Queen's University Belfast
School of Maths and Physics
Department of Applied Mathematics and Theoretical Physics
Centre for Theoretical Atomic, Molecular and Optical Physics
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R-matrix theory is on of the standard techniques to calculate atomic properties. It can be applied to many other systems as well, since it was originally developed to describe problems in nuclear physics. Cold collisions, of the type appearing in Bose-Einstein condensates, can also be described in R-matrix theory. Although the basic equations of motion for the systems are different, they share a common philosophy known as the R-matrix approach.
The basic idea is easiest to explain for electron scattering problems, in which the initial state contains of an atom in the ground state, and an electron far away headed for the atom (figure to the left). Since the electron is very far away from the atom, its indistinguishability with any of the atomic electrons can be neglected. This indistinguishability would appear as a probability that it can 'swap' its identity with one of the atomic electrons. When the electron is very far from the atom that probability is effectively zero. This zero probability simplifies the description of the outer electron enormously, and we can solve the problem exactly.
On the other hand, when the outer electron gets very close to the atom (figure to the right), the electron can no longer be distinguished from the other electrons, and we must take the quantum character of the electrons into account. This is a far more difficult problem, and a large-scale calculation is required here to calculate all the interactions between all the electrons. Such a large-scale calculation can involve the full diagonalization of matrices with sizes beyond 10000.
We have now
illustrated that we can split the scattering
type problem into two separate problems : a simple
problem for large distances, and a hard, but solvable,
one for small distances. The question now becomes,
how do we link these two regions? Here, the name of
the R-matrix becomes apparent, because we link the
two regions through the so-called R-matrix. The
R-matrix essentially gives the information how the
atomic electrons want an outer electron to enter and
how they want an outer electron to leave the inner
region. The atomic properties can now be calculated
by linking this information with the information from
the outer electron: how it wants to enter and leave the
inner region. This link then provides us with all the
important atomic properties, such as energies and
lifetimes of atomic states, and various transition
R-matrix theory can also be employed to study photoionization: one light particle in, one
electron out. In this case, the wavefunction of the initial state of the atom has a different
but known behaviour at long distances. The problem can then again be solved by
matching R-matrices at the boundary. The R-matrix technique is one of the most
widely used approaches for the determination of photoionization properties of a wide variety of systems.
For the more mathematically interested, the R-matrix theory is extensively described in various books and physics journals. See the recommended reading list for a more mathematical treatment of R-matrix theory.
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