The size of Rydberg states

One of the most important aspects of Rydberg states is their rapid increase in size as the principal quantum number n is raised. 

The mean radius of a Rydberg state is approximately aon 2, where ao is the Bohr radius. Although the atom in its ground state is a quantum object, it eventually ceases to be so as n increases for n 100, the diameter of the atom is 104 x 2ao which is roughly 1 /xm, at which point the atom is definitely reaching macroscopic size. For n 1000, the diameter of the atom is l/10th of a millimetre. 

Of course, such large states are rather fragile as stressed above, Rydberg states can only develop if there is enough free space around the atom for the wavefunction of the electron to extend well outside the atomic core. This aspect seems to have been appreciated most clearly in the early days of quantum theory by Sommerfeld and Welker [33], who considered a H atom enclosed in a hollow sphere,10 and showed that, if the radius of the sphere is less than 1.835 ao, then the energy of the system becomes positive, i.e. the electron attempts to escape by exerting a pressure on the inner surface of the sphere. 

This might seem to be an academic problem, with little chance of practical verification. In fact, individual atoms can be trapped in a rare-gas matrix, and the size of the lattice site in which they are confined is large enough that the first Rydberg states can be observed, albeit weakly, whilst higher members are quenched. 

A related situation occurs in cluster physics an aggregate of atoms may possess a hollow spherical shape (an example is buckmasterfullerene, made up of 60 C atoms in a symmetrical football pattern arrangement). It is possible for a single atom to be trapped inside the spherical cage, and the situation is then rather similar to the one considered by Sommerfeld and Welker [33]. It is now possible to capture atoms inside hollow fullerene structures, and this has become a subject of study (see chapter 12.4). It does not seem that any Rydberg excitations of such atoms have yet been observed. 

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Many properties in a Rydberg series scale in different ways. For example, the level spacing scales as n -1/3, which turns out to be an essential property when we come to discuss K-matrix theory in chapter 8. The same is true for core penetration, and all the properties which depend on the overlap between the core and excited electron wavefunctions (see chapters 4 and 6). The size of Rydberg states (discussed in section 2.14) scales as n 2, while transitions between adjacent levels in the Rydberg manifold, which depend on the overlap between adjacent excited states, scale as n 4. Yet more scaling rules for Rydberg series in external fields will emerge in chapter 10. 

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