More on the Quantum Defect

Explanation: JWKB Approach

It must always be kept in mind that the quantum defect is a phenomenological result. To explain how such a simple result arises is obviously an interesting challenge, but it is not to be expected that the solution of this problem will lead to great new physical insight. The only new results obtained from understanding quantum defects in one electron systems are the connection between the quantum defect and the electron scattering length for the same system (Mott and Massey section III 6.2) which may be used to predict low energy electron scattering cross sections (ibid. XVII 9&10), and the simple expressions relating $\delta _{L}$ to the polarizability of the core for larger $\delta _{L}$ \cite{Freeman1976}. The principal use of the quantum defect is to predict the positions of higher terms in a series for which $\delta _{L}$ is known. Explanations of the quantum defect range from the elaborate fully quantal explanation of Seaton \cite{Seaton1966} to the extremely simple treatment of Parsons and Weisskopf \cite{Parsons1967}, who assume that the electron can not penetrate inside the core at all, but use the boundary condition $R(r_{c})=0$ which requires relabeling the lowest $ns$ state 1s since it has no nodes outside the core. This viewpoint has a lot of merit because the exclusion principle and the large kinetic energy of the electron inside the core combine to reduce the amount of time it spends in the core. This is reflected in the true wave function which has $n$ nodes in the core and therefore never has a chance to reach a large amplitude in this region. To show the physics without much math (or rigor) we turn to the JWKB solution to the radial Schrodinger Equation (see Messiah Ch. VI). Defining the wave number

{EQ_expone}

$k_{\ell }(r)={\sqrt {2m[E-V_{\rm {eff}}(r)]\hbar }}$

(remember $V_{\rm {eff}}$ depends on $\ell$) then the phase accumulated in the classically allowed region is

$\phi _{\ell }(E)=\int _{r_{i}}^{r_{o}}k_{\ell }(r)dr$

where $r_{i}$ and $r_{o}$ are the inner and outer turning points. Bound state eigenvalues are found by setting

{EQ_expthree}

$[\phi _{\ell }(E)-\pi /2]=n\pi$

(The $\pi /2$ comes from the connection formulae and would be 1/4 for $\ell$ = 0 state where $r_{i}=0$. Fortunately it cancels out.) To evaluate $\phi (E)$ for hydrogen use the Bohr formula for $n(E)$,

$\phi _{H}(E)=\pi (hcR_{H}/E)^{1/2}+\pi /2~~~{\rm {(independent~of~}}\ell )$

In the spirit of the JWKB approximation, we regard the phase as a continuous function of $E$. Now consider a one-electron atom with a core of inner shell electrons that lies entirely within $r_{c}$. Since it has a hydrogenic potential outside of $r_{c}$, its phase can be written (where $r_{oH}$ and $r_{iH}$ are the outer and inner turning points for hydrogen at energy $E$):

${\begin{aligned}\phi _{\ell }(E)&=\int _{r_{i}}^{r_{oH}}k_{\ell }(r)dr=\int _{r_{i}}^{r_{c}}k_{\ell }(r)dr+\int _{r_{c}}^{r_{oH}}k_{H}(r)dr&=\int _{r_{i}}^{r_{c}}k(r)dr-\int _{r_{iH}}^{r_{c}}k_{H}(r)dr+\int _{r_{iH}}^{r_{oH}}k_{H}(r)dr\end{aligned}}$

The final integral is the phase for hydrogen at some energy $E$, and can be written as $\phi _{H}(E)=(n*+1/2)\pi$. Designating the sum of the first two integrals by the phase $\delta \phi$, then we have

$\delta \phi +(n*+1/2)\pi =(n+1/2)\pi$

or

$n*=n-\delta \phi /\pi$

Hence, we can relate the quantum defect to a phase:

$\delta _{\ell }=\left[\int _{r_{i}}^{r_{c}}k(r)dr-\int _{r_{iH}}^{r_{c}}k_{H}(r)dr\right]{\frac {1}{\pi }}$

since it is clear from Eq.\ \ref{EQ_expone} and the fact that the turning point is determined by $E+V_{\rm {eff}}(r_{i})=0$ that $\delta _{\ell }$ approaches a constant as $E\rightarrow 0$. Now we can find the bound state energies for the atom with a core; starting with Eq.\ \ref{EQ_expthree},

${\begin{aligned}n\pi &=\phi (E)-\pi /2\\&=\pi \delta _{L}(E)+\pi (hcR_{H}/E)^{1/2}\\&\Rightarrow E=hcR_{H}/[n-\delta _{L}(E)]^{2}\equiv {\frac {hcR_{H}}{[n-\delta _{\ell }^{(0)}+\delta _{\ell }^{(1)}hcR_{H}/n^{2}]^{2}}}\end{aligned}}$

thus we have explained the empirical so-called Balmer-Ritz formula. If we look at the radial Schrodinger equation for the electron ion core system in the region where $E>0$ we are dealing with the scattering of an electron by a modified Coulomb potential (Mott & Massey Chapter 3). Intuitively one would expect that there would be an intimate connection between the bound state eigenvalue problem described earlier in this chapter and this scattering problem, especially in the limit $E\rightarrow 0$ (from above and below). Since the quantum defects characterize the bound state problem accurately in this limit one would expect that they should be directly related to the scattering phase shifts $\sigma _{\ell }(k)$ ($k$ is the momentum of the free particle) which obey

$\lim _{k\rightarrow 0}\cot[\sigma _{\ell }(k)]=\pi \delta _{\ell }^{o}$

This has great intuitive appeal: $\pi \delta _{\ell }^{o}$ as discussed above is precisely the phase shift of the wave function with the core present relative to the one with V = $e^{2}/r$. On second thought Eq.\ \ref{EQ_expeight} might appear puzzling since the scattering phase shift is customarily defined as the shift relative to the one with $V=0$. The resolution of this paradox lies in the long range nature of the Coulomb interaction; it forces one to redefine the scattering phase shift, $\sigma _{\ell }(k)$, to be the shift relative to a pure Coulomb potential.

Quantum defects for a model atom

Now we give a calculation of a quantum defect for a potential $V(r)$ which is not physically realistic, but has only the virtue that it is easily solvable. The idea is to put an extra term in the potential which goes as $1/r^{2}$ so that the radial Schrodinger equation (Eq.\ \ref{EQ_secpten}) can be solved simply by adjusting $\ell$. The electrostatic potential corresponds to having all the core electrons in a small cloud of size $r_{n}$ (which is a nuclear size) which decays as an inverse power of $r$.

${\begin{aligned}\phi _{\ell }(r)&={\frac {e}{r}}+{\frac {(Z-1)er_{n}}{2r^{2}}}\\&\Rightarrow E(r)={\frac {e}{r^{2}}}+{\frac {(Z-1)er_{n}}{r^{3}}}\\&\Rightarrow Q_{\rm {inside}}(r)=e+{\frac {(Z-1)er_{n}}{r}}\\&\Rightarrow \rho (r)={\frac {dQ/dr}{4\pi r^{2}}}=-(Z-1)er_{n}/4\pi r^{4}\end{aligned}}$

At $r=r_{n}$, $Q_{\rm {inside}}(r_{n})=Ze$. We presume $r_{n}$ is the nuclear size and pretend that it is so small we don't have to worry about what happens inside it. When the potential V (r) = $-e\phi (r)$ is substituted in Eq.\ \ref{EQ_rehone}, one has

${\frac {d^{2}y}{dr^{2}}}+{\left[{\frac {2\mu }{\hbar ^{2}}}{\left[E_{n}+{\frac {e^{2}}{r}}\right]}-{\frac {\ell (\ell +1)-A}{r^{2}}}\right]}y=0$

${\mbox{where}}~~~A={\frac {2\mu (Z-1)e^{2}r_{n}}{\hbar ^{2}}}$

If one now defines

$\ell ^{\prime }(\ell ^{\prime }+1)=\ell (\ell +1)-A$

then $\ell ^{\prime }<\ell$ since $A>0$, and one can write

$\ell ^{\prime }=\ell -\delta _{\ell }$

Substituting Eq.\ \ref{EQ_qdancfour} in Eq.\ \ref{EQ_qdanctwo} gives the radial Schrodinger equation for hydrogen, (Eq.\ref{EQ_secpten}), except that $\ell ^{\prime }$ replaces $\ell$; eigenvalues occur when (see Eq.\ \ref{EQ_rehfourteen})

$v=n^{\prime }+\ell ^{\prime }+1=n^{*}~~~~~n^{\prime }=0,1,2,\cdots$

where $n^{\prime }$ is an integer. Using $n=n^{\prime }+\ell +1$ as before, we obtain the corresponding eigenvalues as

$E_{n}=hcR_{H}/n^{*2}=hcR_{H}/(n-\delta _{\ell })^{2}$

The quantum defect is independent of $E$

$\delta _{\ell }=\ell -\ell ^{\prime }=n-n^{*}$

Eq.\ \ref{EQ_qdancfour} may be solved for $\delta _{\ell }$ using the standard quadratic form. Retaining the solution which $\rightarrow 0$ as $Z\rightarrow 1$, gives

$\delta _{\ell }=(\ell +1/2)-{\sqrt {(\ell +1/2)^{2}-A}}\approx {\frac {A}{2\ell +1}}~{\mbox{for}}~~A<<\ell +1/2$

This shows that $\delta _{\ell }\rightarrow 0$ as $\ell \rightarrow \infty$. In contrast to the predictions of the above simple model, quantum defects for realistic core potentials decrease much more rapidly with increasing $\ell$ [for example as ($\ell +1/2)^{-3}]$ and generally exhibit $\delta _{\ell }^{(0)}$ close to zero for all $\ell$ greater than the largest $\ell$ value occupied by electrons in the core.