The Zeeman Effect.pdf

7/27/2019 The Zeeman Effect.pdf

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TFionn Fitzmaurice

[email protected]

8.12.2009

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School of Physics, Trinity College Dublin

The Zeeman EffectFionn Fitzmaurice

[email protected]

08.12.2009

Abstract

The Zeeman effect, a phenomenon in quantum mechanics, was studied in the courseof this experiment. It is the lifting of degeneracy of spectral lines through the

application of a magnetic field.

In this discourse, the wavelength of the cadmium red line was determined to

be0= 649.7 nm and the Bohr magneton determined to beB= 9.274 10-24JT-1.

The split spectral lines were discovered to be linearly polarised in the transverse

orientation of the magnetic field and circularly polarised in the longitudinal

orientation.


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Introduction

Pieter Zeeman (25 May 1865 9 October 1943) shared the Nobel Prise in 1902 forhis discovery of the effect whos name lends itself to this lab report. The Zeeman

Effect is the splitting of otherwise degenerate spectral lines in the presence of a

magnetic field. This occurs via interaction between the external magnetic field and theatomic magnetic moment. The atomic magnetic moment is related to the total angular

momentum, which is given by the vector

operator equation

Ji=Li+ Si

where the total orbital angular momentum

L = r p

where L , r and p are position and linear

momentum operators,

rr , i

p .

The operator L obeys the canonical commutation relation for angular momentum

[Li,Lj] = iijkLk

where ijkis the antisymmetric Levi-Civita tensor.

Siis the total spin angular momentum, for which the angular momentum

canonical commutation relation still holds. This criterion also holds for the total

angular momentum. Both the orbital and spin operators have eigenvalue equations

L2

| = 2l (l+ 1) |

S2 | =

2s (s+ 1) |

for some eigenfunction |. The value of lmay only be integer, whilesmay take on

integer and half-integer values.

Part 1. To determine the wavelength of the cadmium red line

From this point on, errors will be denoted by the lower case rather than to avoid aclash of notation.

The objective was to measure the wavelength of the cadmium red line in zero

magnetic field, this wavelength denoted by0.

22

2

212

fnrndm m

A visualisation of the Levi-Civita tensor, where red

represents 1, blue; 1 andempty cubes represent 0.


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Therefore, 0can be calculated by measuring the ring radius, rm, as a function of the

ring order, m, since n, d,fare known quantities.The magnet and lamp assembly was rotated such that its orientation was

appropriate, with the two electromagnets facing the detector, rather than in line with it.

The lenses L1and L2were adjusted in order to properly focus the image. This was theeffect of moving the two lenses. Once the ring pattern was sharp, uniformly bright and

centered, the eyepiece was replaced by 2048 pixel CCD. 2048 pixels is the maximum

resolution for the CCD and was therefore the resolution used during the experiment,

as it provided the highest level of accuracy.

rmwas determined for the first 10 rings using computer software in tandemwith the CCD. This was possible as the software gives a value for the pixel number,

which I will denoteN. This can be related to the separation, x, of two points fromthe pixel number difference, N, via the simple relation

x= N14 m.

Results and Analysis

The values of rm2and mwere plotted against each other, see graph 1.1 rm

2vs. m

appended. The slope of this graph,

sL= 5.3247 10-6

and

02 Lsnf

d

and therefore, using the values for d, n,f, we arrive at

0

3

)0225.0(457.1

)104(

Ls

0= 649.7 nm

The accepted value is 0= 643.85 nm. This is well within experimental error, as thesevalues differ by a factor of 1.01.

Part 2. The transverse normal Zeeman effect

The objective of experiment 2 was to measure the wavelength shift, of the outer

triplet lines and show the relation B, to determine the value of the BohrmagnetonBand to determine the polarisation state of each of the triplet lines.

22 fn

rr


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This shows that can be calculated by measuring rand r, since , n,fare knownquantities.

hc

BB2

0

From this we can therefore determine experimentally a

value forB.The magnet and lamp assembly was left in its

position as per experiment 1. The ring pattern was

viewed through the eyepiece as the field was increased to

7 A, which was not exceeded. At high currents the

electromagnetic coils heat and this changes their

resistance. In order to compensate for this and keep the

current (and by extension field) constant, the voltage

must increase in tandem. This can be set to happen automatically by setting the powersupply to constant current mode. This will only operate effectively at currents

below 7 A, hence the operational upper limit.The eyepiece was then replaced with the VideoCom CCD and the fringes

sharpened. Intensity distributions forI= 4, 5, 6, 7 A were recorded, as were the

corresponding values of rand r, up to about 6 rings.

Finally the polarisation state of each spectral line was determined using the

linear polariser. This was done by looking through the eyepiece while rotating the

polariser by a known angle and noting the change in intensity in each line.


The magnetic field varies with current, but this variation is non-linear, because ofremnant fields. ForI= 0,B0. The following table contains values forIandB.

I(A) B(mT) forI B(mT) forI

4 394 407

5 485 493

6 552 555

7 594 594

The values for r, rand rrare tabulated below for different values ofI.

I= 4 Ar(m) r(m) rr(m2)

0.003332 0.00039 1.29948 10-6

0.005782 0.000225 1.30095 10-6

0.007462 0.00018 1.34316 10-6

0.008806 0.00015 1.3209 10-6

0.009954 0.0001275 1.26914 10-6

I= 5 A

r(m) r(m) rr (m2)

0.003374 0.0004875 1.64483 10-6

p otograp .Zeeman took of hiseponymous effect.


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0.00581 0.000285 1.65585 10-6

0.007462 0.0002175 1.62299 10-6

0.008806 0.0001875 1.65113 10-6

0.009982 0.000165 1.64703 10-6

I= 6 Ar(m) r(m) rr(m

2)0.003374 0.00054 1.82196 10

-6

0.005782 0.0003075 1.77797 10-6

0.007462 0.0002325 1.73492 10-6

0.008806 0.0002025 1.78322 10-6

0.009982 0.000165 1.64703 10-6

I= 7 A

r(m) r(m) rr(m2)

0.003374 0.0006 2.0244 10-6

0.00581 0.000345 2.00445 10-60.007462 0.000255 1.90281 10

-6

0.008806 0.0002175 1.91531 10-6

0.009982 0.00021 2.09622 10-6

r= 10-5

m, r= 10-5

mrr= 2r 10-5m2

The mean value of rrfor each current is tabulated below:

I(A) Average (rr) (m2)

4 1.30673 10-6

5 1.64436 10-6

6 1.75302 10-6

7 1.98864 10-6

I= 0.01 Arr= 2r 10-5m2

The graph of these averages versus the magnetic field was plotted, see graph 2.1 rr

vs.B appended, showing a direct proportionality between andB. The slope of this

graph is

sL= 3 10

-6

and

BL

fn

hcs

0

22

B= 1.92 10-23JT-1 1 10-6JT-1

The recognised value of the Bohr magneton isB= 9.274 10-24

JT-1

. Theexperimental value deviates from the expected value by a factor of 2.07.

The polarisation state for each of the triplet lines was examined. It was foundthat the interior and exterior rings had the same polarisation as each other, observed


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when the polariser had rotated 0 radians. The ring between these two rings was foundto be polarised by radians with respect to the other rings, i.e. the polariser was

rotated by in order to achieve a maximum in intensity for this ring. The quarter-wave plate was not required to make these judgements.

Part 3. Longitudinal normal Zeeman effect

The objectives of experiment 3 were to determine the change, , in , to compare

this with the change in wavelength for the transverse Zeeman effect explored in

experiment 2 and to once again determine the polarisation state of the spectral lines,

this time a doublet.

The magnet and lamp assembly was rotated by such that the light wasviewed in the direction of the field. The current was set to 5 A and 7 A and in each

case the wavelength shift in the doublet lines was measured and compared with thetransverse orientation.

The polarisation state of each of the doublet lines was also determined, using alinear polariser and a quarter-wave plate.


AtI= 5 A andI= 7 A, the following values were measured in a similar manner as

before.

I(A) Average (rr) (m2)

5 2.25 10-6

7 1.63 10-6

I= 0.01 A

rr= 2r 10-5m2

As before, from

22 fn

rr

we can determine the value of , which can be compared to the value of in thetransverse case. In each case, the values of rrshould remain roughly constant, to

within experimental error. Unfortunately, this is not the case, particularly with theresults in part 3. I am unable to explain this large deviation; I believe it must boil

down to human error.

Taking the average of the average of the values of rrin part 2, the value of

which, denoted 2= 1.67 10-6m2, and discounting the value of rrobtained for

I= 5 A in part 3 due to the obvious inaccuracy of the value, such that for part 3,

3= 1.63 10-6m2, the ratio of which being the dimensionless quantity

.98.02

3

Therefore the values of in each case are in 98 % agreement.


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Upon introducing the linear polariser, the intensity of both the lines reduced.However, upon rotation, this intensity remained unchanged and uniform between the

lines, indicating that the lines were not linearly polarised.The quarter-wave plate was then placed between the source and the linear

polariser. In this setup, with both polarisers in their initial positions (i.e. zero rotation),

the exterior ring exhibited a maximum intensity, whilst the interior exhibited aminimum.

Upon rotating either polariser (the quarter-wave plate or the linear polariser)

by , this relation was reversed such that the inner ring exhibited maximum

intensity and the exterior, a maximum. This implies that this doublet is circularly

polarised and the polarisation states for each ring are orthogonal to each other.

Conclusion

The longitudinal and transverse Zeeman effect was observed in this experiment. It

was found that the splitting of energy levels was directly proportional to the intensityof the magnetic field. The wavelength of the cadmium red line was experimentally

measured, as was the value of the Bohr magneton. Polarisation states for the split

spectral lines were also determined experimentally for different orientations of the

magnetic field, and this orientation was seen to determine the type of polarisation

occurring.

References

1) Zeeman, P. (11 February 1897). The Effect of Magnetisation on the Nature of Light

Emitted by a Substance. Nature 55: 347

2) Spencer, J. Brookes (1970). On the Varieties of Nineteenth-Century Magneto-Optical

Discovery. Isis 61 (1): 3451.

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