positronium

python tools pertaining to positronium

Dependencies

Tested using Anaconda (Continuum Analytics) with Python 2.7 and 3.5. Examples written using IPython 4.0.1 (python 3.5.1 kernel).

Package dependencies:

• scipy, numpy, attrs

IPython examples dependencies:

• matplotlib

Installation

via pip (recommended):

pip install positronium

alternatively, try the development version

git clone https://github.com/PositroniumSpectroscopy/positronium

and then run

python setup.py install

About

This package containes useful bits of code relating to the positronium atom (an electron bound to its antiparticle, the positron). The functions are generally simple approximations that give roughly the right values.

The package only contains a few very simple modules.

constants

Useful constants in SI units, including:

const	description
m_Ps	2 * mass_electron
Rydberg_Ps	Rydberg value for Ps
a_Ps	Bohr radius for Ps
decay_pPs	decay rate of para-Ps (S=0)
decay_oPs	decay rate of ortho-Ps (S=1)
lifetime_pPs	lifetime of para-Ps (S=0)
lifetime_oPs	lifetime of ortho-Ps (S=1)
frequency_hfs	frequency of the ground-state hyperfine splitting
energy_hfs	energy interval of the ground-state hyperfine splitting
frequency_1s2s	frequency of the 1s2s transition
energy_1s2s	energy interval of the 1s2s transition

Example usage,

>>> from positronium.constants import lifetime_oPs, frequency_hfs
>>> print("The mean lifetime of ortho-Ps is", lifetime_oPs)
The mean lifetime of ortho-Ps is 142.037 ± 0.036 ns

>>> print("The ground-state hyperfine splitting is", frequency_hfs)
The ground-state hyperfine splitting is 203.38910 ± 0.00074 GHz

Where appropriate constants are stored in a class called MeasuredValue, which is a subclass of float, with extra attributes [uncertainty, unit, source, url]. For example

>>> lifetime_oPs
1.4203738423953184e-07

>>> lifetime_oPs.uncertainty
3.631431333889514e-11

The value and uncertainty are in SI units. When calling print(MeasuredValue), the class attempts to format the result using a suitable metric prefix.

>>> print(lifetime_oPs)
142.037 ± 0.036 ns

To see the value's source,

>>> print(lifetime_oPs.source)
R. S. Vallery, P. W. Zitzewitz, and D. W. Gidley (2003) Phys. Rev. Lett. 90, 203402

>>> lifetime_oPs.article()

The final line opens a url to the source journal.

Bohr

Estimate the principle energy levels of positronium using the Rydberg formula.

For instance, the UV wavelength (in nm) needed to excite the Lyman-alpha transition can be found by:

>>> from positronium.Bohr import energy
>>> energy(1, 2, unit='nm')
243.00454681426382

This accepts numpy arrays for the initial (n1) and/ or final (n2) energy level, e.g.,

>>> import numpy as np
>>> n1 = np.arange(1, 10)
>>> np.array([n1, energy(n1, unit='eV')]).T
array([[ 1.        ,  6.8028465 ],
       [ 2.        ,  1.70071163],
       [ 3.        ,  0.75587183],
       [ 4.        ,  0.42517791],
       [ 5.        ,  0.27211386],
       [ 6.        ,  0.18896796],
       [ 7.        ,  0.1388336 ],
       [ 8.        ,  0.10629448],
       [ 9.        ,  0.08398576]])

Ps

This attrs class can be used to represent a particular atomic state of positronium using the quantum numbers

n	principal
l	orbital angular momentum
S	total spin
J	total angular momentum

And can be used to estimate the energy of the state,

>>> from positronium import Ps
>>> x1 = Ps(n=2, l=1, S=1, J=2)
>>> x1.energy(unit='eV')
-1.7007156831792944

It uses an equation described in

Richard A. Ferrell (1951) Phys. Rev. 84, 858 http://dx.doi.org/10.1103/PhysRev.84.858

which includes fine structure but not radiative corrections.

A representation of the state using Latex code can be made using,

>>> x1.tex()
'$2^{3}P_{2}$'

For further examples see the IPython/ Jupyter notebooks,

https://github.com/PositroniumSpectroscopy/positronium/tree/master/examples