Lecture 11: Periodic systems and Phonons Aims: Mainly: Vibrations in a periodic solid Complete the discussion of the electron-gas Astrophysical electrons Degeneracy pressure White dwarf stars Compressibility/bulk modulus of metals Introduction to periodic systems: Waves in periodic systems Electron bands (not for examination) Lattice vibrations Modes of a 1-dimensional, harmonic chain Monatomic chain with nearest neighbour forces May 0 Lecture 11 1
Astrophysical electrons Typical star Mainly composed of ionised hydrogen M ~ x10 0 Kg, R ~ x10 7 T m ~ x10 8 K T<T T ~ 10 7 K Electrons form a degenerate gas. Nuclear energy keeps the star hot and inflated. White dwarf stars What happens when the nuclear reaction of hydrogen stops? The star is then mainly helium. Star contracts under gravity until balanced by a degeneracy pressure due to the electrons Recall the energies of electrons in box side a k m ( ) l + m + n It takes energy to compress the box, in order that the single-particles energies rise. Hence there is an outwards pressure. π m a May 0 Lecture 11
May 0 Lecture 11 Lecture 10, p4 Lecture 10, p4 Degeneracy pressure Degeneracy pressure Pressure due to a ermi gas at Pressure due to a ermi gas at T0 0. Compressing the gas, we have U -PV and hence irst, calculate <U>, the mean energy of a ermi electron for N electrons V U P ( ) ( ) g g U d d d d 0 1 0 0 0 V N V U P N U N U Degeneracy pressure Degeneracy pressure ( ) V N ( ) R M V N n P A
White dwarf stars N.B. m p Gravitational forces: In equilibrium the degeneracy pressure balances gravitational attraction. Taking P to depend only on G, M and R α β γ P G M R ML 1 T L α 1 α + β; M α T α 1 α + γ ; α 4 P GM R Equating gravitation and degeneracy pressures 4 M R GM R inal radius, R, for 1 M R const a star of mass, M. Result derived by owler in the late 190 s is correct in the non-relativistic limit. Best known white dwarf is Sirius B: radius.6x10 6 m (< earth); mass, x10 0 Kg. 0 7 7 N 10 1.67 10 1. 10 N k v V 1. 10 ( π n) k m 1 e 7 A relativistic analysis is needed ( ~00keV) May 0 Lecture 11 4 M β 6 ( 4π.6 10 ).8 10 4. 10 8 ms m -1-1 L γ 1.6 10 6 m >c >c!! -
Chandrasekhar mass limit Chandrasekhar ~190 Re-analysed the problem relativistically. ( ) 1 ck ( ) ( ) 4 k ck + me c mec g() is affected (see problem sheet 1); however, other steps in the analysis are the same as before. The key result is that the degeneracy pressure~r -4 (rather then R - ) It follows that the final radius is independent of the mass i.e. no white dwarfs above a limiting mass (1.4 x mass of sun) Chandrasekhar limit Non-relativistic limit More massive stars contract to neutron stars, and ultimately to black-holes. Won 198 Nobel prize May 0 Lecture 11
Bulk modulus of metals Degeneracy pressure in metals The pressure of the degenerate electron gas also contributes to the mechanical properties of metals. Isothermal bulk modulus, K T is defined as Eq. A, p. K P K T T p V V T N p V V N V N V Calculated values are of the right magnitude. We have neglected the attractive forces, due to the ion cores. Attractive forces make the metal more compressible (Experimental bulk modulus, K exp, is usually smaller than K T, above). Electron contribution to the bulk modulus Metal n(n/v)/m - E /ev K exp /K T Li 4.6E+8 4.7 0.6 Na.E+8.1 0.8 K 1.E+8.0 1.0 May 0 Lecture 11 6
Periodic structures Waves in crystals Waves in crystals So far we have neglected the periodicity (symmetry) of typical, crystalline solids. The periodicity influences the way in which a wave propagates through the medium. A key concept is that the periodicity in real space introduces a periodicity in momentum space, k-space, as we shall see. Start from a familiar example (IB Waves course): Light passing through a periodic structure (e.g. a diffraction grating) May 0 Lecture 11 7
Waves in crystals Periodic structures We can think of the process as an input wavevector, k i, being converted to many output wavevectors defined by k (y) f k (y) i +ng (y) (in general, a vector equation); n is an integer. The lattice symmetry (periodicity) induces a symmetry (periodicity) in any wave propagating through the lattice. G is a reciprocal lattice vector (see later in the section on phonons). In a crystal any k-dependent property (such as energy) must be periodic Electrons E versus k relation (dispersion relation) in the free-electron case is E(k) E( k) k m k May 0 Lecture 11 8
Origin of electron bands (Not for examination) Electrons in a periodic solid. (A quantitative treatment appears in Part II) Consider a solid with periodicity in 1-D. If the periodicity is weak, it only introduces tiny changes to the E(k) dispersion curve. Using the argument that E(k) must be periodic, with period G, we can construct multiple copies of the free electron curve. E(k) G All the information is in the central section known as the irst Brillouin zone (right). Bands are labelled accordingly (separate colours). Note also the potential for k band gaps (at crossings) and, hence, insulating behaviour. (or gaps to appear we need some modification of the free electron curves, i.e. scattering from the lattice ) May 0 Lecture 11 9 4 1 E(k) k Gaps
Lattice vibrations 1-D D harmonic chain The effects of diffraction in a periodic structure are similar for all waves. These effects emerge naturally from a discussion of any one system. We will discuss lattice vibrations. Take identical masses, m, connected by springs (spring constant, α) This is a model limited to nearest-neighbour interactions. Equation of motion for the nth atom is mu n mu n α α {( un+ 1 un ) ( un un 1) } ( u + u u ) n+ 1 n 1 We have N coupled equations (for N atoms) n May 0 Lecture 11 10
Normal modes Look for travelling wave solutions Wave of angular frequency, ω, and wavevector, q (q is the conventional choice for phonons) u exp i nqa ωt u n 0 { ( )} Substitute into equation of motion mω u mω ω 0 α α exp { i( nqa ωt) } iqa iqa ( e + e ) u exp i( nqa ωt) ( cos qa) 4α m qa ( q) sin { } qa These are the normal modes for the system of coupled atoms. Note: the continuum model for compressive waves (1B Physics, Oscillations, waves and optics course) gave dispersionless solutions, which are the same as the above in the limit of q0, (i.e. the long wavelength limit). 0 4α sin Dispersion relation for phonon modes May 0 Lecture 11 11
Phonon dispersion Dispersion curves ω versus q gives the wave dispersion Key points Key points The periodicity in q (reciprocal space) is a consequence of the periodicity of the lattice in real space (c.f. diffraction on slide ). Thus the phonon at some wavevector, say, q 1 is the same as that at q 1 +ng, for all integers n, where Gπ/a (a reciprocal lattice vector). In the long wavelength limit (q0) we expect the atomic character of the chain to be unimportant. May 0 Lecture 11 1
Limiting behaviour Long wavelength limit dispersion formula (p. 4) ω sin leads to the continuum result (see IB waves course) ω q 0 4α qa sin m ( qa ) qa q αa m a ; ω q q0 Y ρ Continuum result Y - Young s modulus ρ - density These are conventional sound-waves. Short wavelength limit Atomic character is evident as the wavelength approaches atomic dimensions qπ/a. λa is the shortest, possible wavelength Here we have a standing wave ω/ q0 ωmax 4α m May 0 Lecture 11 1
Periodicity: 1 st Brillouin zone Periodicity: All the physically distinguishable modes lie within a single span of π/a. irst Brillouin zone (BZ) choose the range of q to lie within q < π/a. This is the 1 st BZ. 1 st st Brillouin zone (shaded) Number of modes (must equal the number of atoms, N, in the chain) the allowed q values are discrete. Each mode (at particular q) is a quantised, simple-harmonic oscillator 1 1 E n + ω ; n exp( ω kt ) 1 with a particulate character (bosons). Energyω, Mom. q, Velocityv g ω/ q. May 0 Lecture 11 14