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 mI of opposite signs. The evaluation of this field has been calculated previously with the use of the low-field quantum numbers F,mF. We show that this calculation is considerably simpler in the high-field mJ,mI representation. A comparison is given with the earlier work.

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 B0 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 B0, 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=±1/2. Above B0, the magnetic field direction is determined by B. At B0 for one of the electron substates, the nuclear magnetic moment μI 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.

The energy E of the J = 1/2 atom with nuclear spin I is given by the Breit-Rabi formula5 in terms of I, J, the total angular momentum F=I+J, and the magnetic quantum number mF. It is simpler for our purpose to express the Hamiltonian H, given by

Hh=aI·J+μB(gJJ·B+gII·B),
(1)

in terms of the I and J decoupled high-field quantum numbers mI and mJ instead of F and mF:

E=amImJ+μB(gJmJB+gImIB),
(2)

where gI=μI/I,gJ=μj/J, and a = Δν/(I+1/2) with Δν the zero-field energy separation of the two hfs levels, F (for hydrogen, F = 1 and F = 0). In this expression, both gI and gJ are expressed in units of the Bohr magneton μB=e/(2mc). From Eq. (2), we can see that for states with mJ = 1/2, there is a magnetic field B0 for which EmI=EmI (these levels cross); the external field just cancels that produced on the nuclear site by the atomic electron, and these two mI states become degenerate, so that

amImJ+μBgImIB0=a(mImJ)+μBgI(mI)B0.
(3)

From this we obtain for the crossing field

B0=amJμBgI.
(4)

This result is valid not only for hydrogen, but also for other J = 1/2 atoms. Putting in the hydrogen values for a=Δν1420.4 MHz, mJ=1/2, the Bohr magneton in frequency units μB1.404MHz/G(1G=104T),gI=μI/I, and recalling that the value of gI has to be expressed in Bohr magnetons, we obtain B0=166,481G16.65T, 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 B0 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.

For completeness, in the following we derive the result given in Eq. (4) in the more cumbersome low-field F,mF representation of the Breit-Rabi equation. We have mF=mI+mJ and we start with the Breit-Rabi equations5 for the hfs energies of the J = 1/2 atom in an external magnetic field B:

E(F,mF)h=Δν2(2I+1)+mFgIμBB±Δν2(1+4mFx2I+1+x2)1/2.
(5)

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

x=(gJgI)μBBΔν=(gJgI)μBBa,
(6)

and the ± signs denote the states F=I±1/2. The crossing mI levels of interest occur only within one of the F levels, characterized here at high field by mJ = +1/2. Since mF±=mJ±mI, and correspondingly we have E+ and E, these two levels become degenerate at a crossing field given by

1h(E+E)=0=2gIμBBmI+a2[ (1+4(mJ+mI)x2I+1+x2)1/2(1+4(mJmI)x2I+1+x2)1/2 ].
(7)

The high field is characterized by x1, so we can replace 1+x2x2 in this expression (which may affect the result by less than one part in 104), and then expand the square roots to obtain

02gIμBBmI+a24mI(2I+1),
(8)

or

B=a(2I+1)gIμB.
(9)

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

B0=amJμBgI,
(10)

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

The F,mF representation used to obtain Eq. (10) is obviously more laborious than the simple calculation in the mJ,mI scheme. We also did not have to make the approximation x1 as done for Eq. (8). The advantage of using the more appropriate high-field Hamiltonian representation at the start is clear.

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.

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