Rydberg atoms - highly excited states

Atomic structure - ground and excited states

In the simplest of atoms, the hydrogen atom, a single electron is bound by a proton to form a neutral species. The bound electron occupies a quantum state belonging to one of the hydrogen atomīs principal shells. The most tightly bound shell has only two quantum states. One has spin up and the other spin down. Otherwise they are identical. These are the ground states of the atom, traditionally labelled 1s. The atom has an infinite number of excited states which belong to a series of excited shells with decreasing binding strengths. This is illustrated below. Each shell is labelled by an integer, n, called the principal quantum number. The ground states belong to the lowest shell with n=1. The binding energies of the shells are given by In= -Un=1/(2n2)I0, their geometrical sizes by an=n2a0, and the number of quantum states in a shell by 2n2, where I0=27.2 eV and a0=0.53 × 10-8 cm are the atomic units of energy and length.


The electron is not well-localized in space. Its position is described by a probability distribution which is often visualized as a cloud, the so-called electron cloud. The 1s electron cloud is shown below. It is spherically symmetric with respect to the proton at the center of the cloud. The first excited shell, with n=2, has eight states. One of them is the 2p-state. The electron cloud of this state is shown next to the 1s-cloud. It is four times larger, and it has the shape of two spherical clouds, one on top of the other.


Rydberg states

Rydberg atoms are highly excited one-electron atoms with a weakly bound electron occupying a state of a high-lying shell, n>>1. Each shell has 2n2 different electronic states and the size of each of them is n2 times the size of the 1s-cloud. We show two examples below. One is the spherically symmetic 5s-cloud which is 25 times larger than the 1s-cloud and has four radial nodes. The other is a strongly polarized state belonging to the seventh shell. The cloud is almost entirely above the proton, which is marked by a black dot. It has six radial nodes. When n is very large this type of cloud resembles an elliptic orbit with an eccentricity of one, a so-called linear cloud.


Coherent elliptic Rydberg states

A special class of Rydberg states are the coherent elliptic states whose electronic clouds have the shapes of elliptic orbits. The above linear cloud belongs to this type of state. Another one is shown below. It has eccentricity less than but close to 1.


The eccentricity is a continuous parameter, so there is an infinite number of coherent elliptic Rydberg states for each shell, and the larger n the more the clouds are concentrated near classical Kepler orbits. The animation below shows clouds for n=25. The nucleus is fixed and the probability distribution within the plane is shown for the full range of eccentricities from 0 to 1.

States of zero eccentricity have circular electron clouds, and are called circular states. Coherent elliptic states, and in particular the circular states, are important in relation to ApSR. An electron bound by a nucleus of finite mass M is described by a reduced mass given by me=mM/(m+M), where m is the true electron mass. For hydrogen one finds me/m=1836/1837= 0.999456.

General description of Rydberg states

A hydrogenic Rydberg state is described fully by two conserved vector quantities, the orbital angular momentum L and the Runge-Lenz vector A (ref.2). These quantities are shown below for the case of a coherent elliptic state.


The orbital plane of the electron e is perpendicular to L, and A points from the nucleus at N towards the perihelion at P and thus determines the orientation of the major axis.

It is adventageous to form two new vector quantities j1 and j2 from L and A by the definitions

j1º (L+A)/2
j2º (L-A)/2.

The components of j1 and j2 satisfy angular momentum commutation relations and j1 and j2 commute. They are thus independent pseudo-spins, and they have the same constant size

j12 = j22 = j(j+1) with j=(n-1)/2,

where n is the principal quantum number.

The pseudospins may be quantized along two arbitrary directions in space, z1 and z2, separated by the angle 2a as illustrated below. The projections i1 and i2 of the pseudospins take either integer or half-integer values within the 2j+1=n numbers

{-j, -j+1 ,..., j-1, j}.

Quantization of j1 and j2

The principal quantum number n, the angle a, and the spin projections i1 and i2 determine the quantum state completely, |a i1 i2>n. The coherent elliptic states mentioned above are particularly simple in this representation. They have maximum spin projections, so four possibilities given by i1= ±i2= ±j exist, and the excentricity e is simply e=sin(a). While the states |a -j +j >n and |a +j -j >n may be degenerate even in the presence of external fields, the states |a +j +j >n and |a -j -j >n are always non-degenerate when an external fields is present. The latter states are therefore of particular interest for ApSR.