It discuss the history and mathematical analysis of Zeeman effect.

1. ZEEMAN EFFECT
L.Bruno Chandrasekar,
Assistant Professor,
Periyar Maniammai Institute of Science and
Technology, Thanjavur

2.  The first principal doublet was observed by Zeeman in 1896
when the source of sodium is placed between a powerful
electromagnet. The splitting of energy levels in the presence of
external magnetic field is said to be Zeeman effect. In the
normal Zeeman effect, the number of splitting is three or lesser.
Soon after, Lorentz gave the theory for normal Zeeman effect
on the basis of classical physics. He theoretically pointed out
that these lines should be polarized. Zeeman observed the state
of polarization using prisms and he experimentally confirmed
the theoretical prediction made by Lorentz. Moreover, he
studied the Zeeman pattern in Zn, Cu, Cd and Sn.

3. WITH THE USE OF THE GREATER DISPERSION AND
RESOLVING POWER INSTRUMENTS, PRESTON IN
1898 OBSERVED MORE THAN 3 LINES IN CR. THE
ZEEMAN PATTERN OF THE PARTICULAR SERIES
WILL BE THE SAME AND IT IS THE
CHARACTERISTICS OF THAT SERIES. THIS IS
KNOWN AS PRESTON’S LAW. IN A WEAK
MAGNETIC FIELD, THE NUMBER OF ZEEMAN
PATERN IS MORE THAN THREE. THIS IS SAID TO BE
ANOMALOUS ZEEMAN EFFECT (AZE).

4. IN 1902, PASCHEN AND RUNGE OBSERVED THE
ANOMALOUS ZEEMAN PATTERN IN NA, CU AND AG.
RUNGE IN 1907 GAVE THE THEORETICAL EXPLANATION
OF AZE AND HE MENTIONED THAT ALL KNOWN ZEEMAN
PATTERNS COULD BE EXPRESSED AS THE RATIONAL
MULTIPLES OF NORMAL TRIPLET SEPARATION. THE
WAVELENGTH SEPARATION IN THE ZEEMAN PATTERN IS
SMALL AND HENCE ONE CAN OBSERVE WITH
INTERFERENCE SPECTROMETERS SUCH AS FABRY-PEROT
INTERFEROMETER OR GRATING SPECTROMETERS WITH
LONG FOCAL LENGTH.

5. THE NUMBER OF LINES OBSERVED IN THE ZEEMAN
PATTERN DEPENDS ON (I) THE APPLIED MAGNETIC FIELD
AND (II) DIRECTION OF VIEW WITH RESPECT TO THE
EXTERNAL FIELD. ZEEMAN EFFECT IS A POWERFUL TOOL
FOR SPECTRAL ANALYSIS AND USED IN MANY PHYSICAL
CONCEPTS SUCH AS OPTICAL PUMPING. GEORGE E. HALE
WAS OBSERVED THE ZEEMAN EFFECT IN THE SOLAR
SPECTRA AND IT WAS USED TO CONCLUDE THAT THERE IS
STRONG MAGNETIC FIELD IN THE SUN SPOT. LATER,
HOUSTON ET. AL. OBSERVED THE ZEEMAN SPLITTING IN
ZINC AND CADMIUM AND HARRISON OBSERVED MORE
THAN 130 SPLITTING COMPONENT IN PRASEODYMIUM.

6. (I) TRANSVERSE MODE: IF THE VIEW OF OBSERVATION IS
PERPENDICULAR TO THE EXTERNAL FIELD, THEN THE
MODE OF OBSERVED PATTERN IS TERMED AS
TRANSVERSE MODE (TM). 3 LINES ARE OBSERVED IN
THE NORMAL ZEEMAN EFFECT AND MORE THAN 3 ARE
OBSERVED IN THE AZE. IN THIS MODE, THE CENTRAL
LINE IS PLANE POLARIZED WITH ELECTRIC FIELD VECTOR
PARALLEL TO THE FIELD WHILE THE OTHER LINES ARE
PLANE POLARIZED WITH THE ELECTRIC VECTORS
PERPENDICULAR TO THE FIELD. (II) LONGITUDINAL
MODE: IN THIS MODE, THE VIEW OF OBSERVATION IS
PARALLEL TO THE FIELD. HERE THE CENTRAL LINE IS
ABSENT WHEREAS THE OTHER LINES ARE CIRCULARLY
POLARIZED IN THE OPPOSITE DIRECTION TO EACH
OTHER.

7. THE NORMAL ZEEMAN EFFECT CAN BE
EXPLAINED BY CLASSICAL AND QUANTUM
THEORY WHEREAS THE AZE CAN BE EXPLAINED
ONLY BY QUANTUM THEORY WHICH INCLUDE THE
SPIN OF ELECTRONS. IF ONE CONSIDERS ONLY
THE ORBITAL MOTION OF THE ELECTRON, THEN
THE AZE BECOMES NORMAL ZEEMAN EFFECT.
NORMAL ZEEMAN EFFECT IS A VERY GOOD TOOL
TO UNDERSTAND CORRESPONDENCE PRINCIPLE.

8. (ii) Classical theory of normal Zeeman effect
The normal Zeeman effect will be explained by the classical theory on the basis of Bohr’s
atomic model. When there is no external field, the centripetal force of the electron is given as
𝐹 =
𝑚 𝑒 𝑣2
𝑟
= 𝑚 𝑒 𝑟𝜔2
The presence of the magnetic field causes the change in the angular velocity of the electron
and the direction of the additional magnetic force depends the orbital direction of the
electron. i.e., either clockwise or anti-clock wise. The net force due to the electron orbital
motion and the external field is 𝐹 ± 𝐵𝑒𝑣 = 𝑚 𝑒 𝜔 + 𝛿𝜔 2
𝑟. This leads to the change in
angular frequency of
𝛿𝜔 = ±
𝐵𝑒
2𝑚 𝑒
= ±
𝜇 𝐵
ℏ
𝐵.

9. When 𝐵 ≠ 0, the electron get accelerated, and the magnitude of the acceleration depends on
the magnitude of the external field. The corresponding frequency change is
𝛿𝜈 =
𝛿𝜔
2𝜋
= ±
𝐵𝑒
4𝜋𝑚 𝑒
The change in frequency as a function of wavelength (λ) can be obtained by differentiating
the equation 𝑐 = 𝜈𝜆 and it is given as
𝛿𝜈 = −
𝑐
𝜆2
. 𝛿𝜆
The change in the wavelength due to the external field is
𝛿𝜆 = ±
𝐵𝑒𝜆2
4𝜋𝑚 𝑒 𝑐
Here it should be noted that the change in frequency depends on the external field, but the
change in wavelength depends on both external field and the wavelength. Here the splitting
of wavelength and hence the energy can be explained by only considering the orbital motion
of the electron and the spinning motion of the electron is neglected here.

10. (iii) Vector model of Zeeman effect
(a) Normal Zeeman effect: Atom in the external field acquires the additional magnetic
interaction energy and it is given as
𝐸𝑖𝑛𝑡 =
𝑒ℏ
2𝑚 𝑒
𝑚𝑙 𝐵 = 𝜇 𝐵 𝑚𝑙 𝐵
In the absence of the external field, only one energy a state can have. In the presence of the
field, ml can have (2l+1) values for a given value of ‘l’. This means that it may have (2l+1)
states with different energy when there is a field. The transition from sublevels of l1 to l2
causes the normal Zeeman effect according to the selection rule.
Let us consider the transition from ‘m’ to ‘n’. ‘E1’and ‘E2’ represents the energy of the states
‘m’ and ‘n’ respectively when there is no magnetic field and the presence of the magnetic
field causes energy as ‘EB1’ and ‘EB2’ respectively. The magnetic interaction energy is
𝑒ℏ
2𝑚
𝑚𝑙 𝐵, when the spin of the electron is neglected. Hence
𝐸 𝐵1 = 𝐸1 + 𝜇 𝐵 𝑚𝑙 𝐵
𝐸 𝐵2 = 𝐸2 + 𝜇 𝐵 𝑚𝑙 𝐵

11. The energy difference between the states ‘m’ and ‘n’ is
𝐸 𝐵1 − 𝐸 𝐵2 = 𝐸1 − 𝐸2 + 𝜇 𝐵 ∆𝑚𝑙 𝐵
The frequency of the corresponding transition is
𝜈 𝐵 = 𝜈0 +
𝐵𝑒
4𝜋𝑚 𝑒
∆𝑚𝑙
For the transverse mode, the selection rule for the transition is given as ∆𝑚𝑙 = 0, ±1. The
frequency of the p-component (∆𝑚𝑙 = 0) and the s-component (∆𝑚𝑙 = ±1) are given below.
p-component: 𝜈 𝐵 = 𝜈0
s-component: 𝜈 𝐵 = 𝜈0 ±
𝐵𝑒
4𝜋𝑚 𝑒
The selection rule ∆𝑚𝑙 = ±1 can be used for the longitudinal mode and hence p-component
cant be observed and the s-components are opposite in direction.
s-compoent: 𝜈 𝐵 = 𝜈0 ±
𝐵𝑒
4𝜋𝑚 𝑒

12. (b) Anomalous Zeeman effect: The inclusion of the spin of the electron in the previous theory
explains the cause of the anomalous Zeeman effect. The magnetic interaction energy is
𝜇 𝐵 𝑔 𝐿mjB. The energy difference between the states ‘m’ and ‘n’ is
𝐸 𝐵1 − 𝐸 𝐵2 = 𝐸1 − 𝐸2 + 𝜇 𝐵 ∆𝑚𝑗 . ∆𝑔 𝐿 𝐵
where ‘E1’ and ‘E2’ represents the energy of ‘m’ and ‘n’ states, respectively when 𝐵 = 0. The
frequency of the corresponding transition is
𝜈 𝐵 = 𝜈0 +
𝐵𝑒
4𝜋𝑚 𝑒
∆𝑚𝑗 . ∆𝑔 𝐿
There are 2𝑗 + 1 states available for both ‘m’ and ‘n’. As ‘g’ different for each sublevel
since ‘l’ differs, the transitions don’t overlap and this causes the anomalous Zeeman effect.
This theory leads to normal Zeeman effect as 𝑠 = 0, i.e., when the spin neglected. If there is
even number of electrons in an atom, then there is even number of electrons in the outer orbit.
For these atoms, the net spin becomes zero. Thus, one can observe only the normal Zeeman
effect. On the other hand, if odd number of electrons in the outer orbit, then the net spin is
non-zero, and anomalous Zeeman effect can be observed.
Transverse
Mode:
∆𝑚 = ±1 Plane polarized perpendicular to the field
∆𝑚 = 0 Plane polarized parallel to the field
Longitudinal
Mode:
∆𝑚 = ±1 Circularly polarized
∆𝑚 = 0 Forbidden