are indistinguishable

crystallographic rotations: 1-fold (identity), 2-, 3-, 4-, and 6-fold rotations Crystallographic symmetry operations are isometric movements in a 3D-periodic space:

1. Translations a = ua1 + va2 + wa3 (u, v, w  ) properties: no fixed point, shift of entire point lattice 2. Rotations: 1 (identity), 2 (rotation angle 180°), 3 (120°), 4 (90°), 6 (60°) properties: line of fixed points which is called the rotation axis 3. Rotoinversions (combination of n-fold rotations and inversion): 1 (inversion), 2 = m (reflection), 3 , 4 , 6 properties: exactly one fixed point

4. Screw rotations nm (combination of n-fold rotations with m/n·a translations ‖ to rotation axis) properties: no fixed point 5. Glide reflections a, b, c, n, …, d (combination of reflection through a plane (glide plane) and translation by glide vectors a1/2, a2/2, a3/2, (a1 + a2)/2, ..., (a1  a2  a3)/4 ‖ to this plane) properties: no fixed point Rotations and rotoinversions are called point symmetry operations because of at least one fixed point. Point symmetry operations rotations rotoinversions

1 = identity 1 = inversion

2-fold = 180°-rotation 2-fold rotation combined with inversion = reflection 5 1 3 6 4 2 Crystallographic symmetry operations are isometric movements in crystals:

120°

1/3a a a 60°

2/6a

31 = 3 + 1/3 a 62 = 6 + 2/6 a

60° a

4/6a

+ 42, 43 and 65 64 = 6 + 4/6 a Crystallographic symmetry operations are isometric movements in crystals:

reflection: mirror plane m  image plane

a a a/2

glide reflection: ‖ to the glide plane a is the glide vector a/2 All possible combinations of point symmetry operations in 3D: 32 crystallographic point groups (crystal classes)

The point group symmetries determine the anisotropic (macroscopic) physical properties of crystals: mechanical, electrical, optical, thermal, ...

In order to understand point group symbols (Hermann-Mauguin symbols), it is essential to understand the concept of symmetry directions:

A maximum of 3 independent main symmetry directions (“Blickrichtungen”) are sufficient to describe the complete symmetry of a crystal. These are specifically defined for the 7 crystal systems.

Basis vectors are conventionally chosen parallel to important symmetry directions of the crystal system, i. e. the twofold rotations axes in the orthorhombic crystal system are aligned along the mutually perpendicular basis vectors a1, a2, a3 . Symmetry directions in the orthorhombic lattice a  b  c;  =  =  = 90°

z z z

y y y x x x

[100] [010] [001] 2 m Symmetry directions in the tetragonal lattice a = b  c;  =  =  = 90° z z z

y y y x x x

[001] [100] [110] 4 2 m m Orthorhombic system ma2 crystallographic point group: mm2

2( )‖ to [001]: a2 symmetry operation   x  1 0 0 x  ma1      represented by  y   0 1 0 y      a rotation matrix  z   0 0 1 z  a -x,-y,z 1 a2 x y x,y,z a1

 x 1 0 0 x m  a :  x  1 0 0 x  ma1  a1:      a2 2       y    0 1 0 y  y  0 1 0 y            z   0 0 1 z   z  0 0 1 z  All possible combinations of point symmetry operations in 3 dim. lead to 32 crystallographic point groups (crystal classes)

crystal systems

Schoenflies-/Hermann-Mauguin symbols

rotation axis perpendicular to mirror plane X  m X  2

X ‖ m

X  m and X ‖ m Already shown: All possible combinations of point symmetry operations in 3D: 32 crystallographic point groups (crystal classes)

Now: All possible combinations of the 32 crystallographic point groups with translations (screw axes, glide planes, the 14 Bravais lattices) lead to exactly:

230 crystallographic space groups

No way to discuss or even list them all here…

See: International Tables for Crystallography Vol. A Conventional graphic symbols for symmetry elements: • symmetry axes (a) perpendicular, (b) parallel, and (c) inclined to the image plane • symmetry planes (d) perpendicular and (e) parallel to the image plane Diffraction experiment  crystal structure

T example: YBa2Cu3O7- C

• ceramic high-TC superconductor with TC = 92 K

• technical application with liquid N2 cooling is possible Result of a structure refinement from single crystal neutron diffraction

atomic positions of YBa2Cu3O6.96 orthorhombic, space group type P 2/m 2/m 2/m a = 3.858 Å, b = 3.846 Å, c = 11.680 Å (at room temperature) atom/ion multiplicity site symmetry x y z Cu1/Cu2+ 1 2/m 2/m 2/m 0 0 0 Cu2/Cu2+ 2 m m 2 0 0 0.35513(4) Y/Y3+ 1 2/m 2/m 2/m ½ ½ ½ Ba/Ba2+ 2 m m 2 ½ ½ 0.18420(6) O1/O2- 2 m m 2 0 0 0.15863(5) O2/O2- 2 m m 2 0 ½ 0.37831(2) O3/O2- 2 m m 2 ½ 0 0.37631(2) O4/O2- 1 2/m 2/m 2/m 0 ½ 0 a2 a3

a1 a1

a3 YBa2Cu3O7- YBa2Cu3O7- …seems it‘s…

Time for Lunch? Chapter 4: Diffraction Georg Roth Institute of Crystallography RWTH Aachen University

http://www.ifk.rwth-aachen.de http://www.frm2.tum.de

The „crystal palace“ in Aachen-Burtscheid Outstation of the IfK at FRM II in Garching Outline Diffraction part 2:

• Diffraction of neutrons: general considerations, elastic, coherent scattering • Diffraction geometry reciprocal lattice, Ewald construction, single crystal / powder experiments • Diffraction intensities Fourier transform, structure factors, intensity corrections • Diffractometers single crystal (HEiDi), powder (SPODI), monochromator, filter, resolution functions, neutron Rietveld analysis. Terminology: Scattering of particles / waves

• momentum • momentum changed changed • energy • energy unchanged changed

elastic inelastic

Diffraction Scattering

coherent incoherent

• waves can • waves do interfere not interfere Neutron waves & Neutron scattering:

The three major probes used for investigating the structure of condensed matter compared: Photons, Electrons, Neutrons:

1meV 1eV 1keV

1010 Photons 25 meV (300K) 108 1mm 106 100 nm: colloids 104 Electrons 1 m

102 1Å: atoms

Wavelength [Å] [Å] Wavelength Wavelength Wavelength [Å] Wavelength 1nm 100

10-2 1pm Neutrons 10-4 10-6 10-4 10-2 100 102 104 106 Energy [eV]

A general (elastic) scattering experiment: …be it neutrons, X-rays, electrons, gamma-rays, whatever…

Q k' k k ' detector source k „plane wave“ „plane wave“ 2

2 sample -1 k k  k''  k  k, k’ and Q are in Å  Incoming: wavevector k, diffracted: wavevector k’, Sample: scattering vector Q 4 Q = k’ - k => Q Q  k22  k'  2 kk 'cos2  Q  sin  Note: ħQ represents the momentum transfer during scattering de Broglie: Momentum of the particle corresponding to the wave with wave vector k is given by p=ħk. Elastic scattering from a continuous object: Geometry Elastic scattering from a non-periodic object, illustrating the phase difference between a beam scattered at the origin of the coordinate system (point A) and a beam scattered at the position r (point D).

Phase difference ∆Φ between a wave scattered at the origin of the coordinate system and at position r :  AB CD  2'  k  r  k  r  Q  r  Assumed: Kinematic scattering approximation: …you know this type of argument No refraction, no attenuation, weak from the derivation of Braggs law… scattering (no multiple scattering) Elastic scattering from a continuous object: Amplitude & Intensity

Fourier transform: A A  r  eiQ r d3 r 0  s   VS r: Vector to the scattering volume element in the sample in “crystal space” Q: Scattering vector in “Fourier space” A: Complex amplitude - with an absolute value & a phase of the scattered wave.

…if we had the complete knowledge of amplitude and phase of all scattered beams we could reconstruct the object from its The scattered amplitude A is related scattering data by numerically inverting to the density of the scatterers ρ(s) in the Fourier transform… the object by a Fourier transform, …sadly (luckily for crystallographers), again under the kinematic scattering our detectors are insensitive to the phase approximation 2 All we can observe is: IA~ Coherent elastic scattering:

The scattering amplitude is obtained iQRi from a Fourier transform (see above): AQ bie i

What we can actually observe is 2 d iQ Ri * iQR j the square of the amplitude: Q ~ A Q  bij e b e d  => Cross section / Intensity ij Scattered wave (structure factor) More precisely: as a vector in the complex plane: The intensity is proportional to i the product of the amplitude and its complex conjugate: I(Q)  A(Q) 2  A(Q) A * (Q) A(Q) 2  C  i SC i S S A(Q)  A(Q) 2 C2 S2   r C The diffraction experiment – at a glance: …applies again to neutrons as well as X-rays…

Object: Fourier-transform: Intensity: (observable) direct space reciprocal space reciprocal space crystal structure factor squared structure factor

|F|2 = F.F*) FT (r) F( ) I() FT-1 phase-problem

FT: Fourier-transform FT-1: inverse FT The experiment involves: (r): scatterer density in the unit cell r: vector of x,y,z: Direct space F(): structure factor (complex) : vector of h,k,l: Fourier space I(): integral reflection intensity Elements of structural analysis:

MONOCHROMATOR DETECTOR

SOURCE CRYSTAL

TITL KNO3 CELL .71069 6.449 9.189 5.430 90.000 90.000 90.000 ZERR 4 .005 .005 .005 .005 .005 .005 LATT 1  SYMM -X,-Y, .50000+Z SYMM .50000+X, .50000-Y, .50000-Z h k l INT. SIGMA SYMM .50000-X, .50000+Y,-Z -5 -1 3 476.30 12.21 SFAC N O K -4 -3 5 418.17 13.60 UNIT 4 12 4 -4 -3 6 13.75 15.89 -4 -2 3 3.31 7.96 -1 -4 4 1442.42 14.80 lattice parameters,  space group, cell content ... integral reflection intensities  

K1 3 .75551 .08365 1.25000 10.50000 .02580 N2 1 .58389 .24479 .75000 10.50000 .02452  O3 2 .58958 .10917 .75000 10.50000 .03636 O4 2 .58483 .31361 .94923 11.00000 .03657

crystal structure of KNO3 atomic coordinates Coherent scattering from atoms / nuclei: X-rays & neutrons Oh no! Not this one again!

- Assumed: This time, our “green blop” is an atom. - X-rays: size of the scatterer -10  wavelength used for the - Scattering of thermal neutrons (=10 m) -14 -15 experiment : r    10-10m from the atom’s nucleus (r=10 …10 m): => Increasingly destructive Point scatterer, no intensity fall off interference with increasing - Magnetic neutron scattering from unpaired scattering angle electrons in the (outer) atoms shells: => X-ray formfactor fall off - Spin- and orbital- moments => magnetic formfactor fall off Scattering amplitude vs. atomic number: Neutrons & X-rays compared

X-rays X-rays sin   Å -1 sin   Å-1

Ni Comparison of coherent nuclear scattering cross sections for X-rays and neutrons for selected elements

Area of the colored circles proportional to the coherent scattering cross section. For neutrons: blue: positive scattering length, green: negative scattering length (phase shift of ), red: X-rays, 1 barn=10-24 cm2. Coherent & incoherent scattering

Assumed: Two different atom types or two different isotopes of the same atom type on the same site

k’ k

2 Scattering from the 2 iQ Ri be regular average lattice i  Interference => coherence

+ + Scattering from randomly 2 distributed defects N b b N x    isotropic íncoherent scattering Example: Neutron-scattering lengths of hydrogen isotopes http://webster.ncnr.nist.gov/resources/n-lengths/

Neutron scattering lengths and cross sections of hydrogen scattabs concen- coh b inc b coh  inc  Isotope total scatt   tration [10-15m] [10-15m] [10-24cm2] [10-24cm2] [10-24cm2] [10-24cm2] H --- - 3.7390 --- 1.7568 80.26 82.02 0.3326 (natural)

1H 99.985 - 3.7406 25.274 1.7583 80.27 82.03 0.3326

2H  D 0.015 + 6.671 4.04 5.592 2.05 7.64 0.000519

3H  T (12.32 a) + 4.792 -1.04 2.89 0.14 3.03 0 Diffraction geometry:

1 ..3: collimators

E = 0: Elastic scattering

Essential components of a (neutron) diffraction experiment Diffraction geometry: The reciprocal lattice:

Reminder: The crystal lattice ('direct lattice') is composed of the set of all lattice vectors generated by the linear combination of the basis vectors a1, a2, a3 with coefficients u, v, and w (positive or negative integers, incl. 0).

a = u a1 + v a2 + w a3. The Fourier-transform which is occurring during a diffraction experiment, transforms this direct infinite lattice into the so called

’reciprocal lattice’, with basis vectors 1, 2, 3 and the integer Miller indices h,k,l as the coefficients.

Diffracted intensity I() is only observed at the nodes of this reciprocal lattice addressed by the vectors:

 = h 1 + k 2 + l 3 of the reciprocal lattice. Example: Monoclinic cell a1, a2, a3,  > 90°

. Direct and reciprocal basis

d001 vectors satisfy the following a3 conditions:

3 1a1 = 2a2 = 3a3 = 1 . d 100 This means that |ai| = 1 / |i|  and

* 1a2 = 1a3 = 2a1 = ... = 0 1 a2||2 a1 This means that each i is perpendicular to aj and ak: * = * =  =  = 90°  = (a a )/V * = 180°–  i j k c with V = a (a a ) as the dhkl: lattice spacing for the c 1 2 3 set of lattice planes (hkl) volume of the direct cell So: What is the reciprocal lattice good for? A neat way of representing diffraction geometry: Ewald construction: Ewald construction for diffraction from single crystals:

Why do we have to do angular scans across reflections?

The crystal is again represented by its reciprocal lattice 

…see the measuring mode at HEiDi: ω-scans...

Similar arguments apply to the wavelength spread of the incoming neutrons Ewald construction for diffraction from powders: Origin of the diffraction cones:

Reciprocal lattice points degenerate to reciprocal lattice spheres for each reciprocal lattice point (hkl)…

Intersection of Ewald- and reciprocal lattice-spheres => diffraction cones

…see measurements on SPODI. Need a brake? Diffraction intensities: The diffraction experiment – once again:

Object: Fourier-transform: Intensity: (observable) direct space reciprocal space reciprocal space crystal structure factor squared structure factor

|F|2 = F.F*) FT (r) F( ) I() FT-1 phase-problem

FT: Fourier-transform FT-1: inverse FT The experiment involves: (r): scattering density in the unit cell r: vector of x,y,z F(): structure factor (complex) : vector of h,k,l I(): integral reflection intensity Diffraction intensities: Reminder: Experiment corresponds to a Fourier-transform followed by taking the square of the complex amplitudes Simple 1D-Analog: The structure factor for nuclear neutron diffraction:

The structure factor F() is the Fourier transform of the scattering density within the unit cell. It contains the complete structural information, including the atomic coordinates rj = xj a1 + yj a2 + zj a3, site occupations and the atomic displacements (thermal vibrations, Debye-Waller factor, contained in Tj ).

F() = 푗 푏푗·(exp[2i(·rj)]) ·Tj() = |F()|·exp[iφ()].

The summation (index j) runs over all atoms in the unit cell.

With the fractional coordinates xj, yj and zj, of each atom j and the Miller indices of the reflection h,k,l, the scalar product in the exponential can be written as

  rj = hxj + kyj +lzj

To re-iterate, our observable quantity is: I()  |F()|2.

…magnetic neutron scattering will be discussed elsewhere… Atomic displacement parameters: “Temperature factor”:

This is what you know from X-ray diffraction: Displacement of atom positions means:

rj  (rj +rj)

exp [2i( ·rj)]  exp [2i( ·(rj +rj))]

= exp [2i( ·rj)]  exp [2i( ·rj)] Assumed: Atoms are independent harmonic oscillators

 Normal distribution of displacements rj: 1 n 2 2  rj = 0, but rj  =  rj  0 n 1

exp [2i( ·rj)] = exp [2i  ·rj] Thermal displacements in Sb(C5H5)3 Ellipsoids: 50% occupation probability   ·rj  averaged in space and time

2 2 2 2 2 2 < ·rj =  rj  = 1/ 4sin rj 

Temperature factor:

2 2 2 2 Tj( ) = exp -[1/ 8 rj sin ] 2 Random, independent thermal displacements With the mean square displacement Uj = rj  of atoms from their average position 2 2 2  Tj( ) = exp -[1/ 8 Ujsin ] 2 or with the Debye-Waller-Factor Bj = 8 Uj 2 2  Tj( ) = exp-[Bj·1/ ·sin ] Atomic displacements: “Debey-Waller factor”: Effect: Reduction of the intensities at large Q (large scattering angles) 2 2 Tj( ) = exp-[Bj·1/ ·sin ]

Neutrons, B=0

Neutrons, B=0.5

X-rays, B=0

X-rays, B=0.5 Symmetry determination: Now that we know there are • 32 different crystallographic point groups and • 230 different crystallographic space groups: How can we distinguish / determine them? • Mainly from diffraction experiments • But also by measuring physical properties • Still: Determination usually not unique! Ingredients: Crystal system: Usually already evident from lattice metric e.g.: a=bc, ===90° => tetragonal Laue class: Symmetry of the intensity data set e.g.: I(hkl)=I(-khl)=I(-h-kl)=I(k-hl) => 4-fold axis Systematic extinctions: Symmetry elements with translations e.g.: observed: (hkl), h+k+l=2n => I-centered lattice By diffraction methods only the 11 Laue classes can be directly determined and not all 32 crystal classes. Diffraction experiments always “mimic” a centre of symmetry!

Crystal system Laue-class Crystal class m3 m 432, 432, m3m Cubic m3 23, m3 6/m 2/m 2/m 622, 6mm, 62m, 6/m2/m2/m Hexagonal 6/m 6, 6, 6/m 3m 32, 3m, Trigonal 3 3, 4/m2/m2/m 422, 4mm, 4 2m , 4/m2/m2/m Tetragonal 4/m 4, 4 , 4/m Orthorhombic 2/m2/m2/m 222, 2m, 2/m 2/m2/m Monoclinc 2/m 2, m, 2/m

Triclinic 1 1, Laue-class plus extinctions  diffraction symbol  Space group determination (often NOT unique!) Example: Monoclinic crystal system, unique axis b  Laue-class 1 2/m 1. Extinctions (for allowed reflections) from experiment: Integral condition: h, k, l: h + k = 2n  C-centered lattice type Zonal condition: h, 0, l: h = 2n  a(010), contained in C-centering l = 2n  c(010), c-glide plane 0, k, l: k =2n  b(100), contained in C-centering h, k, 0: h + k = 2n  n(001), contained in C-centering Seriell condition: h, 0, 0: h = 2n  contained in C-centering 0, k, 0: k = 2n  contained in C-centering 0, 0, l: l = 2n  contained rule for c-glide plane monoclinic Lattice Symmetry elements (along Laue-class unique axis b type directions of view) diffraction symbol 1 2/m 1 C 1 c 1  2 possible space groups: C1c1 (9) or C12/c1 (15) The structure factor and intensity corrections: A number of corrections have to be made to go from |F()|2 to I(). Most of them apply equally to X-ray or neutron diffraction and to powder or single crystal experiments

I(| |) = SCALE  L  A  E  M  P  |F(| |)|2

L: Lorentz factor: Instrument specific geometric correction A: Absorption correction (usually small for neutrons, expt.: 157Gd, 10B, 113Cd etc.) E: Extinction coefficient takes into account a possible violation of the assumed conditions for the application of the kinematical diffraction theory M: Multiplicity of reflection hkl, specific to powders: counts how many symmetry equivalents a lattice plane has, e.g.: M111=8 (octahedron), M100=6 (cube) P: Preferred orientation, specific to powders: corrects the intensities for deviations from the assumption of randomly oriented crystals in the powder sample. Overview: Experimental hall FRM II HEiDi / SR9

SPODI SR9: The only beamport looking at the hot source of FRM II Diffractometers: Single crystal neutron diffractometry (e.g. HEiDi)

4-circle diffractometer:

Disclaimer: There are MANY other concepts of diffractometers (powder and single crystal) out there! …other geometries, polychromatic instead of monochromatic beams, time of flight (TOF) machines… …not enough time to properly account for them! HEiDi: General Description

. 4 circle single crystal diffractometer . In user operation since 2005 . In JCNS instrument pool since 2012 . operated for JCNS by RWTH Aachen, Institut für Kristallographie (G. Roth)

HEIDI: Instrument scheme

Instrument team HEiDi (Meven, Sazonov, Luberstetter)

. Hot Source MLZ: High neutron flux at SR9B λ = 1.17 Å, Ge(311): 1.4 * 107 n/s/cm² HEIDI: Eulerian cradle with cryostat inserted λ = 0.55 Å, Cu(420): 2.5 * 106 n/s/cm² (Factor 10!) HEiDi: Single crystal diffractometer at the hot source of FRM II: HEiDi: „Heißes Einkristall Diffraktometer“

Solution: Hot source @ FRM II

Specification: – graphite block – mass: 14 kg – H x B: 300 * 200 mm² – T(core): 2400°C – T(surface): 2030°C Hot neutrons: 0.2 – 1.1 Å Thermal neutrons: 0.8 – 3.0 Å Cold neutrons: 2.0 – 20 Å Flux-gain > factor of 10 at 0.6 Å.

Thermal spectrum: λmax = 1.8 Å → small flux at small wavelengths Monochromators: Example: HEiDi

Focussing monochomator Monochromator-mounting: • Three different monochromator-crystals

2QM Ge-(311) Cu-(220) Cu-(420) 20° 0.593 0.443 0.280 40° 1.168 0.870 0.552

50° 1.443 1.079 0.680

• Active area per crystal 120*100 mm² • Crystals consist of 7 lamella each • minimum vertical radius of curvature 0.5 m

• Flux-gain at sample position max. 3,5 • x, y, z alignment • max. Flux ca. 9*106 n/s/cm² (1.1 Å) • three angular alignments. The λ/2-problem:

The Problem: …trouble is: The neutron spectrum is continuous

It reflects the velocity spectrum of the Bragg’s law is not only fulfilled for the moderator-atoms at temperature T: fundamental wavelength but also for λ/2, λ/3…

. sinθ = n λ / 2dhkl = (λ/2)/2d2h 2k 2l = (λ/3)/2d3h 3k 3l

 reflections supposedly measured at λ are contaminated by contributions from λ/2, λ/3…

 Even worse: if reflection 2h2k2l is strong, a systematically absent hkl appears to be present

from Furrer et al. 2009  false superstructure reflections, false Particularly with the hot source, there unit cell dimensions, false symmetry, is quite some flux also on λ/2, λ/3… false structure, everything false… So what...? Neutron filters: Be, Graphite, Er…

The Solution: Take a material, that has a large transmission at λ but a small transmission at λ/2, λ/3…

But how?

- By absorption, e.g. Erbium-filter (Er): Some rare earth elements like Er have a pronounced resonance absorption in the range 0.5 to 1.5 Å (kind of similar to Kβ-filters in the X-ray case)

- By scattering, e.g. Be- or Graphite-filter: Polycrystalline filter material scatters neutrons out of the beam For large λ: The Ewald-sphere is so small, Transmission through a graphite that no hkl can be brought into reflection filter as a function of thickness. => no scattering, high transmission (from Furrer et al. 2009). For small λ: Intense Bragg scattering from the powder, low transmission The resolution function: Example: HEiDi

Important characteristic of any diffractometer: Angular resolution.

Resolution function: Reflection half width) as a function of scattering angle for HEiDi Resolution depends on: Experimental setup of HEiDi: - Primary & secondary collimation Right: Collimator - Monochromator type and quality Center: Eulerian cradle - 2 and (hkl) of the monochr.-reflection Left: Detector. Powder diffractometry: Example: SPODI@FRM II:

Sketch of the principal setup of a Actual experimental setup of powder powder neutron diffractometer. neutron diffractometer SPODI. PSD: position sensitive detector Neutrons come from the upper right Monochromator box at lower right PSD at center. The fundamental problem with powder diffraction (Neutrons & X-rays) data: Loss of information:

Single crystal diffraction data is 3D: • Reflection intensity and position on a 3D lattice: ’reciprocal lattice’ • Phases of the reflection are lost anyhow (see phase problem) The powder experiment projects all 3D-reflections onto 1 dimension: • Reflection intensities as a function of scattering angle  • Overlap of reflections appearing at similar  • Most peaks correspond to overlapped reflections: more than 1 (hkl) • Basic requirement for “ab-initio” structure determination not fulfilled • Classical structure determination (almost) impossible • Still feasible: Refinement of an sufficiently good approximate model. • To retrieve as much information about partially overlapped reflections as possible: Refection profile fitting

=> Rietveld refinement Diffractometers: Neutron Rietveld analysis

Results of a Rietveld refinement at the magnetic phase transition of CoGeO3 red: measured intensity, black: calculated from model, blue: difference, green: tick-marks at allowed reflection positions The figure shows the low-angle part of two diffractograms measured at SPODI at 35K and 30K. Note the strong magnetic reflection appearing below the magnetic ordering transition (in the inset). Outlook: Chapter 8: Structural analysis: Applications of neutron diffraction:

• Diffraction contrast variation

Site occupation and magnetic phase diagram of (Mn1-xCrx)1+Sb H ordering: Ferroelectric phase transition in RbH2PO4 (RDP)

• Accurate atomic coordinates and displacement parameters from

neutron diffraction: Co2SiO4

• Magnetic structures from neutron diffraction: Co2SiO4

• Bonding electron density determination from combined X-ray and

neutron diffraction: Co2SiO4

• Magnetization density distribution within the unit cell from neutron

diffraction: Co2SiO4