Eect Zeeman in Mercury
The goal in this experiment is to measure the shift of atomic energy levels due to an external magnetic eld. This eect is known as the Zeeman eect and was rst observed by the Dutch physicist Peter Zeeman in 1896. Measurements of energy shifts are performed for the green line and both yellow lines in the mercury spectrum, and the qualitatively dierent results for the dierent lines are explained in terms of the properties of the initial and nal states for each line. The results for the energy shifts are compared to theoretical predictions.
1
Background
Many texts oer a detailed discussion of the Zeeman eect, including the ones available on reserve for this course. In light of this, we oer here only a brief overview of the theory, and encourage you to consult either of these texts or one of your own choosing for a more comprehensive treatment. Electrons in atoms occupy states with well-dened energies. When an electron transitions from a higher energy state to a lower energy state, a photon with energy equal to this dierence is emitted by the atom. By analyzing the properties of the emitted photons, namely wavelength and type of polarization, one can learn about the states of the electron before and after the transition. For some atoms, application of a magnetic eld will change the properties of the emitted photons, thus providing additional information about the initial and nal states of the electron. Indeed, in the early development of quantum mechanics studying the eects of external perturbations (applied magnetic and electric elds, for example) on these transitions provided major clues leading to a detailed understanding of atomic structure. As an aside, atomic transitions were rst studied in the optical (humanly visible) portion of the electromagnetic spectrum. Transitions were observed using an instrument called a spectrograph which separates the incoming light by wavelength. Transitions at dierent wavelengths appear as distinct vertical lines in the spectrograph vieweld, hence the term line to denote a transition at a particular wavelength. With the apparatus used in this experiment, the dierent lines are actually observed as sets of concentric circles of dierent radii.
1.1
Magnetic moments
Magnetic moments are a key part of understanding the Zeeman eect, so lets briey review their basic properties. When an electrical current I circulates in a loop of wire with area A (= a 2 ), there is an associated magnetic moment of magnitude = IA , (1)
and direction given by the right-hand rule, as shown in Fig. 1. If this loop is placed in an external magnetic eld B, the loop will experience a torque =B . (2)
Work is done by the magnetic eld on the loop when the orientation of the loop changes relative to that of the magnetic eld, and the potential energy associated with the orientation of the loop is Emagnetic = B . 1 (3)
Figure 1: A current loop and its associated magnetic moment.
1.2
Angular momentum and magnetic moments in the atom
Note that in the following discussion we are considering an atom with only one electron outside the closed inner shells. An example of such an atom is sodium. The optical spectrum of such an atom is due to transitions by the outer electron, and such an electron is called optically active. Later we will consider the case for more than one electron outside the closed shells, an example of such an atom being mercury, the subject of investigation in this experiment. 1.2.1 Orbital angular momentum and magnetic moment
Each state that an electron can occupy in an atom is characterized not only by a well dened energy, but also by a specic amount of angular momentum. There are two contributions to the angular momentum, one due to the orbital motion of the electron about the nucleus, and the second due to the intrinsic angular momentum of the electron. The rst is known as orbital angular momentum, and the second as spin angular momentum. Let us consider each one separately, and then see how they combine to form the total angular momentum. The orbital angular momentum of the electron can be thought of as due to circulation of the electron about the nucleus, and as such can be regarded as a current circulating about a loop. Just as in the macroscopic case discussed above, this circulation gives rise to a magnetic moment. The relationship between the orbital angular momentum l and the associated magnetic moment l is l = e l, 2me (4)
where e is the magnitude of the electron charge, and m e is the electron mass; the minus sign is due to the negative charge of the electron. Exercise (Optional) Derive Eq. (4) from Eq. (1) under the assumption that the current I is from a single circulating electron. This relationship is conventionally stated as l = gl e h 2me l h , = gl B l h , (5)
where gl = 1 for a single optically active electron and e /2m e B is known as the Bohr magh neton. The dimensionless quantity g l is known as the g factor and is a measure of the ratio of the magnitude of the magnetic moment in units of B to the magnitude of the orbital angular 2
momentum in units of h; in other words, g l is dened by Eq. (5). Since gl = 1 for a single optically active electron, having such a factor in the equation for l may not seem to make much sense. But as we shall discover, g can take on values quite dierent from unity when we consider all the contributions to the angular momentum. Now lets take a quantum leap over much important physical theory and simply state some results of quantum mechanics: 1. The variable l is known as the orbital angular momentum quantum number and can take on the values 0, 1, 2, . . . The magnitude of the orbital angular momentum is l(l + 1) . h 2. The projection of the orbital angular momentum along a particular axis is quantized in units of h. When the atom is placed in an external magnetic eld, as when observing the Zeeman eect, the axis of quantization is parallel to the eld. By convention the magnetic eld is taken to point in the +z direction. 3. The quantity ml is the quantum number specifying the projection of the orbital angular momentum along the z axis and can take on the values l, l 1, l 2, . . . , l, for a total of 2l + 1 values. It follows that the possible values of the z component of the orbital angular momentum, l z , are just ml h. Figure 2 shows the situation for l = 2 , with the 5 possible values of m l (all magnitudes are in units of h). As an aside we note that quantum mechanics allows for only one component of the angular momentum to be specied exactlythe other two components have indenite value and average to zero over time [1, p.258]. In our scheme the z component has a denite value, so the x and y components can be anywhere on the cone denoted by the dashed line. Since the magnetic moment and orbital angular momentum are proportional, the magnetic moment will also have quantized projections along the z axis. The question arises: what is the potential energy associated with a particular projection of the magnetic moment along the z axis? Let E( B) be the energy resulting from the interaction of the atom with the external eld B. When we turn on B (in the +z direction) the change in energy is E(B) E(0) = Emagnetic = l B = lz B , (6)
where lz is the component of l in the z direction. From Eq. (5) in terms of the z component alone, we obtain lz = gl B ml , (7) lz = gl B h where again the negative sign comes from the charge of the electron. Thus, the energy change from the magnetic eld is Emagnetic = (gl B ml )B = gl ml B B . (8) 1.2.2 Spin angular momentum and magnetic moment
The treatment of the electron spin angular momentum, s, and spin magnetic moment s is similar to the above treatment of orbital angular momentum and its magnetic moment. The quantity s is known as the spin quantum number and can take on only the numerical value of 1/2 when just h one electron is involved. The magnitude of the spin angular momentum is s(s + 1) . 3
Figure 2: Angular momentum l = 2 where | l| = h.
2(2 + 1) . The z-projections of l are in units of h
The projection of the spin angular momentum along a particular axis, again taken to be the z axis, is quantized in units of h, with possible values + /2 and /2. Here, m s is the quantum number h h specifying this projection and takes on the values +1/2 and 1/2. See Fig. 3. When the atom is placed in an external magnetic eld, as when observing the Zeeman eect, the axis of quantization is again parallel to the eld.
Figure 3: The spin angular momentum vector s for a single electron. The relation between the spin angular momentum s and the associated magnetic moment s is similar to the relations for the orbital quantities: s = gs B s h , (9)
4
and likewise, the energy shift for a spin placed in an external magnetic eld B is Emagnetic = gs ms B B . (10)
Is gs = 1, just like gl ? No. It turns out that gs 2, or more precisely, gs = 2.00232 . . . That it is nominally 2 instead of 1 derives from the relativistic treatment of the electron (i.e., the Dirac equation), and the fact that it deviates from 2 is due to the interaction of the electron with uctuations of the electromagnetic vacuum, a concept deriving from quantum electrodynamics. The deviation of gs from 2 has been a topic of great theoretical and experimental interest for many years. The dierence gs 2 has been measured by researchers in the University of Washington physics department to the remarkable precision of almost 12 decimal places, and increasingly rened theories result in calculations that match these very impressive experimental results. 1.2.3 Spin-orbit interaction
Now that we have expressions for the energy shifts due to the interaction of orbital and spin magnetic moments in an external magnetic eld, are we done? The answer is no. This is due to the fact that for small external magnetic elds, typical of the magnitude that our electromagnet can generate, the orbital and spin magnetic moments do not interact independently with the external magnetic eld. Rather, the orbital and spin magnetic moments interact with each other in such a way as to form a combined magnetic moment that interacts with the external eld. One way to think about this is that the orbital motion of the electron about the nucleus results in a magnetic eld at the location of the electron. (From the point of view of the electron, the nucleus appears to be orbiting about it, resulting in a magnetic eld at the location of the electron.) The spin magnetic moment of the electron interacts with this eld so as to couple the spin and orbital magnetic moments together. The magnitude of the eld seen by the electron is approximately 10 4 gauss [1, pp. 278-9], which is much greater than the elds we use in this experiment. Not until the applied external eld greatly exceeds the internal elds in the atom do the orbital and spin magnetic moments decouple and interact independently with the external eld. How do the spin and orbital angular momenta and magnetic moments couple together? It turns out that the angular momenta combine vectorially to form a total angular momentum j, j =l+s . (11)
Now, j is the angular momentum with a denite projection along the z axis, as shown in Fig. 4. h The magnitude of j is j(j + 1) , where the values for j, called the total angular momentum quantum number, are determined by the set of numbers l + s, l + s 1, . . . , l s, where we have assumed l s. Since s = 1/2, the possible values of j are l + 1/2, l 1/2 only. Here, j z is formed from the sum of lz and sz , which leads to a new quantum number m j specifying the projection of total angular momentum along the z axis: jz = lz + sz = (ml + ms ) = mj h . h (12)
For l = 2 and s = 1/2, the two possible ranges for m j and possible values for j (j = 3/2 and j = 5/2) are shown in Fig. 5. (Note that j = 2 is not allowed as this would require that s z = 0, which is not possible with only one electron.) Now that we have combined spin and orbital angular momenta, can we derive a formula for the energy shift in an external magnetic eld similar to the formulas above? That is, can we write 5
Figure 4: Angular momentum vectors l and s combine to form the angular momentum vector j.
Figure 5: Example of two dierent possibilities for j with l = 2 and s = 1/2. Emagnetic = gj mj B B? The answer is yes, provided we know how to nd the value of g j . This g factor is derived from considering the projections of the spin and orbital magnetic moments along the total angular momentum j, and we state the result: gj = 1 + j(j + 1) + s(s + 1) l(l + 1) . 2j(j + 1) (13)
Derivations of this formula can be found in the text by Preston and Dietz [2, pp. 2469] and in the write-up for the Optical Pumping of Rubidium experiment. The formula clearly gives the correct results for cases where either s or l vanishes, g j then being equal to 1 or 2, respectively.
1.3
Multiple optically active electrons
So far we have been considering an atom with only one electron outside the closed inner shells. But mercury, the atom of interest in this experiment, has two optically active electrons. Now what? It turns out that for many species of atoms the individual orbital angular momenta combine to form a total orbital angular momentum, and the individual spin angular momenta combine to form 6
a total spin angular momentum. Note: upper case letters are typically used to denote combined angular momentum states. Let L be the total orbital angular momentum number and S the total spin angular momentum quantum number for the two electrons. If l 1 and l2 are the values for the individual orbital quantum numbers, possible values for L are: l 1 + l2 , l1 + l2 1, . . . , l1 l2 (assuming l1 l2 ). For l1 = 2 and l2 = 1, the possible values of L are 3, 2, and 1. The spins combine similarly, so for two electrons s 1 = s2 = 1/2 and the possible values of S are 0 and 1. Just as l and s combine to form j, L and S combine to form J: J =L+S , (14)
with the magnitude of J being J(J + 1) , and the component along the z axis being specied by h mJ . This scheme of coupling angular momenta for more than one electron is called LS coupling, and is also known as Russell-Saunders coupling. (The common short-hand notation for two-electron states is used in Figs. 6 which shows a particulr set of transitions for the cadmuim atom, and in Fig. 7 which shows a more complete picture of the energy levels of the mercury atom along with initial and nal states for a number of transitions. The ground state of mercury has two 6s electrons, and is designated 1 S0 . A more detailed explanation of this notation system will be presented later.) So for the multi-electron atom the expression for the energy shift in an external magnetic eld is given by the now-familiar formula: Emagnetic = gJ mJ B B . (15)
Figure 6: Energy-level diagram for the 643.8 nm spectral line of the neutral cadmium atom which shows the splitting of the 1 D2 and 1 P1 states caused by the application of a magnetic eld. Figure 6 is a typical diagram showing the eect of an external magnetic eld on the initial and nal states for a particular transition of the neutral cadmium atom which has two optically active 7
electrons. The multiple levels at the right are called sublevels. The arrows depict just the transitions between these sublevels that are allowed by what are called selection rules. The nine arrows represent nine dierent transitions, but note that there are only three distinct energy dierences between the various initial and nal states. One level is unshifted from its value for B = 0, and the other two are shifted an equal amount above and below the unshifted level. (This follows from the fact that gJ = 1 for both the 1 P1 and 1 D2 levels.) With a total of three lines in the presence of an external magnetic eld, this is an example of the normal Zeeman eect. The Zeeman eect was observed long before the advent of quantum mechanics. The interpretation of this eect based on classical electrodynamics allowed for only three lines, one above and one below the unshifted line, and spectra which showed only three lines were called normal. The more common case, as with the mercury green line, is that there are more than three lines, and when this is the case, it is called, somewhat less than appropriately, the anomalous Zeeman eect.
1.4
Selection rules and polarization
A natural question arises: what transitions are possible? Can an electron (or electrons) go from any arbitrary state to any other arbitrary state? The experimental evidence says no and the theory based on this evidence requires that quantities like angular momentum be conserved in transitions from one state to another. The following rules apply to an atom with two optically active electrons such as mercury (the rules are slightly dierent for atoms with only one optically active electron) [2, p. 252]: S = 0 L = 0, 1 J mJ = 0, 1 (but not J = 0 J = 0)
= 0, 1
(but not mJ = 0 mJ = 0 if J = 0)
These rules apply to what are known as electric dipole transitions, which are the most intense kind of atomic transitions. If a transition violates these rules, it will be occurring via some other mechanism (magnetic dipole, for example), all of which result in considerably weaker lines than for electric dipole transitions [1, pp. 288295]. The polarization of the emitted photons is determined by m J . For mJ = 0, the photons are linearly polarized along the direction of B, and the associated lines are termed lines. For mJ = 1, the photons are circularly polarized along the direction of B (i.e., the electric eld vector rotates clockwise or counterclockwise around the direction of B), and the associated lines are termed lines. When viewed at right angles to B, as is done with our apparatus, these photons have linear polarization perpendicular to the eld.
1.4.1
Notation of states
States are specied using a long established notation, 2S+1 XJ , where S is the total spin quantum number, X denotes the total orbital angular momentum number, and J denotes the total angular momentum quantum number. The peculiar notation for the orbital number, S for L = 0, P for L = 1, D for L = 2, F for L = 3, comes from the names originally given to the sets of spectral lines seen on photographic plates: Sharp, Principal, Diuse and Fundamental. A state with 8
Figure 7: Energy level diagram of the neutral mercury atom. The wavelengths of the more intense spectral lines are shown in units of angstroms. The symbols (6p, etc.) written near each level indicate the principal quantum number and the orbital angular momentum quantum number l for the active emission electron.
9
S = 1, L = 2 and J = 2 is denoted as 3 D2 . The quantity 2S + 1 is referred to as the multiplicity; it represents the number of angular momentum combinations for J that can be formed from a given S and L. For example, using the above values of S and L, J has possible values 3, 2 and 1, hence a multiplicity of 3 for this situation. Note that there are two dierent uses for the letter S here: one use represents the total spin, and the second represents the orbital angular momentum state L = 0. Regarding spin, states with S = 0 are called singlets, and states with S = 1 are called triplets. Initial and nal states of the three mercury lines (shown in Fig. 7) to be investigated in this experiment are: Green line: Yellow lines: = 546.07 nm, initial state (7s) 3 S1 , nal state (6p)3 P2 . = 576.96 nm, initial state (6d) 3 D2 , nal state (6p)1 P1 ; = 579.07 nm, initial state (6d)1 D2 , nal state (6p)1 P1 .
Here we note that the 576.96 nm transition in mercury violates one of the selection rules, namely S = 0. This is due to the angular momentum coupling not being purely LS, but a combination of LS and what is known as jj coupling, which eases the restriction on the S = 0 rule [3]. One can still apply the remaining rules in constructing a diagram for this transition. Note from Fig. 7 that a large majority of the transitions do obey the selection rule S = 0, as there are few transitions that cross the singlet/triplet boundary.
2
Measuring the energy shift: Fabry-Perot interferometry
The energy dierence between the initial and nal states in an atomic transition is determined by measuring the wavelength of the light emitted in the transition. Energy is related to wavelength by the famous equation E = h = hc/ , (16) where and are the frequency and wavelength of the emitted light, respectively, h is Plancks constant and c is the speed of light. Since the goal in this experiment is to measure the shift in atomic energy levels due to an external magnetic eld, then how is the shift measured? Given the magnitude of the magnetic eld available with our electromagnet (around 1000 gauss) and the size of the magnetic moments involved (around a few Bohr magnetons), the shift in wavelength one would expect to see would require an absolute measurement of the wavelength of emitted light to have a precision of better than 6 decimal places. Such a measurement is very dicult and would require sophisticated and very expensive equipment beyond the means of this laboratory. Instead of absolute measurements of the wavelength, we make a relative measurement of the wavelength: we measure the shift in wavelength due to the magnetic eld relative to the unshifted wavelength. To nd the energy dierence in terms of a small change in wavelength dierentiate Eq. (16) to obtain hc . (17) E = 2 The wavelength shift is measured with an instrument known as a Fabry-Perot interferometer. For a description of this device see the appendix of this write-up. Further details are given in the text by Preston and Dietz [2] and the optics text of Hecht [4]; several pages of the material from Preston and Dietz can be found on the class website. Be sure you understand the concepts 10
of nesse and free spectral range, and how mirror separation aects the latter parameter. A very brief description of the Fabry-Perot follows. The Fabry-Perot works by having light reect back and forth many times in a space bounded by two very at, highly reecting mirrors. The set of mirrors is called an etalon (see Fig. 8). The intensity of the light exiting the etalon is maximum when the phase shift of the light is an integer multiple of 2 after reecting from both mirrors (this is the requirement for constructive interference). Light of diering wavelengths satises this condition at dierent exit angles from the etalon resulting in visible rings of diering circumference for the dierent wavelengths. The output of the Fabry-Perot is not just one visible ring for each unique wavelength, but rather a repeating pattern of rings, successive patterns representing an additional phase shift of 2.
Figure 8: Schematic illustration of a Fabry-Perot interferometer and the ring pattern it produces on a screen when used with a monochromatic light source. The strength of the Fabry-Perot is its ability to resolve lines with closely spaced wavelengths. If min is the minimum resolvable wavelength dierence at wavelength , then for our instrument, /min 106 .
A limitation of the Fabry-Perot is that it can only work with wavelength dierences up to a certain maximum which turns out to be 2 max = , (18) 2d where is the nominal wavelength of the light entering the interferometer and d is the separation between the mirror surfaces. max is known as the free spectral range of the interferometer. At this wavelength dierence, the ring due to light with wavelength will just overlap the adjacent ring (remember, the pattern repeats!) due to light with wavelength . This kind of overlap will result in a ring pattern that is dicult to interpret, a situation that can usually be avoided by a judicious choice of mirror spacing. Note: The value for d is measured by a traveling microscope. You will need this number to determine the free spectral range, a key parameter in the analysis of your data. To measure the light intensity of a ring we use the central spot scanning technique in which a light detector, in our case a photomultiplier tube, detects the intensity of the central spot as the ring pattern collapses in to the center as we change the mirror separation. The photomultiplier tube 11
output current is converted to a voltage that is recorded by an X-Y plotter, resulting in a plot showing the intensity and separation of the various rings. Details on how to record this pattern follow in the next section. (Note: in our instrument we change the mirror separation by changing the voltage applied to a set of piezoelectric elements that support the mirror. The Fabry-Perot set described by Preston and Dietz changes the optical path length by varying the pressure of air, and thus the refractive index, between two xed mirrors.) Figure 9 shows a diagram of our apparatus. The mercury discharge lamp is located between the pole pieces of the electromagnet. Light from the lamp is collimated by lens L1 and then passes into the Fabry-Perot interferometer. Light exiting the interferometer passes through the focusing lens (L2) and then through the beamsplitter. An image of the ring pattern comes to a focus on the white disk on the detector, and on the entrance to the spectrometer. The beamsplitter arrangement oers the advantage of being able to visually monitor the output of the interferometer at the same time as the light intensity is being detected and recorded. The data acquisition procedure is described in detail in the next section.
Figure 9: Schematic diagram of the Zeeman eect apparatus.
3
Experimental Procedure
The Burleigh RC-150 Fabry-Perot interferometer is a precision instrument capable of resolving wavelength shifts of 0.001 nanometers or less. To realize the full capabilities of this instrument it is necessary to optimize the alignment of the interferometer mirrors, the external optics and of the detector hardware. The challenge of performing a full alignment (from scratch) of the optical system is both time consuming and dicult; it is not reasonable for the uninitiated student to attempt such a procedure. When you come to start the experiment, the setup should be mostly aligned. You will, however, need to ne tune the electronic controls in order to get usable data.
12
3.1
Apparatus alignment
To scan the ring pattern created by the Zeeman splitting of the green and yellow mercury lines, the spacing of the mirrors in the Fabry-Perot interferometer is varied in time. One of the interferometer mirrors is mounted on three piezoelectric elements, 120 apart, and the mirror spacing is changed by applying voltages to the three elements. The RC-42 Ramp Generator creates the voltages applied to these elements. A variety of knobs control how these voltages are applied: Bias 1, 2, 3 controls The oset (baseline position) of each element can be adjusted separately by the Bias 1, 2, 3 controls, and this is how the alignment of the mirrors is ne tuned, i.e., brought into the parallel condition. Ramp Bias control A simultaneous adjustment of the voltage to all three of the piezo elements is made with the Ramp Bias control. Varying this control results in a parallel translation of the mirror and movement of the ring pattern in or out from the center. Ramp Amplitude and Duration controls A scan is made by applying an internally generated ramp voltage to the piezos. This ramp voltage changes the mirror spacing uniformly in time, enabling one to record a steady change in the ring pattern under observation. Each time the ring pattern repeats as the ramp voltage increases, the mirror separation has changed 1/2 wavelength of the light under observation. The total change in mirror separation over one complete ramp cycle is adjusted with the Ramp Amplitude control. The maximum ramp amplitude setting allows for a change in mirror separation of 5 to 6 half wavelengths in the middle of the visible spectrum. For data taking purposes, scanning through 2 to 3 complete ring patterns is sucient for Zeeman eect observations. The rate at which the ramp voltage changes (and repeat cycles are produced) is set by the Ramp Duration controls consisting of a knob and Magnier switch. For our setup, we use set the switch to its 100 position to make scans with the X-Y recorder of about 20 seconds in duration. Reset and Ramp Stop/Go switches Because the scan speed is typically very slow, it is useful to be able to start the ramp manually in synchrony with the X-Y recorder. One can do this by pressing the Reset button which forces the ramp voltage to the beginning of the ramp. We have added another switch, the Ramp Stop/Go switch, which allows the user to stop the ramp at any point in order to ease the mirror alignment process. Trim 1, 2, 3 controls You should avoid turning these controls. The trim controls correct for variations in the response of each piezo element to the ramp voltage, so that the mirrors stay parallel as the ramp voltage is varied. These controls will have been set during the initial alignment of the instrument and should not need to be adjusted. Mirror alignment The procedure outlined below will enable you to bring the interferometer mirrors into alignment provided they are already reasonably close to being parallel. If the mirrors are too far out of alignment, a separate procedure must be followed. This more involved procedure will be performed by the Lab Manager, Prof., or TA. 1. Check that the magnet power supply (to the right and underneath the table) is o. If it is on, make sure the current is turned to zero and then turn the supply o. A ashlight is available to help see the switches and controls on this unit. 13
2. Check to see that the RC-42 ramp generator is on. If it is not, turn it on. Ideally, it should warm up for an hour or two, as the stability of the instrument is poor otherwise. Turn the Ramp Bias control fully counterclockwise (it has about 3 1/2 turns of adjustment, and the stops at the ends of the adjustment range have a very soft feel to them), and then turn it about 1 turn clockwise. 3. Set the Ramp Duration knob to 200 ms and the magnier switch to 100. 4. Set the Ramp Stop/Go switch to Stop, and press the Reset button once. 5. Before turning on the mercury discharge lamp, check that it is positioned between the magnet pole pieces. CAUTION: The voltage on this lamp is 5000 volts and the supply can provide a hazardous amount of current. Do not touch the lamp or holder while the voltage is on. Turn on the discharge lamp and allow approximately 5 minutes for it to warm up. 6. Check that there are no lters or polarizer in the beam path. The holders for these items are right in front of the magnet pole pieces. Check that the prism spectrometer is set to about 550 nm (dial is on right side of spectrometer body). 7. After the Ramp Generator and mercury lamp have warmed up, look through the prism spectrometer. If the interferometer mirrors are not too far out of alignment, you should see a vertical slice of a pattern of concentric rings. If the rings are highly smeared out or not visible at all, the mirrors may be so far out of alignment (i.e., non-parallel) that it will be necessary to go through the complete alignment procedure. Contact Lab Manager, Prof. or TA to perform this procedure. 8. With a ring pattern visible in the spectrometer, set the bias controls to halfway, i.e., 1 1/2 turns from completely counterclockwise. (Note: the limit of the knobs is soft: you will feel the resistance to turning increase suddenly, but it will not be enough to keep the knob from turning.) Look through the spectrometer. You should see some mainly green light, and if you are lucky, a clearly visible slice of a ring pattern. Adjust the focus of the spectrometer to bring the image of the slits (not the rings!) into sharp vertical lines bounding a (probably) fuzzy ring pattern. The reason to focus the spectrometer on the slit edges is to insure that the spectrometer focus is independent of the ring pattern focus: the spectrometer focus is set for your eye, the ring pattern focus is set for the instrument, since the ring pattern is also projected on the detector, and it must be in good focus there to get good data. 9. While you look through the spectrometer, adjust each of the bias 1, 2, 3 controls a little in either direction to see which one or two has the greatest eect on the focus of the green ring pattern. Choose the two most eective controls (typically 1 and 2), and adjust each one alternately to bring the pattern into best focus. When you have done as well as you can with just two knobs, adjust the third to improve the pattern further. Have patience: several iterations of adjustments may be necessary. Please be careful not to touch the three Trim controls underneath the Bias 1, 2, 3 controls. (The Trim controls keep the displacement of all three elements uniform, and therefore the ring pattern in focus, as the ramp voltage changes.) Note, if you try to adjust the pattern by using all three controls in a round robin manner, you may run out of ability to focus the pattern; it is best to make most of the adjustment with only two knobs.
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10. Adjust the Ramp Bias control so that the two faint rings just inside the inner bright green ring are visible. Adjust the Bias 1, 2, 3 controls to bring these faint rings into the sharpest focus possible. Since the data are taken by observing the intensity at the center of the ring pattern, it is important to optimize the focus of the rings closest to the center of the pattern. When these two faint rings are in sharpest focus (at best they will still be a little fuzzy), the rest of the pattern should also be in good focus. Note that as you adjust each of the knobs, you may see dierent parts of the pattern come into focus, but you should work to get the whole pattern into as best a focus as you can. Once you the get mirrors aligned you should see two patterns side-by-side in the spectrometer: the green pattern will be fairly bright, and the pattern from the two yellow lines will be visible to the right of it. If you look closely at the yellow pattern you should be able to see that there are in fact two overlapping sets of yellow rings whose centers are slightly displaced horizontally. Detector alignment check Look at the white disk around the entrance hole to the photomultiplier tube (PMT) housing on the far side of the beamsplitter. You should see a violet strip of light with no ring structure apparent. This strip of light is the result of allowing all the colors of the mercury spectrum to pass through the Fabry-Perot interferometer. Each line in the spectrum produces a ring pattern, and the violet strip is the result of all the ring patterns sitting on top of each other. For measurement purposes, one must isolate out the line of interest, just as the prism spectrometer does very nicely for visual observation purposes. In our apparatus, isolation of the line of interest is achieved with what is known as an interference lter, which is placed close to the discharge tube. We have two such lters, one of which passes the mercury green line (546 nm lter) and the other of which passes the two closely spaced yellow lines (577 nm lter). 1. Your rst measurements will be on the mercury green line, so now put the 546 nm lter in the lter holder (one closest to pole pieces). If you now look at the white disk (magnifying glass helps) you should see a faint vertical slice of the green ring pattern, just as in the spectrometer. To see this clearly, the room lighting must be very dim and you must be accustomed to the low light level. 2. Now follow this procedure to turn the high voltage supply for the PMT on (assumes that Fluke model 412A is in use): (a) Make sure that room light is extremely dim (barely enough light to see at all). Also make sure that a lter (start with green) is in one of the lter holders near the magnet gap. Each person in the group should have access to a ashlight to keep themselves from bumping into things and people. (b) Moving to the high voltage power supply (in the rack back and to your right) for the PMT, make sure that the Polarity switch is set to HV O and that the X100 knob is completely counter clockwise. (c) Turn the power switch (lower right of panel) to on. You will hear a (fairly loud) hum. (d) When the hum quits, ip the Polarity switch to NEGATIVE (). (e) then turn the X100 to 12 (straight up). The reading on the meter should be just shy of 1000 V. 15
3. Turn on the PDA-700 amplier and check that the range is set to .xxxx A (note the position of the decimal point; select this by the small Range knob). The photomultiplier tube output is a current proportional to the intensity of the incident light; the PDA-700 amplier converts this current to a voltage which drives the Y axis of the X-Y plotter. 4. Moving back to the apparatus, adjust the Ramp Bias control so that there is a small bright disk in the center of the ring pattern. This disk of light should fall on the center of the white disk covering the entrance hole to the PMT housing. Use the Ramp Bias control to expand the disk into a small ring and check, by visual observation, that this ring is centered on the hole in the white disk. If it is not close to being centered, consult with Lab Manager, Prof., or TA for assistance in adjusting the detector hardware. 5. The centering of the ring pattern on the entrance hole to the PMT housing can further be checked by observing the PMT output current as the Ramp Bias is adjusted to expand or contract the disk. The current observed on the PDA-700 amplier should peak at about the same time as the brightness of the disk peaks just before it disappears (look in spectrometer to observe disk contracting and then disappearing). If this is not the case, again consult with Lab Manager, Prof., or TA for assistance in adjusting the detector hardware. Getting ready to record a scan of the ring pattern Check that the Magnier switch is set to the 100 position and set the Ramp Duration switch to 1 s. Flip the Ramp Stop/Go switch to Go, and adjust the Ramp Amplitude knob while looking into the spectrometer so that the scan covers 2 to 3 complete cycles of the ring pattern. The rings will contract as the voltage to the piezoelectric elements is ramped up. At the end of the ramp the voltage returns very quickly to its initial value, the ring pattern snaps back to its initial conguration (so quickly you cant see it happening), and the scan begins again. The scan can be reset to the beginning by pushing the RESET button on the RC-42 controller. The position of the rings at the beginning of the scan can be changed with the Ramp Bias control. Note: If the ramp appears to hold at the beginning for a while, the Ramp Bias may be too low. Turn the Ramp Bias knob up enough so that the pattern begins to contract immediately after you press the RESET button. As a last touch up of the mirror alignment before you attempt a scan, stop the ramp (with the Ramp Stop/Go switch) near the midpoint of the cycle, and adjust the Bias 1, 2, 3 controls to focus the pattern as sharply as possible. You should not need to make signicant changes at this point. If you do, you may need to spend some more time adjusting the overall alignment as decribed earlier. In the following exercises where you are to obtain scans of the ring pattern, you should keep your eye on the rings to make sure that they are staying in good focus. The Fabry-Perot set (including the RC-42 ramp generator) tends to drift a lot when it is rst turned on, and somewhat less when it has been running for a few hours. Important: Do not try to adjust the focus when the magnet current is turned up. The Zeeman eect will cause the ring pattern to appear to be out of focus. It is best to ne tune the focus with the current o, and then turn it up shortly before you run a B = 0 scan.
16
3.2
Required observations and data collection
Green lines at B = 0 Record the ring pattern with the green (546 nm) lter in the lter holder with no magnet current on. You will use this plot to determine the nesse of the interferometer and as a check of the instrument alignment quality. With the Ramp Duration set to 1 s and the Magnier switch set to 100, typical (i.e., yours might be dierent) initial settings for the X-Y plotter are: X axis: Y axis: 10 sec/inch 50 mV/inch
Big huge hint: Before you make a scan, run the plotter without a pen and watch how the carriage moves. You will almost certainly need to adjust the controls in order to make a well-positioned curve. You should set the gain of the Y axis (use the VAR knob along with the switches) so that the peak heights are about one third of the page height and the X axis sweep encompasses 23 full cycles of the pattern. The X-Y recorder sweep is initiated by moving the slide switch from RESET to SWEEP. At the end of the sweep the pen will lift automatically and return to the left side of the recorder. You will have to manually coordinate the X-Y plotter sweep with the ramp cycle of the interferometer. Pushing the RESET button on the RC-42 and then initiating the recorder sweep usually works well. Compare your trace with the peaks shown in Fig. 10. If your trace looks like the one at the bottom of the gure, you are in good shape. But if the peaks are broadened so that you cannot see the ner structure, you need to work on the mirror alignment and try again.
Figure 10: Scans of the 546 nm (green) line of mercury with magnet current o with dierent degrees of mirror alignment quality. Bottom scan shows good alignment; note easily resolved ner structure at base of main peak. Middle scan shows acceptable alignment, but not optimum. Top scan shows unacceptably poor alignment.
17
Observe and record the anomalous Zeeman eect for the green line. Check to be sure that the current and voltage adjust knobs on the magnet power supply are set fully counterclockwise. Turn on the supply and the digital meter, and turn the voltage adjust knob fully clockwise. Slowly turn the current adjust knob clockwise and observe the beautiful display as the brightest green ring begins to split. Turn up the current so that all the rings are clearly distinguishable, but not so far that any of the rings produced by the Zeeman splitting overlap. Make sure that everyone in your group has a chance to observe the the ring separation through the spectroscope as the current is turned up. Now place the linear polarizer lter in the holder, and notice that some of the rings disappear, while others remain. Then remove the polarizer, turn it 90 , and reinsert it into the holder and look at how the pattern has changed. Again, everyone should have a chance to see this eect in the spectroscope. You might want to make sketch of the ring patterns that you see with the two dierent orientations of the polarizer. These phenomena are some of the most vivid examples of the consequences of quantized angular momentum. Take a moment to appreciate how the quantum theory that you learn is evident in these observations. When you are satised with the appearance of the pattern, you can run a scan and record the ring pattern on the X-Y recorder. You may very well need to readjust the Bias 1, 2, 3 controls in order to refocus the pattern. This is most easily done with a fairly large magnet current so that the splittings are well separated. IMPORTANT: Watch and record the magnet current as you run the scan. You will need this number in order to analyze the data. After recording this pattern put the linear polarizer in the beam path and separately record the pattern for both orientations of the polarizer, noting which pattern goes with which orientation. The arrow on the polarizer shows the direction of polarization. Again, keep track of the magnet current during each scan. Observe and record the normal Zeeman eect for the yellow lines Now you want to repeat the entire set of observations and measurements that you made for the green line for the yellow lines. First touch up the focus by looking at the green-line patternit is easier to see. Put the 577 nm (yellow) lter in a holder, and then remove the green lter. Record a scan with no magnet current. The yellow lines are not as intense as the green line, and it may be necessary to go to more sensitive scales on the PDA-700 and Y axis of the recorder. Next, observe through the spectrometer how the yellow lines split as the magnet current is turned up under three conditions: no polarizer, polarizer vertically oriented, and polarizer horizontally oriented. The two yellow lines are suciently close together that the 577 nm interference lter passes both of them. This means that a scan of the yellow lines must be done with some care; the yellow lines are shifted (slightly) in position in the spectrometer, but the ring patterns falling on the detector pinhole are not shifted at all. You will need to know which peaks correspond to which principal
18
yellow line. When you look through the spectrometer, you will see two patterns displaced slightly along the horizontal. The yellow pattern that is displaced towards the original green pattern has the shorter wavelength of the two. As the scan is running, watch to see when a given pattern is recorded, and mark the peaks on your scan accordingly. When recording a scan that shows the Zeeman splitting of these lines, it is helpful to keep the magnetic eld low enough so that it is clear which sets of rings go together. Fortunately, the ring pattern is simpler for the yellow lines than for the green line; the Zeeman splittings follow the model predicted by classical physicsthe normal Zeeman eect. Remember to record the magnet current during the scans for the yellow lines, and to get scans with the polarizer in both orientations. Obtain the mirror spacing At this point, you should have a total of eight scans: for each color you should have a scan with B = 0, a scan with B = 0 with no polarizer, and scans with B = 0 with the polarizer in each of the two orientations. Turn the high voltage supply to the photomultiplier tube o by following these steps: 1. Turn the X100 knob fully counterclockwise. 2. Turn the Polarity switch to HV o. 3. Turn the power switch o. Then turn the room lights back on. In order to analyze your data, you need to know the free spectral range of the Fabry-Perot interferometer. The free spectral range can be specied in terms of either wavelength or frequency; the formulas are max = max = 2 , 2d c , 2d (19) (20)
where d is the mirror separation, c is the speed of light, and is the (nominal) wavelength of light. These equations are derived and discussed in the appendix. The mirror separation is measurable with the traveling microscope mounted near the Fabry-Perot set. Place an alignment jig with a white card behind the Fabry-Perot mirrors, opposite the microscope. Peer through the microscope ocular, and you should see a light vertical band separated by two dark regionsthe mirrors. You may need to adjust the focus of the microscope tube so that the mirror edges are sharply dened, and you may also need to adjust the focus of the ocular in order to clearly see the crosshairs. Note that as you move the microscope position by means of the knob, the image will travel (oppositely) across your eld of view. To make a measurement, back the knob o suciently towards zero, so that both mirror edges are o to one side of the crosshair center. Then slowly advance the knob to park the exact center of the crosshairs on one edge. Record the position. Note: each full revolution of the knob is 1 mm, so each 19
small marking corresponds to 0.01 mm. You should be able to measure the position to 0.005 mm. Advance the knob to the other edge, and record its position. Then calculate the mirror separation. Repeat the measurements a couple of times to check for consistency. The mirror separation should be close to 1.7 mm.
4
Shut down
After you have completed your measurements, be sure to turn everything o. Turn the voltage and current controls on the magnet power supply to zero and the power supply o. Remove the lter and polarizer from the beam path and put them back in their storage boxes. Finally, cover up the optics with the plastic covers.
5
Data analysis
The basic data analysis tasks are presented below as a series of exercises. Your overall goal is to understand the scans made by the Fabry-Perot interferometer in terms of the underlying energy levels and transitions evident in the mercury spectrum. To a lesser extent, you should also understand the physics of the Fabry-Perot interferometer itself. Exercise 1 Calculate the free spectral range of the interferometer from your measurement of the mirror spacing. Then calculate the free spectral range in terms of a maximum energy shift E max in units of electron volts. This will simplify later analysis of your scans. Exercise 2 Consult the appendix on Fabry-Perot interferometry and learn the denition of instrument nesse. From your scan of the mercury green line with no magnet current, calculate an estimate (actually, a lower bound, since the line has its own native width) of the nesse of this instrument. Now you want to consider more carefully how the energy levels in the mercury atom relate to the Zeeman spectra. Exercise 3 The spectroscopic descriptions of the initial and nal states for the three mercury transitions of interest are given in Section 1.4.1. Use this information to construct a diagram similar to that in Fig. 6 for each of these three transitions. The diagrams should show the sublevels for the initial and nal states, and which transitions are allowed. Exercise 4 From the diagrams you created in the above exercise, calculate the predicted energy shifts in the Zeeman eect spectra by rst calculating the value g J for the initial and nal states in each transition (the green line and the two yellow lines). From Eq. (15) we nd that the energy shift of the spectral line Emagnetic is given by Emagnetic = (gJ i mJ i gJ f mJ f ) B B , (21)
where the i and f subscripts denote the initial and nal states of the transition. Then make a scale drawing of the predicted spectrum similar to the lower part of Fig. 6, where the position of the lines along the horizontal is proportional to the energy shift and the two polarizations are represented by solid () and dashed () lines. 20
Now you should be ready to analyze your traces. You will need to know the values of the magnetic eld strengths corresponding to the various magnet currents. You can nd these by using the magnet calibration curve in Fig. 11.
14 12 Magnetic Field (B) [kilogauss] 10 8 6 4 2 0 0 0.5 1 2 1.5 2.5 Magnet Current (I) [Amps] 3 3.5 4 B = 0.38639 + 3.3056*I + 0.31865*I - 0.082466*I
2 3
Figure 11: Calibration curve for the Zeeman eect electromagnet. Note the polynomial t equation which allows one to calculate the eld strength for any value of the magnet current. It is a simple matter to nd the wavelength shift of a Zeeman peak in an interferometer scan. You know that the wavelength or frequency shift corresponding to one free spectral range would correspond to the distance between two similar features. Thus, any peak that appears to be shifted less than a full free spectral range will have a wavelength or energy shift that is proportionately less. So to analyze a trace, you rst measure the distance between repeating features (typically the main peaks), and then calculate a conversion factor from the free spectral range. Exercise 5 Use a ruler to measure the locations and separations of the peaks in your scans to the nearest millimeter. Calculate conversion factors based on the free spectral range and the repeat distance between cycles. Then nd the energy shifts of the Zeeman-split peaks. Make tables of the energy shifts of the peaks, labeled in terms of the particular transitions you worked out in Exercises 3 and 4. Finally, you should compare your calculations to see if everything makes sense. Some questions to consider: Do the g factors seem right? In other words, if you use the known value of the Bohr magneton (B = 5.788 109 eV/gauss) do the energy shifts have the correct size? 21
Are the shifts from the dierent scans consistent with each other? If they are, you should be able to calculate B by combining your results into an average, assuming you have good g values. If you followed the prescription for calculating the predicted spectra, you should have found that the 3 D2 1 P1 (576.96 nm) yellow transition should give a more complicated spectrum than what is observed. Can you explain this discrepancy? What would you do, assuming you could go back to the experiment, to try to resolve it?
References
[1] R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, 2d ed., Wiley, New York, 1985. [2] D. W. Preston and E. R. Dietz,The Art of Experimental Physics, Wiley, New York, 1991. [3] H. Haken and H. Wolf, The Physics of Atoms and Quanta, Springer, New York, 1996, pp.304-5. [4] Hecht, E., Optics (4th ed.), Addison Wesley, San Francisco, 2002, pp. 414425.
Appendix: Fabry-Perot interferometry
REFERENCE Hecht, Optics (4th ed.), Addison Wesley, San Francisco, 2002, pp. 414425. To eectively use a Fabry-Perot (FP) interferometer as a spectroscopic tool, you need to understand the concepts of nesse and free spectral range. This section presents these concepts in a descriptive form with emphasis on the underlying physics; see Hecht, pages 416425, for the mathematical details. The Fabry-Perot interferometer uses the phenomenon of multiple beam interference that arises when light shines through a cavity bounded by two reective parallel surfaces. Each time the light encounters one of the surfaces, a portion of it passes through to the other side, and the remainder is reected back; the net eect is to break a single beam into multiple beams which interfere with each other. If the additional optical path length traveled by a (multiply) reected beam is an integer multiple of the lights wavelength, then the reected beams will interfere constructively. Conceptually, the multiple reections in the cavity interfere with each other in the same way th...