L01-Einstein Solid PDF

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L01-Einstein Solid...

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Chapter 1

The Einstein solid

Figure 1.1: Albert Einstein (1879-1955) [nobel.org]

1.1

Introduction

It was known in the early 19th Century that the specific heat capacities of solids consisting of a single element at room temperature are approximately equal. The Law of Dulong and Petit pre-dates the development of the kinetic theory of gases, the widespread acceptance of the atomic hypothesis or the development of quantum mechanics. Even the relationship between atomic weight and atomic number was not understood at that time. It was regarded as a remarkable observation because this property of matter is independent of the element or the structure of the sample, invalidiating some of the theories of matter that were current at that time. As it became possible to perform experiments at lower temperatures, it was found that this law did not hold and that the specific heat capacity tended towards zero as the temperature was reduced. Dulong and Petit were interesting individuals. Alexis Petit was a child prodigy who was appointed Professor of Physics at Ecole Polytechnique Paris at the age of 23. He was the brother in law of Arago, who performed the critical experiment that verified the predictions of Fresnel’s wave theory of light (the Spot of Arago). Petit died of tuberculosis at thirty-one years of age. He was succeeded in his post by Pierre Dulong, a scientist known for his reckless experimental technique: he lost an eye and some fingers performing experiments on the highly unstable compound nitrogen chloride. The modern interpretation of the Law of Dulong and Petit is based on the classical equipartition theorem. Each atom of mass m is assumed to reside in a parabolic trap with a characteristic anular

2

Figure 1.2: The molar heat capacity plotted of most elements at 25C plotted as a function of atomic number. [https://commons.wikimedia.org/w/index.php?curid=54031936]

Figure 1.3: Pierre Dulong (1785-1838), Alexis-Therese Petit (1791-1820 and Francois Arago (1786-1853) [wikimedia]. The names of Dulong and Arago are recorded (along with 70 other French scientists and engineers) on the Eiffel Tower.

oscillation frequency ω. The Hamiltonian, H, is given by H=

p2 2m

1 + mω2 x2 . 2

(1.1)

According to the equipartition theorem, each quadratic degree of freedom in H contributes kT/2 to the average thermal energy; in this case there are three quadratic degrees of translational freedom and three of potential energy, so the average thermal energy per atom is 3kT and the average internal energy per mole of atoms is U = 3RT, where R is the ideal gas constant. Note that the properties of gases have made their way into the description of a solid, because the description is based fundamentally on statistics rather than the specific state of matter. As a consequence the molar heat capacity at constant volume, Cv , which is appropriate if we ignore the small amount of thermal expansion in a solid, is given by Cv

∂U ∂T = 3R.

=

3

(1.2)

Figure 1.4: One-dimensional representation of an atom of mass m in a classical harmonic trap. Each spring, k, is anchored at one end to an immovable scaffold (or an infinite mass) and x is the displacement of the atom √ from its equilibrium position. The effective angular frequency of each atom is given by ω = 2k/m

.

1.2 Einstein model Dulong and Petit observed that the specific heat of solids was approximately constant for many materials composed of a single element, but they had no mechanical model for the origin of this behaviour and, as a consequence, no means to explain the variation of specific heat with temperature. Similarly, the specific heats of several light elements deviate significantly from the Dulong-Petit law at room temperature, indicating that the equipartition theorem is not applicable in these cases. In the absence of an atomistic model of the structures of solids, it was not possible to understand these discrepancies As is well-known, Einstein produced three seminal pieces of work in 1905 that laid the foundations of modern physics: special relativity, the quantum interpretation of the photoelectric effect and the statistical basis of Brownian motion. Planck had introduced the quantum hypothesis in 1900 in his analysis of black-body radiation, but Einstein extended this insight in 1908 to the description of matter by developing a model of the heat capacity of solids. It is amongst the first practical applications of the principles of quantum mechanics, anticipating the Bohr atom by several years (1913). In Figure 1.4 an atom is trapped in an harmonic potential with angular frequency ω. In the Einstein solid, every atom is trapped in the same way and each atom has the same frequency because there is no possibility of coupling the motion of the atoms. Einstein extended the quantized treatment of photons in Planck’s analysis of black-body radiation to the quantized three-dimensional description of mechanical oscillators in a solid, so that each oscillator could possess energies given by  3 εn = n x + n y + n z + ~ω (1.3) 2 = εnx + εny + εnz (1.4) for n = {nx , n y , nz } ∈ {0, 1, 2, . . .}. Note that there is no information about the arrangement of the atoms, such as the crystal structure; we assume only that we know that there are N such independent atoms. In one-dimension, the partition function, z for this system at temperature T is z=

∞ X n=0

 εn  exp − kT

(1.5)

so that the partition function for a single atom is z3 and for the entire solid is Z, where Z = z3N .

4

(1.6)

1.2.1

Properties of the one-dimensional oscillator

Inserting the expression for the quantized oscillator energies, εn , into Equation 1.5 we obtain ! ∞ ! n~ω ~ω X exp − z = exp − (1.7) 2kT n=0 kT which is of the form of sum of a geometric progression with ! ~ω a = exp − 2kT ! ~ω r = exp − kT a z = 1−r and so, in various equivalent forms,

z =

  ~ω exp − 2kT   ~ω 1 − exp − 2kT 1

= exp =



~ω 2kT

1

2 sinh =

(1.8)

1 2 sinh





  ~ω − exp −2kT

~ω 2kT



 β~ω  .

(1.9) (1.10) (1.11)

2

with β = 1/kT . The average energy of a single oscillator, hei, is given by hei

∂ ln z ∂β ) (  ∂ β~ω "  = − − ln 1 − exp −β~ω − 2 ∂β ~ω ~ω + . = 2 exp(β~ω − 1) = −

(1.12) (1.13) (1.14)

The first term corresponds to the zero point energy and the second to the thermal population of excited states. In the limit T → 0, the second term vanishes, so that the oscillator possesses only zero-point energy. The average value of n is a measure of the population of quantum states; we may calculate hni in two ways. On purely statistical grounds we may evaluate hni directly from the normalized distribution function,   ∞   X 1 n exp − n + β~ω 2 n=0 (1.15) hni = ∞    . X 1 exp − n + β~ω 2 n=0 Noting that ∞ X

n exp(−nβε) =

n=0

5

d dw

∞ X n=0

exp(−nw)

where w = βε, we may identify hni as a logarithmic derivative   ∞  X d     ln hni = − exp(−nw) dw  n=0 ! 1 d ln = − dw 1 − exp(−w) 1 = exp(w) − 1 1 . = exp(βε) − 1

(1.16) (1.17) (1.18) (1.19)

Alternatively, we may note that hei

= =

*

+ ~ω + hn~ωi 2 ~ω + ~ωhni. 2

(1.20) (1.21)

Rearranging we find, as before, hni =

1 exp(βε) − 1

(1.22)

with ε = ~ω.

1.3 Properties of the Einstein solid Considering the solid as an uncorrelated collection of N three-dimensional oscillators, we may immediately obtain the relevant thermodynamic properties from those of the one-dimensional quantum oscillator. The partition function for the solid, Z, is similarly given by Z = z3N . The Helmholtz free energy, F is just F

= −kT ln Z

(1.23)

= −3NkT ln z !!! ~ω ~ω = 3N . + kT ln 1 − exp − 2 kT Similarly, hEi = 3Nhei

) ~ω ~ω . E = 3N + exp(β~ω) − 1 2 (

(1.24) (1.25)

(1.26)

If we now consider the high-temperature limit such that kT ≫ ~ω we first write exp so that

!2 ! ~ω ~ω 1 ~ω + + ... ≃1+ kT 2 kT kT ~ω ≃ kT. exp(β~ω) − 1

(1.27)

(1.28)

Hence, in the high-temperature limit, we have hEi

≃ 3NkT

= 3RT, 6

(1.29) (1.30)

Figure 1.5: Heat capacities of a few elements scaled by the Einstein temperature, ΘE .

. which is consistent with modern interpretations of the Dulong-Petit law. All of the energy is internal energy and so ∂U Cv = = 3R. (1.31) ∂T In considering the temperature dependence of Cv it is convenient to introduce the Einstein temperature, ΘE , defined by ~ω . (1.32) ΘE = k Within this model, the oscillation frequency, ω, is a property of the material, so different elements possess a characteristic ΘE ; what we now mean by “high temperature” is T ≫ ΘE . If we rescale the heat capacities by the Einstein temperature (Figure 1.5) we see an apparently “universal” behaviour, at least in the high-temperature limit. For low temperatures, T ≪ ΘE , we find that Cv = 3R



ΘE T

2

 ΘE  exp − . T

(1.33)

This is a step in the right direction, since the heat capacity vanishes for T → 0, but it vanishes too rapidly; experimentally it is known that Cv ≃ T 3 for T → 0, but the Einstein model predicts exponential behaviour. The main shortcoming in the model is the assumption that each atom moves independently over every other atom with the same zero-order frequency, ω. We will see in the next lecture that we can overcome this problem by treating the solid as a system of coupled oscillators, following the refinement introduced by Debye in 1912.

1.4 Exercises and Problems 1. Graphite is unusual, in that it consists of tightly bound two-dimensional layers held together by rather weak forces between layers. It was found comparatively recently that a single layer, graphene, can be obtained from graphite simply using Cellotape, which led to an explosion in graphene research. A simple model of graphite utilizes different oscillation frequencies for the motion of atoms in the two-dimensional planes (ωk compared with oscillations perpendicular to the planes, ω⊥ . Assuming that ~ωk ≫ 300 K and making a reasonable assumption regarding the relative value of ω⊥ , determine the molar heat capacity of graphite and compare your estimate with experimental data. 7

2. A crude estimate of the melting point of a solid, Tm , is that it is the temperature at which the average vibrational amplitude is 10% of the inter-atomic separation, a. Using this estimate, show that the Einstein temperature is related to Tm by ΘE ∝ where M is the atomic mass.

8

1  Tm  a M

(1.34)...