Handbook of Industrial Diamonds and Diamond Films

Mark A. Prelas, Galina Popovici, Louis K. Bigelow, Mark A. Prelas, Galina Popovici, Louis K. Bigelow

• Publisher: CRC Press
• Year: 2018
• Topic: Technology & Engineering
• eBook ISBN:
  9781351442480
• Subtopic: Chemistry
• Index: Chemistry

Examines both mined and synthetic diamonds and diamond films. The text offers coverage on the use of diamond as an engineering material, integrating original research on the science, technology and applications of diamond. It discusses the use of chemical vapour deposition grown diamonds in electronics, cutting tools, wear resistant coatings, thermal management, optics and acoustics, as well as in new products.

Chapter 5

HEAT CAPACITY, CONDUCTIVITY, AND THE THERMAL COEFFICIENT OF EXPANSION

V. I. Nepsha

Almazy Rossii-Sakha Co., Ltd., Almazny Center, 14 Ul. 1812 Goda 121170, Moscow, Russia

Contents

1. Heat Capacity

2. Thermal Expansion

3. Thermal Conductivity

3.1 SCATTERING PROCESSES

3.2 TYPE IIa AND TYPE IIb DIAMONDS

3.3 TYPE Ib DIAMONDS

3.4 THERMAL CONDUCTIVITY IN DIAMONDS WITH VARIABLE ISOTOPE CONTENT

3.5 HYDRODYNAMIC BEHAVIOR OF PHONONS (POISEUILLE FLOW, THE SECOND SOUND)

3.6 THE THERMAL CONDUCTIVITY OF DIAMOND FILMS

3.7 MATERIALS SINTERED FROM DIAMOND POWDERS

3.8 TWO-COMPONENT DIAMOND COMPOSITES

4. References

1. Heat Capacity

Heat capacity is given by the ratio of heat amount δθ absorbed (given back) by the body to the deviation of its temperature, when the last is negligible:

$C=\delta \theta /\delta T$

Heat capacity depends not only on the initial and final states (in particular, the body temperature), but also on the path of the transition between them. One distinguishes the heat capacity at a constant volume (C_v) and the heat capacity at a constant pressure (C_p), if in the process of heating, the volume of a body or the pressure remains constant. In dielectric crystals, the heat capacity is determined by the heat capacity of the crystal lattice. At a constant volume, heat is spent only on the change of vibrational energy of the crystal lattice, and the body heat capacity C_v coincides with the phonon heat capacity. If one imagines the crystal lattice as a multitude of independent harmonic oscillators with frequencies corresponding to the normal vibrations of the lattice, then it is possible to calculate the energy of the system using the methods of quantum statistics.

Differentiation of the energy with respect to the temperature gives an expression for heat capacity of a monoatomic crystal lattice:

$C_v=3n_{o}k{\displaystyle \int \frac{{\left(\hslash \omega /kT\right)}^2{e}^{\hslash \omega /kT}}{{\left(e^{\hslash \omega /kT}-1\right)}^2}D\left(\omega \right)d\omega }$

(1)

where n₀ is the number of atoms per unit volume, ω is the frequency of normal vibrations of the crystal lattice (i.e. phonons), D(ω) is the spectral density of lattice vibrations, and k and ħ are the Boltzman and Plank constants respectively.

Since heat capacity is an integral quantity, the exact form of the function D(ω) is not of great importance and Debye theory [1] gives suitable results. According to this theorone takes into account only the acoustic vibrations of the lattice. At the same time, one assumes that independently of the polarization, the phonons are characterized by the same speed of spreading s (the velocity of sound) and by the linear dispersion dependence, ω_q=sq, where q is the wave number. This condition is actually fulfilled only at low temperatures. In addition, in Brillouin zones, where there are allowed values of the vector q, a Debye sphere of the same volume is substituted in inverted space. From these condition, one can imagine that there is a maximum wave number, q_D, and a corresponding maximum frequency of normal vibrations (phonons), ω_D, which is called the Debye frequency.

If one uses cyclic boundary conditions and takes into account that the full number of normal vibrations is equal to the number of atoms in the volume, corresponding to the chosen boundary conditions, then

$q_D={\left(6{\pi }^2{n}_o\right)}^{1/3}\mathrm{and}{\omega }_D=s{\left(6{\pi }^2{n}_o\right)}^{1/3}$

The spectral density according to the Debye approximation is given by

$D\left(\omega \right)d\omega =3{\omega }^2/{\omega }_D{}^3$

Then from equation (1)

$C_v=9n_{o}k{\displaystyle \underset{0}{\overset{{\omega }_D}{\int }}\frac{{\left(\hslash \omega /kT\right)}^2{e}^{\hslash \omega /kT}}{{\left(e^{\hslash \omega /kT}-1\right)}^2}\frac{{\omega }^2}{{\omega }_D^3}d\omega =9n_{0}k{\left(T/\theta \right)}^3{\displaystyle \underset{0}{\overset{T/\theta }{\int }}\frac{x^4{e}^x}{{\left(e^x-1\right)}^2}dx}}$

(2)

Here x = ħω/kT, θ = ħω_D/k, the Debye temperature. At low temperatures (T≪θ) one gets from (2) with good accuracy

$C_v=\left(12{\pi }^4/5\right)n_{o}k{\left(T/\theta \right)}^3$

That is, the heat capacity is proportional to T³ (Debye relation of T³ dependence). At high temperatures (T≪θ)

$C_v\approx 3n_{o}k$

That is, the heat capacity does not depend on temperature (the law of Dulong and Pti). In practice, heat capacity temperature dependence is expressed by equation (2) with a single parameter θ for all temperatures, only in general outline which is related to the above mentioned assumptions of Debye theory. The effective Debye temperature θ_ef(T) is the value at which divergence with Debye theory occurs. θ_ef(T) equals the value of θ in equation (2) at which equation (2) agrees with the experimentally determined heat capacity C_v at the given ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Contributors
• Properties
• Thermal Properties
• Characterization
• Mined Diamond
• Theory
• Modeling and Diagnostics of Plasma Reactors
• Methods of Chemical Vapor Deposition Growth
• Structural Modification of Diamond
• Applications of Industrial Diamonds
• Economics
• Index