Diffraction:
Scattering of light on a 3D grating and diffraction of X-rays on a crystal use the same analytical
description with the same equations.
Bragg law:
Bragg (father and son) used the analogy of the reflection of visible light from a family of planes – set
of Bragg planes. Thereby you can imagine the crystal as composed of thousands of mirrors – Bragg
planes – that reflect the incoming X-rays. Therefore diffraction spots are also called reflections.
2dsin = n
d = the distance between the individual planes of the set of Bragg planes.
 = the angle of diffraction/incidence (between the incident/outgoing X-ray and the corresponding
Bragg plane)
In practice a maximum angle exists above which no diffraction is observed anymore. This angle is
related to the resolution 2 (the level of detail of the final crystal structure).
nℤ
 = Wavelength of the incident X-ray
d = n/(2sin)
Concept of reciprocal space: The distance between the Bragg planes decreases as the angle 
increases.
-  = 0°: sin = 0 and d = ∞ => no diffraction
-  = 90°: sin = 1 and d = n/2
Changing the angle  can be achieved by rotating the object or changing the angle of the incoming
radiation itself.
A single diffraction event tells us about the position of an object relative to the Bragg planes:
- Object on plane => object diffracts in phase
- Object between planes => object diffracts out of phase
The resulting phase shift of a wave diffracting from an object is proportional to the distance of the
object from one Bragg plane as a fraction of the distance from the first Bragg plane to the next.
Mathematically it is the dot (scalar) product between the position vector r of the object and the
plane normal with length equal to the reciprocal of the Bragg spacing (= reciprocal space vector S ).
r*S = |r||S|cos with the angle  between the two vectors.
- |r|cos = the projection of r on S perpendicular to the Bragg planes.
- |S| = d* = 1/d so that r*S is the fraction of the distance from one Bragg plane to the next that the
vector r has travelled from the origin.
Since r = xa + yb + zc and s = ha* + kb* + lc* the dot product r*S = (xa + yb + zc) (h/a + k/b + l/c) = hx +
ky + lz
Therefore:
- Bragg plane passing through the origin: A wave diffracted from this plane has a phase of zero.
- Next Bragg plane: A wave diffracted from this plane has a phase of 2π = 0
- Any point in between the planes: A wave diffracted from this point has a phase of 2πr*S
Since changing the angle  changes the distance d between the Bragg planes, two objects at different
positions in the crystal can scatter in or out of phase depending on the angle of incoming radiation.
- All objects sat on planes -> very strong diffraction
- Half of the objects on planes, other half between the planes -> no diffraction
Therefore the intensity of a wave scattered on an atom decreases as the angle  increases.
The crystal works as an amplifier
In directions where the unit cells diffract in phase it amplifies the diffraction pattern.
(In all other directions it eliminates the diffraction pattern.)
<= The unit cell diffracts in phase if the Bragg planes pass through the same points in all unit cells of
the crystal (depends on θ). That means they are separated by one unit cell edge or an integral
fraction(i.e. one-third) of it. Bragg planes have to cut all cell edges an integral number of times.
Otherwise the planes divide at least one unit cell edge by a non-integral number and therefore don’t
pass through equivalent atoms in the different unit cell. The same atoms would in one unit cell sat on
a Bragg plane and in the next unit cell somewhere between the planes. There would be no
diffraction, no contribution of this atom to the diffracted rays. Summarized: the different unit cells
will all diffract out of phase and waves will cancel out.
Miller indices (h,k,l) indicate how often each unit cell edge is cut. A Bragg plane (4 2 7) cuts a four
times (h), b two times (k) and c seven times (l). I(4 2 7) is the corresponding X-ray reflected by this
plane: F(hkl) = |F|e^(i). Miller indices can also be negative (next plane in counterdirection: 1 -1 0
means one unit cell edge back in b direction)
- 1 0 0 plane is divided by d = a and are parallel to bc planes (so that it doesn’t cut them).
- 1 1 0 plane is parallel to c and 1 1 1 plane isn’t parallel to any unit cell edge.
Concept of reciprocal space:
The higher the angle , the closer the planes (the smaller d), the higher the miller indices.
=> finer sampling along the unit cell edges
=> For one direction with a certain θ-value the unit cells scatters out of phase and for another with
another θ-value the unit cells scatters in space.
Reciprocal lattice/space
The reciprocal space/lattice vector = the (Bragg) plane normal with length equal to the reciprocal of
the Bragg spacing (1/d = d*).
The reciprocal lattice/space is defined by the reciprocal space vectors of any possible set of Bragg
planes:
1. Choose an arbitrary origin O (for both the real and the reciprocal lattice).
2. Imagine the planes (110), (120), (130), … through the neighboring lattice point N.
3. Draw the reciprocal space vector of each plane beginning at the origin:
The bigger the miller indices the smaller d the longer the corresponding reciprocal space vector.
=> The points at the other end of the reciprocal space vectors lie on a straight line. They are points of
the reciprocal lattice.
4. Repeat for the planes (210), (310), (410), ….
The resulting reciprocal lattice points lie on a line perpendicular to the firstly formed line.
5. Continue with each possible plane (hkl) and form each possible point (hkl).
> A point in the reciprocal lattice is described with Miller indices (hkl)
> A point in the real space is described with fractional coordinates (fractions of the unit cell axes with
regard to the unit cell origin).
Concept of reciprocal space:
The higher the angle , the closer the planes (the smaller d), the higher the miller indices.
=> finer sampling along the unit cell edges
The smaller d, the bigger d* => The reciprocal lattice points are widely spaced and vice versa.
The reciprocal unit cell with the unit cell edges a*, b* and c*:
- a* lies along a (along the scattering S for reflection hkl = 100), length = 1/a
- b* lies along b (along the scattering S for reflection hkl = 010), length = 1/b
- c* lies along c (along the scattering S for reflection hkl = 001), length = 1/c
=> V* = 1/V
Any point in the reciprocal lattice can be defined as the sum of integral multiples of the reciprocal
axes: s = ha* + kb* + lc*
Ewald sphere/construction:
r = 1/ = length of all depicted beams, origin of the crystal
1: incoming beam
2: unaffected beam (parallel to/elongation of the incoming beam – passing the crystal)
3: diffracted beam (in any direction in 3D -> tip anywhere in the spere)
Angle between 2 and 3 = 2 = 0° - 180°-> imaging a Bragg plane between the beams with an angle 
from each beam.
The reciprocal space vector runs from the tip of 2 to the tip of 3 (OP – the subtraction of both):
(Ewald) OP/(2/) = sin and =2dsin (Bragg with n = 1), sin = OP/(2/2dsin) = OP/(1/dsin) =
OP*dsin
=> OP = sin/(dsin) = 1/d is d*.
A reflection occurs only when Bragg’s law is satisfied. Introducing the Bragg condition into the Ewald
derivation shows that OP equals d*. When we choose O as arbitrary origin P is a reciprocal lattice
point formed by the reciprocal space vector of a corresponding set of Bragg planes with the spacing
d. Therefore reflection includes that P is a reciprocal lattice point. Every time a reciprocal lattice point
lies on the Ewald spere it originates a diffracted ray. => Diffraction is confined to reciprocal lattice
points. Planes of this points intersect the Ewald sphere to give a circle of points fulfilling the
diffraction condition.
Proof that Ewald sphere is a graphical representation of Bragg’s law:
Angle between 2 and 3 = 2 = 0° - 180°-> Imaging a Bragg plane between the beams with an angle 
from each beam. The Bragg plane cuts the reciprocal space vector in two halfs forming two internal
right angled triangles: sin = 1/2d : 1/ -> sin = /2d ->  = 2dsin q.e.d.
Rotation methode:
Rotating the crystal rotates the reciprocal lattice through the Ewald sphere producing many
reflections seen as lunes in the diffraction pattern.
- Before rotation reciprocal lattice points which meet the Ewald sphere are found on planes
perpendicular to the X-ray beam (circles of spots surrounding the direct beam).
- Upon rotation of the reciprocal lattice this planes are not perpendicular anymore. But new lattice
points meet the Ewald sphere. (circles of spots forming lunes).
Scattering on two electrons, one atom, the unit cell
Scattering on two electrons:
Two electrons:
- e1 at the origin
- e2 at a position r with respect to e1
Ewald sphere shows the reciprocal space vector S of the diffraction on e2. We can calculate its phase
2 = 2πrS. The phase of e1 is 1 = 0.
The sum of the two waves scattered from the two electrons is T = |c1|e^(i0) + |c2|e^(i2πrS) = |c1|
+ |c2|e^(i2πrS)
Sum of waves scattered by different electrons: F(S) = Σi exp(2πriS)
Scattering factors are expressed in terms of number of electrons:
A single electron at a point has an amplitude of one electron F(S) = 1e exp(2πirS).
Electron at the origin: scattering factor of 1 electron because phase = zero
Scattering on an atom:
An atom:
- origin in the nucleus
- Electron cloud of the atom scatters the X-ray beam dependent on number and position of electrons
in the cloud. The electron density at the position r is ρ(r). Since the cloud is centrosymmetric around
the origin ρ(r) = ρ(-r).
Atomic scattering/form factor:
f = ∫𝑟 𝜌(𝑟)𝑒 2𝜋𝑟𝑆 dr
- replace summation by an integral
- Instead of an electron at a point we now use a continuous function, the electron density ρ(r)
Scattering by an atom is always real:
If we assume centrosymmetry of the electron cloud the imaginary part of every scattering vector is
compensated (in the integral) by the imaginary part of a vector with equal length but a phase angle
of opposite sign. The resulting vector of the scattered wave is directed along the real axis in the
Argand diagram.
The sum of the two waves is the
double value of their real
component.
Ci (xyz -> x-y-z): f = ∫𝒓 𝝆(𝒓)(𝒆𝟐𝝅𝒓𝑺 + 𝒆−𝟐𝝅𝒓𝑺)dr = 2 ∫𝒓 𝝆(𝒓) 𝐜𝐨𝐬(𝟐𝝅𝒓𝑺)dr
because exp(i2πrS) = cos(2πrS) + i sin(2πrS) and exp(-i2πrS) = cos(2πrS) - i sin(2πrS)
A = Σficos[2π(hxi + kyi + lzi)] + Σficos[2π(-hxi - kyi - lzi)] = 2 Σficos[2π(hxi + kyi + lzi)] since cos(x) = cos(-x)
B = Σfisin[2π(hxi + kyi + lzi)] + Σfisin[2π(-hxi - kyi - lzi)] = 0 since sin(x) = -sin(-x)
 = arctan(B/A) = 0, π
The atomic scattering factor is independent of the direction of S but dependent on its length:
|S| = (2sin)/ see Ewald sphere and Bragg’s law: 2dsin =  and |S| = d* = 1/d = (/2sin)^-1
You can plot the atomic scattering factor against |S| = (2sin)/. By this way you express f as
electron number:
f(carbon atom,  = 0) = 6 because of 1s^2 2s^2 2p^2
f(carbon atom,  > 0) < 6
That is because the X-rays scattering at different points (electrons) in the atom interfere with each
other.
- low scattering angle: d = /(2sin) >> atom dimensions => Electrons will scatter in phase.
- higher scattering angle: Different parts of the electron density distribution will scatter partly out of
phase.
𝑠𝑖𝑛 2
−𝑏 ∗(
)
f°(sin/) = ∑4𝑖=1 𝑎𝑖 ∗ 𝑒 𝑖  + c
9-parameter equation by Don Cromer and J. Mann
We can calculate f() if we know the nine parameter, a(i), b(i) and c as well as the wavelength.
The atomic scattering factor as function of (2sin)/ can be looked up in international tables of
crystallography as well as the diffraction pattern of an atom centered on the origin. It depends on the
type of isolated atom.
Summarizing the atomic scattering factor is
- proportional to the number of electrons
- inversely proportional to the angle of diffraction 2θ
- real and positive if normal scattering
- has a complex component if anomalous scattering
Thermal motion of atoms:
Atoms vibrate around an equilibrium position in successive unit cells. You can think of an X-ray
meeting a smeared atom an a fixed position. The size of the atom is larger if the thermal vibration is
stronger.
The intensity of the scattered X-ray is diminished especially at high scattering angles. That is because
vibration perpendicular to a reflecting plane (hkl) causes diffraction out of phase (as atoms in the
plane would diffract in phase). Therefore we multiply the atomic scattering factor by a temperaturedependent factor.
We use the isotropic thermal factor or Debye-Waller temperature factor B(iso) = 8(π^2)(<u>^2)
because we imagine the simple case of isotropic vibration that includes that the component
perpendicular to the reflecting plane and thus along S is equal for each (hkl).
- u is the mean amplitude of displacement
fB = f°(sin/) * 𝑒
−𝐵(
𝑠𝑖𝑛 2

)
The internal disorder in a crystal and the thermal motion have the same effect on diffraction
intensity.
Scattering by a Unit Cell:
You can imagine a unit cell as a group of n atoms at the positions rj (j ℕ) with respect to the origin
of the unit cell. But the atomic scattering factor of each atom of the unit cell chooses its nucleus as
origin. Transferring this origin to the origin of the unit cell changes the phase by 2πrjS:
fj new= fj exp(2πirjS) = a vector in the Argand diagram.
The total scattering by the unit cell in a certain direction (-> reflected from a certain Bragg plane with
the reciprocal space vector S) is expressed by the structure factor F(S). It depends on the
arrangement (structure) of the atoms in the unit cell:
F(S) = ∑𝑛𝑗=1 𝑓𝑗 exp(2𝜋𝑖𝑟𝑗 𝑆) = |F|exp(i) = A + iB
- summation over all (n) atoms (j)
- amplitude |F|
- phase  with respect to the origin
- A = Σfjcosj = Σfjcos[2π(hxj + kyj + lzj)]
- B = Σfjsinj = Σfjsin[2π(hxj + kyj + lzj)]
We could calculate A and B (and thereby the structure factor) if we would know the coordinates of all
atoms in the unit cell (the complete structure of all molecule in the crystal – asymmetric unit).
- xyz are the fractional coordinates of atom i with regard to the origin of the unit cell.
- hkl are the Miller indices determining the direction of the scattered beam. Diffraction from a crystal
cancels out in all directions except those specified by the Miller indices.
Summarizing we could say that the structure factor is the sum of contributions from individual
electrons or the sum of contributions from the electrons comprising individual atoms. Every atom
contributes to every diffracted ray. Therefore every scattered beam is a sum of contributions of all
the beams that are scattered on individual atoms.
Since the atomic scattering factor is defined as f = ∫𝑟 𝜌(𝑟)𝑒 2𝜋𝑟𝑆 dr we can define the structure factor
as F(hkl) = V ∫ 𝜌(𝑥𝑦𝑧)exp[2πi(hx + ky + lz)]
- with the electron density distribution ρ(xyz) of the unit cell
- and the total phase 2π(hx + ky + lz) of an X-ray scattered at the point (xyz) in the electron cloud
- integration over the unit cell space
- *V: the bigger the unit cell, the bigger F(hkl)
Structure factor can be expressed in terms of number of electrons:
F(000) = number of electrons in the unit cell.
Friedel law: I(hkl) = I(-h –k –l) … reflection in the real axis
- A(hkl) = A(-h –k –l) because cos(x) = cos(-x)
- B(hkl) = -B(-h –k –l) because sin(x) = -sin(-x)
Because I is proportional |F(hkl)|^2 and |F(hkl)| = (A^2 + B^2)^(1/2) we can recognize that |F(hkl)| =
|F(-h –k –l)| since B(hkl)^2 = B(-h –k –l).
Friedel law isn’t true for anomalous scattering:
fa(-h, -k, -l) ≠ fa(h, k, l) and αa(-h, -k, -l) ≠ -αa(h, k, l) <= iΔf’’(h, k, l) = -iΔf’’(-h, -k, -l)
normal scattering: λ1
-> Friedel’s law is true
anomalous scattering: λ2
-> Friedel’s law is not true
Transforming the electron density into a structure factor: Fourier
transform
A crystal is a periodic structure and we have the possibility to describe a periodic function with
Fourier series.
=> Therefore the periodic 3D electron distribution in a crystal can be described with 3D Fourier
series.
A Fourier summation/transformation is the representation of any contiuous function in terms of a set
of sine waves:
We can describe this function as summation of enough sine waves of appropriate frequency,
amplitude and phase.
Example:
1D crystal with a unit cell of three atoms (two carbons and one oxygen): The electron density curve
has two small peaks at 2 and 3 representing the two carbon atoms and one big peak at 5
representing the oxygen atom. Therefore we need three sine waves of the frequencies 2, 3 and 5.
The wave with ν = 2 has two repeats across the unit cell and therefore we see two peaks. Upon
summation with the two further waves the first peak changes into the two carbon-peaks and the
second peak expands to the big oxygen-peak.
Inversion theorem: G is Fourier transform of F then F is Fourier transform of G
The diffraction pattern is a Fourier transform of the electron density and vice versa.
The structure factor is the Fourier transform of a single unit cell. Exactly the structure factor is a
Fourier transform of the electron density and vice versa. Therefore we can regenerate the electron
density from the structure factor with an inverse Fourier transform.
- Both functions are circular symmetric.
- A function and its Fourier transform are reciprocal related.
The inverse Fourier transform of a crystal (a periodic object) involves summation:
Structure factor: F(h) = ∫𝑐𝑒𝑙𝑙 𝜌(𝑥) exp(2𝜋𝑖ℎ𝑥) 𝑑𝑣
It can be calculated if ρ(x) the distribution of electrons in the unit cell is known.
Distribution of electron density in the unit cell: ρ(x) = 1/V ΣhF(h)exp(-2πihx)
It can be calculated if F(h) is known.
Recover the density on the correct scale: if we would not big unit cells would have more electron
density (more structure factors) than small ones.
Phase problem:
ρ(xyz) = (1/V)ΣF(hkl)exp[-2πi(hx + ky + lz] =
= (1/V)Σ(A + iB)exp[-2πi(hx + ky + lz] =
= (1/V)Σ|F|exp(2πi)exp[-2πi(hx + ky + lz] =
= (1/V)Σ|F|exp[-2πi(hx + ky + lz-]
We cannot measure the phases with our detectors.
Electron density = real function because a complex number (A + iB) is multiplied with a complex
exponential exp[-2πirS].
Difference electron density ρ(xyz)
ρΔF(xyz) = (1/V)Σ[|F|obs - |F|calc]exp[-2πi(hx + ky + lz)] -> neg. atoms in the model at wrong position,
positive at positions of missing atoms, zero for correct model
𝟏
Convolution: C(x) =∫𝜼=𝟎 𝒇(𝜼)𝒈(𝒙 − 𝜼)𝒅𝜼 = f*g
The Fourier transform of C(x) is the product of the Fourier transforms of f(η) and g(x-η):
Example: You can imagine a crystal as a convolution of a molecule with a infinite grid. Since the
Fourier transform of the infinite grid (the crystal lattice) is just another infinite grid (the reciprocal
lattice and the Fourier transform of the molecule is a smooth continuous function (like the transform
of an atom is a broad smooth function but itself is a sharp feature), their product is a smooth
function sampled at reciprocal lattice points (= the diffraction pattern).
Example 2: You could also think of a molecule as a convolution of the point atomic structure and the
atomic shape. Its Fourier transform is also a broad, smooth function.