Positronium
Positronium is an exotic atomic system consisting of an electron and its antimatter counterpart, the positron, bound together by their electrostatic attraction. It resembles a hydrogen atom in structure but lacks a nucleus. The mass of positronium is essentially the sum of the masses of an electron and a positron, approximately 2m_e = 1.022 \text{ MeV/c}^2, where m_e is the mass of one electron (approximately 0.511 \text{ MeV/c}^2).
The mass of positronium, while being approximately the sum of the electron and positron masses, is slightly less than double the electron mass due to the binding energy of the system. This binding energy slightly reduces the total mass, but the difference is typically quite small compared to the sum of the electron and positron masses. For practical purposes, the mass of positronium is often approximated as 2m_e, although the exact value depends on the binding energy specific to the energy state of the positronium (either ortho- or para-positronium).
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Para-positronium:
- Spins of the electron and positron are antiparallel (spin singlet state, total spin S = 0).
- Annihilates predominantly into two photons.
- Lifetime in vacuum: approximately 125 \text{ picoseconds}.
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Ortho-positronium:
- Spins of the electron and positron are parallel (spin triplet state, total spin S = 1).
- Annihilates predominantly into three photons.
- Lifetime in vacuum: approximately 142 \text{ nanoseconds} (longer due to conservation laws requiring three-photon decay).
The energy states of positronium vary primarily due to differences in the spin configurations. The total energy includes contributions from kinetic energy, potential energy due to electrostatic interaction, and spin-spin interaction, with ortho-positronium having slightly higher energy due to its triplet spin state.
The binding energy of positronium can be estimated using the Bohr model adapted for the electron-positron system (a detailed derivation is available here). Since both particles have the same mass, the reduced mass \mu is half of the electron mass m_e, which changes the calculation for the energy levels.
For positronium, the energy levels are given by:
E_n = -\frac{1}{n^2} \left[\frac{\mu}{2} \left(\frac{e^2}{4\pi\epsilon_0 \hbar}\right)^2\right] = -\frac{1}{n^2} \left(\frac{\mu e^2}{8 \epsilon_0^2 h^2}\right) = -\frac{1}{n^2} \left(\frac{m_e e^2}{16 \epsilon_0^2 h^2}\right)
Where: - m_e is the electron mass, - e is the elementary charge, - \epsilon_0 is the permittivity of free space, - \hbar is the reduced Planck constant, - n is the principal quantum number (for the ground state, n = 1).
For the ground state (n = 1), the binding energy (the energy needed to dissociate the positronium into a free electron and positron) is approximately:
E_1 = -6.8 \text{ eV}
m_e = 9.10938356e-31 # Electron mass in kg
e = 1.602176634e-19 # Elementary charge in Coulombs
epsilon_0 = 8.854187817e-12 # Permittivity of free space in F/m
h = 6.62607015e-34 # Planck's constant in J*s
# Reduced mass for positronium (mu)
mu = m_e / 2
# Formula for the ground state energy n=1 using the reduced mass
E_1 = - (mu * e**4) / (8 * epsilon_0**2 * h**2)
# Convert energy from Joules to electron volts (1 J = 1 / e eV)
E_1_ev = E_1 / e
E_1_ev
# -6.802846449317475
This value represents the magnitude of the binding energy, indicating how much energy is released when positronium is formed from an electron and a positron.
To correct the mass of positronium by accounting for the binding energy, we use the energy-mass equivalence E = mc^2. The binding energy of 6.8 \text{ eV} can be converted to mass units by dividing by c^2, where c is the speed of light in vacuum.
Given: - Binding energy E = 6.8 \text{ eV} (or approximately 1.09 \times 10^{-18} \text{ J}) - Speed of light c \approx 2.998 \times 10^8 \text{ m/s} - Conversion from joules to electronvolts: 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}
First, convert the energy from eV to kg using E = mc^2:
m_\text{bind} = \frac{E}{c^2} = \frac{6.8 \times 1.602 \times 10^{-19} \text{ J}}{(2.998 \times 10^8 \text{ m/s})^2} \approx 1.21 \times 10^{-35} \text{ kg}
So there is a slight reduction in mass and the mass is:
m_{p} = 2m_e - m_\text{bind} \approx 1.8218645898734655 \times 10^{-30} \text{ kg}
This shows just a slight reduction from the simple sum of electron and positron masses (The mass change for positronium, compared to 2m_e, is approximately 0.000665\% due to the binding energy).
The spin correction in positronium, which includes the spin-spin interaction between the electron and positron, is a fine structure effect that can be estimated using quantum electrodynamics (QED). The fine structure corrections are typically smaller in magnitude compared to the main energy term provided by the Bohr model and are of the order of the fine structure constant, \alpha, times the Bohr energy.
In more specific terms, the energy correction due to spin-spin interaction in positronium is generally of the order of:
\Delta E \approx \alpha^2 E_0
In the case considered E_0 is the leading order energy term of the positronium, which is approximately -6.8 \text{ eV} for the ground state (as calculated by the Bohr model). The fine structure constant \alpha is approximately \frac{1}{137}.
Therefore, the order of magnitude for the spin correction can be estimated as:
\Delta E \approx \left(\frac{1}{137}\right)^2 \times 6.8 \text{ eV} \approx 0.00036 \text{ eV}
This correction significantly impacts the energy levels and contributes to the difference between ortho-positronium and para-positronium, influencing their respective quantum states and observable properties such as lifetimes and decay modes. In the context of many practical applications, the spin-spin interaction energy correction of approximately 0.00036 \text{ eV} in positronium can be considered negligible compared to the overall energy scale and other energy scales encountered in atomic physics and material science. This small value reflects the fact that, while quantum mechanical effects such as spin interactions are crucial for precise theoretical descriptions and for understanding fundamental properties, they often result in very small numerical corrections to the main energy terms. Therefore, in scenarios where extremely high precision is not required or where these small corrections do not significantly influence the phenomena being studied, such corrections can be treated as practically zero, and in the following decay analysis won’t be considered.
Para-Positronium decay
Since the positron is the electron’s antiparticle, after a short period of time they will annihilate each other and produce two photons; using relativistic equations it is possible to compute the energy of these two photons.
It worth noting that in physics before the theory of relativity, this analysis wouldn’t make sense as the principle of conservation of mass applies: in this particular cases, however, the positronium decays into two photons, which have mass equal to zero.
In relativistic physics, what is conserved isn’t mass, but it is the energy and the momentum.
Starting from the momentum and considering a frame of reference in which the positronium is at rest and, in this frame of reference, the momentum is zero, and, since the product of the decay are two photons, as a consequence these two photons are moving in opposite direction with momentum equal in magnitude but of different sign, so that their sum is still zero:
\mathbf p_1 + \mathbf p_2 = 0
For the energy conservation, the initial energy in the rest frame is:
E_i = m_{p}c^2
This must be equal to the energy after the interaction:
E_f = \sqrt{p_f^2c^2 + m_f^2c^4} = E_{f1} + E_{f2} = 2c|p|
Combining these equations gives the value of the momentum:
|p| = \frac{m_{p}c}{2}
Using the energy-wavelength relation for a photon E = \frac{hc}{\lambda}, where h is the Planck constant, we can find the wavelength \lambda of each photon (which carries half of the energy):
\lambda = \frac{hc}{\frac{E_i}{2}} = \frac{2h}{m_pc} = 2.426 \times 10^{-12} m