Lectures accompanying the book: Solid State Physics: An Introduction, by Philip Hofmann (2nd edition 2015, ISBN-10: 3527412824, ISBN-13: 978-3527412822,

lectures accompanying the book: Solid State Physics: An Introduction, by Philip Hofmann (2nd edition 2015, ISBN-10: , ISBN-13: , Wiley-VCH Berlin.

Ingredients for solid state physics
The electromagnetic interaction Quantum Mechanics Many particles Symmetry to handle it all Microscopic picture for a zoo of different phenomena: conductivity, superconductivity, mechanical properties, magnetism, optical properties. Funny new “quasi” particles: ‘electrons’ with unusual mass and charge, phonons, Cooper pairs, particles which are neither bosons or fermions, magnetic monopoles,.... in principle just write down Schr. Eq. for all the particles. encounter macroscopic phenomena and try to find a microscopic explanation what we put in is quite simple. and what we get out

Perfect Crystalline Solids
bismuth NaCl the solids we care about are perfect crystals. Not such a bad approximation 2 we use the Gibbs free energy (fixed T,P,N) gas: entropy is dominant factor, especially for an ideal gas liquid: energy more important, some obvious interaction, manifestation in surface tension... the energy is very important in the solid state. We can forget about the entropy for the time being and consider mostly T=0 what makes the solid hold together so tightly? Can we understand the strength of these bonds? Can me make some prediction as to the melting of the state? small building blocks of atomic size ,unit cells also explains things like cleaving the fact that the solids form crystals is very important. This will become apparent solid made from small, identical building blocks (unit cells) which also show in the macroscopic shape

Crystals and crystalline solids: contents
at the end of this lecture you should understand.... Bravais lattice, unit cell and basis Miller indices Real crystal structures X-ray diffraction: Laue and Bragg conditions, Ewald construction The reciprocal lattice Electron microscopy and diffraction How do we describe crystal structures mathematically? What structure do solids typically have? How do we measure this?

Crystals and crystalline solids
The total energy gain when forming crystals most be very big because the ordered states exists also at elevated temperatures, despite its low entropy. The high symmetry greatly facilitates the solution of the Schrödinger equation for the solid because the solutions also have to be highly symmetric. Crystals are pretty because of their regular, symmetric shape. key problem: too many particles. Schr-equ. can not be solved. Crystal symmetry helps to find approximate sol. However, most of the solids we deal with do not appear to be crystals.

Crystals and crystalline solids
2: because it is 3D: for 100x100x100 atom domain only 6% is close to the boundary. For 1000x1000x1000 it is only 6%% Or N side length. N^3 volume N^2 surface most real solids are polycrystalline still, even if the grains are small, only a very small fraction of the atoms is close to the grain boundary the solids could also be amorphous (with no crystalline order)

Some formal definitions
several things which are seemingly over the top. Meaning will become apparent later.

The crystal lattice: Bravais lattice (2D)
A Bravias lattice is a lattice of points, defined by m, n integers The lattice looks exactly the same from every point

A Bravias lattice is a lattice of points, defined by there is a certain freedom in choosing the vectors or if I translate the entire lattice by ANY vector of Rnm, I get the same (but this is only necessary, not sufficient!) The lattice looks exactly the same from every point

Not every lattice of points is a Bravais lattice impossible to define a1 and a2 which take care of this

A Bravias lattice is a lattice of points, defined by This reflects the translational symmetry of the lattice

Bravais lattice (2D) grouping BL of fundamentally different symmetry
(more precisely, two are the same when having isomorphic symmetry groups) two kinds: translational symmetry, point symmetry important for classification, not so important for us. Just the result for 2D not everything is possible: it must still be possible to have BL one could argue that one can kill all 5 with the 5th, the oblique rectangular. But this one has only two symmetry operations which leave it invariant: rotation around 180 and 360 deg The others are defined such that one keeps special symmetry operations. Explain that the a’s are not a Bravais lattice for the centred cell. Θ The number of possible Bravais lattices (of fundamentally different symmetry) is limited to 5 (2D) and 14 (3D).

The crystal lattice: primitive unit cell
Primitive unit cell: any volume of space which, when translated through all the vectors of the Bravais lattice, fills space without overlap and without leaving voids a2 having the Bravais lattice, we can really define the unit cell, i.e. the building block of crystals! read definition: examples the intersections are the lattice points the primitive unit contains only one lattice point. the non-primitive unit cells contain more than one HOW MANY? a1

The crystal lattice: primitive unit cell
Primitive unit cell: any volume of space which, when translated through all the vectors of the Bravais lattice, fills space without overlap and without leaving voids a2 we can put these unit cells wherever we want the primitive unit contains only one lattice point. the non-primitive unit cells contain more than one. Symmetry: translational is ok but all the others are not for all unit cells! a1

The crystal lattice: Wigner-Seitz cell
Wigner-Seitz cell: special choice of primitive unit cell: region of points closer to a given lattice point than to any other. The Wigner-Seitz cell has the (point) symmetry of the lattice. which one is that? In 3D this is not at all easy to see!

The crystal lattice: basis
We could think: all that remains to do is to put atoms on the lattice points of the Bravais lattice. But: not all crystals can be described by a Bravais lattice (ionic, molecular, not even some crystals containing only one species of atoms.) BUT: all crystals can be described by the combination of a Bravais lattice and a basis. This basis is what one “puts on the lattice points”. we know already that not x-tals can be described by Bravais lattice

The crystal lattice: one atomic basis
The basis can also just consist of one atom. maybe copper or so

Or it can be several atoms. maybe NaCl or so note: same Bravais lattice

Or it can be molecules, proteins and pretty much anything else. x-mas tree plantation: as many as possible, evenly distributed on a limited area. This results in a 2D hexagonal structure

The crystal lattice: one more word about symmetry
The other symmetry to consider is point symmetry. The Bravais lattice for these two crystals is identical: translational symmetry is the same four mirror lines 4-fold rotational axis inversion no additional point symmetry

The crystal lattice: one more word about symmetry
The Bravais lattice vectors are We can define a translation operator T such that In the beginning, we mentioned the importance of symmetry. this will be very useful for the electronic properties. We get information about the general shape of the wave functions without any calculations. we have done this in the case of complete spherical symmetry: we have chosen the wave functions such that they are also eigenfunctions corresponding to the rotational symmetry. This gave l and m. Note that the wave functions don’t have to have full rotational or translational symmetry. Only the observables have to have. We will introduce k as the quantum number for the translational symmetry. This operator commutes with the Hamiltonian of the solid and therefore we can choose the eigenfunctions of the Hamiltonian such that they are also eigenfunctions of the translation operator.

Nasty stuff... adding the basis keeps translational symmetry but can reduce point symmetry usually point symmetries and translational symmetries can be combined: the total group describing the symmetry of the crystal is just the sum of the two groups. But there are two nasty examples: glide planes and screw axis. but it can also add new symmetries (like glide planes here)

Real crystal structures
discuss some, at least for elemental solids and for some ionic solids and molecular

crystal structure: lowest energy
What structure do the solids have? Can we predict it? Consider inert elements (spheres). This could be anything with no directional bonding (noble gases, simple and noble metals). Just put the spheres together in order to fill all space. This should have the lowest energy. A simple cubic structure? crystal structure: lowest energy not simple to predict: vary the positions of all atoms in solid (impossible) in a crystal: vary positions in only one uint cell since they are all the same: well possible but not aim here here just start with one of the simplest structures we can think of

Simple cubic is BL -> one atom basis.
The simple cubic structure is a Bravais lattice. The Wigner-Seitz cell is a cube The basis is one atom. So there is one atom per unit cell. is BL -> one atom basis. This translated gives all the other atoms a2 a1 a3

Simple cubic We can also simply count the atoms we see in one unit cell. But we have to keep track of how many unit cells share these atoms. 1/8 atom another way of viewing this: count all the atoms in the unit cell keep track of atom sharing! also gives one atom easier: construct cubic box around one atom! 1/8 atom 1/8 atom 1/8 atom 1/8 atom 1/8 atom 1/8 atom 1/8 atom

Simple cubic Or we can define the unit cell like this another way of viewing this: count all the atoms in the unit cell keep track of atom sharing! also gives one atom easier: construct cubic box around one atom!

Simple cubic A simple cubic structure is not a good idea for packing spheres (they occupy only 52% of the total volume). Only two elements crystallise in the simple cubic structure (F and O). the 52% are not obvious here. What I mean of course is that this applies for making the spheres as big as possible.

Better packing just put one atom in the middle: much better for filling space easy to see that the cubic lattice is not the lattice describing the bcc lattice as a Bravais lattice. We can’t reach all the points! YET this is a Bravais lattice if you only choose the right vectors. Can you? In the body-centred cubic (bcc) structure 68% of the total volume is occupied. The bcc structure is also a Bravais lattice but the edges of the cube are (obviously) not the Bravais lattice vectors.

Better packing also much more popular: 17 elemental solids, including the alkali metals not the way for ideal packing. So what is the best way of packing? In the body-centred cubic (bcc) structure 68% of the total volume is occupied. The bcc structure is also a Bravais lattice but the edges of the cube are not the correct lattice vectors.

Close-packed structures
greengrocers all over the world know that simple cubic is not a good idea but: Kepler, 1611, easy to believe, very hard to prove. no proof so far.

Close-packed structures: fcc and hcp
ABABAB... fcc ABCABCABC... Let’s start in a plane. There it’s really simple fcc face centred cubic hcp hexagonal close-packed

ABABAB... fcc ABCABCABC...

ABABAB... fcc ABCABCABC... first hcp, second fcc very, very similar! Same packing density of 74% funny mixtures between these would also be possible but are to my knowledge never observed for elemental solids why is fcc a cubic structure? The hexagonal close-packed (hcp) and face-centred cubic (fcc) and structure have the same packing fraction

The fcc structure only spheres on front face shown!
In the face-centred cubic (fcc) structure 74% of the total volume is occupied (slightly better than bcc with 68%) This is probably the optimum (Kepler, 1611) and grocers.

The fcc lattice: Bravais lattice (3D)
The fcc lattice is also a Bravais lattice but the edges of the cube are not the correct lattice vectors. The cubic unit cell contains more than one atom. easy to see that the cubic lattice is not the lattice describing the fcc lattice as a Bravais lattice. We can’t reach all the points! HOW MANY?

The fcc lattice is also a Bravais lattice but the edges of the cube are not the correct lattice vectors. The cubic unit cell contains more than one atom. 8*1/8 = 1 6 * 1/2 = 3 3+1=4 1/8 atom 1/2 atom

The fcc and bcc lattices are also Bravais lattices but the edges of the cube are not the correct lattice vectors. When choosing the correct lattice vectors, one has only one atom per unit cell. the correct vectors look like this

The fcc and bcc lattices are also Bravais lattices but the edges of the cube are not the correct lattice vectors. When choosing the correct lattice vectors, one has only one atom per unit cell. 1/8 atom and then we have 1 atom per unit cell choosing the right unit cell matters: not so much from the point of view of maths but from that of physics. Artificially too big unit cells give typically funny results (we see this in the exercises this week) 1/8 atom

Close-packed structures: hcp
we can try this. Vectors in the plane obvious. Vector to the next plane. And then one more to the next. But it doesn’t hit a lattice site there. If we were to put the next layer there, then we would in fact have fcc. The hcp lattice is NOT a Bravais lattice. It can be constructed from a Bravais lattice with a basis containing two atoms. the packing efficiency is of course exactly the same as for the fcc structure (74 % of space occupied).

Close-packed structures: hcp
this is not a Bravais lattice, hence the different colours The hcp lattice is NOT a Bravais lattice. It can be constructed from a Bravais lattice with a basis containing two atoms. the packing efficiency is of course exactly the same as for the fcc structure (74 % of space occupied).

Close-packed structures
Close-packed structures are indeed found for inert solids and for metals. For metals, the conduction electrons are smeared out and directional bonding is not important. Close-packed structures have a big overlap of the wave functions. Most elements crystallize as hcp (36) or fcc (24). go away from pure elements and consider alloys. For metallic alloys, the same arguments should apply

Close-packed structures: ionic materials
In ionic materials, different considerations can be important (electrostatics, different size of ions) In NaCl the small Na are in interstitial positions of an fcc lattice formed by Cl ions (slightly pushed apart) The negative ions are in general much bigger than the positive ions because of the smaller nuclear charge. So the positive ions can be fit into a close-packed lattice of negative ions as an interstitial alloy. Can you see the fcc crystal / two fcc into each other?

Close-packed structures: ionic materials
In NaCl the Na ions are so small that they fit in the interstitial sites in CsCl they are not. two simple cubics into each other

Non close-packed structures
covalent materials (bond direction more important than packing) diamond (only 34 % packing) graphene graphite sp2 and sp3 bonds are shown as lines

graphene this is how to make graphene

Non close-packed structures
molecular crystals (molecules are not round, complicated structures) α-Gallium, Iodine only a total of 19 elements chooses not to crystallize in the simple cubic, bcc, hcp or fcc structure.

Crystal structure determination
important because comes again and again, not only relevant for physicist but also for things like structural biology advanced mathematical treatment for which we lay the ground here. Some things are difficult to understand and probably impossible in the first lecture. Or first course. Notably the reciprocal lattice.

X-ray diffraction ... and we are looking at sub-nm structures
The atomic structure of crystals cannot be determined by optical microscopy because the wavelength of the photons is much too long (400 nm or so). So one might want to build an x-ray microscope but this does not work for very small wavelength because there are no suitable x-ray optical lenses. The idea is to use the diffraction of x-rays by a perfect crystal. Here: monochromatic x-rays, elastic scattering, kinematic approximation ... and we are looking at sub-nm structures monochromatic: usually not elastic: usually yes kinematic: usually yes

X-ray diffraction The crystals can be used to diffract X-rays (von Laue, 1912). tube: accelleration of electrons on an anode, x-ray emission. crystal diffracts x-rays in different direction. most x-rays pass through pattern emerges: false color (for a photographic plate) dark is high intensity collimator: must be a pinhole really easy what happens without: all spots are broadened in angle. von Laue has also described this theoretically in 1912

The Bragg description (1912): specular reflection
Before we look at the description of von Laue: Simpler but equivalent description by William Lawrence Bragg (1912) The crystal is made of reflective lattice planes. In the specular geometry, this is really easy. Thousands of layers contribute -> reflection very sharp in angle, like optical grating with many lines. WHY IS THAT? IF YOU HAVE TWO LAYERS AND ADD ONE, DO YOU ALSO GET CONSTRUCTIVE INTERFERENCES? CAN WE GET AROUND THE SMALL WAVELENGTH? and this only works for The Bragg condition for constructive interference holds for any number of layers, not only two (why?)

X-ray diffraction: the Bragg description
The X-rays penetrate deeply and many layers contribute to the reflected intensity The diffracted peak intensities are therefore very sharp (in angle) The physics of the lattice planes is totally obscure! What do the planes mean? Uniform plane only meaningful if the wavelength of the x-rays is much longer than the atomic distances. But it is surely not! Now we approach this much more formally What makes the reflection of x-rays? Wiggled and re-emitting electrons

A little reminder about waves and diffraction
λ = 1 Å hν = 12 keV better description needs waves formalism plane, electromagnetic waves: (A is E or B field or the vector potential) definition of k transversal plane waves, explain why write as complex but really of course real, take real part complex notation useful because of interference / propagating waves complex notations (for real )

X-ray diffraction, von Laue description
source of x-rays, electrons, neutrons or whatever spherical waves crystal really far away, so use the plane waves just discussed the wave vector k is the same for all atoms in the crystal rho is a scattering density. For x-rays basically electron density but in general complex. k’ different direction but same length (because elastic) also kinematic: only one scattering process

incoming wave at r absolute E0 of no interest source of x-rays, electrons, neutrons or whatever spherical waves crystal really far away, so use the plane waves just discussed the wave vector k is the same for all atoms in the crystal rho is a scattering density. For x-rays basically electron density but in general complex. k’ different direction but same length (because elastic) also kinematic: only one scattering process outgoing wave in detector direction

the first exponential only due to the geometry of the setup nothing to do with the interference from different positions in solid, so drop it final result for field but observe intensity field at detector for one point and for the whole crystal

final result for field but we can only see intensity: introduce scattering vector K Expression is final but not very helpful. We could try to measure I(K) and infer the structure but this is not smart. We don’t see in which directions we actually have constructive interference so the measured intensity is with

so the measured intensity is discuss rho also oscillating function - and complex. Simplicity in the picture: real, positive and only peaked at atoms (like a charge density) later when doing the proper maths, we see that this assumption is not necessary. exp is a plane wave We multiply a plane wave with rho and then integrate over the whole crystal. This will have roughly the same positive and negative contributions and vanish. ρ(r) in Volume V

so the measured intensity is with 55 try to be smart: if we choose K=0 then the integral is only over the lattice periodic rho, and this does maybe not vanish not very useful to have all the info hidden in the beam which goes straight through the crystal anyway! ρ(r) in Volume V k k’

so the measured intensity is try to be even smarter: It is easy enough to draw a plane wave for which this would not vanish. This will work if the wave has the periodicity of the lattice. NOTE we have so far not exploited that we have a crystal, i.e. a periodic structure. We do this now and find a better expression for rho. We need the so-called reciprocal lattice. Short detour before we come back to x-rays ρ(r) in Volume V

The reciprocal lattice
for a given Bravais lattice the reciprocal lattice is defined as the set of vectors G for which this is now really formal it is also difficult to understand what this means but it is central to the whole solid state physics We go on formal and try to fill it with life later The relation must be correct for any G and R! It will become clear by construction that G is also a Bravais lattice dimensions of reciprocal space: inverse angstroms could construct the b1,b2,b3 out of this requirement. I give you the answer or The reciprocal lattice is also a Bravais lattice

construction of the reciprocal lattice explain Kronecker delta not so carful about the dot here... show that the relation is true (or leave to listener) use a(b x c) = b(c x a) = c(a x b) geometrical meaning of the scalar triple product: “directed” volume of the parallelepiped a useful relation is with this it is easy to see why

example 1: in two dimensions when doing this in 2D, the relation with the delta function is especially useful because we can directly construct using it! |a1|=a |a2|=b |b1|=2π/a |b2|=2π/b

if we have The vectors G of the reciprocal lattice give plane waves with the periodicity of the lattice. In this case G is the wave vector and 2π/|G| the wavelength. then we can write So what is the meaning of the G’s? so if we have such a wave and we add ANY lattice vector to r, we get the same

so the measured intensity is so we had that in general the plane wave times the periodic function (charge density) averages out. ρ(r) in Volume V

so the measured intensity is and obviously waves with the periodicity of the lattice are exactly what we want! We can guess that if we choose K=G we will get a non-vanishing solution Volume V K=G

Lattice waves real space reciprocal space a b 2π/a 2π/b look at waves corresponding to the reciprocal lattice vectors. if we change the place we look at by ANY real lattice vector, we have to get the same Here fore G=0, infinite wave length. (0,0)

Lattice waves Here fore G=2pi/b, lambda=b real space reciprocal space
(0,0)

Lattice waves Here fore G=4pi/b,lambda=b/2 real space reciprocal space
(0,0)

Lattice waves real space reciprocal space a b 2π/a 2π/b (0,0)

The reciprocal of the reciprocal lattice
is again the real lattice in 2D and this example, this is especially obvious! |a1|=a |a2|=b |b1|=2π/a |b2|=2π/b

The reciprocal lattice in 3D
example 2: in three dimensions bcc and fcc lattice not so easy to see but straight forward calculation The fcc lattice is the reciprocal of the bcc lattice and vice versa.

Applications of the reciprocal lattice
1D chain of atoms example of charge density 1 reciprocal lattice good for other things than xrd (that’s why we use it so much) key to solving many physical problems is the symmetry. atomic physics: spherical symmetry, basis for using spherical harmonics and the entire classifying scheme. Here new symmetry: translation (also point symmetry but not so important right now). Using rec lat is the way to exploit this symmetry. example of charge density 2 Greatly simplifies the description of lattice-periodic functions (charge density, one-electron potential...).

example: charge density in the chain Fourier series the constant part must stay out of this series more compact writing using exponential function with complex coefficients rho coefficients must be fulfilling this condition so that the charge density is real SHOW examples of this for rho_0=0, rho_-1=rho_1=1 and rho_0=0, rho_-1=-i, rho_1=i and e^ix = cos x + i sin x this exercise also shows why I have to allow for the coefficients to be complex: must be able to reproduce cos and sin parts and anything in between. alternatively

1D reciprocal lattice crucial point: n2pi/a are the reciprocal lattice points for the 1D lattice! I specify rho_n at the reciprocal lattice points instead of saying what it is everywhere is space (in the unit cell) What’s the gain? In both cases infinitely many points. But in reality not: For describing a fairly complicated charge density, we need only few coefficients.

1D and this finally brings us back to x-ray diffraction! write scattering density as a series using the reciprocal lattice. 3D

so the measured intensity is with Now the expression is really useful: we can make G-K=0, then the exponential is 1 we prevent the whole thing from averaging out. Only in this case do we get constructive interference! Laue Condition in simple picture we had assumed that rho is positive and real but this is not necessary here anymore Each spot in the image has a “name” which is its reciprocal lattice vector The intensity at each spot is propto rho_G^2 V^2. This seems strange: how can it be prop to V^2? The reason is that we look at the top of the peak at K=G. The peak has also a width which decreases as V increases. The total scattered intensity is propto V, as it must. use constructive interference when Laue condition

so the measured intensity is with The most annoying thing is that measuring the intensity of every spot only gives rho_G^2, not the rho_G because these are complex and we cannot take the root. If we could measure the field instead of the intensity we would get the rho_G ie everything. This is the phase problem in xrd. There are several smart ways of overcoming this. make a model of the crystal, calculate rho(r) and rho_G, compare to the measured rho_G^2 and refine the model until this comparison is fine. This is basically how it is done. In principle, we don’t know how big rho_G^2 is but if it is really zero, then we have probably chosen the wrong (too big) unit cell. use for a specific spot

so the measured intensity is seen that it doesn’t matter if rho is complex. Come back to our simple picture! Volume V

so the measured intensity is so the wave the smallest G in the direction would pick up only the tops Note that K=k’-k, a difference wave. So this would be Bragg. K Volume V -k k k’

so the measured intensity is this corresponds to the shortest G/2 (double the wavelength of the shortest G) and would lead to destructive interference Volume V

so the measured intensity is less extreme is a slight deviation which will not work either. Volume V

so the measured intensity is but double would be fine! Volume V

The Ewald construction
Laue condition if G is a rec. lat. vec. Draw (cut through) the reciprocal lattice. Draw a k vector corresponding to the incoming x-rays which ends in a reciprocal lattice point. Draw a circle around the origin of the k vector. The Laue condition is fulfilled for all vectors k’ for which the circle hits a reciprocal lattice point. the Laue condition re-written. It is fulfilled if deltaK=G where G is any vector of the reciprocal lattice. In this way, one can immediately see, in which direction one expects to see diffracted x-rays. and this is a very exclusive condition!!!

Labelling crystal planes (Miller indices)
1. determine the intercepts with the axes in units of the lattice vectors 2. take the reciprocal of each number 3. reduce the numbers to the smallest set of integers having the same ratio. These are then called the Miller indices. We conclude this review of crystal structure with the Miller indices. this is somewhat dull. So we do it briefly. In any case, this can be used to define a plane of atoms. This might be useful but mathematically more convenient is the definition thought normal vector to the plane! The meaning will become apparent later. step 1: (2,1,2) step 2: ((1/2),1,(1/2)) step 3: (1,2,1)

Example cuts a1 at 1, a2 and a3 at infinite
take the reciprocal value which is 1,0,0 multiply to get integers already done

Relation to lattice planes / Miller indices
The vector is the normal vector to the lattice planes with Miller indices (m,n,o) something to show as an exercise. So these lattice planes are quite important because any set of such planes gives Bragg reflections.

Why does the Bragg condition appear so much simpler?
Laue condition automatically fulfilled parallel to the surface (choosing specular reflection) to appreciate how ingenious this is, let’s compare to the Laue description Basically, we kill two of the three conditions in the Laue condition so this leaves us with an effectively 1D problem and we have just shown with the Miller indices that there always is a reciprocal lattice vector perpendicular to a set of planes. 1D reciprocal lattice in this direction: define vector d connecting the planes k k’

x-ray diffraction in practice

Laue Method the spots here can occur from x-rays with different energies! Using white x-rays in transmission or reflection. Obtain the symmetry of the crystal along a certain axis.

reciprocal lattice point
Powder Diffraction reciprocal lattice point compare positions with expected positions with observed positions calculate intensity and compare intensity

advanced X-ray diffraction
The position of the spots gives information about the reciprocal lattice and thus the Bravais lattice. An intensity analysis can give information about the basis. Even the structure of a very complicated basis can be determined (proteins...) every spot turn molecules into crystals before doing this obviously the crystalline order is essential here! the length of the reciprocal lattice vectors is of the order 1/a. Small lattice (like in elemental solids, big reciprocal lattice, points well separated). Big lattice (like protein crystal) spots VERY close to each other. you need a VERY collimated monochromatic beam crystallize protein remember

advanced x-ray sources: synchrotron radiation
SPring-8 ASTRID A highly collimated and monochromatic beam is needed for protein crystallography. This can only be provided by a synchrotron radiation source.

What is in the basis? with we have N unit cells in the crystal and write this as sum over cells the step with the n unit cells is not really obvious. One has to turn the integral into a sum over all unit cells displaced by vectors R and then make use of the fact that G*(r+R)=Gr (sum over the j atoms in the unit cell, model this)

Inelastic scattering Gain information about possible excitations in the crystal (mostly lattice vibrations). Not discussed here.

Other scattering methods (other than x-rays)
electrons (below) neutrons x-rays are good but anything else with the same de Broglie wavelength will also work. electrons are great: easy to generate with the right wave length... big disadvantages: too surface sensitive, multiple scattering (not at all discussed here and irrelevant for XRD) neutrons: difficult to make (reactor or spallation source), difficult to handle (because charge neutral), certainly bulk sensitive. Very important: charge neutral but magnetic moment. Interact with magnetic ordering in the crystal

Electron microscopes / electron diffraction
Electrons can also have a de Broglie wavelength similar to the lattice constant in crystals. For electrons we get for real microscopy, we want to have the wavelength of the electrons MUCH smaller than the size of the objects we want to look at. diffraction effects is precisely NOT what we want and of course even smaller lambda for higher E This gives a wavelength of 5 Å for an energy of 6 eV.

Lectures accompanying the book: Solid State Physics: An Introduction, by Philip Hofmann (2nd edition 2015, ISBN-10: 3527412824, ISBN-13: 978-3527412822,

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Lectures accompanying the book: Solid State Physics: An Introduction, by Philip Hofmann (2nd edition 2015, ISBN-10: 3527412824, ISBN-13: 978-3527412822,

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Presentation on theme: "Lectures accompanying the book: Solid State Physics: An Introduction, by Philip Hofmann (2nd edition 2015, ISBN-10: 3527412824, ISBN-13: 978-3527412822,"— Presentation transcript:

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