THE THEORY OF THE QUADRATIC ZEEMAN EFFECT

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THE THEORY OF THE QUADRATIC ZEEMAN EFFECT

A. Edmonds

To cite this version:

A. Edmonds. THE THEORY OF THE QUADRATIC ZEEMAN EFFECT. Journal de Physique

Colloques, 1970, 31 (C4), pp.C4-71-C4-74. �10.1051/jphyscol:1970411�. �jpa-00213866�

Collocl~rr

C4,

uu 1 1 - 12, Tome 3 1, Noc.-Dec. 1970, page C4-7 1

THE THEORY OF THE QUADRATIC ZEEMAN EFFECT

by A. R. EDMONDS

Physics Department, Imperial College, L o n d o n SW7

ResurnC.

Des experiences recentes B haute resolution de Garton et Tomkins ont montre des nouvelles structures dans les spectres des atonies en presence d'un champ magnetique.

On peut attribuer ces structures a u ter~iie Hamiltonien du deuxieme degre dans le champ

tique, donnant un problenle forn~ellenient inseparable. Les paramitres pertinents sont tels que les approximations ordinaires sont inefficaces.

Le but de cette etude est de sur~nonter ces dificultks, et les resultats prelirninaires ont montrk des accords encourageants avec les donnees.

Abstract.

Recent work at high resolution carried ont at Argonne has exhibited previously unobserved structures in the spectra of atoms in magnetic fields.

These structures may be attributed to the term in the Hamiltonian q ~ ~ a d r a t i c in the magnetic field. giving rise to a formally non-separable problem. The relevant parameters are such that conventional approximations are invalid. The work now reported has been directed to overconiing the resulting diffic~~lties, and preliminary r e s ~ ~ l t s have shown promising agreement with the data.

1. Introduction.

Tlie existence of a term in the expression for t h e Zeeman shift o f spectral lines which is q u a d r a t i c in the magnetic field has been known for m a n y years. Lt was, for cxamplc, d ~ s c u s s e d by Burgers [ I ] in t h e framework of the old q u a n t u m theory. However, most textbooks - eg : C o n d o n &

Shortley [ 2 ] dismiss it a s in practice n small a n d u n i m p o r t a n t correction t o the linear effect. In solid state physics, o n the other h a n d , the quadratic Z e e n ~ a n effect has been recognised t o be of importance Sor some time : for even witli a moderate magnetic field the frequently small effective mass of an e l e c t ~ o n in a crystal magnifies t h e influence o f t h e field. (Cf. : Haide- m e n a k ~ s [3].)

Nevertheless, tlie fact that the quadratic effect is proportional t o the f o u r t h power of tlle principal q u a n t u m n u m b e r is linked with ~ m p r o v e m e n t s in instrumental technique t o ~ n d u c e a new development Long spectral series c a n now be resolved a n d the q u a - dratic Z e e ~ n a n effect can tl1en have a major influence

on spectral structure. We shall see that not only are the predictions of existing theory confirmed, but t h a t i n regions o f t h e spectrum where n-mixing occurs spectral patterns a r e produced by a riiagnetic ficld, f o r which there is a s yet no theoretical basis a t all.

2. D i s c ~ ~ s s i o n of the Results of Carton and Tom- kins 151.

These a u t h o r s have p ~ i b l i s l ~ e d Zeeman spectra of the principal series o f Ba 1 in absorption

tlie field strength was 24 070 gauss. 1 a m grateful t o Professor C a r t o n a n d D r T o m k i n s for informatioll o n unpublished measurements of the plates concerned.

Both the a a n d n spectra may be divided into I-egions with strikingly diKerent characteristics (see Fig. I ) . In tlie lower region, running from a b o u t

30 t o n

37, distinct groups of lines a r e seen wliich may be identified with successive principal quantun7 n u n ~ b e r s . In addition to the so-called quadratic shift.

we have a breaking o f the I-degeneracy characteristic of hydrogenic spectra. T h e lines produced a r e easily

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970411

CI-72 A.

R.

distinguished and have, within each group, a roughly equal spacing. The members of each group in the n spectrum have roughly equal intensities, while in the a spectrum intensities within a group fall off rapidly, the strongest line being that with maximum displacement from the field-free position.

As we move up the series thesc groups start to run together and above n

40 there is little trace of the Rydberg structure remaining. There are, however, striking regularities in the observed spectrum. I n two distinct regions of the

spectrum we observe sequences of regularly spaced lines, the spacing being close to j hw, where w is the so-called cyclotron fre- quency - ie : I e I Blpc. Another system of very broad lines extends from a little below the field-free series limit into the continuum. The spacing is again regular and approximately equal to $ Aw. Other regularities exist, particularly in the n spectrum, but these are still under investigation. We may compare these evenly spaced line sequences with the so-called Landau spectrum of a free electron in a magnetic field, where the level spacing is h a . Thus, it seems natural t o call these structures quasi-La~?(/au levels.

The results discussed above will shortly be aug- mented ; the use of the heat-pipe (Vidal & Cooper [9]) and the forthcoming introduction of superconducting magnets at Argonne will improve the quality of the data from Zeeman experiments still further.

3. The Theory of the Quadratic Zeeman Effect.

We shall assume in the following the complete Paschen- Back regime ie : that the spin of the optical electron may be ignored. This is certainly the case for the Ba I spectrum under consideration. The assumption that the effect of the nucleus and electronic core may be represented by a point charge of z = 1 is clearly also justified, provided that the quantum defects of p and possibly f orbits are taken into account. For we shall always be dealing with an optical electron with large effective principal quantum number-ie : with n > 20.

I t is convenient t o choose a vector potential

where the uniform magnelic field B is along !he z axis.

Then, since the kinetic momentum

(where p is the canonical momentum), the Hamiltonian for the optical electron is

The electrostatic potential

We shall use a frame of reference rotating with the Larmor frequency

about the z axis.

In this frame the Hamiltonian becomes

where pZ

x 2 + Y 2 . The effective potential V is sketched in figure 2.

The energy of the system E* with respect to the Larmor frame is given by

where E is the energy i n the fixed frame.

The system described is non-separable, and thus there is no easy solution in either classical or quantum mechanics. I t resembles in this way the well-known three-body problem.

Gajewski [4] has studied the general problem of a n electron in superimposed Coulomb and magnetic fields, but his results are of limited relevance to the highly excited states characteristic of the atomic system under discussion.

Surfaces of zero velocity in the Larmor frame are

given by

T H E THEORY OF

THE

Z E E M A N

These surfaces are sketched in figure 3 for different energies, and are the bounding surfaces of classical motion. Thus as the (negative) energy is increased, the initially spherical shape of the bounding surface becomes more and more elongated, approximating to a prolate ellipsoid. F o r positive energies the surface is spindle-shaped, extending t o infinity along the posi- tive a n d negative z axes.

The term in the Hamiltonian which mediates the quadratic shift a n d the breaking of I-degeneracy is clearly

For relatively small n we may attempt a solution by diagonalizing the matrix of Q o n a basis of field-free states 1

>. F o r the case being considered we need only

0, $. 1 and odd I. It has been suggested (cf. Scliiff & Snyder [6]), that an adequate approxi- mation may be gained by taking only subniatrices of Q diagonal in

Experience shows that this is the case only for values of n so small as to be of little interest.

We may renlark that the distortion of the orbits by the magnetic field, as shown by the classical bounding surfaces, is such as to need several values of

to give a n adequate representation of the departure from spherical symmetry.

Thus it becomes necessar)I to find the eigenvalues of quite large matrices, which with present computillg facilities is no great problem. However, the difficulty lies in the colnpt~tation of the matrix elements, parti- cularly when the non-integer values of n arising from quantum defects are taken into account. Methods of interpolation from known results have been devised which will be reported elsewhere.

Although results using this approach have u p to date been promising, it cannot be expected to deal adequately with that part of the spectrum where

is no longer significant.

An alternative is to set up a system of coupled differential equations for the radial functions, when the wave-function of the optical electron is expanded in spherical harmonics :

OC

Vl,(r, 8, cp)

1

Xm(& cp) .

1 odd

Then the Schrodinger equation becomes

where

Attempts are under way to produce numerical solu- tions of this system, although the computational problems seem formidable.

I n the region around the field-free series limit (E

O), we may expect that a legitimate approximation may be made t o the solution of the quantum-mechanical problem. The semi-classical frequency of motion of tlie electron along the z axis (proportional to

is small compared with ^o and the acliahatic

ti011

is a natural choice. This has been considered in

detail by solid state theorists. F o r in crystals the effective mass of an electron may be only a few per cent of the free value, so extreme cases of quadratic Zeeman effect can occur even with moderate magnetic fields (:::).

However-. computations of classical orbits indicate that the adiabatic approximation must necessal-ily fail in tlie region of the origin ; for here the velocit)]

along the r axis becomes large compared with the orbiting v e l o c i t ~ ~ . Moreover, this region is importnnt.

for we may consider it to be closely associated with radiative transitions : it is here that the selection rule against transitions other than at c);clotron I'rccl~~cnc!, is broken.

Nevertheless, it is often thc

i n cluanti~rn theor!, that a n unsatislilctvry : i l ~ l ~ ~ - r ~ \ i l n a ~ i c , ~ ~

point to useful conclusions. A xmi-cl;~i\ic.;~l c : ~ l ~ , ~ ~ i . : ; i

therefore. been madc within the I.I.:I

I, adiabatic approximation. Tn the fnll:ls>

(*) Cf. Haidcmenahic 131.

C4-74 A. R. EDMONDS

a s s ~ ~ m e d close to zero, as a free parameter. We use the Bohr-Sommerfeld condition to compute the energy spectrum of an electron in a nearly plane orbit :

We seek those values of E which make N an integer.

The results of numerical integration are as follows : ror large

as might be expected, we get the pure Landau spectrum, with level spacing of hw. As

approaches zero the uniform level spacing is preserved, at least over a range of ten levels or so. The spacing, however, increases to a limiting value of 1.58 fiw i n the region of E = 0. This may be compared with the experimental spacing in this region of approxirna- tely 1.5 hw. Thus, the effcct of the electrostatic field in inducing a higher effective orbital frequency

analogous to the modification of the atomic number Z by screening in atoms and molecules.

We may use the above result along with the following argument to give a qualitative explanation of the structure of broad lines in tlie region of the series limit.

The motion of tlie electron may be regarded as having two modes

parallel to the magnetic field, and in a plane normal to the field respectively. We associate energies Ell and EL with these modes. Cou- pling between the modes occurs through the term U(I.).

An electron which contributes to the a spectrum will have a large proportion of its energy in EL. If this is greater than EL for I ;I

co it will be trappcd ternpur-arily in the magnetic bottle, although the total energy may be positive.

We see that resonances may occur, although the electron is in principle unbound. The theory underlying the widtli of the observed lines seems con-rplicated and needs further work.

The picture outlined has one glaring shortcoming : at a distance about 5 h o ~ below the series limit in the Bar a spectrum, tlie diffuse quasi-Landau levels just dealt with start to merge with a sharply-defined quasi-Landau sequence with a spacing of about 4 h o The adiabatic hypothesis would seem to imply a gradual change of level spacing, not the observed sudden transition. The question also arises of whether the nearness of the quasi-Landau ratios to 4 and 3 is of special significance.

References

[ I ] BURGERS (J. M.), Het Atoommodel vat! Rurherford- [4] GAJEWSKI (R.), Plzysica, 1969, 41, 456.

Bolrr (Haarlem : De Erven Loosjes), 1919. [5] GARTON (W. R. S.) & TOMKINS (F. S.), Astrophys. J . , [2] CONDON (E. U.) & SHORTLEY (G. S.), Tlzeory of Atomic 1969, 158, 839.

Spectra (Cambridge : Cambridge University [6] SCHIFF (L. I.) & SNYDER (H.), Phys. Rev., 1939, 55, 59.

Press), 1935. [7] SCHWINGER (J.), Phys. Rev., 1949, 75, 1912.

[3] HAIDEMENAKIS (E. D.) (ed.), Physics of Solids in It~tense [8] LANDAU

D.), Zs. f. Plzysik, 1930, 64, 629.

Magneric Fields (New York : Plenum Press), [9] VIDAL (C. R.) & COOPER (J.), J. App. Phys., 1969, 40,

1969. 3370.