PHY2009: Answers to problems: Lectures 1 to 6
Lectures 1 and 2
1)
2)
1, 3, 4 and 5 are primitive unit cells; 2 is non primitive, 6 is not a unit cell.
2nd array is not a lattice c.f. layer of graphite. It is a triangular lattice plus a
two-atom basis.
3)
A = n(r ) exp(-ik.r )d 3r
blob
The integral can be broken down to a sum of integrals for each atom
A= =
ra (atom a )
n(r ) exp(-ik.(r + ra )d 3r
Ri r j (atom j) Ri Ri
n(r ) exp -ik. r + Ri + r j d 3r
(
(
)
= exp(-ik.Ri ) exp -ik.r j
rj rj
(
)
n(r ) ex
PHY2009 Physics of Crystals: Problems Sheet for lectures 16 to 18
1) (a) (b) (c) (d) (e) Potassium is a monovalent free-electron metal with an atomic weight of 39. Calculate the number density of electrons, given that the density of potassium is 860 kg m3
PHY2009 Physics of Crystals: Problems Sheet for lectures 11-15
1) Show that the dispersion relation for the lattice vibrations of a chain of identical atoms of mass M, in which each is connected to its first and second nearest neighbours by bonds of sprin
PHY2009 Physics of Crystals: Problems Sheet for lecture 10
1) Draw the lattice points (in real space) for a (001) plane for the hexagonal-P lattice. Calculate (or construct geometrically) the reciprocal lattice for this plane. Using the Ewald Sphere Const
Problems Sheet for lectures 9 and 10
1) Show that the spacing d between the ( hkl) planes in a cubic lattice of side a is a d= (h2 + k 2 + l2 )1 2 (a) For a simple cubic lattice with cube side a, write down the Miller indices for the eight most widely spa
PHY2009 Physics of Crystals: Problems Sheet for lectures 7 and 8
1) A 2D direct lattice has primitive lattice vectors: a = ai b = b(cos )i + (sin ) j) [note: these are just vectors of lengths a and b with angle between them] (a) (b) Sketch a portion of th
PHY2009 Physics of Crystals: Problems Sheet for lectures 4 to 6
1) Prove that the ideal ratio of c/a for a hexagonal close packed structure is 1.633. (the symbols have their conventional meanings) Write down the fractional co-ordinates for the atoms in th
PHY2009 Physics of Crystals: Problems Sheet for lectures 1 to 3
1) The diagram shows a portion of a 2D lattice. Which of the outlines represent (i) primitive unit cells, (ii) non-primitive unit cells; (iii) not unit cells at all.
1 2 3 6 4
5
2)
Of the two
I. II. III. IV. V.
Review of Bonding in Solids Crystal Lattices Elastic Scattering of Waves Atomic Vibrations Electrons in Crystals
1
To look at past examination questions. The examination paper will last for 1 hr 30 mins There will be a choice of 3 fro
Free electron gas model: application to a metal Fermi energy Heat capacity Electrical conductivity Successes and failures
1
Nearly free electron model: Bragg reflection a the BZ boundary Appearance of energy gaps Understanding metals and insulators
2
Failure of the classical gas model heat capacity The free-electron Fermi gas confinement within crystal density of states function Paul exclusion principle Fermi energy Fermi distribution function
1
Free electron gas model: application to a metal Ferm
The diatomic chain
final comments
Next level of complexity: quantisation PHONONS dispersion curves in three dimensions Measuring phonon dispersion inelastic scattering Phonon momentum some weird aspects
1
Failure of the classical gas model heat capaci
The diatomic chain
interpreting the dispersion relation:
> behaviour at special points > motion of atoms
Acoustic: k 0 k = /a
Optical:
1
Next level of complexity: quantisation PHONONS dispersion curves in three dimensions Measuring phonon dispersio
Summary of last lecture
The diatomic chain _ _ equations of motion wave solution dispersion relation
1
Aims of this lecture
The diatomic chain interpreting the dispersion relation: > behaviour at special points > motion of atoms
2
12 *# m + M & 2 # ka &
Summary of last lecture
Monatomic chain normal modes calculation of motion of atoms: > at zone centre > at zone boundary group and phase velocities
1
Quiz on last lecture
For a monatomic linear chain with atom spacing a, a wave at the edge of the first B
Summary of last lecture
Lattice vibrations atoms oscillate about equilibrium positions (even at absolute zero) The monatomic linear chain
Its reciprocal lattice and first Brillouin zone The dispersion relation normal dispersion periodic repeats every 2/
Summary of last lecture
Role of the basis in scattering of waves the structure factor some examples application to a scattering problem
1
S(hkl) = $ f j exp "i2#(hu j + kv j + lw j )
Quiz
j
(
)
! The structure of Caesium Chloride is simple cubic, with a
Summary of last lecture
Examples of reciprocal lattices: more cubic Bravais lattices derivation, Miller indices of diffracting planes Introduction to the structure factor
1
Aims of this lecture
Role of the basis in scattering of waves the structure fact
Summary of last lecture
More about reciprocal lattices: Relationship with diffracting planes (second derivation) Relationship with Bragg condition Brillouin zones
1
Quiz on last lecture
The lattice shown is the reciprocal lattice of a certain 2D real spa
Summary of last lecture
More about reciprocal lattices: Geometric interpretation Mathematical derivation Relationship with diffracting planes and Miller indices:
G hkl has direction perpendicular to the planes, and magnitude 2" d, where d is perpendicula
Summary of last lecture
Elastic scattering of waves in periodic structures The Bragg condition - simple derivation: 2d sin " = n# More sophisticated derivation: "k # R = 2n$ for all ! lattice vectors R ! Introduction to the reciprocal lattice: the set of
Summary of last lecture
Cubic Lattices: Description, primitive lattice vectors and unit cells, and examples of structures based on: The simple cubic lattice The body-centred cubic lattice The face-centred cubic lattice
1
Quiz on last lecture
What is the a
Summary of last lecture
More calculations on hexagonal close-packed structure Packing fraction, coordination number
Cubic Lattices: Description, primitive lattice vectors and unit cells, and examples of structures based on: The simple cubic lattice The bo
Summary of last lecture
How to define and describe positions, directions and planes in crystals Close-packed structures how they form two distinct types The hexagonal close-packed structure
1
b
Quiz on last lecture
For the tetragonal lattice, o a = b " c
Summary of last lecture
Introduction to crystal structure: structure = lattice + basis Primitive lattice vectors Primitive and non-primitive unit cells space filling when translated through lattice vectors The Wigner-Seitz primitive unit cell region clos
Summary of last lecture
Reviewed the origin of the following bonds: Ionic Covalent Metallic van der Waals Hydrogen and discussed the consequent properties of solids possessing these bonds Discussion of the origin of central-core repulsion
1
Quiz on last
PHY2009 Physics of Crystals
22 lectures by Rob Hicken (Room 306, [email protected])
1
Text Books
Core text book is: Kittel C., Introduction to Solid State Physics, Wiley, ISBN 0-471-11181-3
(12 copies, in library, various editions)Very complete, cov
Aims of this lecture
Nearly free electron model: Bragg reflection a the BZ boundary Appearance of energy gaps Understanding metals and insulators
1
2)
The nearly-free-electron model
Electrons in crystals are waves travelling in a periodic potential We ha