Short-‐course on symmetry and crystallography. Part 4: Fourier transform,.
Diffrac5on pa(erns. Michael Engel. Ann Arbor, June 2011 ...
Short-‐course on symmetry and crystallography Part 4: Fourier transform, Diffrac=on pa>erns Michael Engel Ann Arbor, June 2011
Famous diffrac=on images
Max von Laue: First X-‐ray diffrac=on (1912)
Rosalind Franklin: DNA structure (Photo 51, 1952)
Dan Shechtman: Quasicrystal (1984)
Purpose of diffrac=on experiments: • Solve crystal structure (mostly with X-‐rays) • Study crystal dynamics (with neutrons) and defects • Understand local structure (with electrons), in combina=on with electron microscopy
Types of sca>ering experiments • X-‐ray sca)ering (Nobel prize 1912): By Thompson sca>ering from electrons (good for heavy atoms). • Electron sca)ering (Nobel prize 1937): By charged par=cles (protons, electron clouds). By Lorentz force if magne=c field present. • Neutron sca)ering (Nobel prize 1994): By the atomic nuclei (good for light atoms). By magne=c fields of uncharged electon pairs.
Wavelengths … should be in the range of the structure to be inves=gated (i.e. hundred picometer or Ångstrom). à X-‐rays, 1keV electons, thermal neutrons à Light sca>ering is possible from colloidal crystals, from laser gra=ngs, and from photonic and phononic crystals
Single-‐slit experiment
Double-‐slit experiment
Experimental setup The interac=on of the wave (X-‐ray, electrons, neutrons) with ma>er is locally described by the sca>ering power ρ(x), which we will call the distribu)on of ma0er in the following. Incoming wave vector: qin (or k0) Outgoing wave vector: qout (or k1) Transferred wave vector:
Δq = qout - qin Note: We assume elas@c sca)ering
Sca>ering • The wave amplitude as a func=on of the transferred wave vector is given by the Fourier transform: � ρ(q) = d3 xe−iq·x ρ(x) • The wave intensity is equal to: S(q) = �ρ(q)�2 This quan=ty is known as sta@c structure factor or the diffrac@on image. • [Note: The dynamic structure factor S(q, ω) is obtained by a four-‐dimensional Fourier transform from the =me-‐dependent density ρ(x,t). It can be measured in inelas=c Neutron diffrac=on experiments.]
Elementary proper=es of the Fourier transform
Symmetry and Fourier transform • If {A,b} is a symmetry of the crystal, then A is a symmetry of the Fourier transform: ρ(x) = ρ({A, b}x) ⇔ ρ(q) = exp[i(Aq) · b]ρ(Aq)
• If {A,b} is a symmetry of the crystal, then ±A is a symmetry of the diffrac=on image. • à The symmetries of the diffrac=on image are equal to point symmetries “up to inversion”. [The absolute square norm always makes the inversion a symmetry of the diffrac=on image]. • The symmetry group of the diffrac=on pa>ern is called the Laue group.
Reciprocal lajce • The reciprocal lajce is defined as:
• The reciprical lajce is the dual la7ce. • The Voronoi cell (= Wigner-‐Seitz cell) of a given lajce point consists of all points in space that are closer to this lajce point than to any of the other lajce points. • The Voronoi cell (=Wigner-‐Seitz cell) of the reciprocal lajce is called the Brillouin zone. Both are dual to each other.
Examples of Brillouin zones
Self-‐dual
ß Dual à
Decomposi=on • The integral can be decomposed into a convolu=on consis=ng of an integral over the unit cell Z and a sum over the lajce Γ: � � �� � � ρ(q) = e−iq·b ∗ d3 x0 e−iq·x0 ρ(x0 ) t∈Γ
Z
• The first term has delta peaks on a lajce, the second has finite support (i.e. it is zero outside of a finite region of space). • According to the convolu=on theorem, we can then write the Diffrac=on pa>ern as a product: �� � �� � � � � � � � S(q) = � F e−iq·b F d3 x0 e−iq·x0 ρ(x0 ) � � � Z t∈Γ
Dirac combs • A (1D) Dirac comb is a periodic (Schwartz) distribu=on constructed from Dirac delta func=ons: for some period T. • The Fourier transform of a Dirac comb is a Dirac comb:
• A 3D Dirac comb is a (periodic) la7ce of Dirac delta func)ons. • The Fourier transform of a 3D Dirac comb is a Dirac comb on the reciprocal la7ce.
Finite support • Consider a func=on with finite support Z [i.e. there is a radius r0, such that Z completely contained in the sphere with this � radius]: ρ(x) for x ∈ Z, ρZ (x) = 0 otherwise. • The Fourier transform of this func=on is oscilla=ng and rapidly decaying, for example the rectangle func=on:
←F →
Form factor • The Fourier transform of the sca>ering power around a single atoms/par=cles etc. is called the form factor: It describes the sca>ering from the atom/par=cle. • The Fourier transform of the “unit cell content”, i.e. the sum of all the par=cles in the �unit cell, is: fZ (q) = F d3 x0 e−iq·x0 ρ(x0 ) Z
Pujng it all together Let’s look again at the diffrac=on image: �� � �� � � � � � � � 3 −iq·b −iq·x0 S(q) = F e F d x e ρ(x ) � 0 0 � � � Z t∈Γ Now we know that sca>ering of a crystal can be understood as follows: 1. A set of Bragg peaks posi=oned on the reciprocal lajce, given by the first part of the equa=on above. 2. A decay of the peak intensi=es is determined by the second part of the equa=on, basically the unit cell content. Thermal vibra=on lead to a decay of the Bragg peak intensity (à Debye Waller factor).
Ex=nc=on condi=ons • For a periodic crystal: where: * the sum is over all the atoms/par)cles in the unit cell, * fj is the form factor of the j-‐th atom/par=cle, * (xj,yj,zj) its posi=on, * and (h,k,l) is a point of the reciprocal lajce. • The decora=on of the unit cell leads to ex=nc=on condi=ons. • For example:
Finite sample size If the sample is finite, then sca>ering happens only from a finite region A. This means: �� � �� � � � ρ(q) = χA e−iq·b ∗ d3 x0 e−iq·x0 ρ(x0 ) Z t∈Γ here χA is the characteris=c func=on: � 1 for x ∈ A, χA (x) = 0 otherwise. The diffrac=on pa>ern is then: 1. A set of Bragg peaks posi=oned on the reciprocal lajce. 2. A decay of the peak intensi@es from thermal mo=on/form factor. (Debye Waller factor) 3. A broadening of the peaks due to the finite sample size. 4. (see later) Diffuse sca)ering in the background.
Far-‐field approxima=on Small-‐angle sca>ering Incoming wave vector: qin (or k0) = (0, 0, qin) Outgoing wave vector: qout (or k1) = (qx, qy, qz)
Assume: (i) Elas=c sca>ering and (ii) sca>ering angle small Then: Transferred wave vector: Δq = qout ‒ qin ≈ (qx, qz, 0)
�� �2 � � Structure factor: 3 −i(q x+q y) x y � S(qx , qy ) = � d xe ρ(x) � �
Numerical details of the diffrac=on image calcula=on (as done in “injavis”) Parameters: viewing direc=on (i.e. z), Grid size (2n x 2n) Repeat steps 1-‐3 for every atom/par=cle: • Step 1: Project the atom along z. • Step 2: Map the projected atom posi=on into a square box by a shear opera=on following periodic boundary condi=ons. • Step 3: Resize and discre=ze on grid by placing Gaussian. Next: • Step 4: Fourier transform the grid data. • Step 5: Undo the shearing.
Example • Interac=ve simula=on and diffrac=on of the square lajce.
Diffrac=on reciprocity “Big in real space” means “small in Fourier space” “Small in real space” means “big in Fourier space” Hexagonal superstructures of twin boundaries
Spectral theory (3D) Mathema=cal treatment of sca>ering: Spectral theory à Fourier transform is a unitary operator à Look for eigenvalues: plane waves à Spectrum (= distribu=on of eigenvalues, i.e. structure factor) has three parts: (i) point spectrum (Bragg peaks) (ii) con=nuous spectrum à In general there is a singular spectrum, but it is “not physically relevant” Peak width: • Sharp (Bragg) peaks means long (infinite) correla=on length in real space • Broadened peaks means finite correla=on length
Correla=ons in arbitrary dimensions Start with the N-‐par=cle Hamiltonian (harmonic approxima=on): The sta=c structure factor (absolute square of the Fourier transform) can be wri>en as: � N � 3 −iq·x −iq·xj ρ(q) = d xe ρ(x) = e j=1 N or � 1 2 2 ρ(q) = e−iq·Rl0 exp(− 2d q σl (0)) l=1 ul (t) = xl (t) − Rl with the dislacements: and the correla=on func=on: σl2 (t) = �[ul (t) − u0 (0)]2 �
Displacement correla=on func=ons General solu=on of the equa=ons of mo=on are a linear combina=on of harmonic modes: If we put this in σ l2 (t) = �[u l (t) − u 0 (0)] 2 � then we get: With the dispersion rela=on: Note: Use the correla=on func=ons for amplitudes:
Infinite system size (hydrodynamic) limit For N → ∞ : And in the con=nuum limit l → ∞ : Exponen@al decay short-‐range order Algebraic decay “Smeared peaks” No decay Bragg peaks N � −iq·Rl0 1 2 2 ρ(q) = e exp(− [Remember: 2d q σ l (0)) ] l=1
Thermal diffuse sca>ering • Sharp peak = long correla=on length • Smeared out peak = reduced correla=on length At low temperature (room temperature = 25 meV) only long wave length phonons are excited à long correla=on length Phonon band structure of wurtzite-‐type GaN
Diffuse sca>ering
Several origins: • Thermal diffuse sca>ering • Diffuse sca>ering from defects • Random =lings have diffuse sca>ering All have in common reduce correla3on length! See: neutrons.ornl.gov/conf/nxs2009/pdf/Gene_Ice_Diffuse09.pdf Example: Bragg contrast imaging of disloca=ons:
Examples of diffuse sca>ering X-‐ray image of diffuse structure of a natural mordenite single crysta.l [J. Appl. Cryst. (2004). 37, 187–192]
X-‐ray diffuse sca>ering in (CH3)4NCdCl3, different reciprocal layers. (A. Pietraszko)
Welberry et al. ISIS
Two-‐dimensional high-‐resolu=on small-‐angle X-‐ ray sca>ering pa>erns of an aligned ca=onic lipid -‐ DNA complex: a) in the Lß phase, and b) in the L phase. [Science, 275, 810 (1997).]
Grain boundaries • If the sample consists of several grains, then the diffrac=on pa>ern is a linear combina=on of the diffrac=on pa>erns of all grains • Special grain boundary: Crystal twinning occurs when two separate crystals share some of the same crystal la7ce points in a symmetrical manner. Twinned pyrite crystal
Five-‐fold twinning of gold nanopar=cle
Twin of austenite steel
Fractal =lings
Random )lings have “cool” diffrac)on pa0erns: A chair =ling (top right), its diffrac=on image (right), a table =ling (top, blue) and its diffrac=on image (lew) [Dirk Fre>löh, Bielefeld]
Kevin’s random =lings
What you should (hopefully) be able to do NOW: Determine a point group of an object by hand Determine a space group using internet resources Understand symmetry nota=ons in papers Read Wikipedia ar=cles like “Group_(mathema=cs)”, “Space_group”, … • Extract the informa=on contained in diffrac=on pa>erns • • • •
