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1、IV. Vibrational Properties of the Lattice,Heat CapacityEinstein Model The Debye Model Introduction A Continuous Elastic Solid 1-D Monatomic Lattice Counting Modes and Finding N() The Debye Model Calculation 1-D Lattice With Diatomic Basis Phonons and Conservation Laws Dispersion Relations and Brillo
2、uin Zones Anharmonic Properties of the Lattice,A. Heat CapacityEinstein Model (1907),Having studied the structural arrangements of atoms in solids, we now turn to properties of solids that arise from collective vibrations of the atoms about their equilibrium positions.,For a vibrating atom:,Classica
3、l statistical mechanics equipartition theorem: in thermal equilibrium each quadratic term in the E has an average energy , so:,Classical Heat Capacity,For a solid composed of N such atomic oscillators:,Giving a total energy per mole of sample:,So the heat capacity at constant volume per mole is:,Thi
4、s law of Dulong and Petit (1819) is approximately obeyed by most solids at high T ( 300 K). But by the middle of the 19th century it was clear that CV 0 as T 0 for solids.,Sowhat was happening?,Einstein Uses Plancks Work,Planck (1900): vibrating oscillators (atoms) in a solid have quantized energies
5、,later QM showed is actually correct,Einstein (1907): model a solid as a collection of 3N independent 1-D oscillators, all with constant , and use Plancks equation for energy levels,occupation of energy level n: (probability of oscillator being in level n),classical physics (Boltzmann factor),Averag
6、e total energy of solid:,Some Nifty Summing,Using Plancks equation:,Now let,Which can be rewritten:,Now we can use the infinite sum:,To give:,So we obtain:,At lastthe Heat Capacity!,Differentiating:,Now it is traditional to define an “Einstein temperature”:,Using our previous definition:,So we obtai
7、n the prediction:,Limiting Behavior of CV(T),Low T limit:,These predictions are qualitatively correct: CV 3R for large T and CV 0 as T 0:,High T limit:,But Lets Take a Closer Look:,High T behavior: Reasonable agreement with experiment,Low T behavior: CV 0 too quickly as T 0 !,B. The Debye Model (191
8、2),Despite its success in reproducing the approach of CV 0 as T 0, the Einstein model is clearly deficient at very low T. What might be wrong with the assumptions it makes?, 3N independent oscillators, all with frequency Discrete allowed energies:,Details of the Debye Model,Pieter Debye succeeded Ei
9、nstein as professor of physics in Zrich, and soon developed a more sophisticated (but still approximate) treatment of atomic vibrations in solids. Debyes model of a solid: 3N normal modes (patterns) of oscillations Spectrum of frequencies from = 0 to max Treat solid as continuous elastic medium (ign
10、ore details of atomic structure),This changes the expression for CV because each mode of oscillation contributes a frequency-dependent heat capacity and we now have to integrate over all :,C. The Continuous Elastic Solid,We can describe a propagating vibration of amplitude u along a rod of material
11、with Youngs modulus E and density with the wave equation:,for wave propagation along the x-direction,By comparison to the general form of the 1-D wave equation:,group velocity,D. 1-D Monatomic Lattice,By contrast to a continuous solid, a real solid is not uniform on an atomic scale, and thus it will
12、 exhibit dispersion. Consider a 1-D chain of atoms:,For atom s,p = atom label p = 1 nearest neighbors p = 2 next nearest neighbors cp = force constant for atom p,1-D Monatomic Lattice: Equation of Motion,Thus:,Now we use Newtons second law:,Or:,So:,Now since c-p = cp by symmetry,1-D Monatomic Lattic
13、e: Solution!,The result is:,The dispersion relation of the monatomic 1-D lattice!,Often it is reasonable to make the nearest-neighbor approximation (p = 1):,The result is periodic in k and the only unique solutions that are physically meaningful correspond to values in the range:,Dispersion Relation
14、s: Theory vs. Experiment,In a 3-D atomic lattice we expect to observe 3 different branches of the dispersion relation, since there are two mutually perpendicular transverse wave patterns in addition to the longitudinal pattern we have considered.,Along different directions in the reciprocal lattice
15、the shape of the dispersion relation is different. But note the resemblance to the simple 1-D result we found.,E. Counting Modes and Finding N(),A vibrational mode is a vibration of a given wave vector (and thus ), frequency , and energy . How many modes are found in the interval between and ?,# mod
16、es,We will first find N(k) by examining allowed values of k. Then we will be able to calculate N() and evaluate CV in the Debye model.,First step: simplify problem by using periodic boundary conditions for the linear chain of atoms:,We assume atoms s and s+N have the same displacementthe lattice has
17、 periodic behavior, where N is very large.,First: finding N(k),This sets a condition on allowed k values:,So the separation between allowed solutions (k values) is:,independent of k, so the density of modes in k-space is uniform,Since atoms s and s+N have the same displacement, we can write:,Thus, i
18、n 1-D:,Next: finding N(),Now we know from before that we can write the differential # of modes as:,We carry out the integration in k-space by using a “volume” element made up of a constant surface with thickness dk:,N() at last!,A very similar result holds for N(E) using constant energy surfaces for
19、 the density of electron states in a periodic lattice!,Rewriting the differential number of modes in an interval:,We get the result:,This equation gives the prescription for calculating the density of modes N() if we know the dispersion relation (k). We can now set up the Debyes calculation of the h
20、eat capacity of a solid.,F. The Debye Model Calculation,We know that we need to evaluate an upper limit for the heat capacity integral:, 3 independent polarizations (L, T1, T2) with equal propagation speeds vg continuous, elastic solid: = vgk max given by the value that gives the correct number of m
21、odes per polarization (N),If the dispersion relation is known, the upper limit will be the maximum value. But Debye made several simple assumptions, consistent with a uniform, isotropic, elastic solid:,N() in the Debye Model,First we can evaluate the density of modes:,Next we need to find the upper
22、limit for the integral over the allowed range of frequencies.,max in the Debye Model,Since there are N atoms in the solid, there are N unique modes of vibration for each polarization. This requires:,The Debye cutoff frequency,Giving:,Now the pieces are in place to evaluate the heat capacity using th
23、e Debye model! This is the subject of problem 5.2 in Myers book. Remember that there are three polarizations, so you should add a factor of 3 in the expression for CV. If you follow the instructions in the problem, you should obtain:,And you should evaluate this expression in the limits of low T (T
24、D).,Debye Model: Theory vs. Expt.,Universal behavior for all solids!,Debye temperature is related to “stiffness” of solid, as expected,Better agreement than Einstein model at low T,Debye Model at low T: Theory vs. Expt.,Quite impressive agreement with predicted CV T3 dependence for Ar! (noble gas so
25、lid),(See SSS program debye to make a similar comparison for Al, Cu and Pb),G. 1-D Lattice with Diatomic Basis,Consider a linear diatomic chain of atoms (1-D model for a crystal like NaCl):,In equilibrium:,Applying Newtons second law and the nearest-neighbor approximation to this system gives a disp
26、ersion relation with two “branches”:,-(k) 0 as k 0 acoustic modes (M1 and M2 move in phase) +(k) max as k 0 optical modes (M1 and M2 move out of phase),1-D Lattice with Diatomic Basis: Results,These two branches may be sketched schematically as follows:,gap in allowed frequencies,In a real 3-D solid
27、 the dispersion relation will differ along different directions in k-space. In general, for a p atom basis, there are 3 acoustic modes and p-1 groups of 3 optical modes, although for many propagation directions the two transverse modes (T) are degenerate.,Diatomic Basis: Experimental Results,The opt
28、ical modes generally have frequencies near = 1013 1/s, which is in the infrared part of the electromagnetic spectrum. Thus, when IR radiation is incident upon such a lattice it should be strongly absorbed in this band of frequencies.,At right is a transmission spectrum for IR radiation incident upon
29、 a very thin NaCl film. Note the sharp minimum in transmission (maximum in absorption) at a wavelength of about 61 x 10-4 cm, or 61 x 10-6 m. This corresponds to a frequency = 4.9 x 1012 1/s.,If instead we measured this spectrum for LiCl, we would expect the peak to shift to higher frequency (lower
30、wavelength) because MLi MNaexactly what happens!,H. Phonons and Conservation Laws,Collective motion of atoms = “vibrational mode”:,Quantum harmonic oscillator:,Energy content of a vibrational mode of frequency is an integral number of energy quanta . We call these quanta “phonons”. While a photon is
31、 a quantized unit of electromagnetic energy, a phonon is a quantized unit of vibrational (elastic) energy.,Associated with each mode of frequency is a wavevector , which leads to the definition of a “crystal momentum”:,Crystal momentum is analogous to but not equivalent to linear momentum. No net ma
32、ss transport occurs in a propagating lattice vibration, so the linear momentum is actually zero. But phonons interacting with each other or with electrons or photons obey a conservation law similar to the conservation of linear momentum for interacting particles.,Phonons and Conservation Laws,Lattic
33、e vibrations (phonons) of many different frequencies can interact in a solid. In all interactions involving phonons, energy must be conserved and crystal momentum must be conserved to within a reciprocal lattice vector:,Compare this to the special case of elastic scattering of x-rays with a crystal
34、lattice:,I. Brillouin Zones of the Reciprocal Lattice,Remember the dispersion relation of the 1-D monatomic lattice, which repeats with period (in k-space) :,Each BZ contains identical information about the lattice,Wigner-Seitz Cell-Construction,For any lattice of points, one way to define a unit ce
35、ll is to connect each lattice point to all its neighboring points with a line segment and then bisect each line segment with a perpendicular plane. The region bounded by all such planes is called the Wigner-Seitz cell and is a primitive unit cell for the lattice.,1-D lattice: Wigner-Seitz cell is th
36、e line segment bounded by the two dashed planes,2-D lattice: Wigner-Seitz cell is the shaded rectangle bounded by the dashed planes,1st Brillouin Zone-Definition,The Wigner-Seitz cell can be defined for any kind of lattice (direct or reciprocal space), but the WS cell of the reciprocal lattice is al
37、so called the 1st Brillouin Zone. The 1st BZ is the region in reciprocal space containing all information about the lattice vibrations of the solid. Only the values in the 1st BZ correspond to unique vibrational modes. Any outside this zone is mathematically equivalent to a value inside the 1st BZ.
38、This is expressed in terms of a general translation vector of the reciprocal lattice:,1st Brillouin Zone for 3-D Lattices,For 3-D lattices, the construction of the 1st Brillouin Zone leads to a polyhedron whose planes bisect the lines connecting a reciprocal lattice point to its neighboring points.
39、We will see these again!,bcc direct lattice fcc reciprocal lattice,fcc direct lattice bcc reciprocal lattice,I J. Anharmonic Properties of Solids,Two important physical properties that ONLY occur because of anharmonicity in the potential energy function:,Thermal expansion Thermal resistivity (or fin
40、ite thermal conductivity),Thermal expansion In a 1-D lattice where each atom experiences the same potential energy function U(x), we can calculate the average displacement of an atom from its T=0 equilibrium position:,I Thermal Expansion in 1-D,Evaluating this for the harmonic potential energy funct
41、ion U(x) = cx2 gives:,Thus any nonzero must come from terms in U(x) that go beyond x2. For HW you will evaluate the approximate value of for the model function,Now examine the numerator carefullywhat can you conclude?,independent of T !,Why this form? On the next slide you can see that this function
42、 is a reasonable model for the kind of U(r) we have discussed for molecules and solids.,Do you know what form to expect for based on experiment?,Lattice Constant of Ar Crystal vs. Temperature,Above about 40 K, we see:,Thermal Resistivity,When thermal energy propagates through a solid, it is carried
43、by lattice waves or phonons. If the atomic potential energy function is harmonic, lattice waves obey the superposition principle; that is, they can pass through each other without affecting each other. In such a case, propagating lattice waves would never decay, and thermal energy would be carried w
44、ith no resistance (infinite conductivity!). Sothermal resistance has its origins in an anharmonic potential energy.,Classical definition of thermal conductivity,Phonon Scattering,There are three basic mechanisms to consider:,2. Sample boundaries (surfaces),3. Other phonons (deviation from harmonic b
45、ehavior),1. Impurities or grain boundaries in polycrystalline sample,To understand the experimental dependence , consider limiting values of and (since does not vary much with T).,Temperature-Dependence of ,The low and high T limits are summarized in this table:,How well does this match experimental
46、 results?,Experimental (T),Phonon Collisions: (N and U Processes),How exactly do phonon collisions limit the flow of heat?,2-D lattice 1st BZ in k-space:,No resistance to heat flow (N process; phonon momentum conserved), Predominates at low T D since and q will be small,Phonon Collisions: (N and U Processes),What if the phonon wavevectors are a bit larger?,2-D lattice 1st BZ in k-space:,Two phonons combine to give a net phonon with an opposite momentum! This causes resistance to heat flow. (U process; phonon momentum “lost” in units of G.),More likely at high T D since and q will be larger,