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SOLID STATE PHYSICS :fire: - Coggle Diagram
SOLID STATE PHYSICS
:fire:
Lattice Vibration II
Nur Najiha binti Ahmad Rasid - A20SC0262
DERIVATION OF FORCE CONSTANT FROM EXPERIMENT
The displacements of the atoms from their equilibrium position are Un
Force to the right :
Force to the left :
The total force = Force to the right – Force to the left = ma
Equation of motion for an atom
=
Wavelike solution in which all atom oscillate with a same amplitude a and frequency ω :
First differentiate :
Second differentiate :
FIRST BRILLOUIN ZONE
Equation :
ω-k relation : Dispersion Relation
K= 2π / λ
λmin =2a
Kmax=π /a
-π/a<K<π/a
What if the physical significance of wavenumbers outside this range?
Explanation:
Fig. 1 (a) shows the displacements for k = π/a, which gives the maximum frequency 2(K/M)1/2 that we have calculated.
Now consider the displacements, shown by the full curve in Fig. 1 (b), for the slightly larger value 8π/7a of k corresponding to the point A in Fig. 2.
The displacements can also be represented by the longer wave, shown broken in Fig. 1 (b), for which Ikl = 6π/7a; this corresponds to points B or C in Fig. 2.
Thus, points A, B and C correspond to the same instantaneous atomic displacements as well as the same frequency.
1D- MONOATOMIC LATTICE
Monoatamic Chain
Simplest crystal is 1-D chain of identical atom, with identical masses M connected by identical springs of spring constant K.
Atom are separated by a distance of 'a'
Atom moves only in a direction parallel to the chain
Only nearest neighbours interact ( short range forces )
Nur Wardina Syahirah binti Mohamad Fadil - A20SC0274
PHASE VELOCITY & GROUP VELOCITY
In a travelling wave, the velocity of a particular feature of the wave (such as its maximum) is its phase velocity,
Equation :
Specifically, for the dispersion relation of the monatomic chain, this means
Equation :
The cos term forces the group velocity down to zero at the edge of the first Brillouin zone (k=π/a),
i.e. there is a standing wave with no net propagation in this limit.
However, energy is transported by the wave at a generally slower speed, the group velocity,
Equation :
LONG WAVELENGTH LIMIT
Long wavelength unit equation :
λ >>a
q=2π / λ << 2π / a
qa <<1
small q- close to the center of Brillouin zone
Linear dispersion:
So it is evident that for long wavelengths or small frequencies the lattice behaves as a continuum or no dispersion takes place
Lattice Vibration I
:<3:
Nurul Athirah Husna Binti Zarazal (A20SC0303
) :<3:
-atoms are very heavy
-forces between atoms are too large
-prevents the atoms from moving.
lattice point is also :red_cross: moving (static).
-in reality, atoms are not very heavy
-force acting between the atoms is not an infinite force.
-static lattice models are valid only at zero temperatures
vibration around the equilibrium lattice position.
LATTICE VIBRATION :star:
When a wave propagate along the crystal,entire planes of atoms move in phase with displacement (parallel/ perpendicular) to the direction of wavevector. :fire:
atoms or ions are physically linked together through the bonding system -the vibrational motion is in general ( :red_cross: isolated)
IMPORTANT :!:
modes and frequencies of vibration of the atoms in a crystal about their positions of equilibrium
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atoms are in a permanent position.
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CRYSTAL DYNAMICS
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Physical properties of solid can be divided by electrons and movement of atom about equilibrium position
(sound velocity and thermal properties).:check:
characteristics vibrations of a crystalline solid
- vibrational movement of atoms in the solid state
Vibrations Spectrum:
A knowledge of the frequencies of atomic vibration and of the manner in which the available degrees of freedom are distributed amongst them
:pencil2:
consideration of the conditions for wave propagation in a periodic lattice,
the energy content
the specific heat of lattice waves,
the particle aspects of quantized lattice vibrations (phonons)
consequences of an harmonic coupling between atoms
Coupling
: In classical mechanics, coupling is a connection between two oscillating systems :check:
Harmonic
: A harmonic is a signal or wave whose frequency is an integral (whole-number) multiple of the frequency of wave :check:
:!!:
Atoms vibrate about their equilibrium position at absolute zero
:!!:
zero point energy (ZPE):
:warning:
lowest possible energy that a quantum mechanical system may have.
:red_flag:
Quantum mechanics differs from classical physics
in energy, momentum, angular momentum, and other quantities of a bound system- restricted to discrete values (quantization), objects have characteristics of both particles and waves
The atoms in a crystal at absolute zero temperature with zero point motion of atoms with displacement from the equilibrium position as a1
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- The atoms in a crystal at room temperature with increased amplitude of vibrations of atoms with displacement from the equilibrium position as a2.:check:
:explode:
HOOKE'S LAW
:explode:
properties of elasticity
it takes about twice as much force to stretch a spring twice as far. That linear dependence of displacement upon stretching force is called Hooke's law.
When a force pulls a spring, the spring stretches. The extension of the spring is directly proportional to the pulling force
:!!:
When a
spring is stretched, there is a restoring force that is proportional to the displacement.
Elastic Region ends
is called the inelastic limit, or the proportional limit. In actuality, these two points are :red_cross: same.
The inelastic Limit is the point at which permanent deformation occurs
if the force is taken off the sample,
it will :red_cross: return to its original size and shape, permanent deformation has occurred.
The Proportional Limit
is the point at which the deformation is no longer directly proportional to the applied force:warning:
:pencil2:
Nur Nadirah binti Mohd Nor:pencil2: (A20SC0261)
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SIMPLE PERIODIC MOTION
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:!!:
WHAT IS SIMPLE PERIODIC MOTION :!!:
That motion in which a
body moves back and forth
over a fixed path
Returning to each
position
and
velocity
after a definite interval of time
Period, T
is the time for one complete oscillation.
(seconds,s)
Frequency, f
is the number of complete oscillations per second.
Hertz (s^-1)
Amplitude, A
is the amplitude is simply the maximum displacement of the object from the equilibrium position
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IN THE PERIODIC MOTION
The displacement of the object may or may
not be in the direction
of the restoring force
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IN SIMPLE HARMONIC MOTION
The displacement of the object is always in the
opposite direction
of the restoring force
:warning:
Motion
that repeats itself is
periodic motion
A particular kind of
periodic motion
is known as
simple harmonic motion
When an object is
disturbed from equilibrium
, its
motion
is probably
simple harmonic motion
:explode:
All simple harmonic motions are periodic motions but all periodic motions are not simple harmonic motions
:explode:
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SIMPLE HARMONIC MOTION
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:!!:
WHAT IS SIMPLE HARMONIC MOTION:!!:
Is periodic motion in the
absence of friction
and produced by a restoring force that is
directly proportional to the displacement
and
oppositely directed
SHM
is a special type of periodic motion or oscillation where the
restoring force is directly proportional to the displacement
Acts in the
direction opposite to that of displacement
A restoring force, F
acts in the direction opposite the displacement of the oscillating body
:star:
F=-kx
:star:
Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear
elastic
restoring force given by
Hooke's Law
The motion is sinusoidal in time and
demonstrates a
single resonant frequency
:red_flag:
Mass on spring resonance
A mass on a spring has a single resonant frequency determined by its
spring constant k
and the
mass m
.
Using Hooke's law and neglecting damping and the mass of the spring, Newton's second law gives the equation of motion:
In physics, resonance describes the phenomena of amplification that occurs when the frequency of a periodically applied force is in harmonic proportion to a natural frequency of the system on which it acts.
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THERMAL ENERGY AND LATTICE VIBRATIONS
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Atoms vibrate about their equilibrium position
They produce vibrational waves
This motion is increased as the temperature is raised
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In a solid
, the energy associated with this
vibration
and perhaps also with the
rotation
of atoms and molecules is called as
thermal energy
.
:check:
In a gas
, the translational motion of atoms and molecules contribute to this energy
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Heat capacity from Lattice vibrations
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:explode:
The energy given to lattice vibrations is the dominant contribution to the heat capacity in most solids.
In non-magnetic insulators, it is the only contribution
Other contributions:
In metals -> from the conduction electrons.
In magnetic materials -> from magneting ordering
Atomic vibrations leads to band of normal mode frequencies from zero up to some maximum value.
Calculation of the lattice energy and heat capacity of a solid therefore falls into two parts:
i) the evaluation of the contribution of a single mode, and
ii) the summation over the frequency distribution of the modes
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Heat Capacity
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:fire:Heat capacity is the
amount of heat
(measured in Joules or Calories) needed to raise an
unit amount of substance
(measured in grams or moles) by an
unit in temperature
(measured in C or K)
This ‘heating’ (addition of energy) can be carried out at
constant volume
or
constant pressure
At
constant pressure
, some of the heat supplied goes into doing work of expansion and less is available with the system (to raise it temperature)
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Heat capacity at constant Volume (CV):
It is the slope of the plot of internal energy with temperature
:check:
Heat capacity at constant Pressure (CP):
It is the slope of the plot of enthalpy with temperature
:warning:
Units: Joules/Kelvin/mole, J/K/mole, J/C/mole, J/C/g
:red_flag:If a substance has higher heat capacity, then more heat has to be added to raise its temperature. As
T->0K
, the
heat capacity tends to zero
U
is energy and
T
is temperature
Cv
is heat capacity at constant volume
Cp
is heat capacity at constant pressure
Clat
is lattice heat capacity
:check:
1) Heat capacity C can be found by differentiating the average phonon energy
2) Lattice heat capacity: It treats the vibrations of the atomic lattice (heat) as phonons.
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EXPERIMENTAL RESULTS
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:warning:
The heat capacity at constant volume, CV as a function of temperature for solid
:star:The constant value of the heat capacity of many simple solids is sometimes called
Dulong-Petit law
In 1819 Dulong and Petit found experimentally that for many solids at room temperature, c,
~ 3R = 25 JK^-1mol^-1
:check:This is consistent with equipartition theorem of
classical mechanics:
Energy added to solids takes the form of
atomic vibrations
and
both kinetic
and
potential energy
is associated with the three degrees of freedom of each atom.
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Heat Capacity of the lattice
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:red_flag:
Classical statistical mechanics explained the heat capacity of insulators at room temperature fairly well but failed for lower temperatures and it totally failed for metals
:red_flag:
Metals were expected to have much higher heat capacity than insulators because of many free electrons but it turn out that a metal's heat capacity at room temperature is similar to that of insulator
Nor Aina Munirah Binti Mad Yasin
(A20SC0197)
Classical gas theory
:check: Only have translational kinetic energy :
Temperature dependent known as Dulong and Petit law
Three Models of Lattice Vibration
Einstein Model (1906)
Assumed that all the oscillators oscillate with a common frequency
The high temperature behavior is good, the behavior at T=0
The low temperature is not very good
Debye Model (1912)
Just like Einstein model, it also recovers the Dulong-Petit law at high temperature
Accurate at low temperature
Bon Vorn Karman Model (1912)
Boundary conditions are periodic boundary conditions which impose the restriction that a wave function must be periodic on a certain Bravais lattice
This condition is often applied in solid state physics to model an ideal crystal
The condition can be stated as
