Doctoral Dissertation PDF - Atomic Spectra and Heat Capacity

# Chapter One: Atomic Spectra ## Sommerfeld Relativistic Model - Sommerfeld successfully explained the fine structure of spectral lines by expanding on Bohr's theory. - He introduced the idea of elliptical orbits for the electron around the nucleus, in addition to circular ones, and considered the...

# Chapter One: Atomic Spectra ## Sommerfeld Relativistic Model - Sommerfeld successfully explained the fine structure of spectral lines by expanding on Bohr's theory. - He introduced the idea of elliptical orbits for the electron around the nucleus, in addition to circular ones, and considered the relativistic change in electron mass with velocity. - The new model takes into account two essential variables: - The principal quantum number *n*, which determines the electron's energy level. - The azimuthal quantum number *l*, which describes the shape of the orbit and is related to the angular momentum of the electron. - As the electron's velocity changes, both its mass and angular momentum change. - The electron's orbit is elliptical, with the nucleus at one focus. ### Shape of the Orbits - Sommerfeld's relativistic model allows us to determine the shape of the orbits based on *n* and *l*. - The shape is described by two parameters: *a* and *b*. - *a* = *n*. - *b* = *n* + *l*. - For example: - If *n* = 1 and *l* = 0, then *a* = 1 and *b* = 1. The orbit is a circle. - If *n* = 2 and *l* = 2, then *a* = 2 and *b* = 3. The orbit is an ellipse. ### Principle Quantum Number *n* - The principle quantum number *n* determines the electron's energy level. - It can take on positive integer values, with higher values corresponding to higher energy levels. - Each energy level can hold a maximum of 2(*n*)² electrons. - We can represent the energy levels using the letters K, L, M, and N, which correspond to *n* = 1, 2, 3, and 4 respectively. ### Azimuthal Quantum Number *l* - The azimuthal quantum number *l* determines the shape of an electron's orbit and is related to its angular momentum. - It can take on integer values from 0 to *n* - 1. - Different values of *l* correspond to different orbital shapes. - *l* = 0 corresponds to an s-orbital, which is spherical. - *l* = 1 corresponds to a p-orbital, which is dumbbell shaped. - *l* = 2 corresponds to a d-orbital, which has a more complex shape with four lobes. - *l* = 3 corresponds to an f-orbital, which has an even more complex shape. ### Total Angular Momentum Quantum Number *J* - The total angular momentum quantum number *J* is the sum of the orbital angular momentum (*l*) and the spin angular momentum (*s*) of an electron. - It can take on values from *l* - *s* to *l* + *s*. - Each value of *J* corresponds to a different energy level, resulting in the fine structure of spectral lines. ### Magnetic Quantum Number *m_j* - The magnetic quantum number *m_j* determines the orientation of an electron's total angular momentum in space. - It can take on values from -*J* to +*J*, including 0. - Since each value of *J* can have (2*J* + 1) values of *m_j*, the total number of possible energy levels for a given *J* value is (2*J* + 1). ### Spin Quantum Number *m_s* - The spin quantum number *m_s* describes the intrinsic angular momentum of an electron, which is called its spin angular momentum. - It is associated with an intrinsic magnetic dipole moment of the electron. - The spin quantum number can only take on two values: - + ½ for spin up. - - ½ for spin down. ### Distribution of Electrons in Sub-Shells - To understand how electrons are distributed in different sub-shells (energy levels with a given *n* and *l*), we need to use the following rules: - The principle quantum number *n* defines the energy level. - The angular momentum number *l* describes the shape of the orbital. - Each sub-shell can hold a maximum of 2(2*l* + 1) electrons. - Pauli's exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers (*n*, *l*, *m*_j, *m_s*) # Chapter Two: Heat Capacity of Solids ## Lattice Specific Heat Capacity - This chapter discusses the heat capacity of solids and how it relates to the statistical physics of particles. ## Types of Particles - There are two main types of particles: - Distinguishable particles: - These are particles that can be identified individually. - Their probability distribution is given by the classical Maxwell-Boltzmann distribution. - Indistinguishable particles: - These are particles that cannot be identified individually. - They are further categorized based on their spin. - Particles with half-integer spin (e.g., electrons, protons) follow Fermi Dirac statistics. - Particles with integer spin (e.g., photons, phonons) follow Bose-Einstein statistics. ## The Classical Model - The classical model of heat capacity assumes that the atoms in a solid are distinguishable particles and behave as classical harmonic oscillators. - This model predicts that the heat capacity of a solid should be 3R, where R is the ideal gas constant. - This is known as the Dulong-Petit law, which predicts that the molar specific heat capacity of a solid is approximately 25 J/mol K at high temperatures. ## The Einstein Model - The Einstein model introduces the idea of quantized energy levels for the atomic oscillators. - It assumes all atoms in the solid oscillate with the same frequency, and the energy of each oscillator is quantized in multiples of *hw*, where *h* is Planck's constant and *w* is the angular frequency of oscillation. - This model predicts that the heat capacity of a solid will decrease as the temperature decreases, eventually approaching zero at very low temperatures. - The Einstein model successfully explains the decrease in heat capacity at low temperatures, but it fails to accurately predict the heat capacity at all temperatures. ## The Debye Model - The Debye model improves on the Einstein model by considering a distribution of frequencies for the atomic oscillators. - It assumes that the vibrational modes of the solid are quantized and form a continuous spectrum of frequencies up to a maximum frequency *w*_D, known as the Debye frequency. - The Debye model predicts a more accurate heat capacity curve that agrees better with experimental data, particularly at low temperatures. ## Experimental Heat Capacity Curves - Experimental measurements of heat capacity as a function of temperature show a behavior that is qualitatively consistent with the theoretical models. - At high temperatures, the heat capacity is approximately constant and close to the Dulong-Petit value. - As the temperature decreases, the heat capacity starts decreasing, approaching zero at very low temperatures. - The Debye model provides a better fit to the experimental data than the Einstein model.