In hydrogen, an external magnetic field, which we calculate to be ≈16.65 T, cancels the internal field caused by the electron motion in the magnetic sublevels with $mJ=+1/2$. This results in an energy-level degeneracy between states with nuclear magnetic sublevels *m _{I}* of opposite signs. The evaluation of this field has been calculated previously with the use of the low-field quantum numbers $F,\u2009mF$. We show that this calculation is considerably simpler in the high-field $mJ,\u2009mI$ representation. A comparison is given with the earlier work.

## I. INTRODUCTION

Level crossings in atoms have been the subject of study for nearly one century. The polarization behavior of radiating atoms near zero magnetic field, in a region where level separation into distinct magnetic quantum substates is being established, known as the Hanle effect,^{1} is a textbook subject.^{2} This study of the Hanle effect allows the determination of spectral linewidths and shapes, and hence parameters such as lifetimes and collision cross sections. Atomic level crossings at higher (laboratory) magnetic fields have been exploited in hyperfine structure (hfs) studies.^{3} Hyperfine structure denotes the small energy splittings in an electron energy level caused by the interaction of the atomic electron with nuclear magnetic (and possibly electric) multipole moments. Another type of hfs level crossing, e.g., in atoms with electron (spin plus orbital) angular momentum *J* = 1/2, occurs when the internal magnetic field *B*_{0} created by the atomic electron is just equal and opposite to an externally applied field *B*. This situation has been studied formally by Dickson and Weil.^{4} When the applied field is below *B*_{0}, the field seen by the nucleus is essentially that produced by the electron; it is in opposite directions for the two sets of electron magnetic substates $mJ=\xb11/2$. Above *B*_{0}, the magnetic field direction is determined by *B*. At *B*_{0} for one of the electron substates, the nuclear magnetic moment *μ _{I}* is in zero magnetic field and the energies of the nuclear magnetic sublevels become degenerate. We show that this “crossing field” is simple to calculate in the high magnetic field representation.

## II. J = 1/2 ATOM IN HIGH *B*-FIELD: $mI,\u2009mJ$ REPRESENTATION

The energy *E* of the *J* = 1/2 atom with nuclear spin *I* is given by the Breit-Rabi formula^{5} in terms of *I*, *J*, the total angular momentum $F=I+J$, and the magnetic quantum number *m _{F}*. It is simpler for our purpose to express the Hamiltonian $H$, given by

in terms of the *I* and *J* decoupled high-field quantum numbers *m _{I}* and

*m*instead of

_{J}*F*and

*m*:

_{F}where $gI=\mu I/I,\u2009gJ=\mu j/J$, and *a* = $\Delta \nu /(I+1/2)$ with $\Delta \nu $ the zero-field energy separation of the two hfs levels, *F* (for hydrogen, *F* = 1 and *F* = 0). In this expression, both *g _{I}* and

*g*are expressed in units of the Bohr magneton $\mu B=e\u210f/(2mc)$. From Eq. (2), we can see that for states with

_{J}*m*= 1/2, there is a magnetic field

_{J}*B*

_{0}for which $EmI=E\u2212mI$ (these levels cross); the external field just cancels that produced on the nuclear site by the atomic electron, and these two

*m*states become degenerate, so that

_{I}From this we obtain for the crossing field

This result is valid not only for hydrogen, but also for other *J* = 1/2 atoms. Putting in the hydrogen values for $a=\Delta \nu \u22481420.4$ MHz, $mJ=1/2$, the Bohr magneton in frequency units $\mu B\u22481.404\u2009MHz/G\u2009(1\u2009G=10\u22124\u2009T),\u2009gI=\mu I/I$, and recalling that the value of *g _{I}* has to be expressed in Bohr magnetons, we obtain $B0=166,481\u2009G\u224816.65\u2009T$, in accord with the value obtained in Ref. 4 (and presented in Sec. III). It is suggested in Ref. 4 that with present-day technology one may reach this field in the laboratory. We should note, however, that the approach to the crossing magnetic field has such a weak dependence on

*B*, that

*B*

_{0}would be difficult to determine. Otherwise one would have a new tool to determine the spatial dependence of the hfs interaction on the nuclear magnetic structure.

## III. *J* = 1/2 ATOM IN HIGH *B* FIELD: $F,\u2009mF$ REPRESENTATION

For completeness, in the following we derive the result given in Eq. (4) in the more cumbersome low-field $F,\u2009mF$ representation of the Breit-Rabi equation. We have $mF=mI+mJ$ and we start with the Breit-Rabi equations^{5} for the hfs energies of the *J* = 1/2 atom in an external magnetic field *B*:

Here, the magnetic field parameter *x* for *J* = 1/2 is

and the ± signs denote the states $F=I\xb11/2$. The crossing *m _{I}* levels of interest occur only within one of the

*F*levels, characterized here at high field by

*m*= $+1/2$. Since $mF\xb1=mJ\xb1mI$, and correspondingly we have $E+$ and $E\u2212$, these two levels become degenerate at a crossing field given by

_{J}The high field is characterized by $x\u226b1$, so we can replace $1+x2\u2248x2$ in this expression (which may affect the result by less than one part in 10^{4}), and then expand the square roots to obtain

or

Inserting *I* = 1/2 and $mJ=1/2$, we obtain

identical to the result obtained in the “high-field” representation given by Eq. (4).

## IV. CONCLUSION

The $F,\u2009mF$ representation used to obtain Eq. (10) is obviously more laborious than the simple calculation in the $mJ,\u2009mI$ scheme. We also did not have to make the approximation $x\u226b1$ as done for Eq. (8). The advantage of using the more appropriate high-field Hamiltonian representation at the start is clear.

## ACKNOWLEDGMENTS

K.M.L was supported by the FWO Pegasus Marie Curie Fellowship No. 267216; K.T.F. had support from the STFC Continuation Grant No. ST/J000159/1 and Advanced Fellowship Scheme Grant No. ST/G006415/1. H.H.S. gratefully acknowledges support of our work by Edward Schneiderman.

## References

^{193m}Hg