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PS204 - Solid State Physics (2023R) - Coggle Diagram
PS204 - Solid State Physics (2023R)
X-ray Generation for Diffraction Experiments:
Bremsstrahlung X-ray Emission:
Electrons accelerated towards a target material undergo deceleration.
Deceleration causes emission of electromagnetic radiation, including X-rays.
Continuous spectrum of X-rays produced due to varying deceleration.
Characteristic (Line Spectra) Emission:
Incident electrons interact with inner-shell electrons of target atoms.
Electrons knock out inner-shell electrons, leaving vacancies.
Electrons from higher energy levels fill these vacancies, emitting X-rays with energies characteristic of the atom.
Discrete lines in the X-ray spectrum correspond to transitions between specific energy levels.
Overall Process:
X-ray tube generates X-rays through both bremsstrahlung and characteristic emission.
Resulting X-rays used for diffraction experiments to probe crystal structures.
Bragg's Law:
Definition: Relates the angle of incidence, the wavelength of X-rays, and the inter-planar spacing of a crystal lattice to constructive interference in X-ray diffraction.
Terms in Bragg's Law:
n: Integer representing the order of the diffraction peak.
λ: Wavelength of incident X-rays.
θ: Angle between incident X-ray beam and the crystal lattice planes.
d: Inter-planar spacing of the crystal lattice.
Explanation of Terms:
n: Order of diffraction peak; indicates which diffracted beam (1st, 2nd, etc.) is being considered.
λ: Wavelength of X-rays; determines the spacing between adjacent peaks in the diffraction pattern.
θ: Angle of incidence; the angle at which X-rays strike the crystal lattice planes.
d: Inter-planar spacing; distance between adjacent crystal lattice planes.
Models
Classical Dulong-Petit Model:
Predicts a constant molar specific heat capacity (Cv) for crystals at high temperatures.
Assumes each vibrational mode in the crystal lattice contributes kT/2 to the heat capacity, based on equipartition theorem.
Agrees well with experimental data at high temperatures but fails at low temperatures.
Einstein Model:
Treats atoms as independent harmonic oscillators.
Assumes all oscillators have the same frequency (Einstein frequency) and contribute equally to heat capacity.
Accurate at low temperatures but overestimates heat capacity at high temperatures.
Debye Model:
Treats atoms as part of a continuous elastic medium.
Considers the vibrational modes of the entire crystal lattice.
Introduces a characteristic frequency (Debye frequency) that varies with temperature.
Accurate at both low and high temperatures, capturing the temperature dependence of heat capacity.
Main Difference:
Classical Model: Predicts a constant heat capacity regardless of temperature.
Einstein and Debye Models: Account for temperature dependence of heat capacity due to varying vibrational modes.
Agreement:
Einstein and Debye models agree with the classical model at high temperatures when thermal energy is large enough to excite all vibrational modes.
Drude Theory
Thermal Velocity (vt):
Definition: Average velocity of charge carriers due to thermal energy.
Units: meters per second (m/s).
Drift Velocity (vd):
Definition: Average velocity of charge carriers under the influence of an electric field.
Units: meters per second (m/s).
Carrier Mobility (μ):
Definition: Measure of how easily charge carriers can move through a material under the influence of an electric field.
Units: square meters per volt-second (m^2/Vs).
Conductivity (σ):
Definition: Measure of a material's ability to conduct electric current.
Units: Siemens per meter (S/m) or reciprocal ohm-meter (Ω^-1*m^-1).
Hall Coefficient (RH):
Definition: Measure of the voltage generated across a conductor transverse to an electric current in the presence of a magnetic field.
Units: cubic meters per coulomb (m^3/C).
Diamagnetism and Paramagnetism
Diamagnetism
Underlying Physics:
Results from the orbital motion of electrons in atoms.
Induced magnetic moment opposes an applied magnetic field.
Alignment of Induced Magnetic Moment:
Opposite to the direction of the applied magnetic field.
Relative Size of Response:
Typically weak and decreases with increasing magnetic field strength.
Materials:
Observed in materials containing unpaired electrons, such as transition metals and rare earth metals.
Paramagnetism
Underlying Physics:
Induced magnetic moment aligns with an applied magnetic field.
Due to the presence of unpaired electrons in atoms or ions.
Alignment of Induced Magnetic Moment:
Parallel to the direction of the applied magnetic field.
Relative Size of Response:
Stronger response compared to diamagnetism.
Increases with increasing magnetic field strength.
Materials
Present in all materials.
However, typically overshadowed by paramagnetic or ferromagnetic effects in most materials.
Magnetic Ordering Behaviour
Exchange Interaction:
Definition:
Arises due to the overlap of electron wave functions and the Pauli exclusion principle.
Quantum mechanical effect describing the interaction between neighboring magnetic moments in a material.
Concept:
Electrons in adjacent atoms or ions influence each other's spins.
Leads to alignment or anti-alignment of neighboring magnetic moments depending on the strength and sign of the exchange interaction.
Types
Ferromagnetic:
Definition: Adjacent magnetic moments align parallel to each other.
Result: Strong net magnetization even in the absence of an external magnetic field.
Example: Iron, cobalt, nickel.
Antiferromagnetic:
Definition: Adjacent magnetic moments align antiparallel to each other.
Result: Net magnetic moment is zero, but there is magnetic ordering.
Example: Manganese oxide (MnO).
Ferrimagnetic:
Definition: Similar to antiferromagnetic, but with unequal magnitudes of magnetic moments.
Result: Net magnetic moment exists due to unequal cancellation of moments.
Example: Magnetite (Fe3O4).