16.1 Introduction1. It is important in the study of condensed matter2. This is another example that classical kinetic theorycannot provide answers that agree withexperimental observations.3. Dulong and Petit observed in 1819 that the specificheat capacity at constant volume of all elementarysolids is approximately 2.49*104J .kilomole-1K-1i.e.3R.4. Dulong and Petit’s result can be explained by theprinciple of equipartition of energy via treatingevery atom of the solid as a linear oscillator with sixdegrees of freedom.
5. Extensive studies show that the specific heatcapacity of solid varies with temperature,becomes zero as the temperatureapproaches zero.6. Specific heat capacities of certain substancessuch as boron, carbon and silicon are foundto be much smaller than 3R at roomtemperature.7. The discrepancy between experimentalresults and theoretical prediction leads tothe development of new theory.
16.2 Einstein’s Theory of The HeatCapacity of a Solid• The crystal lattice structure of a solidcomprising N atoms can be treated as anassembly of 3N distinguishable one-dimensional oscillators!• The assumption is based on that each atom isfree to move in three dimensions!
From chapter 15:the internal energy for N linear oscillatorsisU= Nkθ(1/2 + 1/(eθ/T-1)) with θ = hv/kThe internal energy of a solid is thusHere θ is the Einstein temperature and can bereplaced by θE.)1121(3−+=TE EeNkU θθ
The heat capacity:TeNkNkTUCTEEvvE∂−+∂=∂∂=)13213( θθθ
Case 1: when T >> θEThis result is the same as Dulong & Petit’s
Case 2:As discussed earlier, the increase of is outpowered by the increase ofAs a result, whenT << θE
If an element has a large θE , the ratio will belarge even for temperatures well aboveabsolute zeroWhen is large, is small
SinceA large θE value means a biggerOn the other handTo achieve a larger , we need a large k or asmall u (reduced mass), which corresponds tolighter element and elements that producevery hard crystals.khvE =θukvπ21=
• The essential behavior of the specific heat capacityof solid is incorporated in the ratio of θE/T.• For example, the heat capacity of diamondapproaches 3Nk only at extremely high temperaturesas θE = 1450 k for diamond.• Different elements at different temperatures willposes the same specific heat capacity if the ratio θE/Tis the same.• Careful measurements of heat capacity show thatEinstein’s model gives results which are slightlybelow experimental values in the transition range of
16.3 Debye’s theory of the heat capacity ofa solid• The main problem of Einstein theory lies in theassumption that a single frequency of vibrationcharacterizes all 3N oscillators.• Considering the vibrations of a body as a whole,regarding it as a continuous elastic solid.• In Debye’s theory a solid is viewed as a phonon gas.Vibrational waves are matter waves, each with itsown de Broglie wavelength and associated particle• De Broglie relationship: any particle travelling with alinear momentum P should have a wavelength givenby the de Broglie relation:
For quantum waves in a one dimensional box,the wave function iswithSince where is the speed
Considering an elastic solid as a cube of volumev = L3whereThe quantum numbers are positive integers.Let f(v)dv be the number of possible frequenciesin the range v to v + dv, since n is proportionalto v, f(v)dv is the number of positive sets ofintegers in the interval n to n + dn.
In a vibrating solid, there are three types ofwavesAfter considering one longitudinal and twotransverse waves,Note that: since each oscillator of the assemblyvibrates with its own frequency, and we areconsidering an assembly of 3N linearoscillators, there must be an upper limit to thefrequency, so that
is determined by theaverage inter atomicspacing
The principle difference between Einstein’sdescription and Debye’s modelThere is no restriction on the number ofphonons per energy level, therefore phononsare bosons!
• Because the total number of phonons is notan independent variableThe internal energy of the assembly
• Example I: (problem 16.1) The partition function ofan Einstein solid iswhere θE is the Einstein temperature. Treat thecrystalline lattice as an assembly of 3Ndistinguishable oscillators.(a) Calculate the Helmholtz function F.(b) Calculate entropy S.(c) Show that the entropy approaches zero as thetemperature goes to absolute zero. Show that athigh temperatures, S ≈ 3Nk[1 + ln(T/ θE )]. SketchS/3Nk as a function of T/ θE .
Solution (a)Follow the definitionThe value of U is known asTo solve F, we need to know S (as discussed inclass)
For distinguishable oscillatorstherefore, for distinguishable oscillators (orparticles)
since we have 3N oscillators(this is the solution for b)