19 The Classical Zeeman Effect

The Classical Zeeman Effect In 1846, Michael Faraday discovered that by placing a block of heavy lead glass between the poles of an electromagnet and passing a linearly polarized beam th

The Classical Zeeman Effect

In 1846, Michael Faraday discovered that by placing a block of heavy lead glass between the poles of an electromagnet and passing a linearly polarized beam through the block in the direction of the lines of force, the plane of polarization of the linearly polarized beam was rotated by the magnetic medium; this is called the Faraday effect Thus, he established that there was a link between electromagnetism and light It was this discovery that stimulated J C Maxwell, a great admirer of Faraday, to begin to think of the relation between the electromagnetic field and the optical field

Faraday was very skillful at inverting questions in physics In 1819, H Oersted discovered that a current gives rise to a magnetic field Faraday then asked the inverse question of how can a magnetic field give rise to a current? After many years of experimentation Faraday discovered that a changing magnetic field rather than a steady magnetic field generates a current (Faraday’s law) In the Faraday effect, Faraday had shown that a magnetic medium affects the polarization of light

as it propagates through the medium Faraday now asked the question, how, if at all, does the magnetic field affect the source of light itself ? To answer this question, he placed a sodium flame between the poles of a large electromagnet and observed the

D lines of the sodium radiation when the magnetic field was ‘‘on’’ and when it was

‘‘off.’’ After many attempts, by 1862 he was still unable to convince himself that any change resulted in the appearance of the lines, a circumstance which we now know was due to the insufficient resolving power of his spectroscope

In 1896, P Zeeman, using a more powerful magnet and an improved spectro-scope, repeated Faraday’s experiment This time there was success He established that the D lines were broadened when a constant magnetic field was applied

H Lorentz heard of Zeeman’s discovery and quickly developed a theory to explain the phenomenon

The fact has been pointed out that, even with the success of Hertz’s experi-ments in 1888, Maxwell’s theory was still not accepted by the optics community, because Hertz had carried out his experiments not at optical frequencies but at

microwave frequencies; he developed a source which operated at microwaves For Maxwell’s theory to be accepted by the optical community, it would be necessary to prove the theory at optical frequencies (wavelengths); that is, an optical source which could be characterized in terms of a current would have to be created There was nothing in Fresnel’s wave theory which enabled this to be done Lorentz recognized that at long last an optical source could be created which could be understood in terms of the simple electron theory (sodium has only a single electron in its outer shell) Therefore, he used the simple model of the (sodium) atom in which an electron was bound to the nucleus and its motion governed by Hooke’s law With this model

he then discovered that Zeeman’s line broadening should actually consist of two or even three spectral lines Furthermore, using Maxwell’s theory he was able to predict that the lines would be linearly, circularly, or elliptically polarized in a completely predictable manner Lorentz communicated his theoretical conclusions to Zeeman, who investigated the edges of his broadened lines and confirmed Lorentz’s predic-tions in all respects

Lorentz’s spectacular predictions with respect to the splitting, intensity, and polarization of the spectral lines led to the complete acceptance of Maxwell’s theory Especially impressive were the polarization predictions, because they were very com-plicated It was virtually impossible without Maxwell’s theory and the electron theory even remotely to understand the polarization behavior of the spectral lines Thus, polarization played a critical role in the acceptance of Maxwell’s theory

In 1902, Zeeman and Lorentz shared the Nobel Prize in physics for their work The prize was given not just for their discovery of and understanding of the Zeeman effect but, even more importantly, for the verification of Maxwell’s theory

at optical wavelengths It is important to recognize that Lorentz’s contribution was

of critical importance Zeeman discovered that the D lines of the sodium were broadened, not split Because Lorentz predicted that the spectral lines would be split, further experiments were conducted and the splitting was observed Soon after Zeeman’s discovery, however, it was discovered that additional spectral lines appeared In fact, just as quickly as Lorentz’s theory was accepted, it was discovered that it was inadequate to explain the appearance of the numerous spectral lines The explanation would only come with the advent of quantum mechanics in 1925 The Zeeman effect and the Faraday effect belong to a class of optical phenom-ena that are called magneto-optical effects In this chapter we analyze the Zeeman effect in terms of the Stokes vector We shall see that the Stokes vector takes on a new and very interesting interpretation InChapter 20we describe the Faraday effect along with other related phenomena in terms of the Mueller matrices

MAGNETIC FIELD

To describe the Zeeman effect and determine the Stokes vector of the emitted radia-tion, it is necessary to analyze the motion of a bound electron in a constant magnetic field, that is, determine x(t), y(t), z(t) of the electron and then the corresponding accelerations The model proposed by Lorentz to describe the Zeeman effect was a charge bound to the nucleus of an atom and oscillating with an amplitude A through the origin The motion is shown in Fig 19-1; is the polar angle and is the

azimuthal angle In particular, the angle describes the projection of OP on to the

xyplane The significance of emphasizing this will appear shortly

The equation of motion of the bound electron in the magnetic field is governed

by the Lorentz force equation:

where m is the mass of the electron, kr is the restoring force (Hooke’s law), v is the velocity of the electron, and B is the strength of the applied magnetic field In component form (19-1) can be written

We saw in the previous chapter that for a constant magnetic field directed along the positive z axis (B ¼ Buz), (19-2) becomes

Equation (19-3) can be rewritten further as

€

x

x þ !20x ¼  eB

m

 _

€

y

y þ !20y ¼  eB

m

 _

where !0¼ ffiffiffiffiffiffiffiffiffi

k=m

p

is the natural frequency of the charge oscillating along the line OP

Figure 19-1 Motion of bound charge in a constant magnetic field; is the polar angle and is the azimuthal angle In particular, the angle describes the projection of OP on to the xy plane

Equation (19-4c) can be solved immediately We assume a solution of the form z(t) ¼ e!t Then, the auxiliary equation for (19-4c) is

so

The general solution of (19-4c) is then

To find a specific solution of (19-6), the constants c1and c2must be found from the initial conditions on z(0) and _zzð0Þ FromFig 19-1we see that when the charge is

at P we have

Using (19-7) we find the solution of (19-6) to be

Next, we solve (19-4a) and (19-4b) We again introduce the complex variable:

In the same manner as in the previous chapters (19-4a) and (19-4b) can be written as

a single equation:

€

 þ ieB m

_

Again, assuming a solution of the form z(t) ¼ e!t, the solution of the auxiliary equation is

! ¼ i eB 2m

i !20 eB

2m

2!1=2

ð19-11Þ

The term (eB/2m)2in (19-11) is orders of magnitude smaller than !20, so (19-11) can

be written as

where

!L¼eB

The frequency !Lis known from the Larmor precession frequency; the reason for the term precession will soon become clear The solution of (19-10) is then

zðtÞ ¼ c1ei!þ tþc2ei! t

ð19-13Þ where !þis given by (19-12a)

To obtain a specific solution of (19-13), we must again use the initial condi-tions FromFig 19-1we see that

so

_

After a little algebraic manipulation we find that the conditions (19-14c) and (19-14d) lead to the following specific relations for x(t) and y(t):

xðtÞ ¼Asin

!0 ½!0cosð þ !LtÞcos !0t þ !Lsinð þ !LtÞsin !0t ð19-15aÞ yðtÞ ¼Asin

!0 ½!0sinð þ !LtÞcos !0t  !Lcosð þ !LtÞsin !0t ð19-15bÞ Because the Larmor frequency is much smaller than the fundamental oscillation frequency of the bound electron, !L !0, the second term in (19-15a) and (19-15b) can be dropped The equations of motion for x(t), y(t), and z(t) are then simply

In (19-16) we have also included z(t) from (19-8) as (19-16c) We see that !Lt, the angle of precession, is coupled only with and is completely independent of

To show this precessional behavior we deliberately chose to show inFig 19-1.The angle is completely arbitrary and is symmetric around the z axis We could have chosen its value immediately to be zero However, to demonstrate clearly that !Ltis restricted to the xy plane, we chose to include in the formulation We therefore see from (19-16) that, as time increases, the factor increases by !Lt Thus, while the bound charge is oscillating to and fro along the radius OP there is a simultaneous counterclockwise rotation in the xy plane This motion is called precession, and we see !Lt is the angle of precession The precession caused by the presence of the magnetic field is very often called the Larmor precession, after J Larmor, who, around 1900, first pointed out this behavior of an electron in a magnetic field The angle is arbitrary, so we can conveniently set ¼ 0 in (19-16) The equations then become

We note immediately that (19-17) satisfies the equation:

This result is completely expected because the radial motion is due only to the natural oscillation of the electron The magnetic field has no effect on this radial motion, and, indeed, we see that there is no contribution

Equations (19-17) are the fundamental equations which describe the path of the bound electron From them the accelerations can then be obtained as is done in the following section However, we consider (19-17a) and (19-17b) further If we plot these equations, we can ‘‘follow’’ the precessional motion of the bound electron as it oscillates along OP Equations (19-17a) and (19-17b) give rise to a remarkably beautiful pattern called a petal plot Physically, we have the electron oscillating very rapidly along the radius OP while the magnetic field forces the electron to move relatively slowly counterclockwise in the xy plane Normally, !L !0and

!L’!0/107 Thus, the electron oscillates about 10 million times through the origin during one precessional revolution Clearly, this is a practical impossibility to illus-trate or plot However, if we artificially take !Lto be close to !0, we can demonstrate the precessional behavior and still lose none of our physical insight To show this behavior we first arbitrarily set the factor A sin to unity Then, using the well-known trigonometric sum and difference formulas, (19-17a) and (19-17b) can be written as

xðtÞ ¼1

yðtÞ ¼1

We now set

so (19-19) becomes

xð
0Þ ¼1

yð
0Þ ¼1

To plot the precessional motion, we set
L ¼
0/p, where p can take on any integer value Equation (19-21) then can be written as

xð
0Þ ¼1

2 cos

p þ1 p

0þcos p 1

p

0

ð19-22aÞ

yð
0Þ ¼1

2 sin

p þ1 p

0sin p 1

p

0

ð19-22bÞ

where we have dropped the subscript L As a first example of (19-22) we set !L ¼

!0/5, so
L ¼0.2
0 InFig 19-2,(19-22) has been plotted over 360
for
L¼0.2
0

(in which time the electron makes 5  360 ¼ 1800 radial oscillations, which is equivalent to
taking on values from 0 to 1800
The figure shows that the electron describes five petals over a single precessional cycle The actual path and direction taken by the electron can be followed by starting, say, at the origin, facing the three

o’clock position and following the arrows while keeping the ‘‘surface’’ of the petal to the left of the electron as it traverses the path

One can readily consider other values of !L In Fig 19-3through Fig 19-6 other petal diagrams are shown for four additional values of !L, namely, !0, !0/2,

!0/4, and !0/8, respectively The result shows a proportional increase in the number

of petals and reveals a very beautiful pattern for the precessional motion of the bound electron

Equations (19-21) (or (19-19)) can be transformed in an interesting manner by

a rotational transformation The equations are

where
is the angle of rotation We now substitute (19-21) into (19-23), group terms, and find that

where

Inspecting (19-24) we see that the equations are identical in form with (19-21); that is, under a rotation of coordinates x and y are invariant In a (weak) magnetic field (19-24) shows that the equations of motion with respect to axes rotating with an angular velocity !L are the same as those in a nonrotating system when B is zero This is known as Larmor’s theorem The result expressed by (19-24) allows us to Figure 19-2 Petal diagram for a precessing electron; !L¼!0=5,
L¼
0=5

describe x0and y0in a very simple way If we set
¼
L
0then
0¼
0and (19-24a) and (19-24b) reduce, respectively, to

Figure 19-3 Petal diagram for a precessing electron; !L¼!0,
L¼
0

Figure 19-4 Petal diagram for a precessing electron; !L¼!0=2,
L¼
0=2

Thus, in the primed coordinate system only
0, the natural oscillation angle, appears The angle
L can be eliminated and we find that

which is a circle of unit diameter with intercepts on the x0 axis at 0 and 1

Figure 19-5 Petal diagram for a precessing electron; !L¼!0=4,
L¼
0=4

Figure 19-6 Petal diagram for a precessing electron; !L¼!0=8,
L¼
0=8

A final observation can be made The petal diagrams for precession based on (19-21) and shown in the figures appear to be remarkably similar to the rose dia-grams which arise in analytical geometry, described by the equation:

where there are 2N petals if N is even and N petals if N is odd We can express (19-27) in terms of x and y from the relations:

so

where we have used the sum and difference formulas for the cosine and sine func-tions

We can show that the precession equations (19-21a) and (19-21b) reduce to either (19-27) or (19-29) by writing them as

where

Equation (19-30) can be transformed to polar coordinates by squaring and adding (19-30a) and (19-30b)

We now set
0to

so

Thus,

Substituting (19-32) into (19-30) and (19-33) into (19-31) then yields

and substituting (19-33) into (19-31) yields,

or

We see that (19-36) (or, equivalently, (19-34)) is the well-known rose equation of analytical geometry Thus, the rose equation describes the phenomenon of the pre-cession of a bound electron in a magnetic field, an interesting fact that does not appear to be pointed out in courses in analytical geometry

We now determine the Stokes vector for the Zeeman effect We repeat Eqs (19-17), which describe the path of the oscillating electron bound to an atom

where

!L¼eB

Equations (19-17) can be represented in complex form by first rewriting them by using the trigonometric identities for sums and differences:

xðtÞ ¼A

yðtÞ ¼A

where

Using the familiar rule of writing (19-37) in complex notation, we have

xðtÞ ¼A

yðtÞ ¼ i A

2



so

The general solution of (19- 4c) is then

To find a specific solution of (19- 6), the constants... (19- 17) are the fundamental equations which describe the path of the bound electron From them the accelerations can then be obtained as is done in the following section However, we consider (19- 17a)... for the cosine and sine func-tions

We can show that the precession equations (19- 21a) and (19- 21b) reduce to either (19- 27) or (19- 29) by writing them as

where

Equation (19- 30)