Lecture 19 19.11.2019 Bose-Einstein Distribution, Planck Distribution Debye Model for Thermal Vibrations of Crystals

Lecture 19 19.11.2019 Bose-Einstein distribution, Planck distribution Debye model for thermal vibrations of crystals

Fys2160, 2019 1 Bose-Einstein distribution

Each quantum state can be occupied by any number of bosons, hence the grand partition function for a single energy state is 1 � �, �, � = � = , �� > � 1 − � The probability that there are N bosons in the same quantum state is () () � = (1 − � )� � The average occupation number � = ∑ � � = , �� > � � ��(��)�

Classical limit: as high-T limit: �(�) ≪ � � �(��) = ≈ �� = �(��) � ��(��) − � �

Fys2160, 2019 2 Photon gas and the black body radiation

At thermodynamic equilibrium the photon gas is also called thermal radiation or black body radiation. Blackbody: Idealized body that absorbs/emits photons of any wavelength and reflects none. Absorbed energy = emitted energy (at each light frequency) Blackbody radiation: Backbody radiation is the radiation emitted by a blackbody at temperature T At finite temperature, a blackbody will glow with a «colour» depending on T

Planck distribution: shows how much energy is contained at a given light frequency as a function of temperature. We can derive it using that the photon gas is a bose gas that satisfies the Bose-Einsteind distribution for the occupation number

Fys2160, 2019 3 Photons and Bose-Einstein distribution

The chemical potential of photons is zero.

Photons are created/destroyed in any quantity – their number is not conserved. We say that there are uncountable. What this means is that the Helmholtz free energy must be independent of the number of photons, �. It should not be affected by changes in N, and this means by definition that � = 0 �� = = 0 �� , Thus, the Bose-Einstein distribution reads as � � = , �� > � � �� − � What are the energy levels � of a photon?

Fys2160, 2019 4 Energy levels of photons Blackbody radiation: standing electromagnetic (EM) waves • EM modes are described by Ø wavevector � , which is restricted to discrete values � = �, � = 0,1, ⋯ ,, Ø Quantized momentum �⃗ = ℏ� Ø Frequency, �, of an EM mode is 2�� = � � = � |�| • EM mode has two transverse modes � ⋅ � = 0, � ⋅ � = 0 Photons: quanta of light • Each EM mode at frequency � is populated by photons, each with a quanta of energy: ℎ� � = �� = ℎ� = |�| 2�

Fys2160, 2019 5 Bose-Einstein distribution of photons For a given EM mode at frequency �, the grand partition function is a sum over all the photons at that frequency 1 � �, � = 1 + � + � + ⋯ = 1 − � The probability of having N photons at frequency � is � = (1 − � )�

The average occupation number per frequency at a given temperature � � (�) = ∑ � � = ��

The average energy of a given EM mode E �, T = − = This is also equivalent to the energy of a photon occupying a given mode at frequency � times the average number of photons occupying that mode E �, T = ℎ� �(�) How many modes can there be at the same frequency?

Fys2160, 2019 6 Density of states in 1D 1D box of size L, standing waves: � � = � , � = 1,2 ⋯ 2 Frequency � = = � → �� = ��

This corresponds to one wave, but EM waves have 2 polarizations – two degrees of freedom

How many EM modes per frequency? Density of states in a frequency interval: � � �� = 2 �� → � � = 2

Fys2160, 2019 7 Density of states in 3D

3D box of size L, standing waves: � � = |�| , � = 1,2 ⋯ 2 ,, Frequency � = � = � → �� = �� The �-vector can point in any direction in 3D, which means that the EM waves at a given frequency can oscillate in different planes. So, there are many more waves at the same frequency. How many EM modes per frequency in 3D? Density of modes in a frequency interval: (2 for the polarization, for the positive «quadrant» in the state space) 1 1 2� 2� 8�� = 2× ×4�� → � � = 2× ×4� � → �(�) = � 8 8 � � �

Fys2160, 2019 8 Planck distribution The average energy of a given EM mode E �, T = ℎ�×� � =

Planck distribution: The average energy per EM mode at frequency � (Spectral energy density ) �� ℇ �, � = �(�)� �, � = �� − � Spectral energy density is equal to the number of EM modes at a given frequency (density of states) times the energy of a photon (ℎ�) times the average number of photons occupying that mode (Bose-Einstein distribution �(�, �)) �� ℇ �, � = = � � ×��×�(�, �) �� − �

Fys2160, 2019 9 �� Planck distribution ℇ �, � = �� = � � ×��×�(�, �) �� − �

��ℎ � �� : �� ℇ �, � ≈ �� = � � �� Each oscillatory mode get a kT energy from the equipartition of energy

Fys2160, 2019 10 Planck distribution over wavelengths

Spectral energy density per frequency �� ℇ �, � = �� − � By a change of variables, � = ℇ �, � �� = � � �� Spectral energy density per wavelength 8ℎ�� 1 � � = � �/ − 1 This distribution is used to find e.g. the temperature of cosmic background radition and of glowing far way stars. The spectral light intensity is fitted by the Plank distribution, with � = as fitting parameter.

Fys2160, 2019 11 Temperature of Cosmic Background Radiation

• Cosmic microwave background radiation is the leftover thermal radiation from the early state of the universe. • The universe was then filled with ionised gas interacting with EM radiation – temperature was then 3000 K. • Expansion of the universe Doppler shifted the wavelength. • The spectral intensity of the measured background radiation is very well fitted by the Plank distribution at the temperature � = 2.7260 ± 0.0013 �.

Cosmic Background Explorer (NASA, 1989-93) Wilkinson Microwave Anisotropy Probe (NASA, 2001-10) Planck (ESA, 2009-13)

Fys2160, 2019 12 ThermalThermalvibrationsvibrationsinin solids:ain crystal solids: Einstein Einstein crystal crystal • Collection of identical harmonic • Collection oscillatorsof identical harmonic Vibratingoscillators atom in 3D = 3× 1� harmonic oscillators. • Each atom in 3D has 3 one- dimensional harmonic oscillators 1 We• Eachcan modelatom in 3D has 3 onea crystal by a collection- of identical H = G?9, K = 0 4 2 vibratingdimensionalatoms• Classical occupyingharmonicharmonicspecificoscillators latticeoscillator sites 1 H = G?9, K = 0 • Internal energy of a single classical oscillator 4 2 ( ) 8 • Classical harmonic7 oscillator 5 9 9 � • 4 == ++ <= (�?−−�? ) 5 9: 9 A D9 H = 0, K = C 2< 8 • Frequency = = 7 5 9 :9 D9 1 • 45 = + <= (? − ?A) 4 = + G?9 9 • Frequency9:: � 9= Fys2160 2018 2< 2 23D H = 0, K = C 2< • Frequency = = : D9 1 4 = + G?9 Fys2160 2018 2< 2 23 Fys2160, 2019 13 Debye theory for thermal vibrations in crystals

When vibrating atoms are coupled with each other through potential Thermalinteraction energy invibrations the lattice, this leads to thein thermal solids: vibrations Einstein crystal propagating like elastic waves. • Collection of identical harmonic Debye proposed a model assuming that the crystal is like a continuum oscillatorselastic medium. The elastic waves generated by thermal vibrations can then be treated by analogous to electromagnatic radiation in vacuum. • Each• Theatom in 3D has 3 one normal modes of oscillations in -an elastic medium are sound dimensionalwaves. Howeverharmonic, unlike the speedoscillators of light, the sound speed �� depends on the material properties (stiffness and density). It is 1 much, much smaller (by factor 105) than the speed of light. H = G?9, K = 0 4 2 • Classical• Sound harmonicwaves can have 3oscillator polarisations: 2 longitudinal and 1 transversal. • The shortest wavelengts are limited by the interatomic distance. 78 5 • 4 = + <=9(? − ? )9 5 9: 9 A D9 H = 0, K = C 2< • Frequency = = Fys2160, 2019 14 : D9 1 4 = + G?9 Fys2160 2018 2< 2 23 Einstein crystal One quantum harmonic oscillator

%̂& 1 %̂& 1 "! = + +-,& = + (.&-,& Phonons: quantized thermal vibrations2( 2 2( 2 1 Quantized energy levels / = 0 + ℏ. &

Energy level relative to the ground state • Elastic waves in a crystal are quantized, much like the EM waves in a box Δ/ = 0ℏ. Total energy of N harmonic oscillators 5 4 = ∑5 / = ∑5 0 ℏ. + ℏ. are also quantized. 5 781 7 781 7 &

Energy units for N quantum harmonic oscillators at frequency 9 • Each vibrational mode at frequency � can be occupied by a phonon, < ;< − ℏ9 : = > ℏ9 which has a quantum of energy ℎ�. Phonon are directly analogous to Fys2160 2018 24 photons. Suppose we have a crystal of finite size � Then the elastic waves are quantized in a very similar way that EM waves in a box get quantized. We consider standing elastic waves when the crystal is a given temperature. The elastic waves are generated by thermal vibrations of atoms at a given temperature. One normal mode is frequency � at thermal equilibrium at temperature T can be populated by quanta of energy called phonons. The average occupation number of phonons of a normal mode at � is given by the Planck distribution: � � = �� How many normal modes are at a given frequency?

Fys2160, 2019 15 Density of states for normal modes How many normal modes are at a given frequency? Density of states per frequency interval is simular to the density of states of EM modes, except that we now have 3 polarizations 1 2� 2� 12�� = 3× ×4� � → � � = � 8 � � � Density of states per mode number, � = |�| 1 � � = 3× ×4� � 8 Frequency of a mode is related to the quantization number, n by the dispersion relation: � � 2� � � = � = = � 2� 2� � 2� Unlike with the EM modes which in principle can have any frequency �, for the vibrational modes we have a minimum wavelength corresponding to a maximum frequency. This is called the Debye frequency �. How do we determine the Debye frequency?

Fys2160, 2019 16 Debye frequency �

Debye frequency depends on the number of vibrating atoms in the crystal lattice, �.

The total number of harmonic oscillators is 3×�. This number of oscillator must be equal to the total number of modes ∫ �� = ×4�� .

This allows us to determine �: � 6� 3×� = � → � = 2 � From the dispersion relation this corresponds to a maximum frequency (Debye frequency) � � = � 2�

� � �� = � � ��

Fys2160, 2019 17 Total internal energy of thermal crystal, �(�, �, �)

The average energy at a given temperature is computed as sum of the average energy of each normal mode The average energy of one mode is the energy of a phonon times the average number of phonons occupying that mode �� , � = ��×� = � �� − �

Total energy: � = ∑ ∑ ∑ E �, T �(�, �, �) = �� E �, T → �(�, �, �) = �� (�)E �, T � � � � � �� , �, � = � �� , �� = �� − � ��

This expression allows us to compute the heat capacity �� = ��/�� ,� as function of temperature

Fys2160, 2019 18 Total internal energy of thermal crystal, �(�, �, �)

∑ ∑ ∑ Total energy: � = E �, T �� (�, �, �) = � �� − �

After a change of variables, � = , the internal energy can be written as �� /� �� = � �� − � Where � = is the Debye temperature.

Fys2160, 2019 19 Heat capacity of thermal crystals � = ��/�� ,

� ��/� � �� ℎ� � = � �� , � = �� − � �

High temperature limit � ≫ � is the classical limit (equipartition of energy and Dulong-Petit law)

�� ≈ � �� = � = �� → �� = ��

Low temperature limit T ≪ � � �� ≈ � �� = � × → �� = �� − � ��

Fys2160, 2019 20 Experimental verification of the heat capacity 12⇡4 T 3 C = Nk V 5 T ✓ D ◆

Fys2160, 2019 21 Universal scaling of the heat capacity 12⇡4 T 3 C = Nk V 5 T ✓ D ◆

Fys2160, 2019 22 Debye temperature • Heat capacity reaches 95% of its maximum value at TD • Can be found from sound speed – which is the metal specific parameter. • The higher the sound speed and the density of ions, the higher the Debye temperature

• Easiest to treat TD as an experimental fitting parameter.

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