Optical Pumping of Rb 85 & Rb 87 Fleet Admiral Tim Welsh PhD. M.D. J.D. (Dated: February 28, 2013) In this experiment we penetrate the mystery surrounding the hyperfine structure of Rb 85 and Rb 87. We employ the use of optical pumping to explore the atomic energy states and transitions for each isotope concurrently. We demonstrate experimentally the low field and quadratic Zeeman effects, as well as qualitatively and quantitatively confirm the predictions of the Breit-Rabi equation. The nuclear spins I are determined for each isotope and found to be 2.5 ±.1 for Rb 85 and 1.5 ±.1 for Rb 87. I. INTRODUCTION II. THEORY The advent of high resolution spectroscopy in the form of the Michelson interferometer revealed new phenomenon in physics. A shining example of experiment informing theory; The optical fine structure was observed in 1881 by Albert Abraham Michelson [1]. He found that under sufficient resolution, spectral lines previously thought to be single lines, or even doublets, split further into many lines. This conundrum was made more enigmatic by the fact that not all spectral lines split further under increasing resolution. It could, however, only be explained in terms of quantum mechanics when Wolfgang Pauli proposed the existence of a small nuclear magnetic moment in 1924 [2]. Dubbed The Zeeman Effect the hyperfine structure is understood to be a result of the interaction between the magnetic moments of the nucleus and an external magnetic field. Perturbing the atoms with a uniform magnetic field allows the experimenter to study how energy level spacings change with changing magnetic field strength. This leads directly to the Quadratic Zeeman Effect, wherein the the change in energy level spacings, and according transmission spacing, ceases to be linear with changing B. The emergence of double quantum transitions is also observed in the quadratic domain, here the emission peaks become erratic and seemingly disordered. Optical pumping is a technique used to prepare a system in a desired state. In gas lasers optical pumping is used to produce a population inversion, in which many atoms have electrons trapped in meta-stable states. This population may be stimulated to decay all at once, producing an intense coherent stream of photons. In this experiment we use optical pumping to destroy undesirable transitions, with the effect that the transitions of interest are easier to observe. The pumping mechanism utilizes a discharge lamp, interference filter, linear polarizer, and circular polarizer in order to disallow certain electric dipole and magnetic dipole transitions. We will examine the Zeeman effect and the apparatus more closely in the following sections. a. Just a taste of quantum. Bound electrons are diffuse things, they may inhabit certain energy states but have no definitive position within an atom, and adjacent energy states do not overlap spatially. The energy state of an electron is more usefully described in terms of angular momentum, but we ll get to that latter. In addition to being stuck in an energy state, electrons also have 1 2 spin. The spin can be thought of as the electrons intrinsic magnetic moment alignment; it can be positive or negative. An electron may transition between bound states by absorbing or releasing energy in one form or another. Typically this is through the absorbtion or transmission of a photon (in this experiment for instance), but an electron may be promoted to a higher energy state during an atomic collision (if Feynman is to be believe this is just a photon interaction as well!). There are restrictions on what transitions are allowed however, and this depends strongly on the type of spin coupling involved, but we wont go in to that, instead we will consider one type of coupling, described shortly hereafter. An obvious fact, not unique to quantum mechanics is the property of conservation of energy and momentum. When a transition occurs, in this case we will refer to a photon emission, the total change in energy affected on the electron must be equal to the amount of energy carried away by the photon it emits. This must account for any change in angular momentum, potential, and kinetic energy. This is useful for spectroscopy (indeed I fail to imagine how it would exist without this fact!) in that atomic transitions may be directly observed through their absorption and emission spectra. b. Angular Momentum and Spin. The hyperfine structure is impossible to understand without first discussing the role of angular momentum and spin, and how they interact. As you may have realized, both bound electrons and the nucleus have their own angular momentum and spins. This leads to increasingly complex (read: tedious) interactions, but we need only talk about the electron for now. A bound electron has spin angular momentum S, and orbital angular momentum L. The total angular momentum is denoted J. As such the total angular momentum is simply the sum of the spin and orbital angular momentum:
2 J = S + L. (1) The nucleus only has spin angular momentum we ll call I. Now we have that the total angular momentum of the nucleus-electron system to be: F = J + I. (2) c. Rubidium is basically hydrogen with more protons, and also neutrons, and also electrons. As such it is a good choice to show that the basic assumptions that apply to the simplest quantum system, also apply to a complex quantum system. Rubidium s outer most electron acts very much like hydrogens. They both have L = 0 (S states are L = 0). Additionally because there is only one spin 1 2 electron in each, S = 1 2. Now by (1), J = 1 2. d. Z-Man As explained previously The hyperfine structure is the result of the interactions between the magnetic moments of the nucleus and the electrons. The strength of these moments is related, in the case of the electron to µ J, and for the nucleus, µ I. The resultant splitting can be seen in figure 1. H = A(I J) µ J J (J B) µ I (I B), (3) I where B is the applied magnetic field, J and I are angular momentums and µ J and µ I are the electron and nuclear magnetic moments. Solving the eigenvalue problem for this equation will yield the energy levels of the system. Proving this for atoms other than rubidium is left as an exercise for the reader. E = g F µ B M F B, (4) where µ B = 9.274 10 24 J T is the Bohr magneton. It will become useful later to find resonance frequencies in terms of g F and B, a substitution can change (4) into just such a form: v = g F µ B B/h (5) Neglecting I, the total hamiltonian (3) can be solved resulting in the Landé g-factor, g F. We then have that the Landé g-factor is: g J = 1 + J(J + 1) + S(S + 1) L(L + 1). (6) 2J(J + 1) But perhaps it was a foolish thing to neglect the nucleus. No matter we can fix that and attain a new term g F : FIG. 1. Hyperfine and Zeeman Splitting of the ground state and first excited state of Rb Imposing an external magnetic field increases the spacing between the levels, which while unperturbed is on the order of 10 8 smaller than between the 2S1 / 2 and 2P1 / 2 states. Each level has 2F + 1 degeneracy, meaning each state may decay 2F + 1 ways. This is easy to see from the figure. The Hamiltonian in the case of a applied external field (the Zeeman hamiltonian is negligible otherwise) will take the form of a simple sum. Just add the hyperfine and Zeeman hamiltonians like so: g F = g J F (F + 1) + J(J + 1) L(L + 1) 2F (F + 1) (7) This allows us to calculate nuclear spins from a known g F value, which is important for our final result. Now because the energy level equation (4) was so general, we may simply substitute g F for g J to gain the energy levels while accounting for the nucleus and the electrons, which is important. But we are still not done. The energy levels given by (4) are for the ideal, unperturbed case, which is never the case. Basically anything serves to perturb the hamiltonian, quantum effects, relativistic effects, gravitational, you name it. In this case we shall only consider the perturbation cause by the external magnetic field. This effect is seen quite quickly in experimentation, (4) tells us that the spacing between energy levels due to the low field Zeeman effect should change linearly with increasing B forever. This is not what is observed, at high field strengths the spacing between energy levels becomes nonlinear and perverse. Fear not however as perturbation theory was conceived to solve just such problems, in this case the solution is the Breit-Rabi equation: W (F, M) = W 2(2I + 1) µ I I BM± W 2 1 + 4Mx 2I + 1 + x2. (8)
3 With W and W being the interaction energy of the nucleus and electrons, and the hyperfine splitting energy, respectiely. Deriving this result is again left as a simple exercise for the reader. Note in (8): and x = (g J g I )µ B B, (9) W g I = µ I µ B I. (10) Figure 2 shows a plot derived from the Breit-Rabi equation (8), detailing W and W vs. B: light and pumping scheme such that only M = ±1 is allowed. This allows us to observe the Zeeman effect and determine the rubidium nuclear spin. More on that in a moment. III. EXPERIMENT A. Apparatus A schematic of the apparatus can be seen in Fig.(3). FIG. 3. Optical Pump. FIG. 2. Breit-Rabi diagram. Energy level spacing becomes nonlinear in the high field region. You can see quite clearly from Fig.(2) the energy spacing between levels is essentially the same for each level and increases linearly with increasing B in the low field region. Eventually the field becomes so strong that the space between energy levels no longer scales linearly with B. One level goes so far as to peel off and begin descending with the levels below it, which is quite peculiar behavior. This is the Quadratic Zeeman Effect, correctly predicted by the Breit-Rabi Equation (8). e. Optical Pumping A bound electron may only undergo a transition if it is an allowed transition. There are a multitude of transitions such as electric dipole, electric quadropole, magnetic dipole, and magnetic quadropole, but the probability that the electron will undergo such a transition is hierarchical in nature. There are a number of selection rules which determine what is an allowed transition, and the set of selection rules depends on the type of spin coupling involved in a system. A typical set of rules for a system like this (LS coupling) would be: M = 0, ±1; L = ±1; S = 0; J = 0, ±1 (11) These rules are relatively self explanatory. However it is important to note that the optical pumping apparatus used in this experiment manipulates the incident An RF discharge lamp shines through an interference filter(if), linear polarizer and quarter wave plate in that order. The IF acts as a notch filter, only letting through our wavelength of interest, which is 795nm. This line is specifically chosen because it stimulates the transition from 2 S frac12 to 2 P frac12 but not to 2 P frac32. Which is a good thing, because 2 P frac32 has a level where M F = 3. This would destroy the optical pumping effect as electrons in the 2 P frac12 with M F = 2 could transition to the 2 P frac32 M F = 3 state. So all the electrons eventually get trapped in the 2 P frac12 M F = 2 state and become transparent to incident photons. The linear polarizer is necessary, as the half wave plate circularly polarizes all incident light, if the light has many directional components the effect will be lost. The circularly polarized light establishes the special selection rule M = ±1 which is rather important for our experiment. After the light is polarized it impinges on the rubidium cell, and finally a photodiode for collection. There are a few sets of helmholtz coils which are used to nullify the Earth s magnetic field, and record the zero field transition. The Earth s B field is in essence an imposed magnetic field, thus is causes slight hyperfine splitting via the zeeman effect. Nullifying this field destroys the level splitting and thus the optical pumping, the gas is no longer transparent to the discharge lamp. The helmholtz coils are also used to push the rubidium gas into the high field region, for study of the quadratic Zeeman effect. Finally one of the coils is oscillated in the RF range and allows for the observation of double quantum transitions. f. Weak Field Zeeman Effect The lamp was heated to 45 C, and the zero field transition was observed by canceling out the Earth s magnetic field. The horizontal set of coils was swept at RF frequencies between 64kHz and 114kHz in 10kHz increments. As the field was swept
4 in strength a series of dips were observed corresponding to Rb 85 and Rb 87. Using the voltage ramp shown on the oscilloscope as well as attaching a multimeter to the pump controller allowed us to calculate the field at the center of each dip as follows: B(Gauss) = 8.991 10 3 IN R, (12) where I is current, N is the turns in the coil, and R is the radius of the helmholtz coils. The current is easily got from ohms law, and consulting the lab manual for intrinsic resistances. From the coil current it is straightforward to calculate the magnetic field from (12). A table of values can be found in the results section. gives two slopes, the ratio of these slopes comes out to be 1.45, which is quite close to the theoretical value of 1.5. Calculating g F from (5) for both nuclear spins I = 5 2 and I = 3 2 we find values of.34 ±.01 and.49 ±.01 which are very close to the theoretical values of 1 3 and 1 2 found with (7) and within uncertainty. B. Quadratic Zeeman Effect IV. TABLES TABLE I. Low field Zeeman transitions B Field Frequency a 85 Rb b 87 Rb b 64 0.127 0.011 74 0.144 0.038 84 0.166 0.071 104 0.204 0.121 114 0.227 0.155 124 0.249 0.183 a Frequency is in KHz b Magnetic field B is in Gauss V. RESULTS FIG. 5. Simulation of the Breit-Rabi equation for Rb 85 (red) and Rb 87 (blue). If given more than 18 hours to completely redo this paper the author may have had time to plot experimental data on this simulation. A. Weak Field Zeeman Effect FIG. 4. The slope of each line is that isotopes g F value. B is the calculated for each transmission peak in the weak field region calculated using (12).Plotting this table
5 VI. CONCLUSION We show good agreement between theory and experiment for the low field Zeeman effect, calculating accurate nuclear spin values of 2.5 ±.1 for Rb 85 and 1.5 ±.1 for Rb 87. As well as an appropriate ratio of g F s. The qaudratic zeeman effect is demonstrated and simulated but calculations relating to the agreement of theory and experimentation is not forthcoming. [1] http://www.aip.org/history/gap/michelson/michelson.html [2] The Historical Development of Quantum Theory Jagdish Mehra, Helmut Rechenberg (2000)