The Zeeman Effect - McGill Undergraduate Physics Lab and Myles - 2 Introduction The Zeeman Effect is...

Horrocks and Myles - 1

The Zeeman Effect

by

Brett Horrocks and

Caitlin Myles

0=Br

kGB 100=r

PHYS 439 Professors F. Buchinger and R. Bennewitz

December 2, 2005 The Zeeman Effect was studied using a sample of Neon as the strength of an external magnetic field was varied. The spectrum of the sample was split into vertical dots using a Lummer-Gehrecke plate interferometer. The 626.6nm spectral line was identified both by its characteristic Normal Zeeman Effect pattern as well as by the polarization of the satellite (σ) and main (π) dots. The magnetic field strength was varied between (2.86 ± 0.09)kG and (7.45 ± 0.09)kG, and the resulting energy split (∆E) of the satellite and main dots was calculated and plotted against magnetic field strength (B); this plot yielded a linear fit of ∆E=(9.4 ± 0.7)×10-24 B. According to theory, the slope of this plot is the Bohr magneton, µB = 9.27…×10-24 J/T, thus the experimental value agrees with the theoretical value within error. The Anomalous Zeeman Effect was then investigated by examining the spectral lines 597.6nm, 638.3nm and 650.7nm with the highest magnetic field strength applied to them. Because of the relatively poor resolution of the apparatus, only the π dots were observed. The experimental patterns were found to agree relatively well with the accepted values.


Introduction The Zeeman Effect is named after its Dutch discoverer, Pieter Zeeman (depicted on the front cover of this report). In 1896 he began to study the effect of an external magnetic field on light. He noticed that spectral lines split under the influence the field, and although he could not then fully explain the phenomenon (due to the fact that quantum physics had not yet been devised), he was able to prove the theory of electromagnetic radiation put forth by Lorentz. Zeeman was awarded the Nobel Prize in Physics in 1902 for his discovery. Furthermore, a crater on the Moon has been named in his honour1. In this lab report, we begin by exploring the quantum mechanical phenomena behind the Zeeman Effect which were unknown to its discoverer. We then examine in detail the functioning of two crucial apparatuses, the Lummer-Gehrcke plate interferometer and the CCD camera. In the experiment, the first step is to identify the spectral lines of our sample of Neon. This is done by comparing their intensities and relative positions to accepted values (Figure 10). It is confirmed once the external magnetic field is switched on and the splitting patterns resulting from the Zeeman Effect are scrutinized. The effect of the magnetic field strength on the splitting of the 626.6nm line is the first thing that is studied, since this line displays the Normal Zeeman Effect. The resulting linear fit allows us to determine an experimental value for the Bohr magneton. We then examine the anomalous splitting patterns at the highest magnetic field strength used in this experiment for the 597.6nm, 638.3nm and 650.7nm spectral lines. These patterns are compared to the accepted values. Both the experimental result for the Bohr magneton and the anomalous splitting patterns agree relatively well with the accepted values.

1 “Pieter Zeeman.” Wikipedia, the free encyclopedia. 08 Nov 2005 (02 Dec 2005). <http://en.wikipedia.org/wiki/Pieter_Zeeman>.


1. Atomic Spectra We know from quantum physics that a particle’s energy can only assume very specific values. An atom’s electrons are therefore limited to certain energy levels; furthermore, no two electrons can occupy exactly the same state simultaneously, according to the Pauli Exclusion Principle. An electron’s energy level may change if an external source disturbs it in some way, for example interaction with light or a change in temperature. If an electron jumps to a higher energy state, it absorbs energy, and if it jumps to a lower state, it emits energy, usually through electromagnetic radiation. In a given sample, electrons are constantly making transitions to different states, therefore energy is constantly being absorbed and emitted. In many atoms the transition of electrons to lower states emits energy in the form of visible light, creating a spectrum. The energy of the emitted photos is equal to the energy difference between the electrons’ initial and final states. The Planck formula related a photon’s energy (Eγ) with its frequency (ν) by Planck’s constant, (h hπ2= ) [1]:

νγ hE = (1)

Furthermore, we know [1]:

νλ c= (2)

Thus a relationship can be established between the energy difference between the initial and final states of an electron (E) and the frequency of the light emitted by this transition [1]:

λhcE = (3)

Thus by analyzing a sample’s spectrum it is possible to determine its electrons’ energy states; this in turn allows one to determine the nature of the sample, since each atom has a unique spectrum [1]. 2. The Zeeman Effect 2.1. Quantum Numbers The Zeeman Effect is related to the spin of electrons. As was mentioned above, no two electrons can occupy the same state, and an electron’s state is described by its quantum numbers. Thus two electrons can have the same energy as long as their quantum numbers are different. The first three quantum numbers we are interested in are n, l, and ml, or the principal, orbital and magnetic quantum numbers, respectively. These are also known as the spatial quantum numbers. The principal quantum number, n, describes the energy level of the electron. It is for this reason that the shells of electrons are denoted by n in the periodic table, where a higher n indicates a


higher energy level. The orbital quantum number, l, determines an electron’s orbital angular momentum L by the following relationship [3]:

22 )1( h+= llL (4)

The orbital quantum number depends on n in the following manner [3]:

1,,2,1,0 −= nl K (5) Finally, the magnetic quantum number, ml, describes the quantization of the z-component of the electron’s orbital angular momentum, Lz [3]:

hlz mL = (6)

As its subscript suggests, there is a relationship between ml and l [3]:

lllml ++−−= ,,1, K (7)

In order to fully understand what is known as the Anomalous Zeeman Effect, we must also include s, the angular momentum quantum number, or spin. This is not a spatial quantum number because it does not refer to any physical movement or position of the particle in question. Instead it describes what is called the intrinsic angular momentum, S

r, which is unrelated to its orbital

angular momentum, described by l. S is related to s by the following equation:

( ) 22 1 h+= ssS (8)

This leads to yet another quantum number, ms, which describes the projection of the total angular momentum on the z-axis (this is analogous to the relationship between l and ml) [3]:

hsz mS = (9)

And

sms ±= (10)

Thus for the same n, l, and ml values, there are two possible overall states, one for each value of s. 2.2. Orbital and Intrinsic Angular Momentum It is possible to represent an electron’s orbital angular momentum L

r by a vector precessing

about a fixed direction, which we shall conveniently call the z-direction. The angular momentum is related to the orbital quantum number l by equation 4. Quantum mechanics demands that the


position of this vector be quantized. We know from equation 7 that for every value of l, there are 2l+1 possible values for ml. The magnetic quantum number in fact describes the projection of L

r

onto the z-axis. Thus for every value of l (which describes the angular momentum), there is a specific number of directions in which L

r can point. This is illustrated in Figure 1, for l=2 (and

therefore ml = -2, -1, 0, 1, 2).

Figure 1: Quantization of Orbital Angular Momentum [3] A similar representation can be made of an electron’s intrinsic angular momentum, S

r. First, it is

important to note that experiments such as the Stern-Gerlach experiment have determined that an electron’s spin has only one value, ½, thus for an electron [3]:

h23

=S (11)

Thus for an electron, ms has only two possible values, ± ½. We refer to the positive value of ms as “spin up,” and to its negative value as “spin down.” Figure 2 shows the vector representation of S

r.

Figure 2: Quantization of Intrinsic Angular Momentum [3]


2.3. Total angular momentum Finally we can combine an electron’s orbital and intrinsic angular momentums to define the total angular momentum, J

r, which can be described as the vector addition of L

r and S

r. We introduce

a final pair of quantum numbers, j and mj, the total angular momentum quantum number and the number describing the projection of J

r on the z-axis. The relationships between these are

expressed in the following equations [3]:

h)1( += jjJ (12)

21

±=±= lslj (for an electron) (13)

hjz mJ = (14)

jjjjm j ,1,,1, −+−−= K (15)

It is important to note that the reason we’re so interested in the projection of angular momentum along the z-direction is because this is the component of angular momentum that is actually measurable; because of the precession of the vector about the z-axis, the x- and y- components cancel out over time. 2.4. Magnetic moments Classically, an electron in orbit around the nucleus of an atom can be seen as a tiny loop of current I, generating an orbital magnetic moment µl [3]:

IAl =µ (16)

A is simply the area of the electron’s orbit. The current can be written in terms of the charge –e of the electron and the period of its orbit, T; this in turn can be expressed as a function of the electron’s speed, v, and the distance traveled in one period, 2πr [3]:

rev

TeI

π2=

−= (17)

If we multiply the numerator and denominator by the mass of the electron, me, and then substitute into equation 16, we obtain [3]:

Lme

el 2

−=µ (18)


If we now apply equation 4, we obtain the following quantization for the orbital magnetic moment [3]:

( )h12

+−

= llme

elµ (19)

We now introduce the Bohr magneton, µB [3]:

( ) Ble

B llTJme µµµ 1/10...274.9

224 +=⇒×== −h (20)

The z-component of µl is quantized according to ml [3]:

Blzl m µµ −=, (21)

The intrinsic angular momentum also produces a magnetic moment, µs, whose z-projection is given by the following equation [4]:

Bszs gm µµ =, (22)

The total magnetic moment is simply given by the vector addition of µl and µs; its z-projection is given by [4]:

Bjjzj gm µµ −=, (23)

gj is called the Landé factor, and is given by the following equation [4]:

( ) ( ) ( )( )12

1111+

+−++++=

jjllssjjg j (24)

2.5. The Normal Zeeman Effect If an external uniform magnetic field B

r is applied to an electron, a torque will be exerted as its

orbital magnetic moment tries to rearrange itself to the lowest possible potential energy, opposite and parallel to the magnetic field lines. Thus the magnetic potential energy is [2]:

BV lB

rr⋅= µ (25)

If we let the positive z-direction be aligned with B

r, we can use equation 18 to rewrite the

magnetic potential energy [2]:


BLmeV ze

B 2−

= (26)

Since Lz is quantized, the magnetic potential energy is as well. We now change the notation of VB to ∆E, the energy difference which occurs as a result of the applied field. Furthermore, applying equation 6, we obtain the following relationship for the energy gap which comes about when a magnetic field is applied to the orbital magnetic moment [2]:

Bl BmE µ=∆ (27)

This energy gap is what causes the Zeeman Effect. Each value of l has 2l+1 values of ml associated with it. Thus, when an external magnetic field is turned on, the electrons with the same value for l split into their different values for ml, raising or lowering their total energy as described in equation 27. Since there are more states for the electrons to be in, there are more possible transitions that they can make, and therefore there are more spectral lines. This is described schematically in Figure 2.

Figure 2: Quantization of Intrinsic Angular Momentum [modified from 5]

It is important to point out that there are certain selection rules which apply to electron transitions, that is, certain transitions are “forbidden” (that is they are extremely unlikely). Transitions are governed by the following transition rules [3]:


1,01

±=∆±=∆

lml

(28)

This is why in the case of the transition from l=2 to l=1 (as in Figure 2), there are only three spectral lines once the magnetic field is applied. The central line, with ∆ml = 0, is called the main line or the π line, and the lines for which ∆ml = ±1 are the satellites or σ lines. For reasons which will not be elaborated on in this lab report, the π spectral line is polarized parallel to the magnetic field lines, and the σ lines are polarized perpendicularly to the magnetic field lines [4]. 2.6. The Anomalous Zeeman Effect We now take into account the effect of the magnetic field on the intrinsic magnetic moment, µs. The procedure is much the same as in 2.5, but this time we must take into account the Landé factor, gj, mentioned previously. The total change in energy (when both the orbital and intrinsic magnetic moments are taken into account) is [3]:

Bjj BmgE µ=∆ (29)

This causes what is known as the “Anomalous” Zeeman Effect. Much like in the Normal Zeeman Effect, the electrons with the same value for j split off into their respective values of mj, with their energy gain or loss described by equation 29. When spin is taken into account, the transition rules are the following [3]:

1,01,0

±=∆⇒±=∆

jmj

(30)

The fact that we’re taking spin into account changes how the spectrum appears once an external magnetic field is applied. First, there are more possible values of mj for a given j than there are values of ml for a given l (see equations 13 and 15). Thus, it is possible to see more than two satellites. Second, the Landé factor, gj, means that the spacing of the satellites and the main lines is not necessarily even, but rather depends on the values of j, l, and s. The Anomalous Zeeman Effect therefore describes strange spectral patterns which appear when an external magnetic field is applied. Indeed, the Anomalous Zeeman Effect is in reality the only Zeeman Effect; the Normal Zeeman Effect is but a special case of this effect. 2.7. The Convention for Describing Anomalous Zeeman Patterns Many of the spectral patterns which result from the Anomalous Zeeman Effect have been determined experimentally for given atoms and wavelengths, thus it is not necessary to calculate ∆E for every transition. Because the spacing of the split lines depends on the strength of the


external magnetic field, the spectral patterns are not described by fixed positions. Rather, we are provided with ratios of their position with respect to the position of the π and σ lines which would occur under the Normal Zeeman Effect with the same external magnetic field. Specifically, zero is set to be where the π line would appear, and ±30 is set to be where the σ lines would be. It is best to describe the conventional notation with an example. Let us consider the 588.2nm line for Neon. Its anomalous pattern is described by the following [6]:

30504540)5()0(

61098)1()0(

= (31)

Note that in this example the ratio has been simplified. For clarity, I will explain the example using 30 in the denominator. The numbers on the numerator in parentheses refer to the π lines, which in this case, can be found at 0/30 and ±5/30 units. The σ lines can be found at ±40/30, ±45/30 and ±50/30. Figure 3 shows what this would look like schematically. In the figure, the circles (at 0 and ±30) represent the Normal Zeeman Effect π and σ lines. The Anomalous Zeeman Effect π lines are represented by the vertical bars above the horizontal scale, and the σ lines are those below it.

Figure 3: Schematic of the Anomalous Zeeman Pattern for Ne 588.2nm

3. Setup, Procedure and Apparatus 3.1 Setup As seen in Figure 4, the setup consists of a neon emission tube between the poles of an electromagnet, which is controlled by a power supply. The light from the emission tube, coming perpendicular to the magnetic field from the electromagnet, passes through a slit and collimator, and then through a Lummer-Gehrcke plate (more on this in section 3.2). Optionally, one can place a polarizing filter between the collimator and L-G plate, which can be used to verify the polarization of the emitted light. After passing though the L-G plate, the light is passed through a constant deviation prism. The combination of the L-G plate and prism has the effect of splitting the normal prism-generated set of spectral lines into lines of dots, which allows the Zeeman splitting to be observed more easily. Finally, the light is collected either in a telescope to be viewed by the observer, or by a CCD camera (which shall be examined in section 3.3). The image from the camera is then seen immediately on the computer, where the software allows us to take still images with varying exposure times.


Figure 4: Diagram of the Experiment Setup [6]

3.2 Lummer-Gehrcke Plate [Reference for this section: 6] A Lummer-Gehrcke (L-G) plate is a glass plate whose surfaces are fully plane and parallel to each other [6]. Monochromatic plane parallel light is incident on the upper surface at a very slight angle, and after being reflected into the plate, undergoes multiple internal reflections. At each reflection point, a small amount of light emerges from the surface, which is in phase with the light emerging from all the other reflection points, and which can be focused to a single point along with the other emergent rays via a lens, as seen below in Figure 5.


Figure 5: The Lummer-Gehrcke Plate [6]

From the diagram above, it can be seen that the phase difference (∆ζ) between the emergent rays can be found as follows, with t being the thickness of the plate, and µ being the index of refraction:

)sin(2

cos2)sin(tan2)(sec2

)(

22 φµ

θµφθθµ

µξ

−=

=−=−+=∆

t

ttt

ADBCAB

(32)

In order for the rays to be in phase, the above ∆ζ from (32) must be equal to nλ where n is an integer representing the order of the spectrum. Thus, the result is the following:

)sin(4 22222 φµλ −= tn (33)

Due to the above, after passing the emergent light through the constant deviation prism, each spectral line, being monochromatic is split into a series of vertical dots, each corresponding to a different order of the spectrum. To obtain the angular separation between two orders (meaning ∆n = 1), we simply differentiate (33) respect to φ, which results in:

)2sin()sin(

)2sin(2

)2sin(2

22

2

2

22

φφµλ

φλφ

φφλ

t

tn

dtdnn

−−=

−=∆⇒

−=

(34) Note that ∆φ replaces dφ. In order to determine the difference in wavelength between two lines in a single order, one must instead differentiate (33) with respect to λ, resulting in:


⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛∂∂

−⎟⎠⎞

⎜⎝⎛=

λφφ

λµµλ )2sin(22 22

ddtn

)2sin(2

4

2

22

φ

λλµ

λφ

t

nddt −⎟

⎠⎞

⎜⎝⎛

=∂∂

⇒ (35)

However, substituting (34) into (35), we obtain:

λλλµµλ ∆⎥

⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛∂∂

=− 222 4 ntn (36)

In this case, ∆λ is the difference between lines in a single order, which we will subsequently refer to as ∆λm. With that minor alteration, (36) becomes:

⎟⎠⎞

⎜⎝⎛∂∂

−=∆

λµµλ

λλ22

2

4tn

nm

(37)

Now, if we substitute (34) into (38), we obtain:

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛∂∂

−−

−=

∆

λµλµφµ

φµλλ

22

22

2

sin2

)sin(

t

m (38)

This equation will be used to calculate the Bohr magneton later on. The resolving power, R, of the L-G plate can be found be taking the angular separation of two lines just clearly resolved (equation 39 below) and substituting into (35).

φλφ

cosL=∆ (39)

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛∂∂

−−=∆

=λµλµφµ

φλλλ 22 sin

sinLR (40)

The L-G plate is used to resolve spectral lines which are very close together, or to determine the wavelength of a satellite line from the wavelength of a known line [6].


Figure 6: Depiction of s and m

We call the separation between two orders of a known π line the distance m, and the separation between the π line and its σ lines s. This is shown in Figure 6. The ratio of s:m is used in equation 41 to define the relationship between ∆λs and ∆λm, where ∆λs is the absolute difference in wavelength between the π line and its satellites.

22 λλ

λλ ms

ms ∆

=∆

(41)

However, substituting (38) into (41) above gives the following:

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛∂∂

−−

−=

∆

λµλµφµ

φµλλ

22

22

2

sin2

)sin(

tmss

(42)

For the above equations, it was necessary to determine the index of refraction of the L-G plate. From the manufacturer’s specifications, the index of refraction of the glass, at wavelengths between 404 and 1014 nm can be given by the following equation:

85

64

43

22

210

2 −−−− +++++= λλλλλµ AAAAAA (43) Despite the fact that the coefficients Ai were all listed in the specifications, we fit the given indices of refraction to a simple second-order polynomial of the form , as seen in Figure 7, and the results obtained were as follows:

CBA ++= λλµ 2

Coefficient Value Error

A 9.62x10-08 2x10-10

B -0.0001566 3x10-07

C 1.57563 9x10-05

Table 1 - Fit for the Index of Refraction

This allowed us to determine the index of refraction of the glass for all the desired wavelengths, when making calculations for the Normal and Anomalous Zeeman effect. Also, from this, it is

evident that BA +=∂∂ λλµ 2 .


1.513

1.514

1.515

1.516

1.517

1.518

1.519

1.52

1.521

1.522

480 500 520 540 560 580 600 620 640 660 680 700

Inde

x of

Ref

ract

ion

Wavelength (nm)

Data

Figure 7: Index of Refraction as a Function of Wavelength for the L-G Plate

3.3 CCD Camera and Software A charge-coupled device (CCD) is a device which consists of a silicon core, which has been divided into an array of light-sensitive units called photosites, each of which is surrounded by a layer of insulating material serving to isolate it from the other sites. Whenever a photon strikes the silicon surface an electron is released due the photoelectric effect. The electron is kept inside the boundary of the insulating material and remains there until it is collected when the image is formed. Each photosite represents one pixel in the final image, and the intensity of the pixel is directly proportional to the charge built up in the photosite over the exposure time. Thus, the longer the exposure to a given light source, the higher the intensity of the picture. However, a drawback to CCD cameras is the fact that they are not only sensitive to light, but also to heat. Any electrons generated by heat in the device will show up in the image as anomalous pixels or noise; however, it is not possible to average out this anomaly because the electrons generated by the heat are not generated uniformly and do not effect all the photosites to the same extent. If the device were kept at a low enough temperature, it would be possible to keep the noise to a minimum, but to eliminate it completely, the CCD would have to be kept at 0 K [7]. From the first images taken, it was obvious that a balance needed to be struck between intensity and width of the lines; a longer exposure time resulted in more intense lines, but they were wider. Ideally, very thin, very intense lines would make analysis simpler by accentuating the difference between the light of the lines and the dark of the background. It was eventually determined that an exposure time of 30 seconds gave results that were sufficiently intense, and thin enough to be of use. Another property of the camera and optical setup was the fact that the lower part of the images showed a tendency to curve to the left and be more compact. The latter


was most evident when analyzing the spaces between the dots, which was obviously greater toward the top of the image than at the bottom, as seen in Figure 8.

Figure 8: Spectrum with 7.45kG Field, Demonstrating the

Curvature of the Image Due to the CCD Camera The software used to capture the images was EyeSpy, which was also the program used to obtain the intensity pattern for the images. It allowed us to bisect the image at any angle with a line, and provided an intensity pattern along that line. This featured was used to determine the wavelength of the peaks, when compared to the theoretical spectrum as seen in Section 3.5 below. It was also used to measure the spacing between the main dots and the satellites to determine the Bohr magneton, as in Figure 12. 3.4 Calibration of the Electromagnet The first step in the experiment was to calibrate the magnet with respect to the power supply. We recorded the magnetic field between the poles of the magnet with a Hall probe for varying voltages of the power supply. We then fit this to a linear equation, as seen in Figure 9, and the result is shown in equation (44):


)0002.01104.0()00001.091787.0( ±+±= VB (44)

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 30 35

Mag

netic

Fie

ld (

kG)

Voltage (V)

Data

Figure 9: Magnetic Field as a Function of the Voltage of the Power Supply

From this, it was possible to determine the magnetic field without having to resort to the Hall probe, making it easier to take measurements in the dark without disturbing the emission tube. It was also noted, using the Hall probe, that the magnetic field was sufficiently constant to within approximately 1cm, radially outward from the center of the poles of the electromagnet. 3.5 Spectral Line Identification Determining the correct lines to use in the experiment was the first challenge we faced. The first step was to try and identify the 626.6 nm using the milled ring to rotate the platform on which the constant deviation prism rested. The line that was in the middle of the image, when the milled ring read 626.6 nm, was tested first using a magnetic field, and then a polarizing filter. A magnetic field of approximately 0.5 T was applied to the emission tube and the pattern of splitting was observed. However, the splitting of the line was not consistent with the expectations, nor was the image when viewed with a polarizing filter. In the latter case, it was expected that the central dot would not appear when the polarizer was set perpendicular to the magnetic field (as discussed in section 2.5); instead, all of the dots still appeared. From these tests, it was apparent that the line given by the milled ring was not the 626.6 nm line. In order to find the desired line, we then compared an intensity plot of a line bisecting an image of the spectrum to multiple theoretical spectra of Ne, one of which is seen in Figure 10. Eventually, a few possible candidates were decided upon and tested in the manner described above. Only one line split as expected in the magnetic field, and it also appeared correctly when viewed under the polarizing filter, with the main dot not appearing. This line is marked in Figure


11 below as 626.6 nm. Other lines of interest marked on the Figure below include the 597.6 nm, 638.3 nm and 650.7 nm lines used to discuss the Anomalous Zeeman Effect, and the 640.2 nm line which served as a valuable reference point for the identification of all the lines because it is the most intense. It should be noted that the intensities of the various lines seen in Figure 10 do not correspond to the intensities seen in our experiment. This is due to the differing wavelength sensitivities of the instruments used to gather the data.

Figure 10: Experimentally Determined, Accepted Emission Spectrum of Neon2

2Astrosurf.org. (02 Dec 2005). <http://astrosurf.org/buil/us/spe2/calib2/neon1>.


Figure 11: Intensity Plot of the Obtained Neon Spectrum

The choice of the 626.6 nm line of neon was made because it has an Anomalous Zeeman pattern corresponding to the Normal Zeeman Effect, that is,

11)0( (45)

4 Data and Analysis 4.1 Bohr Magneton (Normal Zeeman Effect) In order to determine the Bohr magneton, we start with equation (3) and differentiate it with respect to λ, as follows.

s

hcEλ

=

ss

hcE λλ

∆−=∆ 2 (46)

However, from equation (27), we know that BE Bµ=∆ . Therefore,


hcBB

s

s µλλ

=∆

2 (47)

Substituting equation (41) in equation (46) results in:

EhcmsB m

B ∆=∆

= 2

2

λλ

µ (48)

We measured the ratio s/m by finding the pixel location of all the peaks of the 626.6 nm line using the intensity plot in EyeSpy. We took s to be the average separation from the main dot to the satellites in a triplet, while m was the average distance between two orders of the spectrum. This method is seen in Figure 12.

Figure 12: Theoretical Emission Spectrum of Neon

Using those values, as seen below in Table 2, and the values obtained from equation (38), we plotted the energy split against the magnetic field, in order to obtain the Bohr magneton, which would be the slope of that plot according to equation 27 (note that ml=1).


Br

(T) Average S/M Energy Split (10-24 J) 0.286 ± 0.009 0.11 ± 0.04 2.1 ± 0.8 0.378 ± 0.009 0.17 ± 0.04 3.1 ± 0.7 0.470 ± 0.009 0.19 ± 0.04 3.6 ± 0.8 0.562 ± 0.009 0.23 ± 0.04 4.4 ± 0.7 0.654 ± 0.009 0.26 ± 0.04 4.9 ± 0.8 0.745 ± 0.009 0.30 ± 0.04 5.7 ± 0.8

Table 2 - Data for Calculation of the Bohr Magneton Note that the large error on the energy split come from the error on the analysis of the peaks on the intensity spectrum when determining the ratio of s/m, whose minimum error is one pixel. From the above table, the energy split was plotted as a function of magnetic field, as seen below in Figure 13.

1

2

3

4

5

6

7

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

Ene

rgy

Spl

it (J

)

Magnetic Field (T)

Data

Figure 13: Energy Split as a Function of Magnetic Field

The data from Table 2 was then fitted to the linear equation BE Bµ=∆ , and the value for Bµ was found to be (9.4 ± 0.7)x10-24 J/T. This agrees, within error, with the accepted value of 9.27…x10-24 J/T seen in equation (20). The images used in this part of the analysis are in the Appendix; it is easy to see that the 626.6nm line does indeed split into a triplet, and that the spacing of the triplet increases with the increasing magnetic field.


4.2 The Anomalous Zeeman Effect The resolution of our pictures is poor enough that it is impossible to make out more than two satellites for any given line. Also, as discussed previously, if the resolution were increased by decreasing the exposure time, the intensity would have to be sacrificed, and many of the satellites would not show up at all. Despite this resolution problem, however, it is possible to study the Anomalous Zeeman Effect at least approximately for several wavelengths. We chose to examine the wavelengths 597.6nm, 638.3nm and 650.7nm simply because they appeared to be well-defined and relatively bright triplets or doublets, as can be seen in Figure 14. The wavelengths were determined by counting peaks using Figure 10. Furthermore, all the analysis was done using the image taken with the maximum external magnetic field we used, that is 7.45kG.

Figure 14: Identification of the Three Spectral Lines Used

to Study the Anomalous Zeeman Effect


Using the same technique as in the Normal Zeeman Effect analysis, we determined the average S/M ratios and the resulting energy gaps to be the following:

Wavelength (nm) Average S/M ∆E (10-24 J) 597.6 0.18 ± 0.04 3.3 ± 0.7 638.3 0.19 ± 0.04 3.6 ± 0.7 650.7 0.10 ± 0.03 1.8 ± 0.5

Table 3: Analysis results Note that for the 638.3nm line (the doublet), the position of the non-existent main dot was simply taken to be mid-way between the two visible satellites. The expected patterns corresponding to each of the studied lines are the following:

Wavelength Expected Pattern

597.6 2

432)1()0(

638.3 15

22)12(10

650.7 15

221712)5()0(

Table 4: Expected patterns [6] The observed energy splits are compared to the expected Normal Zeeman Effect split (using the energy split from Ne 626.6nm). The pattern is simply the ratio of observed ∆E to the Normal Zeeman Effect ∆E.

Wavelength (nm)

Observed ∆E (10-24 J)

Normal Zeeman Effect ∆E (10-24 J)

Observed pattern out of 30

Expected pattern out of 30 [6]

597.6 3.3 ± 0.7 5.7 ± 0.8 17 ± 6 (0), (15), 30, 45 60638.3 3.6 ± 0.7 5.7 ± 0.8 19 ± 6 20,(24),44 650.7 1.8 ± 0.5 5.7 ± 0.8 10 ± 4 (0),(10),24,34,44

Table 5: Observed Anomalous Effect patterns Note that the large errors on the observed pattern are due to the propagation of the relatively large errors on the ∆E values, which, as discussed above, are a consequence of the error of one pixel on every measurement of the peak position. The observed patterns agree relatively well (even if we don’t take into account the inflated error) with the pattern expected for the π lines. Because the π lines have a higher intensity than the σ lines, it is likely that the points we are able to distinguish are indeed the main lines of the anomalous pattern.


Conclusion Overall, this experiment was quite successful, despite the initial difficulty of identifying the spectral lines of our Neon source. In the first part of the experiment, we looked at the Normal Zeeman Effect, using the 626.6nm line. We used a linear fit of the plot of energy split of versus magnetic field strength to calculate the Bohr magneton. The experimental result was (9.4 ± 0.7)×10-24 J/T, which agrees within error to the theoretical value of 9.27…×10-24 J/T. In the second part, we examined the Anomalous Zeeman Effect at the highest magnetic field strength we used, (7.45 ± 0.09)kG, as it affected three lines, 597.6nm, 638.3nm and 650.7nm. The patterns of the π dots of these three lines agreed relatively well with the accepted values. The main shortcoming of the apparatus was with regards to resolution, which is limited by the fact that all analysis is done in pixels. If the pixel resolution were improved, the intensity peaks could be more accurately measured. Another major drawback to the apparatus is the problem encountered when trying to calibrate it. Because it is highly sensitive, this is almost impossible to do, and so the identification of spectral lines must be determined rather qualitatively, looking at various effects and plots in order to determine their wavelength. Unfortunately, with the current apparatus it is not possible to improve upon this issue. Acknowledgements As usual, we would like to thank Mark Orchard-Webb for his invaluable technical input. We would also like to thank Professors Bennewitz and Buchinger, who frequently came to our rescue over the course of this experiment. Finally, we would like to thank Vanela Bushi and Naomi Kemeny for the use of their lab report and their knowledge of the Zeeman experiment, as well as Raphael Kapfer and Vincent Pelletier, whose lab report also elucidated several aspects of the experiment.


Works Cited 1. Griffiths, David J. Introduction to Quantum Mechanics. 2nd Ed. Upper Saddle River (NJ): Pearson Prentice Hall; 2005. 2. Vachon, Brigitte. “Quantum Physics” Physics 446. McGill University, Montreal, QC. 3. Zeeman Effect. Ed. Carl R. Nave. Hyperphysics, Georgia State University. 02 December 2005. <http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/zeeman.html>. 4. Haken, Hermann and Hans Christoph Wolf. The Physics of Atoms and Quanta. New York:Springer-Verlag Berlin Heidelberg, 2000. 5. "Selection Rules.” Schonland Research Institute for Nuclear Sciences. 2004. 02 December 2005. <http://www.src.wits.ac.za/pages/teaching/Connell/phys284/2005/lecture-03/lecture_03/img155.png> 6. Lab Instructions, Zeeman Lab Manual 7. Bushi, Vanela and Naomi Kemeny. The Zeeman Effect Lab Report. Physics 439. McGill University, Montreal, QC. 03 October 2005.

http://www.src.wits.ac.za/pages/teaching/Connell/phys284/2005/lecture-03/lecture_03/img155.png

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