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Determining g-factors in Rubidium-85 and Rubidium-87 using optical pumping Will Weigand,1 Adam Egbert,1 and Hannah Sadler1 University of San Diego (Dated: 25 September 2015) In this paper we quantify the Zeeman effect in rubidium. We find the g-factor for Rb-85 to be gF = 0.0114 ± 1.0552 ∗ 10−5 and for Rb-87 to be gF = 0.0171 ± 7.0506 ∗ 10−5 . The theoretical values of gF are 13 and 12 respectively. The nuclear magnetic quantum number for Rb-87 is I = 57.98 and for Rb-85 I = 87.22. The accepted values for I are 23 and 52 respectively. I. INTRODUCTION Optical pumping allows us to probe the quantum world and understand the Zeeman effect, the splitting of energy levels when an atom is perturbed with a magnetic field. Alfred Kastler won the Nobel Prize in 1966 for using optical pumping to determine nuclear magnetic moments. The ideas of optical pumping are useful in the designs of MRI and lasers. Conceptually, optical pumping is easy to understand. Suppose we have a two-state system consisting of a set of particles that are distributed evenly between the two states. We can define a set of rate equations that describe the transitions between the two states of this system. Now suppose that the rate of transition from state 1 to state 2 is greater than the reverse. Over time, all the particles in state one will end up in state 2. This is optical pumping: disturbing the thermodynamic equilibrium of a collection of atoms. From a quantum mechanical viewpoint, the way to force a transition rate to be greater or lesser than another is to have a selection rule that has an associated ’forbidden’ transition. For our experiments the selection rule of concern is 4M = +1 where M is the magnetic quantum number. This rule only applies to absorption of photons which means that there will exist a state that cannot absorb photons but it is not forbidden to decay into this state which is exactly what optical pumping requires. In this paper we will investigate optical pumping in a Rubidium gas. Section II will lay down more of the theory of optical pumping. Section III will explain the experimental apparatus and measurements. In Section IV we present and discuss our results. electrons total angular momentum and the nuclear spin, there is an additional third energy level splitting from the application of an external magnetic field. This is the so-called Zeeman effect and will be the focus of our theoretical discussion. The Zeeman effect can be divided into three different regions: weak, intermediate, and strong, dependening on the relative strengths of the magnetic field interaction and the spin-orbit coupling. The perturbing Hamiltonian including spin-orbit coupling, spin-spin coupling, and the Zeeman effect is: H = H0 + AL · S + BSp · Se + Hz . Our main concern is the Zeeman perturbation which reads: µJ µI Hz = haI · J − J ·B− I · B, (2) J I where µJ is the electron magnetic dipole moment and µI is the nuclear magnetic dipole moment. If we ignore the nuclear magnetic moment, which is a good approximation when B is small, we can define the Lande g-factor to be gJ = 1 + (j(j + 1) + s(s + 1) − l(l + 1))/2j(j + 1). Adding in the interaction with the nucleus gives a new g factor, gF , which is defined as: gF = gJ f (f + 1) + j(j + 1) − i(i + 1) . 2f (f + 1) THEORY The concepts behind optical pumping arise from quantum mechanical perturbation theory.1,2 In addition to the spin-orbit coupling, the interaction of an electron’s intrinsic spin and its orbital angular momentum, and the spin-spin coupling, the interaction between the (3) Both of the above g factors along with the following interaction energies can be determined from either the vector or matrix models of quantum mechanics. When taking into account both the electronic and nuclear magnetic dipole moments, and the magnetic field is small, we can define the interaction energy as: W = gF µ0 BM, II. (1) (4) where M is the component of electron spin along the B field and µ0 is the Bohr magneton. Equation 4 is by far the most useful equation for quantifying the linear Zeeman effect. Suppose we take a sample of rubidium gas and impinge on it a certain frequency of light, as described in Section III, then by the Planck-Einstein relation the interaction energy is simply proportional to 2 the frequency, ν. Thus, one can determine the g factors and finally the nuclear spin quantum number. In the case where the external magnetic field is strong equation 2 can be diagonalized to retrieve the Breit-Rabi equation: 4W µI 4W 4M − BM ± [1 + x + x2 ]1/2 . 2(2I + 1) I 2 2I + 1 (5) Here we define 4W to be the interaction energy due to µ0 B µI . hyperfine splitting, x = (gJ − gI ) 4W and gI = − Iµ 0 The quadratic Zeeman effect splits the energy levels into 2F+1 sublevels with unequal spacing, where F is the atoms total angular momentum of the atom. In addition to the energies involved there are various selection rules that define the different allowed transitions between energy levels. In optical pumping experiments we exploit the 4M = ±1 selection rule. W =− III. METHODS To observe optical pumping we use an optical pumping device supplied by TeachSpin, Inc. A rubidium discharge cell containing equal amounts of Rb85 and Rb87 and Xenon buffer gas is heated to 323K. This gas is simultaneously excited an oscillatory frequency of 100Mhz. This allows for the gas to ionize enabling electrons to excite rubidium atoms by collisions. This produces two resonance lines at 780nm and 795nm. The resonance light then passes through an interference filter that removes the 780nm light. An interference filter is an optical device similar to a low pass or high pass filter built in such a way that it blocks off all light above and below the desired frequency. The light is then right handed circularly polarized through the use of a linear polarizer and a quarter-wave plate. Right handed circularly polarized light forces 4M = +1 meaning that there will exist an energy that that cannot absorb light. After the light has been circularly polarized it enters the rubidium absorption cell. The cell is surrounded by two Helmoltz coils. One coil produces a horizontal magnetic field and the other produces a vertical magnetic field. These are necessary to cancel out the vertical and horizontal parts of the Earth’s magnetic field. We orient the pumping device along the third component of the Earth’s magnetic field. The magnetic field, in units of gauss, can be measured by the equation for the magnetic field in a Helmholtz coil: 8.991 ∗ 10−3 N I , (6) R̄ where N is the number of turns in the coil, I is the current, and R̄ is the mean coil radius. An RF magnetic field is used to examine both the linear and quadratic Zeeman effects. After passing through the absorption cell the transmitted light is collected by a photodiode which converts the intensity of light to a voltage. B= FIG. 1. A schematic of the optical pumping equipment with all parts labeled. We conducted two experiments using the optical pumping apparatus. The first experiment we conducted was used to determine the nuclear spin quantum number. To accomplish this, we first determine at which current the zero field transition occurs. We can calculate the magnetic field produced by this current using equation 6. This is numerically equivalent to the Earth’s magnetic field in that direction. The RF frequency is then varied from 40kHz to 100kHz and the currents where absorption increases are recorded. The g factors, and therefore the nuclear spin quantum number, are calculated from Equation 4. IV. RESULTS AND DISCUSSION We find that the ratio of g-factors is 1.49. The calculated g factor for Rubidium 87 is gF = 0.0171 ± 7.0506 ∗ 10−5 . The calculated g factor for Rubidium 85 is gf = 0.0114 ± 1.0552 ∗ 10−5 . The g factors were calculated from the slope of the plot in Figure 2 of the frequency versus the magnetic field. This slope The magnetic field is defined to be the magnetic field where resonance is determined to occur minus the zero field. These values are approximately 30 times less than the expected gf values of 12 for Rubidium 87 and 31 for Rubidium 85. Through the use of equation 3 and considering that J = 12 we find that the nuclear spin quantum number, I, for Rubidium 87 is I = 57.98. Likewise for Rubidium 85, I = 87.22. These values are about a factor of 30 off from the accepted theoretical values of 23 and 52 respectively. The large variation from the theoretical values is difficult to pinpoint. As stated in the previous paragraph the ratio of the g-factors was 1.49. This is within 0.01 of the theoretical value of the ratio. This leads us to believe that a coil was connectedthat should not have been. If an unwanted coil was on this would mean that the cancellation of Earth’s magnetic field may not have been calibrated correctly leading to the incorrect slope values but a correct ratio. 3 V. CONCLUSIONS Optical pumping is a unique and novel way to determine small magnetic fields. Through our understanding of optical pumping we were able to determine the quantum numbers of Rubidium atoms which teaches us much about the atom. Our results confirm theoretical predictions on the ratio of g-factors but leaves up to debate the correct values. ACKNOWLEDGMENTS The authors would like to thank the USD Physics and Biophysics department for providing the optical pumping equipment. FIG. 2. Plot of resonant frequency versus the magnetic field of transition. 1 Optical Pumping of Rubidium OP1-A. (TeachSpin Inc., Buffalo,2002). 2 D.J. Griffiths. Introduction to Quantum Mechanics. (Pearson Education Inc., Upper Saddle River, 2005). Annalen der Physik, 322(10):891921, 1905.