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"**Maxwell speed distribution**\n",
"```\n",
"\n",
"\\begin{align}\n",
"```{math}\n",
":label: Maxwell_speed\n",
"F(v)\\ dv = 4\\pi Ce^{-\\frac{1}{2}\\beta m v^2}v^2\\ dv.\n",
"\\end{align}\n",
"```\n",
"\n",
"The Maxwell speed distribution as derived from purely classical considerations, where it gives a nonzero probability of finding a particle with a speed greater than $c$. Therefore, it is only valid in the classical limit.\n",
"\n",
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"metadata": {},
"source": [
"## Classical and Quantum Statistics\n",
"The Maxwell speed distribution is only valid in the classical limit, which prohibits its use at exceptionally high temperatures (i.e., gas particle velocities near $c$). An ideal gas is dilute, which means that the gas molecules rarely interact or collide with one another due to their large (relative) separations. When collisions occur, they can be treated as totally elastic, and have no effect on the distributions and mean values.\n",
"\n",
"When the density of matter is higher (i.e., solid or liquids), the assumption of noninteracting particles may no longer be valid. If molecules, atoms, or subatomic particles are closely packed together, the Pauli exclusion principle prevents two particles in identical quantum states from sharing the same s=pace. This limits the allowed energy states of any particle (fermion), which affects the distribution of energies for a system of particles.\n",
"\n",
"### Classical Distributions\n",
"Because energy levels are fundamental in quantum theory, we rewrite the Maxwell speed distribution in terms of energy rather than velcocity. For a monatomic gas, the energy is all translational kinetic energy. We can use the following relations\n",
"\n",
"\\begin{align}\n",
"E &= \\frac{1}{2}mv^2, \\\\\n",
"dE &= mv dv, \\\\\n",
"dv &= \\frac{dE}{\\sqrt{2mE}}.\n",
"\\end{align}\n",
"\n",
"Similar to our derivation of the Maxwell speed distribution (see Eq.{eq}`Maxwell_speed`), we can transform $F(v) \\rightarrow F(E)$,\n",
"\n",
"```{margin}\n",
"**Maxwell-Boltzmann energy distribution**\n",
"```\n",
"\n",
"\\begin{align}\n",
"F(v)\\ dv &= 4\\pi C e^{-\\frac{1}{2}\\beta mv^2}v^2\\ dv, \\\\\n",
"&= 4\\pi C e^{-\\beta E}\\left( \\frac{2E}{m}\\right) \\frac{dE}{\\sqrt{2mE}}, \\\\\n",
"&= \\frac{8\\pi C}{\\sqrt{2m^3}} e^{-\\beta E}\\sqrt{E}\\ dE.\n",
"\\end{align}\n",
"\n",
"to get the **Maxwell-Boltzmann energy distribution**. The factor $e^{-\\beta E}$ is important because Boltzmann showed that this factor is a characteristic of any classical system, regardless of how quantities other than molecular speeds may affect the energy of a given state. We define the **Maxwell-Boltzmann factor** for classical systems as\n",
"\n",
"```{margin}\n",
"**Maxwell-Boltzmann factor**\n",
"```\n",
"\n",
"\\begin{align}\n",
"F_{\\rm MB} = Ae^{-\\beta E},\n",
"\\end{align}\n",
"\n",
"where $A$ is a normalization constant. The energy distribution for a classical system will then have the form\n",
"\n",
"\\begin{align}\n",
"n(E) = g(E)F_{\\rm MB},\n",
"\\end{align}\n",
"\n",
"where $n(E)$ is a distribution that represents the number of particles within a bin of energy from $E$ to $E+dE$.\n",
"\n",
"The function $g(E)$ is known as the **density of states**, or the number of states available per unit energy range. The density of states is an essential element in al distributions. The factor $F_{\\rm MB}$ is the relative probability that an energy state is occupied at a given temperature.\n",
"\n",
"### Quantum Distributions\n",
"In quantum theory, particles are described by wave functions. Indentical particles cannot be distinguished from one another if there is a significant overlap of their wave functions. It is this characteristic that makes quantum statistics different from classical stastics.\n",
"\n",
"**Suppose that we have a system of just two particles.** Each particle has an equal probability (0.5) of existing in either of two energy states. If the particles are distinguishable (e.g., labeled A and B), then the possible configurations are\n",
"\n",
"1. Both in state 1,\n",
"2. Either A or B in each state, but not together, or\n",
"3. Both in state 2.\n",
"\n",
"These configurations can be illustrated using a probability table:\n",
"\n",
"```{table} Probability Table of Two Particles (A & B)\n",
":width: 250px\n",
":align: center\n",
"\n",
"| **State 1** \t| **State 2** \t|\n",
"|-------------\t|-------------\t|\n",
"| AB \t| \t|\n",
"| A \t| B \t|\n",
"| B \t| A \t|\n",
"| \t| AB \t|\n",
"\n",
"```\n",
"\n",
"Each of the four configurations are equally likely, where the probability of each is one-fourth (0.25). If the two particles are indistinguishable, then our probability table changes:\n",
"\n",
"```{table} Probability Table of Two Indistinguishable Particles\n",
":width: 250px\n",
":align: center\n",
"\n",
"| **State 1** \t| **State 2** \t|\n",
"|-------------\t|-------------\t|\n",
"| XX \t| \t|\n",
"| X \t| X \t|\n",
"| \t| XX \t|\n",
"\n",
"```\n",
"\n",
"Now there are only three equally likely configurations, where each have a probability of one-third (${\\sim}0.33$).\n",
"\n",
"Two kinds of quantum distributions are needed because some particles obey the Pauli exclusion principle and others do not. \n",
"\n",
"- Particles that obey the Pauli exclusion princple have half-integers spins and are called **fermions**. Protons, neutrons, and electrons are examples of fermions. \n",
"- Particles with zero or integer spins do not obey the Pauli exclusion principle and are known as **bosons**. Photons and pions are examples of bosons.\n",
"\n",
"```{note}\n",
"Atoms and molecules consisting of an even number of fermions must be bosons when considered as a whole, because their total spin will be zero or an integer. Conversely, atoms and molecules with an odd number of fermions are fermions.\n",
"```\n",
"\n",
"The probability distribution of fermions are given by the **Fermi-Dirac distribution**:\n",
"\n",
"```{margin}\n",
"**Fermi-Dirac distribution**\n",
"```\n",
"\n",
"\\begin{align}\n",
"n(E) &= g(E)F_{\\rm FD},\\ \\qquad \\text{where } \\\\\n",
"F_{\\rm FD} &= \\frac{1}{B_{\\rm FD}\\ e^{\\beta E}+1}.\n",
"\\end{align}\n",
"\n",
"Similarly the **Bose-Einstein distribution** is valid for bosons and is\n",
"\n",
"```{margin}\n",
"**Bose-Einstein distribution**\n",
"```\n",
"\n",
"\\begin{align}\n",
"n(E) &= g(E)F_{\\rm BE},\\ \\qquad \\text{where } \\\\\n",
"F_{\\rm BE} &= \\frac{1}{B_{\\rm BE}\\ e^{\\beta E}-1}.\n",
"\\end{align}\n",
"\n",
"In each case $B_i$ ($B_{\\rm FD}$ or $B_{\\rm BE}$) represents a normalization factor, and $g(E)$ is the density of states appropriate for a particular situation.\n",
"\n",
"```{note}\n",
"The Fermi-Dirac and Bose-Einstein distributions look very similar, where they differ only by the normalization constant and by the sign attached to the $1$ in the denominator.\n",
"```\n",
"\n",
"Both the Fermi-Dirac and Bose-Einstein distributions reduce to the classical Maxwell-Boltzmann distribution when $B_ie^{\\beta E} \\gg 1$ $(\\text{recall }x\\pm 1 \\approx x$ for $x\\gg 1)$ and $A = 1/B_i$. This means that the Maxwell-Boltzmann factor is much less than 1 (i.e., the probability that a particular energy state will be occupied is much less than 1).\n",
"\n",
"```{table} Classical and Quantum Distributions\n",
":width: 600px\n",
":align: center\n",
"\n",
"| Distributions \t| Properties \t| Examples \t| Function \t|\n",
"|-------------------\t|--------------------------------------------------------\t|-------------------------------\t|--------------------------------------------------\t|\n",
"| Maxwell-Boltzmann \t| Identical, <br>distinguishable \t| Ideal gases \t| $F_{\\rm MB} = Ae^{-\\beta E}$ \t|\n",
"| Bose-Einstein \t| Identical, indistinguishable<br>with integer spin \t| Liquid $\\rm ^4He$,<br>photons \t| $F_{\\rm BE} = \\frac{1}{B_{\\rm BE}\\ e^{\\beta E}-1}$ \t|\n",
"| Fermi-Dirac \t| Identical, indistinguishable<br>with half-integer spin \t| Electron gas \t| $F_{\\rm FD} = \\frac{1}{B_{\\rm FD}\\ e^{\\beta E}+1}$ \t|\n",
"```\n"
]
},
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"source": [
"```{exercise}\n",
":class: orange\n",
"\n",
"**Assume that the Maxwell-Boltmann distribution is valid in a gas of atomic hydrogen. What is the relative number of atoms in the ground state and first excited state at: $293\\ {\\rm K}$, $5000\\ {\\rm K}$, and $10^6\\ {\\rm K}$?**\n",
"\n",
"The relative number of atoms between the ground and first excited states can be determined by using the ratio of $n(E_2)$ to $n(E_1)$. Mathematically this is given as\n",
"\n",
"\\begin{align}\n",
"\\frac{n(E_2)}{n(E_1)} &= \\frac{g(E_2)}{g(E_1)} e^{-\\beta(E_2-E1)},\n",
"\\end{align}\n",
"\n",
"where the density of states $g(E_1) = 2$ for the ground state (spin up or down) and $g(E_2) = 8$ for the first excited state (see [Electron shells](https://en.wikipedia.org/wiki/Electron_shell)). For atomic hydrogen, $E_2-E_1 = 10.2\\ {\\rm eV}$. Putting this together we can write\n",
"\n",
"\\begin{align*}\n",
"\\frac{n(E_2)}{n(E_1)} &= \\frac{8}{2} e^{-\\frac{10.2\\ {\\rm eV}}{kT}} = 4e^{-\\frac{10.2\\ {\\rm eV}}{kT}},\n",
"\\end{align*}\n",
"\n",
"for a given temperature $T$. Using the Boltzmann constant in $\\rm eV/K$, we can easily calculate the relative numbers as\n",
"\n",
"\\begin{align*}\n",
"\\frac{n(E_2)}{n(E_1)} &=& 4e^{-404} &\\approx& 10^{-175}, \\quad & \\text{for }T = 293\\ {\\rm K}, \\\\\n",
"&=& 4e^{-23.7}\\quad &\\approx& 2\\times 10^{-10}, \\quad & \\text{for }T = 5000\\ {\\rm K}, \\\\\n",
"&=& 4e^{-0.118} &\\approx& 3.55, \\quad & \\text{for }T = 10^6\\ {\\rm K}.\n",
"\\end{align*}\n",
"\n",
"At very high temperatures, the exponential factor approaches 1, so the result simply approaches the ratio of the density of states.\n",
"```\n"
]
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"execution_count": 5,
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"text": [
"-404.14892253761053\n",
"The relative number of atoms at 293 K is 1.21e-175.\n",
"-23.683126860703975\n",
"The relative number of atoms at 293 K is 2.07e-10.\n",
"-0.11841563430351988\n",
"The relative number of atoms at 293 K is 3.55.\n"
]
}
],
"source": [
"import numpy as np\n",
"from scipy.constants import physical_constants \n",
"\n",
"##For atomic hydrogen only\n",
"\n",
"def energy(n):\n",
" #n = energy level\n",
" return -E_o/n**2 #energy in eV\n",
"\n",
"def relative_states(n1,n2,T):\n",
" #n2 = energy level 2; n2 > n1\n",
" #n1 = energy level 1\n",
" #T = temperature in K\n",
" g2, g1 = 2*n2**2, 2*n1**2\n",
" E2,E1 = energy(n2),energy(n1)\n",
" return (g2/g1)*np.exp(-(E2-E1)/(k_eV*T))\n",
"\n",
"k_eV = physical_constants['Boltzmann constant in eV/K'][0]\n",
"E_o = physical_constants['Rydberg constant times hc in eV'][0]\n",
"\n",
"### Quantum Distributions"
"print(\"The relative number of atoms at 293 K is %1.2e.\" % relative_states(1,2,293))\n",
"print(\"The relative number of atoms at 293 K is %1.2e.\" % relative_states(1,2,5000))\n",
"print(\"The relative number of atoms at 293 K is %1.2f.\" % relative_states(1,2,1e6))"
]
},
{
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