SOLIDS SUBSTANCE

THERMAL PROPERTIES OF SOLIDS

Oleh:

Gde Parie Perdana 1113021059/VIIA

Putu Aya Mahadewi 1113021063/VIIA

JURUSAN PENDIDIKAN FISIKA

FAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAM

UNIVERSITAS PENDIDIKAN FISIKA

SINGARAJA

2014

1

THERMAL PROPERTIES OF SOLIDS

Specific heat of solids is one of the thermal properties of solids are important. In fact as

will be shown in this chapter is very difficult to understand the experimental results of the

specific heat of solids except by introducing quantum statistical mechanics. Therefore, this

section will discuss some of the specific heat theory is based on the development of statistical

mechanics.

1.1 Thermal Oscillations

Atoms in solids are in a steady state thermal vibrations. The frequency and amplitude

of the thermal vibrations of atoms in a solid can be determined by using the rationale that

the solids are composed of discrete atoms, and the discrete nature of these should be

primarily used in the calculation of lattice vibrations. However, when the wavelength of

the waves generated by the thermal vibration is very long then the solids can be viewed as

a continuous medium. wave generated by such vibrations known as elastic waves.

To explain the propagation of elastic waves please review a sample of a long rod as

shown in Figure 1.1.

Suppose on the rod occur a longitudinal wave in the x direction, and stated elastic

displacement at point x is u(x). Strain is defined as;

dx

due .................................................................................. 1.1

and stress is defined as force per unit area which is also a function of x, according to

Hooke's law, the stress depends on the strain in the form;

𝑆 = 𝑌. 𝑒 ....................................................................................... 1.2

where Y is known as the elastic constants Young's modulus. Using Newton's second law,

the dynamics of the stem segment dx experiencing thermal vibration can be derived as

follows;

AxSdxxSt

udxA )(..

2

2

................................................................ 1.3

where ρ the mass density of rods and a cross section through which the wave by writing

x x + dx

Gb. 1.1 Elastic waves in rods

2

x

SxSdxxS

)( and by replacing S according to the equation 1.2 and then

use the equations of the dynamics of the wave equation oA V

3,12 is obtained as;

02

2

2

2

t

u

Yx

u ........................................................................................... 1.4

known as the one-dimensional wave equation. 4.4 Completion of the wave equation is

txkieAtxu ...),( .................................................................................... 1.5

where k is the wave number (

2k ), ω is the wave frequency and A is the wave

amplitude. By inserting equation 1.5 into equation 1.4 is obtained;

vk. ........................................................................................................... 1.6

where

Yv .......................................................................................................... 1.7

Equation 1.6 which connects the frequency and wave number is called dispersion

equation. Equation 1.7 states that the speed of propagation of mechanical waves that occur

in a solid medium properties of solids which are influenced by the nature of the medium

elasticity and the mass density of the medium. Figure 1.2 shows the elastic wave dispersion

relation, which is a straight line whose slope (slope) equal to the speed of sound waves,

where ω linearly related to the k

Equation 1.7 can also be used for Young's modulus. The measurement results show

a certain solids have a mass density ρ = 5 gr/cm3 with a speed of 5 x 105 m/s has Y = 1,25

x 1012 gr/cm.s2.

k

ω=v.k

Gb. 1.2 relation of elastic wave dispersion

3

The above equation is derived for a longitudinal wave, but the same way can also be

applied to transverse waves. In understanding the elastic waves as described above is used

as a solid approach to continuous isotropic medium, but in fact crystalline solids are not

isotropic (anisotropic) and the effect of non isotropic crystal leads to characteristic values

(Young's modulus, conductivity) of the crystal . To further understand the effects of non

isotropic crystal used mathematical approaches Tensor. For simplicity of further discussion

on the discussion of the solids used approach isotropic solids.

1.2 Circumstance Mode of Density

Review the generated elastic waves a stick figure 1.1, in which the wave propagates

in the direction of only one dimension. Completion is generated as equation 1.5 can be

expressed by;

xikeAxu ..)( ................................................................................................ 1.8

Further discussion of the boundary condition (boundary condition) of equation 4.8.

The boundary condition resulting from the application of the external effects of the ends of

the rod. Type of boundary condition that is often applied is the periodic boundary condition,

namely the right end of the rod is restricted so as to have the same state oscillation with the

left end of the rod. Suppose the length of the rod is L by taking a point cloud on the left end

point of the rod, the periodic condition states;

u(x = 0) = u(x = L) ......................................................................................... 1.9

where is the solution of equation 1.8. When inserted into equation 1.9 obtained;

1. Like ......................................................................................................... 1.10

This equation determines a state of acceptable values of k and k values only from

1.10 equation is exactly what allowed. Because 12. ine for every integer n, thus the

value of k that are permitted;

Lnk

2 ......................................................................................................... 1.11

with n = 0, ± 1, ± 2, ± 3, .... When these prices is depicted along the axis k, then

formed an irregular space points one dimension, as shown in Figure 1.3

1.3

4

Each price k equations or 1.11 each image point 1.3 declares a vibrate mode. Suppose

chosen a certain interval dk in k space and can then be determined from the number of

modes of vibration occur in the interval k. By taking L large enough, the points can be

viewed as a quasi-continuous. Since the distance between the points is .................. 2π/L

then the number of modes vibrate in the interval dk is

dkL

dn2

..................................................................................................... 1.12

but k where ω are interconnected through a dispersion equation, can thus be

determined number of modes vibrate in the frequency interval d ω which lies between

ω until ω + dω. Density condition of g(ω) defined such that g(ω) dω stating the

amount of vibration mode frequency intervals that occur between ω until ω + d ω. From

these definitions can be written

g(ω) d ω = dkL

2 .......................................................................................... 1.13

or it can be stated density condition vibrate modes are

d

dkLg . ................................................................................................ 1.14

Equation 1.14 is a general equation for the one-dimensional case, which shows that

the density condition g(ω) determined by the dispersion equation. For linear relationship

equation 1.6 dω/dk = v so that the equation can be expressed;

v

Lg

1.

.................................................................................................... 1.15

which is a constant price does not depend on ω.

The results obtained in the discussion of one-dimensional vibration can be developed

for the case of three-dimensional vibration. Settlement for the case of three-dimensional

wave is expressed by;

k.rieAzkykxki

eAtzyxu zyx .)(

.),,,(

.................................................. 1.16

where wave propagation is described by the vector k indicates arh wave propagation

direction and the magnitude is inversely proportional to wavelength. In the discussion of

three-dimensional waves in the medium once again takes the effect of the boundary

conditions for three-dimensional. For simplification, review the discussion of three-

dimensional rods as a medium in the form of a cube whose sides are L. By applying the

periodic boundary condition, the obtained velue allowable k is eligible;

5

1)(

LkLkLkie zyx

.................................................................................... 1.17

ie value generated

Lzn

Lyn

Lxnzkykxk

2,

2,

2,,

............................................................ 1.18

is a pair of three integers. Value of k is allowed to be three-dimensional waves. In k

space as shown Figure 1.4 each pair of the three value zkykxk ,, express a point in k

space stating an allowable vibration mode.

The volume of each point in k space is (2π/L)3. The number of modes vibrate is the

same as the number of points k price pairs are allowed in the ball room. Volume of a sphere

which radius k is (4π/3).k3, due to the volume of each point is (2π/L)3 then the number of

modes of vibration are permitted in k space

3

3

4

32

3

3

43

2k

Vk

LN

................................................................. 1.19

with V = L3 is the volume of the solid sample. 1.19 equation states that the sum of

all allowable wave that has a price k is smaller than a certain price and spread in every

direction. With 4:19 to lower the k equation is obtained

dkk

VdN 24

32

....................................................................................... 1.20

stating the amount of vibration mode in the spherical shell elements whose fingers

between k to (k + dk)

6

Back to the definition of a state meeting of g (ω) such that g (ω) dω is the number of

modes of vibration frequency lies in the interval ω to ω + ω d. This amount can be

determined from the equation by changing variables k 4:20 into ω by using the dispersion

equation ω = v.k, so get

d

v

Vdg

3

24

32)(

.......................................................................... 1.21

Based on the equation of state of 1.21, the meeting of the vibrating modes allowable

wave frequency ω is

3

24

32)(

v

Vg

..................................................................................... 1.22

From 1.22 equations show that g (ω) increases with ω2, unlike the one-dimensional

case where g (ω) constant value does not depend on ω. This increase is the fact that the

volume of the ball element in Figure 4.4 increases to k2.

1.3 Energy of Thermal Oscillation

To determine the amplitude of thermal vibrations can be determined from the

average energy of a one-dimensional vibration in a state of thermal equilibrium of the

environment. Thus the relative probability of vibration energy E at temperature T is given

by the Boltzmann factor kTEe /

, thus the average energy vibration in thermal equilibrium

expressed

dEkTEe

dEkTEeEE

/

/. ...................................................................................... 1.23

The amount of vibrational energy at any time expressed by

7

2.212.

21 xKvmE ...................................................................................... 1.24

the particle velocity v states, elastic constants K and x the displacement from the

equilibrium position. By inserting equation 1.23 to equation 1.24 obtained an average

energy:

dxdvkTEe

dxdvkTEeE

E

./

./.

........................................................................... 1.25

by including that expressed by the equation E 1.24 to equation 1.25 equations

obtained

dxdvkTKxmve

dxdvkTKxmveKxmv

E

.2/

.2/.2212

21

22

22

dxkTKxedvkTmve

dvkTmvedxkTKxexK

dxkTKxedvkTmve

dxkTKxedvkTmvevm

2/2/

2/2/221

2/2/

2/2/221

22

22

22

22

dxkTKxe

dxkTKxexK

dvkTmve

dvkTmvevm

2/

2/221

2/

2/221

2

2

2

2

................................. 1.26

By exampling

kT

mvy

221

2 and kT

Kxz

221

2

We get 222 ym

kTv

dym

kTdv

2 ................................................................................................ 1.27

And

8

222 zK

kTx

dzK

kTdx

2 ................................................................................................. 1.28

by inserting equation 1.27 and the 1.28 to 1.26 equations resulting equations;

dzze

dzzezkT

dyye

dyyeykT

E2

2

2

2 22

........................................................ 1.29

equation is a form of special functions known as shape functions Gama. 1.29

equations used to complete the following formula;

2

2 2 de

and

de2

............................................ 1.30

thus the result of the equation is 1.29

kTkTkTE 21

21

................................................................................... 1.31

The average kinetic energy equal to the average potential energy is equal to kT21 ,

so that the average vibrational energy of particles is equal to kT.

In statistical mechanics has been discussed that the vibration of a particle in one

dimension has two degrees of freedom, one degree of freedom associated with each of the

two modes is the energy possessed by kT21 . Thus for the three-dimensional vibration each

particle will have three degrees of freedom, each degree of freedom will contribute an

average vibrational energy of kT. So that every vibration of a particle in three dimensions

would contribute an average energy of 3 kT.

The amplitude of a harmonic vibration is the maximum displacement on either side

of the equilibrium position. When x = A, then all the energy in the form of potential energy

kTKAE 221 . With this approach gained an average amplitude of vibration produced

is

K

kTA

2 ..................................................................................................... 1.32

9

Equation 1.32 states amplitude average particle vibration in thermal equilibrium at

temperature T. Equation 1.32 shows that the average amplitude depends on K and T and

not depend on the mass of the particle.

1.4 Specific Heat of Solids

When the temperature of a solid is increased, the energy of these solids will increase.

If the energy of a solid generated by the vibrational energy of the atoms making up the

solids, the specific heat of solids can be determined directly from the results of the previous

discussion of vibrational energy.

Review the molar specific heat of solid at constant volume cv, which is defined as the

energy that must be added to 1 kilo mole a solid substance whose volume is set to raise its

temperature constant 1oC. The specific heat of a solid at constant pressure cp is 3% to 5%

higher than cv, due to the constant pressure process produces attempt to change the volume

in addition to increasing energy in solids.

The vibrations of each atom in a solid can be decompose into three components along

the perpendicular axis. Thus the vibration of the individual atoms making up the solid can

be viewed as three harmonic oscillator, so the average energy oscillator produced by each

atom is 3 kT, because for each oscillator harmonics. Each kilo mole solids containing as

much as N0 = 6.02 x 1026 atoms. Thus the amount of energy per kilo mole solids is:

𝑈 = 3𝑁0𝑘𝑇 = 3𝑅𝑇

With R = N0k = 8.31 x 103 Joule/kmole.K is the universal gas constant. Solids Specific

heat at constant volume expressed by

VT

Uvc

........................................................................................................ 1.34

Thus, the specific heat of the solids obtained

cv = 5.97 kkal/kmole.K ......................................................................................... 1.35

Dulong and Petit then show experimental results that the specific heat of solids at

room temperature and greater temperature is cv ≈ 3R, known as the Dulong-Petit law.

However, the Dulong-Petit law failed to account for the specific heat of light elements such

as boron, beryllium and carbon like diamond, each of which has a specific heat respectively

3.34, 3.85 and 1.46 kcal / kmole.K at room temperature. Even the Dulong-Petit law also

fails to explain the specific heat of all solids were down sharply as a function of T3 at low

temperatures near zero at temperatures near 0 K. Figure 1.4 shows how the change of

10

specific heat against T for several types of solids. The two failures of the Dulong-Petit law

is a very serious failure of the experimental results.

1.5 Einstein Theory

In 1907, Einstein showed fundamental error of the equation 1.35, which is located at

kT overview of the average energy per oscillator in a solid. This error is the same as the

existing errors in the Rayleigh-Jeans formula that considers spectrum is continuous, but in

fact the energy is quantized in multiples hυ. Einstein treat atoms in solids as an oscillator,

which do not depend on each other, the energy of each oscillator expressed in quantum

mechanics differs from classical theory. According to the theory of quantum mechanics the

energy of an isolated oscillator is

En = n hυ = n ................................................................................................. 1.36

Where n is a positive integer (n = 0, 1, 2, 3…). Equations 1.36 isolated oscillator energy

states, but the oscillations of atoms is not isolated solids, the atoms continuously conduct

exchanges thermal energy with the energy of the solid environment. Therefore, the energy

of solids is continuously changing, but the average price of energy in thermal equilibrium

expressed by

CV

(kkal

/km

ole

.K)

Absolute Temperature (K)

Lead Aluminum Silicon

Carbon

(diamond)

0 200 400 600 800 1000 1200

1

2

3

4

5

6

7

Figure 1.6 Temperature changes specific heat of some solids

11

0

/

0

/

n

kTE

n

kTE

n

n

n

e

eE

E ............................................................................................... 1.37

Factor kTnE

e/

known as the Boltzmann factor, which represents the probability of the

oscillator has energy state En. By inserting equation 1.36 to 1.37 and the equation 1.37

expressed in the form

0

/ln

/1n

kTnEe

kTE

Therefore, the summation inside the logarithm will be infinite geometric series. By

summing the series and differentiated then obtained the average energy per oscillator

according to Einstein is

1/

.

kThe

hE

.................................................................................................. 1.38

Thus the overall amount of energy oscillator is

1/

..33

kThe

hoNEoNU

.................................................................................... 1.39

From the equation 1.39 can be determined molar specific heat of solids is

21/

/23

kThe

kThe

kT

hR

VT

Uvc

.......................................................... 1.40

To give an interpretation of the equation 1.40, for the high temperature at which the

prevailing circumstances hυ << kT so

kT

hkThe 1/ ................................................................................................. 1.41

Because ...!3

3

!2

21

xxxxe , so in high temperature 1.40 become kTE . Thus the

molar specific heat obtained Rvc 3 , which corresponds to the Dulong-Petit law. For the

low temperature h >> kT then 1/ kThe so that the equation 1.40 becomes

kThehE /. ...................................................................................................... 1.42

Which shows that the average energy down exponentially with decrease in temperature

solids. From equations 1.42, can be derived molar specific heat of solids, which produce

12

kThekT

hRvc /

23

..................................................................................... 1.43

Which indicates that the molar specific heat of solids down exponentially with decrease in

temperature solids. Einstein's formulations cv down near zero at very low temperatures, as

opposed to the Dulong-Petit law.

Einstein also argued that the possibility of a oscillator harmonic energy is

hnnE .21 (n = 0, 1 , 2, …) ...................................................................... 1.44

Equation 1.44 states that the oscillator in the ground state has an energy ½ h means that

the energy of the ground state is not equal to zero as obtained from the classical discussion

above. The ground state energy ½ h known as zero-point energy. The existence of zero

point energy does not affect the analysis of the specific heat of solids due to the zero-point

energy is not dependent solids temperature.

1.6 Debye Theory

Although Einstein's theory can successfully show the state of the specific heat of solids

at high temperatures corresponding to the Dulong-Petit law and can explain the drop of the

molar specific heat of solids at low temperatures, but Einstein's theory fails to explain the

experimental results show that the decline in the molar specific heat of solids as T3 function.

Einstein in analyzing the specific heat of solids considers that the solids prepared by

the atoms that serve as sources of isolated oscillator not depend on each other and each

oscillator have energy h . Debye in 1912 developed the theory of specific heat by

considering the coupling effect between oscillator of atoms nearest neighbors. Debye

looked solid as a continuum elastic substance. Energy in the solids generated by standing

elastic waves, such as electromagnetic wave system in a black box containing quantized

energy. Quantum energy in solids called the phonons, which propagates at the speed of

light.

The amount of elastic waves standing between the frequency ω to ω + dω is in

accordance with the equation 1.22 is

dv

Vdg

3

24

32)( . Thus the number of

modes of elastic waves standing between the frequency of unity volume υ to υ + dυ is

dv

dn3

2.4 .............................................................................................. 1.45

13

With v is the speed of the wave. In solids, there are two types of waves that can occur are

longitudinal waves and transverse waves, each of which has a different speed rates are vl

and vt, further there are two directions of polarization of transverse waves, so the statement

equation 1.45 becomes

d

tvl

vdn 2

3

2

3

14

............................................................................... 1.46

Average energy of elastic waves that occur can be obtained from the equation 1.36

developed by Einstein. In the calculation of the total energy by the Debye frequency elastic

waves is restricted from 0 until υD known as maximum frequency or the Debye frequency.

Thus the energy in solids is

d

kThe

h

tvl

vVU

D

0 1/

3.

3

2

3

14 ................................................................ 1.47

The upper limit is the maximum frequency of a particular frequency υD who interpret the

fact that it cannot happen to infinity standing waves in solids or solids will have a certain

energy. Debye assumed that the total number of standing waves in solids is 3No. So from

equations 1.46 obtained

D

d

tvl

vVoN

0

2.3

2

3

143

333

.21

3

4D

tl vvV

3

1

33

0

214

9

tl

D

vvV

N

....................................................................................... 1.48

Using formulations equation 1.48 can rewritten into

d

e

hNU

D

kThD

0/

3

30

1.

9 .................................................................................. 1.49

14

To further simplify the form of the integral equation 1.49 is more appropriate change

variables υ to a dimensionless quantities x where kT

hx

which dx

h

kTd , in addition

defined Debye characteristic temperature k

h D . In x and θ equation 1.49 become

dxe

xTkNU

D

xo

0

3

3

4

1.9 dx

e

xTR

D

x

0

3

3

4

1.9 ................................................... 1.50

Thus the molar specific heat at constant volume is

T

TxV

veTe

dxxTR

T

Uc

/

0/

33

1

1

1.49

........................................ 1.51

Equation 1.51 shown that specific heat is function T/θ. Next to the two extremes of

temperature, state of high temperature and low temperature can explained as follows.

For high temperature then T/θ is very small so Te T /1/ then obtained

1

11

1

1

1/

T

TeT T

and

3/

0

2

/

0

3

3

1

1

Tdxx

e

dxxTT

x

Then for high temperature equation 1.51 produce RcV 3 , corresponding to the

experimental results.

For low temperature θ/T then the second term in square brackets equation 1.51

be very small and negligible upper bound integral first term becomes ∞. Because

151

4

0

3

xe

dxx

Then at low temperatures the equation 1.51 produces

34

43

3

12

1549

TR

TRcV ................................................................... 1.52

Results shown equation 1.52 shows that the calculation results according to the

experimental results stating that the specific heat of solids as a function of T3. At moderate

temperature, specific heat of solids according to the Debye formula to calculated

numerically. Interchangeability of specific heat of solids to the T/θ according to the Debye

theory shown in Figure 1.7

15

Figure 1.7 Changes in specific heat of the T/θ according to the Debye theory

CV (k

kal

/km

ole

.K)

0 0,5 1,0 1,5 2,0 2,5

1

2

3

4

5

6

T/θ