Michael Good Advanced Lab I The Zeeman Effect: Splitting of Spectral Lines by a Magnetic Field Introduction Pieter Zeeman, a Dutch physicist discovered the ‘Zeeman Effect’ in 1896. His discovery, led the way for a quantum explanation of spin and its relation to a particle’s magnetic field. The Zeeman Effect is the splitting of a single spectral line into a group of closely spaced lines when the substance producing the single line is subjected to a uniform magnetic field. There are two types of effects, the normal and anomalous Zeeman effects. In the normal Zeeman effect, the spectral line corresponding to the original frequency of the light, in the absence of the magnetic field, appears with two other lines arranged symmetrically on either side of the original line. In the more common anomalous Zeeman effect, several lines appear, forming a complex pattern. Pieter Zeeman The normal Zeeman effect was successfully explained by H. A. Lorentz using the laws of classical physics. Lorentz was Zeeman’s mentor and advisor, and they both shared the 1902 Nobel Prize for their work. The anomalous Zeeman effect was a bit more difficult to account for. It could not be explained using classical physics and had to wait for the development of quantum mechanics. The discovery of the electron's intrinsic spin led to a satisfactory explanation of the anomalous Zeeman effect. Heisenberg himself battled with the anomalous Zeeman effect as a student. All spectral lines arise from transitions of electrons between different allowed energy levels within the atom, the frequency of the spectral line being proportional to the energy difference between the initial and final levels. Because of its intrinsic spin, the electron has a magnetic field associated with it. This magnetic field is affected when an external magnetic field is present. The electron’s field will assume only certain alignments in the presence of an external field. Slight differences in energy are associated with these different alignments, so what was once just a single energy level can become three or more when an external magnetic field is present to align the electron’s field. The Zeeman effect has various practical applications including spectral analysis and measurement of magnetic field strength. Since the separation of the components of the spectral line is proportional to the field strength, the Zeeman effect is particularly useful where the magnetic field cannot be measured by more direct methods, say for instance, measuring the magnetic field of a star or a planet. The purpose of this experiment is to further our understanding of the physics of quantum mechanics by observing the effects that a magnetic field has on the spectral lines emitted by cadmium and mercury. In this experiment the splitting of these lines will be studied and measurements will be made on both the Cd red line and the Hg green line (although the measurements on the Hg lines will be rougher) to give us the results of the Zeeman effects on both substances. The normal Zeeman effect associated with the cadmium and the anomalous Zeeman effect associated with the mercury will further our understanding of quantum mechanics, as we will be introduced to the experimental basis of electron spin, of which there is no classical analog. Goudsmit and Uhlenbeck first postulated the existence of spin in 1925 but the unfortunate term was dubbed by Pauli. It is an intrinsic property of a particle just like mass and charge, and behaves like an angular momentum with an associated magnetic moment. Also, in this experiment we will observe the transverse and circular polarizations of light emitted in the Zeeman effects of both elements. Equipment and Procedure There are relatively few pieces of equipment in this lab, all of which are listed in the handout. Pictures of the equipment where taken in order to clarify the set up and procedure of the experiment.
The smaller accessories include the different filters and lens:
Figure 9: From left to right: Red Filter, Quarter Wave Plate, Lens, Plane Polarizer, Green Filter The Zeeman Effect experimental apparatus is fragile. The LummerGehrcke plate is loosly positioned in the optical system, the CdHg lamp must be handled with care, lest it break, and the lens and filters are prone to attracting finger prints which can make observations and measurements harder. The set up for the optical system is neatly shown as a figure in the lab handout. Figure 10: Optical System Apparatus Diagram ^
Before any observations where made the magnetic field was calibrated. This was done carefully by controlling the changes made in voltage. Slowly raising the voltage from zero up to its maximum value, then running the voltage back down to zero, then back up again to a desired value. Measuring along the way, obtaining data for a B versus V plot, we went up in steps of 1 V and then obtained two more points by going directly to the values of 8 V and 10 V. This is the only time the Gaussmeter is used during the experiment. What this calibration does is allow us to use magnetic field values for the rest of the experiment. The B values found during this calibration will be used throughout the data associated with the different voltages. The plot above is the data of B versus V, distinguishing between points taken while increasing the voltage and while decreasing the voltage. The data of the calibration is presented as a table shown below, the final two points are also included and are indicative of the reproducibility of the magnetic field, using this cycling procedure.
Table 1: Final Two Points of the Magnetic Field Calibration
Table 2: Magnetic Field Calibration, B versus V Procedure: The general procedure was to select three fringes for observation; taking the more widely separated fringes because they will yield better data. With no magnetic field applied, we measured the separation between the outer fringes, the distance 2a. The indicator gauge attached to the optical system was used to measure this distance. Only a ratio of distances was used in the computations and thus we had no need to calibrate the indicator gauge. When the magnetic field was applied, the splitting of the central line was studied. For the lines of interest here, the splitting was symmetrical, and the distance 2s was measured. The distance 2s and 2a are shown below in Figure 12.
The experiment was divided into four sections. The normal Zeeman effect on the Cd red line, with the measurements made perpendicular to B (1) and then made parallel to B (2). Next, the observations of polarization of the anomalous Zeeman effect on the Hg green line, (both orientations of the magnet) (3) and finally a semiquantitative study of the anomalous Zeeman effect on the Hg green line (4).
Observe light emitted (through hole in magnet pole piece) parallel to B. No polarizing filter or quarterwave plate is used. Circular polarization is expected, but use of a filter will destroy the symmetry of the splitting and upset measurements of the splitting 2s. Here we were observing the anomalous Zeeman effect on the Hg green line, and the same general procedure was used. This observation was slightly different than the rest in that limited resolution of the apparatus made precise measurements impossible, and there also existed a field of values for voltages over which only decent measurements where possible. The overall objective is to observe and measure the splitting and polarization of the Cd and Hg lines. We are to record the corresponding voltages and calculate Δλ using the measured splitting of the spectrum. This is easier than it sounds, as the data was made available in a supplementary section that included the equations and values we needed for calculating the Δλ’s. Theoretical Background Polarization: The nature of polarized light allows for one to treat light as a transverse electromagnetic wave. When the orientation of the electric field is constant, light is said to be planepolarized. The electric field resides in a plane of vibration which contains both E and k, the electric field vector and the propagation vector in the direction of motion. To understand the nature of polarization, consider taking two harmonic, linearly polarized light waves of the same frequency, moving through the same region of space in the same direction, and if their electric field vectors are collinear, the superimposing disturbance will simply combine to form a resultant linearly polarized wave. On the contrary, if you take the two light waves and their electric field directions are mutually perpendicular, the resultant wave may or may not be linearly polarized. The two types of polarization we will be concerned about in the Zeeman effect will be linearly polarized light, and circular polarized light. In the case of linear polarization, you can represent the two light waves in the form E_{x}(z, t) = i E_{0x} cos ( kz – ωt) [1] and E_{y}(z, t) = j E_{0y} cos (kz – ωt + з), [2] where з is the relative phase difference between the waves, both of which are traveling in the zdirection. E_{y} lags E_{x} by з > 0, and E_{y} leads E_{x} by з < 0. If з is zero or multiple of 2π, the waves will be in phase. In this case, the total resultant vector sum is E = ( i E_{0x} + j E_{0y} ) cos (kz – ωt). [3] Note that this resultant wave has a fixed amplitude, and is also linearly polarized. This can be observed in a single plane where the waves advance to, and we can see a single resultant E. This process of addiction can be undone, so to speak, allowing us to resolve any planepolarized wave into two orthogonal components. If we where to suppose that the two waves are out of phase, where з is now an odd integer multiple of π, the total resultant vector sum becomes E = ( i E_{0x} – j E_{0y} ) cos (kz – ωt). [4] This wave is also linearly polarized, but the plane of vibration has been rotated. In the case of circular polarization, there are two types, right circular light, and left circular. If both waves have equal amplitudes, and phase difference is increased or decreased from – π/2 by whole number multiples, the wave equations become E_{x}(z, t) = i E_{0} cos ( kz – ωt) [5] and E_{y}(z, t) = j E_{0} sin (kz – ωt). [6] The result wave then becomes E = E_{0 }[ i cos ( kz – ωt) + j sin (kz – ωt) ]. [7] The scalar amplitude is constant, but the direction of E varies with time. Because it varies with time, the direction is not restricted anymore to a single plane. At different times E now lies along different axis. It is happily rotating clockwise at its frequency of ω, as seen by someone who is about to get hit by the light. Rotating clockwise, the light is called right circular. If you simply make the phase shift equal to π/2, 5π/2, 9π/2, etc, than the result wave gets a negative sign, E = E_{0 }[ i cos ( kz – ωt) – j sin (kz – ωt) ], [8] where the amplitude is still unaffected, but now E rotates counterclockwise and the wave is dubbed left circular light. It is interesting to note that you can make a linearly polarized wave from two oppositely polarized circular waves, as long as they have the same amplitudes, just by adding the two. A quarterwave plate, usually made of quartz, like the one that was used in this experiment, is an optical element that introduces a relative phase shift of 90˚. When you have linear light at 45˚ to either principal axis incident on a quarterwave plate, components of the light will have equal amplitudes. When this happens, a 90˚ phase shift converts the wave into circular light. Similarly, an incoming circular beam will emerge linearly polarized. ^ The theory behind the Zeeman effect can be quite formidable to the novice quantum mechanic, but I will give an overview of what is going on without going into too much detail and listing all the relevant equations, of which there are many. Basically, when an atom is placed in a uniform external magnetic field, the energy levels are shifted. This is the Zeeman effect and the splitting depends critically on the strength of the external field in comparison with the internal field. The type of Zeeman effect our experiment dealt with was the WeakField Zeeman effect. The other two regimes of Zeeman effects are called the StrongField Zeeman effect (also known as the PaschenBack effect), and the IntermediateField Zeeman effect. In the weakfield effect the external magnetic field is much less than the internal magnetic field. Fine structure dominates, and only firstorder perturbation theory is needed to find the Zeeman correction to the energy. L and S are not separately conserved, only the total angular momentum, L + S = J (this is due to the presence of spinorbit coupling). Here is an illustration of the precession of L and S. Figure 12: L and S not separately conserved and thus precess around J. The correction to the energy is Mμ_{B}Bg. Where M is the transition, μ_{B }is the Bohr magneton, B is the external magnetic field and g is the Lande gfactor. I’ll list them in order here, (for our experiment, they take on these values): M = 2, 1, 0, 1, 2 [9] μ_{B} = e g = 1 + [J(J+1)+S(S+1)L(L+1)] / [2J(J+1)] [11] For the normal Zeeman effect, S = 0, J = L and g becomes 1. The correction to the energy is found through the analysis of the expectation value of S and solving for the expectation of value L + 2S. The perturbation of a single electron as shown in Griffiths, is H = (μ_{l} + μ_{s}) B [12] Where μ_{s }is the magnetic dipole moment associated with electron spin, eS/m, and μ_{l } is the dipole moment associated with orbital motion, eL/2m. When you plug these values in you get H = (e/2m)(L + 2S) B [13] for the perturbation of the single electron. This is why the expectation value of L + 2S is calculated. This also leads to both the derivation and value of the Lande gfactor and the Bohr magnetron. Data and Experiment Normal Zeeman Effect on Cd Red Line, Perpendicular to B For this section of the experiment, the red filter was used, along with a lens to observe the Cd red lines. Three fringes where selected and the distance 2a was measured to be 6.31 x 10^{4} m. We found that the use of a plane polarizing filter was not helpful in blocking out the undisplaced line. We measured the splitting 2s as a function of voltage applied to the magnet coil and went from a range of 4 V to 12 V, in steps of 1 V. The table showing V, B, 2s, and Δλ is below, and is labeled Table 3.
Table 3: V, B, 2s, Δλ. A plot of Δλ versus B is shown in Figure 13. B values for the V values used where obtained from the calibration curve made in the preliminary part of the experiment shown in Figure 11. Also, the value for the index of refraction for the LummerGehrcke plate, used in the calculation of the Δλ’s, was found in the graph of the index of refraction of quartz versus wavelength, which was supplied to us in the supplemental section. Figure 13: Δλ versus B. Our investigation of the polarization of the displaced line and the undisplaced line was an interesting part of this experiment. In this view the magnet was set perpendicular and plane polarization was expected and observed. Using the plane polarizer by turning it and looking closely and patiently, it was easy to see exactly the shifting from one line to two lines on either side. From ΔM equal to 1 to 0 to +1, the plane polarization of the light from the cadmium was readily observable and it was pleasing to get what was expected. I am choosing to include the chart of polarization intensities (Figure 14) supplied in the supplementary section to further clarify our observations.
Figure 14: Polarization Intensity Chart The experimental guidelines suggested instrumental effects may suggest small circular polarization, but this was fortunately not observed. ^ Here the magnet is reoriented to observe the Zeeman effect parallel to B. No quarterwave plate or planepolarizer was used when measuring the splitting.
Table 4: V, B, 2s, Δλ. To ensure that the points for the two different experiments are distinguishable, I plotted Δλ versus B on two separate graphs, but with the same minimum and maximum axis values to allow for an easy comparison. For the normal Zeeman effect, parallel to B, Figure 15 shows the data, along with a comparison with the data collected for the effect perpendicular to B as shown earlier in Figure 13. Figure 15: Comparison of Data for Normal Zeeman Effect Δλ versus B Plots As is shown on Figure 14, the polarization for the cadmium red lines where circularly polarized and it was incredibly easy to observe the shift from the transition from the left side to the right side, with ΔM = 1 or 1. With the use of the quarterwave plate, and turning the planepolarizer, the transition was unambiguously clear, which turns out to be a rare find, as we will see in the next parts of the experiment. ^ Here we replaced the red filter and lens with the green filter and lens to observe not cadmium, but mercury. As mentioned earlier, this equipment is not suitable to make good studies of the anomalous Zeeman effect, but is adequate to make an attempt at polarization observations. Observing the expected polarization patterns was easier with the field perpendicular. Also, the stronger the field, the easier it was to see the larger visible effect. Here I saw one very fuzzy line, turn into two very fuzzy lines. Figure 14 implies the one very fuzzy line that I saw, is actually three lines, and that they turn into six differently positioned lines. After careful observation and patient examination with no lights on, and sitting in different positions, I still could not observe anything other than the thick, fuzzy one line, making a transition into two other thick, very fuzzy lines. It is assumed that the resolution capabilities of our equipment hindered our ability to see the true anomalous Zeeman polarization patterns on the mercury green light. For the field parallel to B, it was unfortunately even worse. The pattern expected, shown in Figure 14 of course, implies two lines transitioning into three lines, with one having a very low intensity. We observed one line, shifting to another position, as if we where observing the normal Zeeman effect on cadmium. This is somewhat reasonable, considering the relative intensities implied on the chart, but still a disappointment. ^ Rough measurements are possible in this procedure, as we observe light parallel to B. We chose configuration A, as mentioned in earlier. A qualitative investigation of the appearance of the splitting as a function of field yielded a range of voltages from 3 V to 8 V. Above this, lines from adjacent fringes overlap to such an extent as to render further measurements impractical. Table 5 shows the results for this observation.
Table 5: V, B, 2s, Δλ, for Anomalous Zeeman Effect, Parallel to B Figure 16: Plot of Δλ versus B Analysis and Computation Analysis of Zeeman Splitting Data For a best straight line fit to the data points on the plots of Δλ versus B, we ignored the implied point at the origin and fit a best straight line to the actual data points corresponding to B not equal to zero. This is procedure II as suggested in the laboratory guide. Δλ is expressed for both normal and anomalous effects by f(g_{1},g_{2})μ_{B}Bλ_{0}^{2}/hc. The slope for the best fit straight line has the formula: f(g_{1},g_{2})μ_{B} = (hc/λ_{0}^{2 })δ(Δλ)/δ(B). Figure 17: Straight Line Fit for Normal Zeeman Effect The slope value for the effect parallel to B is 2.5370 x 10^{11 }[J/T] with a standard error of plus or minus .04368 x 10^{11 }for determining the slope. This is 1.72% of the expected slope. The slope value for the effect perpendicular to B is 2.7409 x 10^{11 }[J/T] with a standard error reported by Excel as .02137 x 10^{11} for predicting the slope. This is 0.78% of the expected slope. Analysis of Normal Zeeman Effect For the normal effect, the splitting of the displaced lines with f(g_{1},g_{2}) = 1, is shown in the laboratory guide as [14] The experimental values for μ_{B} from the plots perpendicular to B and parallel to B are surprising very close for both the normal Zeeman effects, and have the values: Parallel to B: μ_{B} = 1.217 x 10^{23} ± .021 x 10^{23 }[J/T] [15] Perpendicular to B: μ_{B} = 1.314 x 10^{23} ± .010 x 10^{23 }[J/T] [16] Be reminded that from equation [10] that the value of the Bohr magneton is μ_{B} = 9.273 x 10^{24} [J/T]. From this it is found that the experimental value for μ_{B }perpendicular to B is within 41.7% of the accepted value of the Bohr magneton. As for the calculation for μ_{B }in the set up parallel to B its experimental value is within 31.2% percent of the Bohr magneton. These numbers are certainly within the same ball park, and considering the small size of these numbers, the human eyeballing of the separation distance, it was pleasant to get numbers so close to the fundamental constant, e ^ The product f(g_{1},g_{2})μ_{B }determined from the plot is: 1.773 x 10^{23} [J/T]. This was found using a straight line fit and applying the equation f(g_{1},g_{2})μ_{B} = (hc/λ_{0}^{2 })δ(Δλ)/δ(B). Calculation of g_{1} and g_{2 }for the states involved in the transition that yields the observed Hg green line, give g_{2 }= 2 for the state S_{1 }because these states have the quantum numbers J = 1, L = 0, and S = 1, using the Lande gfactor equation [11]. They also give g_{1 }= 2 for the P_{2} state. Using Table 5 to calculate the g factor for Hg, with hc(Δλ)/ λ_{0}^{2 }= ΔE and then ΔE/μ_{B}BM_{l }= g yields the values: 2.7, 2.1, and 2.2. When I calculate a value for f(g_{1},g_{2})_{ }I get 1.9. So therefore my estimate for a reasonable value that is applicable to the observations is the nice number 2. Computing an expected value for f(g_{1},g_{2})μ_{B} then yields 1.854 x 10^{23} [J/T], as compared with the experimentally determined value of 1.773 x 10^{23} [J/T] mentioned above. Comparing the determined value with the expected yields an error of only 4.4%. Figure 18: Straight Line Fit with Error Analysis of Green Hg Line, Anomalous Effect Parallel to B ^ We are given Δλ = c/f – c/f_{0} and f = f_{0} + ΔE/h, and we are to show that Δλ ≈ ΔE λ_{0}^{2}/hc, where λ_{0} = c/f_{0}. It is understood that the formula is valid provided ΔE << hf_{0}, and the use of the binomial expansion for (1 + ΔE/hf_{0})^{1} is necessary. I will explicitly write out the steps for this verification: Δλ ≈ ΔE λ_{0}^{2}/hc = ΔE (c/f_{0})^{2}/hc = ΔEc/f_{0}^{2}h. [17] Binomial expansion: ( 1 + ΔE/hf_{0 })^{1 }= 1  ΔE/f_{0}h + ( ΔE/f_{0}h )^{2 }+ ... [18] Δλ = c/f – c/f_{0 }= c/( f_{0 }+ ΔE/h ) – c/f_{0 }= ( c/f_{0 })/( 1 + ΔE/f_{0}h ) – c/f_{0 } [19] Δλ = ( c/f_{0 })( 1 + ΔE/f_{0}h )^{1} – c/f_{0 } ≈ ( c/f_{0 })/( 1 – ΔE/f_{0}h ) – c/f_{0 } = – ΔEc/hf_{0}^{2 } [20] Δλ ≈ ΔEc/hf_{0}^{2} = ΔE λ_{0}^{2}/hc. QED. [21] Discussion and Conclusion The shell structure of mercury is 2.8.18.32.18.2. Whereas the shell structure of cadmium is 2.8.18.18.2. In a multielectron atom such as the Hg and Cd atoms used in this lab, the net atomic magnetic moment results from the sum of the spin and orbital magnetic moment contributions of all electrons, as mentioned earlier. The inert core of the atom contributes no net magnetic moment, so the problem simplifies to that of calculating the net magnetic moment due to the valence electrons. Both Cd and Hg have the same number of valence electrons, 2. For the normal Zeeman effect you assume that the interaction between the orbital angular momenta and spins is negligible so that the two different components can be summed separately. Spin is summed to be zero in the normal Zeeman effect. The total angular momentum of the atomic system will either be integral or halfintegral depending on whether the number of electrons is even or odd. If we where to assume that the magnetic moments associated had the same form as that due to orbital angular momentum, then the magnetic levels would be the normal Zeeman effect, making three lines for each transition with the separation μ_{B}B. But the anomalous Zeeman effect comes about because the intrinsic magnetic moment of a single electron is one Bohr magneton, despite the electron having a spin angular momentum of ½. As I understand it, in the classical model, this leads to the magnetic moment being parallel to the total angular momentum, J, with the magnetic moment precessing about the angular momentum vector. The frequency of this precession will depend on the magnitude of the component of the magnetic moment parallel to the total angular momentum. The time averaged perpendicular component will be zero. So what is done, both classically and quantum mechanically is to modify the effective magnitude of the parallel component of the angular moment. And the magnetic splitting that comes from placing the Hg atoms in a magnetic field, changes by the amount ΔE = gMμ_{B}B. [22] Where g, equation [11], is dependent upon the spin. This results in a splitting of what was originally a single transition line in the optical spectrum into multiple lines, which we can see and record. In its entirety, this lab introduced a wealth of new understanding about quantum mechanics and particularly the phenomena of spin. Applying the fundamental principles of quantized angular momenta in an attempt to develop an understanding of the observed phenomena it was found that the energy transitions are split when the situation is complicated by applying a magnetic field. Through the data collection, analysis and research done on both normal and anomalous Zeeman effects, I discovered that interaction of atomic magnetic moments with an external magnetic field is solely responsible for the Zeeman effect. It was observed that the splitting is proportional to the magnitude of the applied field and a reasonable estimate for the Bohr magneton was obtained from the measurements. I chose to study the Zeeman effect as an interpretation of the precession of the orbital angular momentum vector in the magnetic field, similar to a precession of the axis of a spinning top in a gravitational field. Through the careful observation of the polarized light, linear and circular polarization was differentiated from, and connected with the associated transitions of the energy levels. It was observed that the lines corresponding to Zeeman splitting exhibiting polarization effect that are dependent on the direction of the electromagnetic fields. This obviously connects with the Zeeman effect because it has the effect on whether the spectral light can be observed or not. An interesting anecdote I remember reading about Heisenberg and his thoughts on the Zeeman effect goes something like this: A friend saw Heisenberg on the streets of Cophenhagen, around 1920. Heisenberg had a grim expression, and seemed obviously concerned about something. ``Cheer up, Werner, things can't be that bad!'' And Heisenberg replied stressfully, ``How can one be cheerful when one is thinking about the anomalous Zeeman effect?!''
