Crystallography And Diffraction Flashcards
How can we observe crystals?
Regular form of the faces of mineral specimens.
Scattering of light by periodic objects (eg opals).
Diffraction experiments.
High resolution electron microscope images.
What makes a material crystalline?
Ordered arrangement of atoms.
Atoms stack together to form regular networks.
Atomic arrangement is often reflected in the macroscopic geometry.
What makes a material amorphous?
Random arrangement of atoms.
What makes a material polycrystalline?
It is made up of many small crystals.
What is a lattice?
An infinite array of points in space in which the environment of every point is identical.
What is a motif?
A repeating unit of pattern (eg the arrangement of atoms that is placed on each lattice point).
What is a mesh?
The arrangement of lines that joins the lattice points.
What is a unit mesh?
One unit which, when repeated, makes up the mesh.
Where can the motif be placed on the lattice?
The motif can be placed anywhere in relation to the lattice point as long as the same position is chosen for every motif.
Can there be more than one unit mesh for a lattice?
Yes, it is possible, however we generally use the primitive mesh.
What is the primitive mesh?
The unit mesh with only one lattice point.
What is a centred mesh?
A square mesh with a lattice point in the middle.
How can lattice points be reached on a lattice?
With a combination of lattice vectors.
What method can we use to project 3D to 2D?
We can use crystal plans or crystal projections.
How do we draw crystal projections?
We view the cell in a set plane and label elevations for each point not at 1 or 0.
What is the most efficient way to pack in 2D?
Hexagonal packing (91%).
What are the most efficient ways to pack in 3D?
HCP and FCC (74%).
What interstices do both HCP and FCC possess?
Octahedral and Tetrahedral interstices?
What are interstices?
Small spaces within a lattice that may be able to accommodate another atom.
Example of an interstice:
Na+ in a FCC Cl- lattice inside the octahedral interstice.
What symmetry elements are there in 2D?
Rotational and Mirror.
If when does a body have n-fold symmetry?
When the final and initial forms of the body are identical when it is rotated 360º/n around the axis of rotation.
How does mirror symmetry work?
When an object is reflected along a mirror plane and its image in that plane is identical.
Pure rotation operations generate what?
4 2D plane point groups (2-fold, 3-fold, 4-fold 6-fold).
Why can’t 5-fold symmetry exist in crystallography?
5-fold rotation does not cause tessellation of the lattice.
Combining mirror and rotational elements generates how many more 2D crystallographic plane point groups?
5 more: m, 2mm, 3m, 4mm and 6mm.
What is the most basic (asymmetrical) 2D lattice we can draw?
It is oblique ( the haggle between two of its sides is >90º and <180º) and the two adjacent sides have different lengths.
What symmetry does the primitive unit mesh of the asymmetrical shape have?
P21 as it has a diad.
What does the presence of the first diad in the p2 cell cause?
3 more dads to be generated automatically, making it in effect p222, however the nomenclature is p2 as these are automatically generated and implied.
What symmetry does a primitive 90º, unequal adjacent sided cell show?
p2mm.
Why does a primitive, 90º unequal adjacent sided cell show p2mm symmetry?
It has a diad axis on its vertices and generated additional diad axes at the centre of the edges and at the cell centre. It also possesses a horizontal and vertical mirror plane thus p2mm as the second mirror is caused by the first.
If a motif has rotational symmetry 2 and the lattice it is applied to is oblique, whet is the symmetry of the crystal?
p2 as the oblique lattice has symmetry p2.
What happens when you place a motif of lower symmetry on a lattice that has higher symmetry?
The symmetry of the crystal is lowered.
Can you apply a motif of higher symmetry to a lattice of lower symmetry and get a crystal that shows symmetry?
No.
Where to find the list of lattice point groups with possible crystal point groups:
Lecture 2 back page.
Directions are given in what brackets?
[ ]
Sets of directions related by symmetry are given in what brackets?
< >
Sets of symmetrically arranged planes are given in what brackets?
{ }
Planes are given in what brackets?
( )
What is different about directions and planes in the hexagonal system?
They use the Miller-Bravais indices: hkli
Equation for the Weiss Zone Law:
hU + kV + lW = 0
Define the Weiss Zone Law.
If [UVW] lies in the plane (hkl) then hU+kV+lW=0. Alternatively, if [UVW] is parallel to (hkl) then the Weiss Zone Law is satisfied.
A set of planes with a common axis (direction) is known as what?
A zone.
Equations for direction at intersection of two planes:
U = k1l2-k2l1 V = l1h2-l2h1 W = h1k2-h2k1
or use cross product trick.
How to find the plane parallel to two directions:
(hkl) = (V1W2 – V2W1 W1U2 – W2U1 U1V2 – U2V1)
or use cross product trick.
Two planes lie in the same zone if:
h1U+k1V+l1W=0
and
h2U+k2V+l2W=0
Addition rule of planes:
(ph1+qh2)U + (pk1+qk2)W + (pl1+ql2)V = 0
so, p(h1k1l1) + q(h2k2l2) = 0
What is wave number?
Aka the propagation constant, it is 1/λ in radians per metre. Given letter k.
De Broglie’s equation
λ=h/p
General form that describes a travelling wave:
ψ=F(x–vt)
can also be written as:
ψ=F(t–x/v)
where ψ=F(x,t)
Write the equation that describes a sine wave travelling at speed v, with amplitude A.
ψ(x,t)=AsinK(x–vt)
K is constant to avoid taking sine of physical units.
What happens to a periodic wave if it’s position, x is increased by the wavelength, λ?
ψ(x,t)=ψ(x±λ,t)
which for a sine wave alters the argument of the wave by ±2π hence:
sinK(x–vt)=sinK[(x±λ)–vt]=sin[K(x-vt)±2π]
Equation for a harmonic wave.
ψ(x,t)=Asin((2π/λ)(x–vt))
which when k=1/λ and f=v/λ leaves
ψ(x,t)=Asin(2π(kx–ft))
Add the phase of the wave to the general equation for a harmonic wave:
ψ(x,t)=Asin((2π/λ)(x–vt+ε))
Principle of superposition:
The resultant of several waves at any point, x is given by adding their effects on the point, x.
How do you calculate the equation for a standing wave?
Consider two harmonic waves moving in opposite directions with the equations:
ψ1(x,t)=Asin(2π(kx–ft))
ψ2(x,t)=Asin(2π(kx+ft))
Ho do two waves travelling with the same amplitude but different directions superimpose?
ψ1(x,t)=Asin(2π(k1x–f1t)) \+ ψ2(x,t)=Asin(2π(k2x–f2t)) = ψ3(x,t)=2Asin(2π(k3x–f3t))cos(2π(k4x–f4t)) Where k3=(k1+k2)/2 k4=(k1-k2)/2 f3=(f1+f2)/2 f4=(f1-f2)/2
How does the resultant of two waves with the same amplitude travelling in the same direction with different wavelengths appear when superimposed and plotted?
As smaller waves within an envelope. The second wave function controls the envelope.
Derive asinθ=nλ using Huygens construction.
Sketch.
Derive asinθ=nλ using constructive interference knowledge.
Sketch.
Equation for constructive interference when the incident ray is oblique to the diffraction grating:
a(sinθ–sini)=nλ
Where i is angle of incidence between plane and ray and θ is angle of diffraction between normal to plane and exiting ray.
Effect of slit width on diffraction pattern:
Affects the envelope function. Wider slit width means narrower envelope function.
Effect of slit spacing on diffraction pattern:
Affects the maxima spacing. Slits that are further apart result in closer together maxima.
Effect of number of slits on diffraction pattern:
Does not affect the envelope or maxima separation. Changes width of each maxima. More slits means sharper maxima.
How can 2D diffraction patterns be dealt with?
By super imposing their 1D diffraction patterns.
Constructive interference in 2D satisfies which two equations?
a(sinθa–sinia)=hλ
b(sinθb–sinib)=kλ
Constructive interference in 3D satisfies which two equations?
a(sinθa–sinia)=hλ
b(sinθb–sinib)=kλ
c(sinθc–sinic)=lλ
The Laue Equations:
a(sinθa–sinia)=hλ
b(sinθb–sinib)=kλ
c(sinθc–sinic)=lλ
What do the Laue equations tell us?
The directions which an incident wave can travel through the crystal and exit in phase with one another.
Use diagram to show Bragg Law nλ=2dsinθ
Sketch.
How are X-rays scattered?
Scattered predominantly by electrons where electron is excited by the electronic component of the X-ray and falls back to ground state releasing another X-ray.
What is the scattering of X-rays by a unit cell composed of?
The sum of scattering from each atom.
What is the name for scattering when electrons are tightly bound?
Raleigh scattering.
When electrons are tightly bound is scattering coherent or incoherent?
Coherent.
What is the name for scattering when electrons are loosely bound?
Cropton scattering.
When electrons are loosely bound is the scattering coherent or incoherent?
Incoherent.
What is the name for scattering in coherent and incoherent components by a whole unit cell?
Thomson scattering.
How does scattering change with direction?
Amplitude scattered changes with respect to direction (2θ) known as the scattering factor fm(2θ).
What is atomic scattering factor?
The ratio between the amplitude scattered by a single (whole) atom to the amplitude scattered in the same direction by a single electron.
How does electron diffraction work?
Electrons are scattered by the distribution of potential across an atom so are predominantly scattered by nuclei.
Compare scattering factors for electrons and X-rays.
There is a more pronounced fall off in scattering factors for electron diffraction than for X-rays.
Significant multiple scattering events in electron diffraction occur due to what?
Electrons being charged.
Neutron scattering is caused by what part of the atom?
The nucleus.
Neutron diffraction is invariant with respect to what?
Angle as the wavelength of neutrons is significantly larger than the nuclear radius of an atom.
Does the neutron form factor follow a pattern?
No, it varies across the periodic table and between isotopes randomly.
Structure factor is a result of what and takes into account what?
Scattering caused by each atom and the relative position of atoms in a unit cell.
What happens to X-rays scattered in a unit cell?
They destructively interfere except in directions given by Bragg Law (or Laue equations).
Structure factor equation:
F(hkl) = Σmfmcos(2π(hxm+kym+lzm))
Where fm is the atomic scattering factor.
Intensity from scattering factor:
Square the scattering factor to find intensity.
What happens to the scattering factor if there is an inversion centre of symmetry?
The scattering factor is always real.
For bcc, which reflections are allowed?
h+k+l must always be even.
For fcc, which reflections are allowed?
Either h,k and l are all even or all odd.
Reciprocal lattice vectors are what to real space planes?
At normals to real space planes with length 1/dhkl.
The reciprocal lattice of cubic I is:
Cubic F and vice versa.
What is the meaning of reciprocal lattice points intersecting the Ewald sphere?
Bragg’s Law is satisfied so these diffracted beams are present.
Summary of the Ewald sphere construction:
Draw a circle of radius 1/λ
For the Bragg condition 1/λsinθ=1/2 dhkl
If origin is placed on sphere aligned with incident beam direction, any intersection satisfies the Bragg Condition.
For non-primitive lattices it is necessary to remove reciprocal lattice points that correspond to systematic absences.
Powder diffraction works how?
It has crystal lattice in all orientations so sphere is rotated about its origin 360º. Radius is 2/λ
What happens if a range of wavelengths is used to form an Ewald sphere?
If white radiation is used then the range of wavelengths corresponds to spheres with a range of diameters. Any reciprocal lattice points that lie on the nest will diffract.
What is the name for using a range of wavelengths to do X-ray diffraction?
Laue photography.
Two main differences between electron and X-ray diffraction:
Electrons have a much smaller wavelength than X-rays.
The sample is very thin in the direction of the X-ray beam.
What happens to the reciprocal lattice points in electron diffraction due to the thin geometry of the sample?
The lattice points become elongated.
Why are lattice points elongated in real space in electron diffraction?
The large radius of the Ewald sphere intersects many lattice points.
The crystal dimensions are smaller so condition for constructive interference is not as severe.
What equation shows the width of principle maxima with respect to grating width, w?
(Δsinθ)λ/w=width
What are HOLZs?
Higher Order Laue Zones, where the Ewald sphere intersects reciprocal lattice points on adjacent planes parallel planes. Electron diffraction.
Intensity of HOLZs equation:
hu + kv + lw = N
where N is the Laue Zone, [uvw] is direction of incident beam and hkl is the coordinates of the reciprocal lattice point.
Systematic absences are present in electron diffraction but what happens in thicker samples?
Multiple scattering may lead to intensity in positions that correspond to systematic absences.
How does a Coolidge tube work?
An Al cathode is heated and electrons generated, electrons collide with platinum anode emitting electrons.
How does a rotating anode tube work?
The anode in a Coolidge tube is rotated to allow heat to be dissipated and not build up on one focal point.
How does a synchrotron produce radiation?
A charged particle is accelerated centripetally emitting radiation tangentially.
In order to determine angles experimentally that satisfy the Laue equations what factor remain constant?
Wavelength.
In order to determine angles experimentally that satisfy the Laue equations what can be varied?
The single crystal is rotated varying θ.
A variety of wavelengths is used on a fixed θ.
Use a powder sample with fixed θ and λ.
What is a Debye-Scherrer camera?
Where X-rays are transmitted through a sample and a cone of diffracted rays impact a circular film (or other detector). The angle 2θ can easily be calculated as the arc length can be measured.
What is the Laue method of X-ray diffraction?
WWhite radiation incident on sample and is either transmitted or reflected. Results trickier to interpret as many wavelengths and diffraction from plane hkl at λ will be in the same position as 2h2k2l at λ/2.
Key advantages of using electron diffraction:
Extremely bright sources available.
Charged so can be focused into atomic sized probes.
Lens availability.
Many different signals generated.
Interplanar spacing and lattice parameter relationship:
dhkl=a/sqrt(N)
Scattering factor for neutron diffraction is proportional to what?
Atomic number.
What is the neutron source in neutron diffraction?
A nuclear reactor or spallation source.
What are the requirements for neutron diffraction?
A neutron source, a large sample (of material to be studied due to low flux) and a suitable detector.
Due to large sample in neutron diffraction, what kind of diffraction usually takes place?
Powder diffraction.
How does a spallation source produce neutrons?
Proton ‘bunches’ are accelerated in a synchrotron. They then collide with a heavy tungsten target under constant cooling load. The impact cause neutrons to spall off tungsten atoms and are channeled to instruments.
Bragg Law for cubic crystals:
λ=(2asinθ)/sqrt(h^2+k^2+l^2)
Square and rearrange gives:
sin^2(θ)+(λ^2/(4a^2)).(h^2+k^2+l^2)
After rearrangement of Bragg Law for cubic crystal what is true?
sin^2(θ)/R=N where N is constant.
Recipe for indexing:
Measure 2θ from powder diffraction lines.
Calculate sin^2(θ) for each line.
Take ratio of first two sin^2(θ).
Guess first value of N for lowest sin^2(θ).
Calculate sin^2(θ)/N which should be R.
For second sin^2(θ) do sin^2(θ)/R, if integer then N was correct.
If incident beam direction is known how can you index an electron diffraction pattern?
If structure is known, list allowed planes for reflections smallest first.
Consider sets of planes {} and whether they obey the WZ Law for the given direction.
Find the angle between the two smallest reciprocal lattice vectors and the ratio of their lengths.
If the direction of the incident beam is unknown how can you index a diffraction pattern?
Distance between r (000) and reflection (hkl) is
rhkl=λL/dhkl where L is distance to camera.
Angle between h1k1l1, 000 and h2k2l2 is same as angle between [h1k1l1]* and [h2k2l2]*
What is Vegard’s Law?
The change in lattice parameter for a solid solution system is linearly related to the atomic percent of the second element dissolved into the lattice.
What happens diffraction pattern appears in two phase regions?
The diffraction patterns of both phases are present, the relative intensities of each depend on the relative proportions of the phases.
CuxAuy occupies what arrangement?
Random solution of Cu and Au atoms in bcc arrangement.
CuAu has what structure?
Tetragonal on original bcc lattice sites with slight c/a distortion. Change in symmetry results in change in diffraction pattern and change in diffraction angles for planes as tetragonal unit cell as opposed to bcc.
Cu3Au has what structure?
Cubic P. Additional reflections are observed due to the loss of the F lattice systematic absences.
