Download Zeeman Effect - Lab exercises 24

Zeeman Effect - Lab exercises 24
Pieter Zeeman
Franziska Beyer
August 2010
1
Overview and Introduction
The Zeeman effect consists of the splitting of energy levels of atoms if they are situated
in a magnetic field. The distances between these components increases linearly with the
magnetic field and can be used for the estimation of the specific charge e/m of the electron.
The splitting can be observed by electronic transitions. The detection of the splitting requires
the dispersion of the emitted light by usage of e.g. a prism. Thus the components can be
found separately. The Zeeman splitting is rather small that’s why a high resolution is
needed which is realized in our case, by a Fabry-Perot spectrometer. By analyzing the
properties of the emitted photons, namely wavelength and type of polarization, one can
learn about the states of the electron before (initial) and after (final) the transition.
In the mid-nineteenth century, the widening of the atomic spectral lines was observed
for the first time. No satisfactory explanation to this broadening was found until the end
of that century. Already at this time, a connection of this phenomenon to the presence
of magnetic field was proposed. In 1896, a Dutch physicist Pieter Zeeman, succeeded to
partially explain the experimental results. He showed that the spectral line splitting can be
classified in, what is today known as, the normal Zeeman effect and the anomalous Zeeman
effect. While the normal Zeeman effect was in agreement with the classical theory developed
by Lorentz, the anomalous Zeeman effect remained unexplained for the next thirty years.
After the development of quantum theory in the early twentieth century, it turned out that
for the understanding of the anomalous Zeeman effect the concept of quantum theory was
necessary.
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Experimental observation
Observation along the magnetic field vector corresponds to the longitudinal Zeeman effect
and perpendicular to it to the transverse Zeeman effect, respectively, as it can be seen in
figure 1.
B
σ
π
σ-
transversal
σ+
longitudinal
Figure 1: Observed polarization directions in relation to the applied magnetic field, B.
The normal Zeeman effect is characterized by a triplet or doublet splitting of the spectral line in case of transverse or longitudinal observation, respectively. The middle line in
the triplets represents the component of the spectral line which is unaffected by the magnetic field, the other two lines shift by the same amount to higher and lower wavelengths,
respectively, due to the applied field. Depending on the observation direction, the polarization of the split lines is different. In the longitudinal case, circular polarization occurs with
opposite sense of rotation for the two components. Transversally, the middle component of
the triplet is polarized parallel to the field and the other two perpendicular.
For the anomalous Zeeman effect, the splitting is more complicated, even if the shift is
still proportional and symmetrical to the applied field.
During the laboration we are going to study the normal Zeeman effect in transversal
observation direction.
1
3
Classical explanation for the normal Zeeman effect
For singlet states, the spin is zero, S = 0, and the total angular momentum J is equal to the
orbital angular momentum L, J = L. Magnetic moments, µ are inseparably connected to
the angular momentum, L= r × p = me rvn̂. The interaction between the magnetic moment
of an atom and the external magnetic field B causes splitting of the atomic energy levels.
We can look at separate electrons in the electron shell as point charges orbiting around
the nucleus. In such a case, each electron represents a loop carrying a current I. A particle
with electric charge e in a circular orbit with radius r and speed v gives a current (looking
at one loop only using Q = e and v = 2πr
t ):
Q
e
=
v
(1)
t
2πr
The magnetic moment, µ due to such a current loop is (negative sign due to the charge of
the electron):
e
e
L gl = 1
(2)
µ = Iπr2 n̂ = − rvn̂ = −gl
2
2me
I=
In an external magnetic field, the current loop experiences a torque T = µ x B and
subsequently a force with the corresponding potential energy U :
U = −µ · B =
e
L·B
2me
(3)
The angular momentum is conserved by precession, i.e. µ moves on a cone around B with
the Lamour frequency:
ωL = 2πfL =
eBz
2me
(4)
The light emitted by the atom (frequency f0 ) superimposes with this precession frequency
as follows: v0 + vL and v0 − vL . This explains the normal Zeeman effect. Thus the
polarization of the emitted radiation is circular, looking in the direction of the field, due to
the circular motion of the precession. Perpendicular, the precession occurs as linear as we
only see its projection.
B 6= 0
B=0
ml
2
1
l=2
0
−1
−2
1
l=1
0
−1
∆ml = −1, 0, +1
Figure 2: Normal Zeeman effect for the transition between d and p levels.
2
4
Quantum mechanical description the Zeeman effect
Emission of light is now regarded as consequence of an electronic transition from a level of
higher energy (initial state E2 ) to a level of lower energy (final state E1 ). The transition
frequency, f is
E2 − E1
(5)
h
with h the Planck’s constant. The En are determined by the atomic structure. However, a magnetic field (flux density B) interacts with the magnetic moments, µ of the
electrons, which thus possess additional potential energy, equation 3. Thus for an atom in
a weak magnetic field, |B| < 1 T, the total energy is: E = Enl + ESO + U . Orientation
and magnitude of µ in the different states are generally different. The transition frequency
changes to:
f=
E02 − µ2 · B − (E01 − µ1 · B)
(6)
h
The z-component of the orbital magnetic moment µLz is coupled to its angular momentum
Lz :
f=
eh̄
e
Lz = −
ml = −gl µB · ml using : |L| = ml · h̄
(7)
2me
2me
Equation 6 shows that f increases with increasing field and equation 7 indicates that quantum mechanical treatments of the angular momentum have to be taken into account.
µLz = −gl
4.1
The Quantum numbers
An electron in an atom is characterized by quantum numbers, which completely describe
it’s energy {n l ml ms }:
1. principal quantum number n defines the electronic shell, n = 1 . . . nmax ; where nmax
is the electron shell containing the outermost electron of that atom
2. orbital angular momentum quantum number l describes the sub shellp
(0 = s-orbital,
1 = p-orbital, 2 = d-orbital, 3 = f -orbital, etc.), l = 0 . . . n − 1, |l| = l(l + 1)h̄
3. magnetic quantum number (projection of angular momentum) ml describes the specific
orbital (or ”cloud”) within that sub shell ml = −l . . . ml . . . l, total of 2l + 1 values
4. spin projection quantum number ms describes the electron
spin with angular momenp
tum vector s and quantum numbers s = ±1/2, |s| = s(s + 1)h̄
For electrons, as for all fermions (particles with half-integer spin), the Pauli exclusion
principle is valid: In an atom, there cannot exist a pair of electrons with an identical
quadruple of quantum numbers. The total angular momentum of an atom can be estimated
as vector sum of the individual electronicPcontributions.
P The total spin S and the total
angular momentum L are given by: S = | ms |; L = | ml |.
There is a number of possibilities to combine ml and ms . Hund’s Rules helps to find
the ground state:
• completely filled sub shells do not contribute to J
• partially filled sub shells follow the Pauli exclusion principle and:
1. as much as possible electrons have parallel spin (S → max)
2. electrons occupy states to maximize L
• J = L ± S, ”+” for sub shells which are more than half filled and ”−” for less than
half filled sub shells
The atomic states are specified using n2S+1 LJ .
3
4.2
Magnetic moment
Equation 7 can be expressed for electrons by means of the Bohr magneton, µB
µB =
eh̄
= 9.274 · 10−24 J/T
2me
(8)
The spin generates a magnetic moment, µS which is double of that of the orbital angular
momentum, µL :
µL = −gl µB L and µS = −gs µB S gs ≈ 2
(9)
The exact free electron value for gs = 2.00232 can be obtained from quantum electrodynamics. Then the total angular momentum is: µJ = gj · µB J with the scaling factor gj , the
Landé factor, which for free atoms can take values between 1 and 2.
gj = 1 +
J(J + 1) − L(L + 1) + S(S + 1)
2J(J + 1)
(10)
Particular cases:
• S = 0 → J = L → gj = 1 → normal Zeeman effect
• L = 0 → J = S → gj = 2 (e.g. free electrons)
• S 6= 0, L 6= 0 → gj = 1 · · · 2
Equation 10 gives an approximative value of the Landé factor.
4.3
The general (anomalous) Zeeman effect
The general, also known as anomalous, Zeeman effect is present for atoms with non-zero
total spin. Since electronic spin can have only two values, namely + 21 and − 12 , all atoms
with an odd number of electrons posses a non-zero spin. The orbiting electrons in the atom
are equivalent to a classical magnetic gyroscope. The torque applied by the field causes the
atomic magnetic dipole to precess around B (Larmor precession). The external magnetic
field therefore causes J to precess slowly about B. L and S meanwhile precess more rapidly
about J due to the spin-orbit interaction, see Fig. 3. The speed of precession about B is
proportional to the field strength.
B
J
L
S
Figure 3: Addition of angular moments.
spin-orbit interaction
The orbital and the spin magnetic moments do not interact independently with the small
external magnetic field (|B| < 1T). Rather, the orbital and spin magnetic moments interact
with each other in such a way to form a combined magnetic moment µj that interact with
the external field. The orbital motion of the electron about the nucleus results in a magnetic
field at the location of the electron. The spin magnetic moment of the electron interacts
with this field so as to couple the spin and the orbital magnetic moments together. The
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magnitude of the field seen by the electron is approximately 1 T. If the external field is higher
than the internal than the spin and the orbital magnetic moments decouple and interact
independently with the external field.
The angular momenta combine vectorial to form a total angular momentum J = L + S.
J is the angular momentum
with a definite projection along the z -axis , see Fig. 3. The
p
magnitude of J is J(J + 1)h̄ where the total angular momentum quantum number is
determined by J = |L − S|, |L − S| + 1, . . . , L + S. The projection a long the z -axis of the
total angular momentum is the quantum number mj :
Jz = Lz + Sz = (ml + ms )h̄ = mj h̄
(11)
The angular momentum and also the magnetic moments are quantized along the magnetic field, which is here chosen in z-axis. So for a multi-electron atom the expression for
the energy shift or the additional potential energy in the external field is given by:
U = −µJz · Bz = gj µB mJ Bz
(12)
Without a field, the states with different mJ have equal energy (degenerate states). Increasing B, the states split and more transitions are possible. There exist following selection
rules ∆mJ = ±1 or 0, but not 0 → 0 if ∆J = 0 see allowed transitions in figure 4. The
distances between neighboring energy levels amounts to µB gj Bz .
B=0
B 6= 0
mj
Fine structure
3p
3/2
Anomalous Zeeman-effect
3/2
J
3/2
1/2
−1/2
3p
−3/2
l = 1, s = 1/2
3p
1/2
1/2
1/2
−1/2
J =L+S
l = 0, s = 1/2
2s
2s
1/2
1/2
1/2
−1/2
∆S = 0
∆L = −1, 0, +1
∆J = −1, 0, +1( not J = 0 → J = 0)
∆mj = −1, 0, +1( not mj = 0 → mj = 0 if ∆J = 0)
Figure 4: General or anomalous Zeeman splitting for the transition between 3p and 2s
levels (e.g. Na-doublet).
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Figure 5: Ne spectrum
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5
Questions and Exercises
The questions and exercises should be prepared before the laboration takes
place. During the experiments we will discuss the answers.
1. Explain the Zeeman effect and its experimental detection. What means ”transverse”
and ”longitudinal” Zeeman effect?
2. Why does the magnetic field force the magnetic dipoles of atoms to precess, instead
of aligning with the field?
3. In which cases does the classical description fail? Which observations cannot be explained?
4. Why do completely filled shells not contribute to the total angular momentum J of an
atom?
5. Why is a normal Zeeman effect expected for the transition 31 D2 ↔ 21 P1 .
6. Without magnetic field we have only one transition, which gives rise to one spectral
line. With magnetic field present, we get multiple transitions resulting from the splitting of spectral lines. According to Fig. 2, can you imagine the transitions between p
and s levels and between f and d levels?
7. Compare Fig. 2 and Fig. 4. Try to explain how the presence of electronic spin changes
atomic energy levels.
8. What implies setting Landé factor to unity?
9. Why is the general Zeeman effect also known as ”anomalous” Zeeman effect?
10. According to Fig. 3 and Fig. 4, explain what is spin-orbit coupling, fine structure and
how this is formed.
11. During the lab exercises you will observe certain transitions in neon, with wavelengths
5852 Å, 6074 Å and 6164 Å, Fig. 5.
Calculate g for the upper (övre nivå) and the lower (undre nivå) levels. Draw a
transition schema for the possible transitions.
Hint: The value of the quantum numbers are given in Fig. 5 and by 2S+1 LJ . J are
the fine structure levels and J = |L − S|, |L − S| + 1, . . . , L + S
12. The difference in wavelengths ∆λ between Zeeman components as a function of magnetic field is given by the relation (derive this relation):
∆λ =
gj µB Bz ∆mJ λ2
hc
(13)
Our magnetic field is approximately 0.1 T. Take the g values determined in the previous
exercise. Calculate the approximate value of ∆λ at that field for some of the mentioned
transitions in neon, you will later observe during this lab. You should get a few
hundredths of Å for ∆λ .
That also means, that the difference between spectral lines under the influence of the
magnetic field will be a few hundredths of Å. During the lab you will measure this difference
in wavelengths, and see that the Zeeman effect exists in reality. After all, it is nice to be
able to compare measurements and theoretical results. The most important contribution
from the theory is just g-factor, which, as already shown, has a relatively simple expression
(10).
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6
Measurements
1. We start with observing the light from a Na lamp. This light consists only of a small
part in the visible region (yellow) which is composed of two close lying wavelengths,
the famous Na-doublet. You will observe the doublet without the interferometer, to
get an impression of the prism’s capability to separate the neighboring wavelengths
(resolution).
λ(D1 ) =
Å
(14)
λ(D2 ) =
Å
(15)
Out of your result, you can guess the spectrometer’s resolution, which is for 6000 Å
around
Å. (Keep this in mind - this belongs to the general physical properties.)
2. As can be seen from the Fig. 5, the neon spectrum contains a large quantity of lines. In
order to be sure that we are looking at the right lines, we have to check the wavelength
scale on the spectrometer. There is no need to do any precise calibration, because we
will not determine wavelengths in neon. But to be on the safe side, you should check
the gradation. This can be done using a Cd-lamp spectrum – for which you will find
wavelength values for each lines on a paper in the lab.
3. The next step is to increase the wavelength resolution using a Fabry-Perot interferometer. Put it in the beam path and study its function. The Zeeman effect is difficult
to observe since the splitting causes only a very small differences in wavelengths (few
hundredths of Å). From exercise 1 it is clear that the resolution of the spectrometer
prism is far too coarse to distinguish between the Zeeman lines. If the incoming light
is composed of multiple discrete wavelengths, they will be refracted differently by the
prism. Each wavelength will form an image of the light entrance opening in the focal plane of the ocular. This kind of spectrometer can give a resolution of a few Å.
However, by putting the interferometer in front of the collimator lens it is possible
to increase significantly the resolution of the instrument. Let’s see how Fabry-Perot
interferometer is constructed and how it works.
Fabry-Perot interferometer
A Fabry-Perot interferometer works in the same way as an interference filter. It is
composed of two glass plates covered each with semi-reflective coating. The incoming
light beam is partially transmitted through the semi-reflective mirror and partially
reflected as shown in Fig. 6a. The separated light beams coming out of the interferometer differ not only in intensity but also in phase. This phase shift will cause them to
interfere with each other. If the out-coming light from the interferometer is focused by
a lens the resulting interference pattern looks like shown in Fig. 6b. The interference
pattern looks as in Fig. 7, when observed through a slit.
By measuring the distances ∆R and dR between the interference lines and Zeeman
splitting lines, one can calculate the difference in wavelength between atomic transition
lines and Zeeman lines as given by the equation:
∆λ =
λ2
2dR
dR λ2
·
·
=
′
∆R 2d
∆R + ∆R 2d
(16)
∆R and dR are measured by cross-hairs that you can observe in the ocular, moved by
a micrometer screw. d is given as d = 1 cm.
8
d
Semi−reflective mirrors
a)
b)
Figure 6: a) Light path through interferometer; b) Interference pattern.
4. Zeeman splitting observation
The transition, which we are observing, will split into the three lines under the influence
of the magnetic field: one π-component and two σ-components. This results in three
narrow-lying circle segments after passing the interferometer. Because of the very small
splitting, the three lines are placed very close to each other. This makes it difficult
to really detect splitting and to distinguish the lines. Since the σ and π components
are polarized perpendicularly to each other, it is possible to block one component by a
polarizer and transmit the other component. By changing the polarization direction,
one can switch between the two polarization directions and thus between the three
lines.
σ
π
σ
∆R
∆R’
dR
b)
a)
Figure 7: Interference pattern of the spectral lines as seen in the eyepiece. a) Without
magnetic field, or with σ components blocked by the polarizer; b) Zeeman splitting in the
presence of the magnetic field. Interference pattern.
9
Measurement procedure:
• Turn on the neon lamp and focus carefully on the entrance slit; adequate opening
size 0.075 mm.
• Look first at the line at 5852 Å.
• Turn on the power supply to the electromagnet - check before that the outgoing
voltage is zero.
• Increase the voltage, thus the magnetic field and check if you can observe the
widening of the diffraction line – no polarizer on the beam path.
• Place the polarizer on the beam path. Turn it and convince yourself that you can
filter out either π-component or σ-components.
• Change the magnetic field, check how dR changes and observe how two neighboring diffraction orders can overlap with each other, i.e., dR → ∆R.
• Measure ∆R′ , ∆R and 2dR for around eight diffraction orders. The easiest is to
successively measure values for neighboring lying orders. Take care not to change
voltage to the magnet during the measurements, because the distance of the line
splitting is dependent on the magnetic field!
• Measure the magnetic field with a gauss-meter.
5. Finally, you can calculate the experimental values for the actual levels and compare
dR
them to the theoretical values obtained from Landés equation. As a value of ∆R
we
2dR
take ∆R + ∆R′ in each order. The mean value over all measured orders is then put in
the actual formula.
Estimate relative error of the input values. Which factor gives the biggest contribution
to the total error in g in equation (13)?
Try to summarize what you get out of this laboratory exercise. Which new things
have you learnt and which previous knowledge have you improved?
10
Zeeman splitting
in neon
Colour
Intensity
Upper level
Figure 5: Ne spectrum
Lower level