Physical Structure of Matter Physics of the Electron

5.1.10-05/07 Zeeman effect / normal and anomalous version

What you can learn about …

 Bohr’s atomic model
 Quantisation of energy levels
 Electron spin
 Bohr’s magneton
 Interference of
electromagnetic waves
 Fabry-Perot interferometer

Principle:
The “Zeeman effect” is the splitting
of the spectral lines of atoms within
a magnetic field. The simplest is the
splitting up of one spectral line into
three components called “normal
Zeeman effect”. Usualy the phe-
nomenon is more complex and the
central line splits into many more
components. This is the “anomalous
Zeeman effect”. Both effects can be
studied using a cadmium lamp as a
specimen. The cadmium lamp is sub-
mitted to different magnetic flux
densities and the splitting of the red
cadmium line (normal Zeeman effect)
and that of a green cadmium line

What you need:

P2511005 with Electromagnet
P2511007 with Magnetic system, variable
Fabry-Perot interferometer 09050.02 1 1
Cadmium lamp for Zeeman effect 09050.20 1 1
Electromagnet w/o pole shoes 06480.01 1
Pole pieces, drilled, conical 06480.03 1
Rot.table for heavy loads 02077.00 1
Magnetic System, variable 06327.00 1
Power supply for spectral lamps 13662.97 1 1
Variable transformer 25 V~/20 V-, 12 A 13531.93 1
Capacitor, electrolyte, 22000 mic-F 06211.00 1
Digital multimeter 2010 07128.00 1
Optical profile-bench, l = 1000mm 08282.00 1 1
Base for optical profile-bench, adjustable 08284.00 2 2 Interference rings with the anomalous Zeeman effect.
Slide mount for optical bench, h = 30 mm 08286.01 6 5
Slide mount for optical profil-bench, h = 80 mm 08286.02 2 2 (anomalous Zeeman effect) is inves- Tasks:
Lens holder 08012.00 4 4 tigated using a Fabry-Perot interfer- 1. Using the Fabry-Perot interfero-
Lens, mounted, f = +50 mm 08020.01 2 2 ometer. The evaluation of the results meter and a self made telescope
Lens, mounted, f = +300 mm 08023.01 1 1 leads to a fairly precise value for the splitting up of the central lines
Iris diaphragm 08045.00 1 1 Bohr’s magneton. into different lines is measured in
Polarising filter, on stem 08610.00 1 1 wave numbers as a function of the
Polarization specimen, mica 08664.00 1 1 magnetic flux density.
Connecting cord, 32 A, l = 250 mm, red 07360.01 1 2. From the results of point 1. a value
Connecting cord, 32 A, l = 250 mm, blue 07360.04 1 for Bohr’s magneton is evaluated.
Connecting cord, 32 A, l = 500 mm, red 07361.01 1 3. The light emitted within the
Connecting cord, 32 A, l = 500 mm, blue 07361.04 1 direction of the magnetic field is
Connecting cord, 32 A, l = 750 mm, red 07362.01 1 qualitatively investigated.
Connecting cord, 32 A, l = 1000 mm, red 07363.01 1
Connecting cord, 32 A, l = 1000 mm, blue 07363.04 1
CCD camera Moticam 352 for PC, 0.3 megapixels 88037.01 1 1
PC, Windows® XP or higher

Complete Equipment Set, Manual on CD-ROM included

Zeeman effect P25110 05/07
226 Laboratory Experiments Physics PHYWE Systeme GmbH & Co. KG · D - 37070 Göttingen
 LEP
Normal and anomalous Zeeman effect 5.1.10
-05

Related topics Power supply for spectral lamps 13662.97 1

Quantization of energy levels, Bohr’s atomic model, vector Variable transformer, 25 V AC/20 V DC, 12 A 13531.93 1
model of atomic states, orbital angular moment, electron spin, Capacitor, electrolytic, 22000 µF 06211.00 1
Bohr’s magneton, interference of electromagnetic waves, Digital multimeter 07128.00 1
Fabry-Perot interferometer. Optical profile-bench, l = 1000 mm 08282.00 1
Base for opt. profile-bench, adjust. 08284.00 2
Slide mount for opt. profile-bench, h = 30 mm 08286.01 5
Principle
Slide mount for opt. profile-bench, h = 80 mm 08286.02 2
The “Zeeman effect” is the energy shift of atomic states Lens holder 08012.00 4
caused by an magnetic field. This shift is due to the coupling Lens, mounted, f = +50 mm 08020.01 2
of the electron orbital angular momentum to the external mag- Lens, mounted, f = +300 mm 08023.01 1
netic field. The normal Zeeman effect occurs when there is no Iris diaphragm 08045.00 1
spin magnetic moment – states with zero spin are necessary. Polarizing filter, on stem 08610.00 1
In singulett systems the spins of the electrons cancel each Polarization specimen, mica 08664.00 1
other i.e. add up to zero. The energy shift of the atomic states Connecting cord, l = 25 cm, 32 A, red 07360.01 1
in an outer magnetic field can be observed by the wavelength Connecting cord, l = 25 cm, 32 A, blue 07360.04 1
shift of the radiation emitted in atomic transitions between Connecting cord, l = 50 cm, 32 A, red 07361.01 1
these states. Connecting cord, l = 50 cm, 32 A, blue 07361.04 1
Generally there is not only a magnetic moment of the orbit of Connecting cord, l = 75 cm, 32 A, red 07362.01 1
an electron state, but also a magnetic moment of the electron Connecting cord, l = 100 cm, 32 A, red 07363.01 1
spin. This leads to a more complicated behaviour of the atom- Connecting cord, l = 100 cm, 32 A, blue 07363.04 1
ic states in an outer magnetic field. This is called anomalous CDC-Camera for PC
Zeeman Effect and can be observed in atomic transitions incl. measurement software* 88037.00 1
where non-singulett states are involved. PC with USB interface, Windows®98SE / Windows®Me /
Windows®2000 / Windows®XP
Equipment *For classical version of the Zeeman Effect (P2511001), alter-
Fabry–Perot interferometer native to CCD-Camera incl. measurement software:
for 643.847 nm and 508.588 nm 09050.02 1
Cadmium lamp for Zeeman effect 09050.20 1 Sliding device, horizontal 08713.00 1
Electromagnet without pole pieces 06480.01 1 Swinging arm 08256.00 1
Pole pieces, drilled, conical 06480.03 1 Plate holder with tension spring 08288.00 1
Rotating table for heavy loads 02077.00 1 Screen, with aperture and scale 08340.00 1

Fig.1a: Experimental set-up

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5.1.10 Normal and anomalous Zeeman effect
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Tasks Initial adjustment and observation of the longitudinal Zeeman

1a. Normal Zeeman effect: Transversal and longitudinal effect is done without the iris diaphragm. During observation
observation of the splitting of the red 643.847 nm Cd-line of the transverse Zeeman effect the iris diaphragm is illumi-
in the magnetic field showing the normal Zeeman effect. nated by the Cd-lamp and acts as the light source. The lens
1b. Anomalous Zeeman effect: Transversal and longitudinal L1 and a lens of f = 100 mm, incorporated in the étalon, cre-
observation of the splitting of the green 508.588 nm Cd- ate a nearly parallel light beam which the Fabry-Perot étalon
line in the magnetic field showing the anomalous Zeeman needs for producing a proper interference pattern.
effect. For the observation of the normal Zeeman effect the red
2. Observation of the effect of polarization filter and polar- colour filter is to be inserted in the holder of the étalon. For the
ization filter combined with l/4 plate for the splitted green observation of the anomalous Zeeman effect the red colour fil-
and red lines in transversal and longitudinal direction. ter is to be removed from the étalon and the 508 nm interfer-
3. Measurement of the frequency shift with help of the CCD ence filter is to be attached onto the holder of the +300 mm
camera and the supplied measurement software or with lens L2 beneath the lens (so that there are less disturbing
the screen with scale and the sliding device in the classi- reflections between Fabry Perot interferometer and interfer-
cal version for both of the above mentioned spectral lines. ence filter).
The étalon produces an interference pattern of rings which
can be observed through the telescope formed by L2 and L3.
Set-up and Procedure
The ring diameters can be measured using the CCD-camera
The electromagnet is put on the rotating table for heavy loads and the software supplied with it. In the classical version the
and mounted with the two pole-pieces with holes so that a interference pattern is produced within the plane of the screen
gap large enough for the Cd-lamp (9-11 mm) remains. The with a scale mounted on a slide mount which can laterally be
pole-pieces have to be tightened well! The Cd-lamp is insert- displaced with a precision of 1/100th mm. The measurement
ed into the gap without touching the pole-pieces and con- here can be done by displacing the slash representing the “0“
nected to the power supply for spectral lamps. The coils of the of the scale.
electromagnet are connected in parallel and via an ammeter
connected to the variable power supply of up to 20 VDC,12 A.
A capacitor of 22 000 µF is in parallel to the power output to
smoothen the DC-voltage.
The optical bench for investigation of the line splitting carries
the following elements (their approximate position in cm in
brackets):

(80) CDC-Camera
(73) L3 = +50 mm
(68) Screen with scale (only in classical version)
(45) Analyser
(39) L2 = +300 mm
(33) Fabry-Perot Étalon
(25) L1 = +50 mm
(20) Iris diaphragm
(20) Drilled pole-pieces
(0) Cd-spectral lamp on rotating table Fig.1b: Set-up for the classical version of the experiment.

Fig. 2: Arrangement of the optical components.

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Normal and anomalous Zeeman effect 5.1.10
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Initial adjustment: The rotating table is adjusted so that the Remark: For later evaluations the calibration curve of the mag-
centres of the holes in the pole-pieces lie about 28 cm above netic flux density versus the coil current has to be recorded
the table. The optical bench with all elements (except iris previously. This can be done if a teslameter is available.
diaphragm and CCD-camera) mounted, is then moved closer Otherwise the results of Fig. 3 can be used. The curve of Fig. 3
to the electromagnet so that one of the outlet holes of the was recorded by measuring the flux density in the centre of
pole-pieces coincides with the previous position of the iris the gap in the absence of the Cd-lamp. For the evaluations
diaphragm. L1 is then adjusted so that the outlet hole is with- these centre-values were increased by 3.5 % to account for
in its focal plane. All other optical elements of Fig.2. are sub- the non-uniform flux distribution within the gap.
sequently readjusted with respect to their height.
The current of the coils is set to 5 A (increase in light intensity Theory
of the Cd-lamp!) and the ring interference pattern in axial
As early as 1862, Faraday investigated whether the spectrum
direction is observed through L3 by the eye. The splitting of
of coloured flames changes under the influence of a magnet-
the line should be well visible. The pattern must be centered
ic field, but without success. It was not until 1885 that Fievez
and sharp which is eventually achieved by a last, slight move-
from Belgium was able to demonstrate an effect, but it was
ment of the étalon (to the right or to the left) and by displace-
forgotten and only rediscovered 11 years later by the
ment of L2 (vertically and horizontally) and of L3.
Dutchman Zeeman, who studied it together with Lorentz.
Finally the CCD-camera is focussed so that far away things
Here the effect is demonstrated with the light of a Cadmium
are clear and mounted to the optical bench and adjusted in
lamp and the help of a Fabry-Perot interferometer for resolv-
horizontal and vertical position as well as in tilt until a clear
ing a small part of the spectrum preselected by a color filter or
picture of the ring pattern is visible on the computer screen.
an interference filter so only the light of a single atomic transi-
For installation and use of the camera and software please
tion line is observed. Without field the magnetic sub-levels
refer to the manual supplied with the camera.
have the same energy but with field the degeneration of the
In the classical version the screen with scale is shifted in a way
levels with different mJ is cancelled and the line is split.
that the slash representing the „0“ of the scale is clearly seen
Cadmium has the electron structure (Kr) 4d10 5s2, i.e. the outer
coinciding, for instance, with the centre of the fairly bright
shell taking part in optical transitions is composed by the two
inner ring. The scale itself must be able to move horizontally
5s2 electrons that represent a completed electron shell. ((Kr) =
along the diameter of the ring pattern. (Set-up see Fig. 1b.)
1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6.) This is similar to the outer
Hint: best results are achieved when the experiment is carried
electron structure of Helium but also of Mercury. A scheme of
out in a darkened room.
the energy levels of Cd is shown in Fig.4. In a completed shell
The electromagnet is now turned by 90° and the iris dia-
in it's ground state the electron spins always compensate
phragm is inserted for transversal observation.
each other – they are anti-parallel. If the total electron spin is
zero, also the magnetic moment connected to electron spin is
zero. Atomic states with zero total spin are called singulett
states. So in transitions between different singulett states the
magnetic moment of spin does not play a role, as is the case
with the normal Zeeman effect. Electric dipole radiation as in
common optical transitions does not change the electron spin
except in heavy atoms with jj-coupling, so transitions are nor-
mally between different states in the same multiplicity system.
But Fig. 4 shows there is some jj-coupling in Cadmium.
The transition used to demonstrate the normal Zeeman effect
is 3 1D2 S 2 1P1 with 643.847 nm and the transition used to
demonstrate the anomalous Zeeman effect is 2 3S1 S 2 3P2
with 508.588 nm.
In a term like 2 3S1 the first number "2" denotes the main
quantum number of the radiating electron with respect to the
atom's ground state (that is counted as "1"), here this is real-
ly the 6th s-shell since 5s2 is the ground state. (This is why the
2 P – states are below the 2 S – states, 2 3P2 denotes the 5th
p-shell since Krypton has 4p6.) The upper "3" denotes the
multiplicity, that is 2s+1 with s here the spin quantum number.
The lower "1" denotes the quantum number j of the total
angular momentum, i.e. j = l+s, l+s-1, … , l-s with l the quan-
tum number of the angular momentum of the orbit. "S", "P",
"D", "F" denote the actual value of l, i.e. "S" means l = 0, "P"
means l = 1, …
3 1D2 S 2 1P1 is a transition within the singulett system so the
spin magnetic moments have no effect. But in the transition
2 3S1 S 2 3P2 triplett states are involved and the spin magnetic
moment does not vanish in all sub-states.
The selection rule for optical transitions is ∆mJ = 0, ±1 and the
radiation belonging to transitions with ∆mJ = 0 are called
Fig. 3: Magnetic flux density B in the centre of gap without the p-lines and the ones with ∆mJ = ±1 are called s-lines. With the
Cd-lamp (gap width: 9 mm) as a function of the coil magnetic field turned on in the absence of the analyser three
current. lines can be seen simultaneously in the normal Zeeman effect

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen P2511005 3
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5.1.10 Normal and anomalous Zeeman effect
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in transversal observation. In the case of the anomalous

Zeeman effect three groups of three lines appear. Inserting the
analyser in the normal Zeeman effect two s-lines can be
observed if the analyser is in the vertical position, while only
the p-line appears if the analyser is turned into its horizontal
position (transversal Zeeman effect). In the anomalous
Zeeman effect there are two groups of three s-lines in vertical
polarization and one group of three p-lines in horizontal polar-
ization. Turning the electromagnet by 90° the light coming
from the spectral lamp parallel to the direction of the field (lon-
gitudinal) can also be studied trough the holes in the pole-
pieces. It can be shown that this light is circular polarized light
(longitudinal Zeeman effect). Fig. 5 summarizes the facts.
A l/4-plate is generally used to convert linear into elliptical Fig. 5: Longitudinal and transversal Zeeman effect.
polarized light. In this experiment the l/4-plate is used in the
opposite way. With the l/4-plate inserted before the analyser,
the light of the longitudinal Zeeman effect is investigated. If
the optical axis of the l/4-plate coincides with the vertical, it In the normal Zeeman effect with the transition 3 1D2 S 2 1P1
is observed that some rings disappear if the analyser is at an with 643.847 nm the electron spins cancel each other in both
angle of +45° with the vertical while other rings disappear for the initial and final state and the energy of an atomic state in
a position of –45°. That means that the light of the longitudi- a magnetic field depends only on the magnetic moments of
nal Zeeman effect is polarized in a circular (opposed way). The the electron orbit.
p-lines are longitudinally not observable. S
The magnetic moment of the orbital angular momentum l is

S
e S l
ml

S l
 gl mB (*)
2me U

with Bohr's magneton

eU
mB

9.274 · 10 24 Am2
2me

and the gyromagnetic factor of orbital angular momentum

gl = 1.
In the vector model of the atom the energy shifts can be cal-
culated. It is assumed, that angular moments and magnetic
moments can be handled as vectors. Angular moment and the
magnetic moment connected with it are antiparallel because
of the negative electron charge. The amount of the orbital
S
magnetic moment of the orbital angular momentum l , with
quantum number l such that

0 l 0
U 2l 1l  12 , is: ml
mB 2l 1l  12
S

In case of LS-coupling (Russel-Saunders coupling, spin-orbit

coupling) for many electron systems is the amount of the total
angular momentum

0 J 0
0 L  S 0
U2J1J  12 with S
a S
S S S S
si

the sum of the spins of the single electrons and

S S
L
a li

the sum of the orbit angular moments of the single electrons.

Here it is
S
S
0.

So 0 J 0
0 L 0
U2L1L  12 .
S S
Fig. 4: The atomic states of Cadmium, wavelength in
Å = 0.1 nm

4 P2511005 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
 LEP
Normal and anomalous Zeeman effect 5.1.10
-05

The amount of the component of the corresponding magnetic

S and here the (mJf gf – mJi gi)-values are simply equal to ∆mJ. So
moment m J in direction of J is:
S
in case of LS-coupling in the normal Zeeman effect three
equidistant lines are expected in this transition with a distance
m J2J 0
0S
0 1S m L 0
mB 2L1L  1 2
gJ mB 2J1J  12 in frequency or wave number proportional to the magnetic
field strength. The polarization of the transitions with ∆mJ = 0
with gJ = 1. in transversal observation is parallel to the magnetic field (here
horizontal) and of the other transitions the polarization is per-
Observable is only the projection of the magnetic moment on pendicular to that.
S
J
S The anomalous Zeeman effect is the more general case
1S
m J 2 J
 gJ mB
J
where the electron spins do not cancel each other and the
U energy of an atomic state in a magnetic field depends on both
the magnetic moments of electron orbit and electron spin.
with it's quantization with respect to z-axis S
The magnetic moment of the orbital angular momentum l
is as above (see ( * )) and the magnetic moment of the spin S
s
m J 2 J,z
 mJ gJ mB
1S is
e S Ss
with the magnetic quantization number mJ with mJ = J, ms

S s
 gs mB
2me U
J-1, …,-J
The interaction energy with the outer magnetic field B0 along with the gyromagnetic factor of orbital angular momentum
the z-axis is then gs = 2.0023.
Additional to the orbital magnetic moment of the orbital angu-
Vm = –mJ gJ mB B0. S
lar momentum l the amount of the spin magnetic moment
J
S
of the spin s , with quantum number s such that
Here the used transition for the normal Zeeman effect is
0S
s 0
U2s 1s  12 ,
3 D2 S 2 1P1.
So in the initial state is L = 2, S = 0 and J = 2. mJ may have
the values mJ = -2, -1, 0, 1, 2. The gyromagnetic factor is gi = 1
has to be taken into account:
and the energy difference between two neighbouring sub-
ms
0  gs mB 2s 1s  12 0
states of the initial state is then ∆E = –1mBB0.
In the final state is L = 1, S = 0 and J = 1. mJ may have the
values mJ = -1, 0, 1. The gyromagnetic factor is gf = 1 and the
In case of LS-coupling (Russel-Saunders coupling, spin-orbit
energy difference between two neighbouring sub-states of the
coupling) for many electron systems the amount of the total
final state is then ∆E = –1mBB0, too, i.e. for transitions with the
angular momentum is
same ∆mJ between initial and final state the energy shift is for
0 J 0
0 L  S 0
U2J1J  12 with S
a S
initial and final state the same – so they have altogether the S S S S
si
same frequency.
Fig. 6 shows the resulting transition diagram. the sum of the spins of the single electrons and
For electrical dipole transitions the selection rule states ∆mJ =
1, 0, -1. S S
L
a li
The energy shift of a transition between initial state with mJi
and gJi and final state with mJf and gf is then
the sum of the orbit angular moments of the single electrons.
In the vector model it is assumed, that angular moments and
VmJ –VmJ = (mJf gf – mJi gi) mBB0
i f both spin and orbital magnetic moments can be handled as
vectors. So the cosine rule applies for the sum of two vectors
with an angle between them. The amount of the component of
S
the corresponding magnetic moment m J in direction of J is
S

with the approximation gs ≈ 2:

m J2J 0
0S
0 1S m L 0 cos 1 L , J 2  0 S
m S 0 cos 1 S , J 2
SS SS

= mB a 2L 1L12 cos 1 L , J 2  22S 1S1 2 cos 1 S , J 2 b

SS SS

3J1J12 S1S12  L1L12

0 1S
m J2J 0
mB
22J1J12

g Jm B 2J1J1

with
J1J12 S1S12  L1L12
g1
1  .
2J1J12
Fig. 6: Energy shift of the atomic states

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5.1.10 Normal and anomalous Zeeman effect
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Observable is only the projection of the magnetic moment on

S For electrical dipole transitions the selection rule states ∆mJ =
J 1, 0, -1.
S The energy shift of a transition between initial state with mJi
1S
m J 2 J
 gJ mB
J
and gJi and final state with mJf and gf is then
U
VmJ –VmJ = (mJf gf – mJi gi) mBB0
with it's quantization with respect to z-axis i f

The following table shows the energy shifts of the transitions:

m J 2 J,z
 mJ gJ mB
1S

with the magnetic quantization number mJ with mJ = J, No. 1 2 3 4 5 6 7 8 9

J-1, …,-J ∆mj 1 1 1 0 0 0 -1 -1 -1
The interaction energy with the outer magnetic field B0 along
mJf gf – mJi gi -2 -3/2 -1 -1/2 0 1/2 1 3/2 2
the z-axis is then

Vm = –mJ gJ mB B0. So in case of LS-coupling in the anomalous Zeeman effect

J
nine equidistant lines are expected in this transition instead of
Here for the anomalous Zeeman effect the used transition is three without spin magnetism. The polarization of the transi-
is 2 3S1 S 2 3P2. tions with ∆mJ = 0 in transversal observation is parallel to the
So in the initial state is L = 0, S = 1/2 + 1/2 = 1 and J = 1 + 0 magnetic field (here horizontal) and the polarization of the
= 1. mJ may have the values mJ = -1, 0, 1. The gyromagnetic other transitions is perpendicular to the magnetic field.
factor is At observing the s-lines of the transversal Zeeman effect it is
easy to see that the amount of splitting increases with increas-
1111 2 111 2 0101 2 ing magnetic field strength. For a quantitative measurement of
gi
1 
2
2 · 1111 2 this splitting in terms of number of wavelengths, a Fabry-Perot
interferometer is used, the functioning of which has to be
and the energy difference between neighbouring sub-states of explained:
the initial state is then The Fabry-Perot étalon has a resolution of approximately
400000. That means that a wavelength change of less then
∆E = –2mBB0. 0.002 nm can still be detected.
The étalon consists of a quartz glass plate of 3 mm thickness
In the final state is L = 1, S = 1 and J = 2. mJ may have the coated on both sides with a partially reflecting layer (90 %
values mJ = -2, -1, 0, 1, 2. The gyromagnetic factor is reflection, 10 % transmission). Let us consider the two par-
tially transmitting surfaces (1) and (2) in Fig.8 seperated by a
2121 2 111 2 1111 2 3 distance t. An incoming ray forming an angle with the plate
gf
1 

2 · 2121 2 2 normal will be split into the rays AB, CD, EF, etc. the path dif-
ference between the wave fronts of two adjacent rays (e.g. AB
and the energy difference between neighboured sub-states of and CD) is
the final state is then
d = m · (BC + CK)
3
∆E =  mBB0. where BK is defined normal to CD and m is the refractive index
2
of quartz at 509 nm, m = 1.4519. At 644 nm is m = 1.4560. With
Fig. 7 shows the resulting transition diagram.
CK = BC cos 2u

and
BC · cos u = t

Fig. 8: Reflected and transmitted rays at the parallel surfaces

Fig. 7: Energy shift of the atomic states (1) and (2) of the étalon. The étalon spacing is t = 3 mm.

6 P2511005 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
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Normal and anomalous Zeeman effect 5.1.10
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we obtain where n1 is the closest integer to n0 (smaller than n0). In gen-

eral is for the pth ring of the pattern, measured starting from
d = m · BCK = m · BC(1 + cos 2u) the center, the following is valid:
= 2m · BC cos 2u = 2m · t · cosu
np = (n0 - e) – (np – 1) (4)
and for a constructive interference it is:
Combining equation (4) with equations (2) and (3), we obtain
nl = 2m · t · cosu (1) for the radii of the rings, writing rp for rnp,

where n is an integer and l the light's wavelength. Equation (1) 2f2

is the basic interferometer equation. Let the parallel rays B, rp
· 21p12  e (5)
D, F, etc. be brought to a focus by the use of a lens of focal B n0
length f as shown in Fig. 9.
We note that the difference between the squares of the radii of
adjacent rings is a constant:

2f2
r2p1  r2p
(6)
n0

e can be determined graphically plotting rp2 versus p and

extrapolating to rp2 = 0.
Now, if there are two components of a spectral line (splitting
of one central line into two components) with wavelengths la
and lb, which are very close to one another, they will have
fractional orders at the center ea and eb:
Fig. 9: Focusing of the light emerging from a Fabry-Perot
2m · t
étalon. Light entering the étalon at an angle u is ea

n1,a
2m · t · kan1,a
focused onto a ring of radius r = fu where f is the focal la
length of the lens.
2m · t
eb

n1,b
2m · t · kbn1,b
lb

For u fulfilling equation (1) bright rings will appear in the focal where ka and kb are the corresponding wave numbers and
plane with the radius n1,a, n1,b is the interference order of the first ring. Hence, if the
rings do not overlap by a whole order so n1,a = n1,b and the dif-
rn = f tan un ≈ fun (2) ference in wave numbers between the two components is

for small values un, e.g. rays nearly parallel to the optical axis. ea  eb
∆k
ka  kb
(7)
Since 2m · t

cos un
n0 cos un
n0 a 1  2sin2 b

2m · t un
n
Using equations (5) and (6), we get
l 2

with r2p1,a
p
e (8)
2m · t r2p1  r2p
n0

l
Applying equation (8) to the components a and b, yields
we finally obtain
r2p1,a
u2n  p
ea
n
n0 a1  b r2p1,a  r2p,a
2
or
and
21n0  n2 r2p1,b
un
(3)  p
eb
B n0 r2p1,b  r2p,b

If un corresponds to a bright fringe, n is an integer. However is By substituting these fractional orders into equation (7), we
n0 the interference condition for the center (for u = 0) general- get for the difference of the wave numbers:
ly not an integer.
If n1 is the interference order of the first ring, it is n1 < n0 since r2p1,a r2p1,b
a 2 b
1
n1 = n0 cos un. We then let ∆k
 (9)
2m · t rp1,b  r2p r2p1,b  r2p,b
n1 = n 0 - e ; 0 < e < 1

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen P2511005 7
 LEP
5.1.10 Normal and anomalous Zeeman effect
-05

From equation (6) we get the difference between the squares must be equal, regardless of p (the order of interference) and
of the radii of component a: their average may be taken as may be done for the different
2f2 ∆-values. With d (the difference of squares of radii of different
∆p1,p
a
r2p1,a  r2p,a
lines of the same order of interference) and ∆ (difference of
n0,a squares of radii of different orders) as average values we get
for the difference of the wave numbers of the components a
this is equal to (within a very small part) the same difference and b:
for component b
1 d
2f2 ∆k
· (10)
∆p1,p
b
r2p1,b  r2p,b
2m · t ∆
n0,b

Hence we assume Note: Equation (10) shows that ∆k does not depend on the
dimensions used in measuring the radii of the ring system.
∆p1,p
a
∆p1,p
b

for all values of p. Similarly, all values

dpa,b
r2p1,a  r2p1,b

Fig.10: Normal Zeeman effect: Interference pattern without polarization filter for no coil current and for 5 A coil current. On the
left there is one ring per order of interference, on the right there are three rings per order of interference

Fig.11: Anomalous Zeeman effect: Interference pattern without polarisation filter and magnified cut-out of the first completely
visible two orders of interference

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