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The index j is the principal quantum number of the j-th energy level of i-times ionized uranium. The partition function of a hypothetical isolated particle cannot be evaluated exactly since (4.35) involves an infinite number of terms. When the summation extends to j =  , the partition function diverges since Xi,j approaches a constant value (the ionization energy), and gi,j is not monotonically decreasing as j - -. However, due to the effect of the Debye field, the summation can be terminated at a finite cut-off quantum number, nc, corresponding to the highest energy level at which the electron is still bound. The calculational form of the partition function based on bound electron states is

                    n (i)
                    c          - ~~Xi,j/kT         (439)
               ui =j0gi,j e-(.9 j=O

The cut-off quantum number is a function of the effective ionization potential since nc (i) is the largest integer eff                                 Frteit       rnu
value for j for which X. < E.     . For the i-th uranium ion, the cut-off number is expressed as


               nc(i) =    3 Xij < Eief             (4.40)



        The relation between principal quantum number and energy level for a hydrogen-like atom is (45)