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positronium 0.1.8

python tools pertaining to positronium

python tools pertaining to positronium


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

IPython examples dependencies:

  • matplotlib


via pip (recommended):

pip install positronium

alternatively, try the development version

git clone

and then run

python install


This package is designed to collate 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 answers, rather than rigorous quantum mechanical calculations.

The package currently only contains a few very simple modules.


is intended to collect 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", "%.1f ns."%(lifetime_oPs * 1e9))
The mean lifetime of ortho-Ps is 142.0 ns.

>>> print("The ground-state hyperfine splitting is", "%.1f GHz."%(frequency_hfs * 1e-9))
The ground-state hyperfine splitting is 203.4 GHz.

Where appropriate constants are stored in a subclass of float called MeasuredValue, which has a few extra attributes [uncertainty, unit, source, url], for example

>>> lifetime_oPs

>>> lifetime_oPs.uncertainty

>>> 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.


contains an adaptation of the Rydberg formula, which is used to calculate the principle energy levels of positronium, or the interval between two levels. The default unit is ‘eV’, however, this can be changed using the keyword argument ‘unit’.

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

>>> from positronium import Bohr
>>>, 2, unit='nm')

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,, 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]])


This package contains a class called Ps, which can be used to represent a particular atomic state of positronium using the quantum numbers

n principle
l orbital angular momentum
m magnetic quantum number
S total spin
J total angular momentum

This can be used to return estimates of, e.g., the energy level,

>>> from positronium import Ps
>>> x1 = Ps(n=2, l=1, S=1, J=2)

which uses an equation described in

Richard A. Ferrell (1951) Phys. Rev. 84, 858

This includes fine structure but not radiative corrections.

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

>>> x1.tex()

For further examples see the IPython/ Jupyter notebooks,

File Type Py Version Uploaded on Size
positronium-0.1.8.tar.gz (md5) Source 2016-07-16 8KB