PHY1039

Properties of MatterHeat Capacity of Crystalline Solids

March 26 and 29, 2012

Lectures 15 and 16

Body-centred cubic

Face-centred cubic

Why do crystals diffract X-rays?

(100) in grey (110) in grey (111) in grey

(100) in grey (110) in grey (111) in grey

d100

d111

For cubic crystals, with a lattice constant, a: 𝑑h𝑘𝑙=

𝑎

√h2+𝑘2+𝑙2

for parallel (hkl) planes.

Diffraction Peaks Seen When the Bragg Condition Holds

𝑑h𝑘𝑙=𝜆

2 𝑠𝑖𝑛𝜃

= q0

q q

q

2q

In an experiment, can vary either l or q.

Recap: Molar Heat Capacity of Gases

From Tipler’s Physics

n = 1Accessible degrees of freedom

3

5

7

PE = ½ (x – xo)2

• Atoms can vibrate in one of three directions, and have both kinetic and potential energies. During an oscillation, the energy alternates between each type.

• Depending on the local arrangement of atoms in a crystal lattice, the potential energy interactions can vary in the x, y, and z directions.

• For each direction, a spring constant can be defined: Kx, Ky, and Kz. Each direction contributes 1 degree of freedom with ½ kT of thermal energy.

Kinetic and Potential Energies of Atoms in a Crystal

ny

nz n

KE = = nx

Vibrational energy

xo

Higher K and smaller xLower K and larger x

K Affects the Shape of the Harmonic Potential

x

PE = ½ (x – xo)2

PE = ½ (x – xo)2 + ½ (y – yo)2 + ½ (z – zo)2 In 3D:

Molar Heat Capacity of Solids at High T

Dulong-Petit Law for “High Temperature” Molar Heat Capacity

At temperatures where all six vibrational degrees of freedom are accessible in a crystal of a single element, e.g. Cu, the molar heat capacity, cmol, is given as:

= = 3R ~ 25 J mol-1 K-1

This is called the Dulong-Petit Law. Surprisingly, heat capacity does not depend on crystal structure nor on bonding.

Remember that the mass of one mole depends on the molar mass of an element, which increases with the atomic number.

Unlike in gases, in solids CV is only slightly smaller than CP.

At 298 K, Most Solid Elements Have a C of about 3R

Figure from “Understanding Properties of Matter” by M. de Podesta

Heat Capacity of Compounds Containing More Than One Element

The basis consists of a Cl- ion (large green) at (0,0,0) and a K+ ion (small blue) at (1/2, 0, 0) .

One mole of KCl contains 2 NA atoms in total. Compared to one mole of pure potassium (K), there are twice as many atoms contributing to the heat capacity.

Heat Capacity of Compounds Containing More Than One Element

• Two elements in a compound: cmol ≈ 6R = 49.9 J mol-1 K-1

For instance, for NaCl: cmol = 51 J mol-1 K-1

• If three elements (or atoms) in a compound, each mole will have 3 NA atoms: cmol ≈ 9R = 74.8 J mol-1 K-1

For instance, for CaF2: cmol = 72 J mol-1 K-1

Heat Capacity of Solids in the Limit of Low Temperature

This equation predicts that as T 0 K, the second term 0.In the limit when kT >> hfo, however, the second term will increase towards kT.

PE=12h 𝑓 𝑜+

h 𝑓 𝑜

𝑒𝑥𝑝( h 𝑓 𝑜𝑘𝑇 )−1

=

Einstein treated each atom on a lattice in a solid as an independent oscillator with a frequency of fo.

Einstein predicted at thermal energies kT < hf0, some vibrational energy states are not accessible. He derived an equation to describe the thermal activation of the energy of the oscillators as:

m is the mass of the atom.

Figure from “Understanding Properties of Matter” by M. de Podesta

Amplitude of Vibrations are QuantisedEnergy of vibrations take on quantised values: (n + ½)hfo

Lower mass, m Higher mass, m Higher K

PE = ½K (x – xo)2

f=12𝜋 √𝐾𝑚

Einstein Theory Applied to C for Copper

Fit to the data uses fo = 4.79 x 1012 Hz

The fit of the data to the Einstein theory is not good at very low T.

Figure from “Understanding Properties of Matter” by M. de Podesta

Theory successfully predicts that C is lower at low T, as some vibrational states are not accessible.

Atomic vibrations on a lattice are correlated. The vibration of one atom affects its neighbours.

A phonon is a wave-like displacement of atoms in a lattice. Vibrations are described as waves with a frequency f. The amplitude of the waves are quantised such that the energy can only take discrete values:

The Concept of Phonons

(n + ½) hf, where n is an integer

Einstein was not correct to say that all atoms have a single vibration frequency and to ignore coupling between vibrations.

Transverse Phonons in a 1-D Crystal

No phonons

Long l; low f, small displacement (x)

Long l; low f, larger displacement (x)

Short l; high f, small displacement (x)

Constant Wave velocity = l f

l

Assuming a speed of sound in the solid of 4000 m/s, the frequency, f, at the shortest l (= a) must be less than about 1013 Hz.

Lattice spacings, a, are on the order of 10-10 m.

What is the Maximum Phonon Frequency, fmax?

Figure from “Understanding Properties of Matter” by M. de Podesta

4000 m/s = l f

Heat Capacity of Solids at Low Temperature: Debye Equation

Debye derived an equation for heat capacity of solids that assumes internal energy is in phonons (quantised sound waves). A distribution of vibration frequencies at low temperature.

qD is called the Debye temperature. Its value, which depends on the particular solid, is related to the maximum vibration frequency, fmax as:

𝜃𝐷=h 𝑓 𝑚𝑎𝑥

𝑘where h is the Planck constant, and k is the Boltzmann constant.

=

= 1944 J mol-1 K-1

where

Comparisons of Molar Heat Capacity for Several Metals

D

T

hf

kT

max

● = Ag; D = 215 K

= Pb; D = 88 K

X = C; D = 1860 K

Ο = Cu; D = 315 K

Fig. 9.4 from D. Tabor, Gases, Liquids and Solids (1991) Cambridge Univ Press.

3RDulong-Petit limit

Debye Theory Describes the Low-T Heat Capacity Better than Einstein Theory

Figure from “Understanding Properties of Matter” by M. de Podesta

The Debye Temperature Correlates with the Speed of Sound, cs, in a Solid!

Both quantities are related to lattice vibrations.

Figure from “Understanding Properties of Matter” by M. de Podesta

𝜃𝐷=h𝑓𝑘=h𝑐 𝑠𝜆𝑚𝑖𝑛

cs

3R

Cv ~ T3

From Tipler’s Physics

𝑄=∫𝑇 1

𝑇 2

𝐶𝑉 (𝑇 )𝑑𝑇

Heat Requirements to Raise a Solid’s T

T1 T2

Molar heat capacity of iron (Fe)

Molar Heat Capacity of Cu

Fit uses Einstein vibrational frequency of n = 5 x 1012 s-1

Temp (K)

3R

Fig. 9.1 from D. Tabor, Gases, Liquids and Solids (1991) Cambridge Univ Press.

Strong T dependence

Dulong-Petit limit