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Optical Pumping of Natural Rubidium Erick Arabia University of San Diego (Jonathon Garamilla and Enrique Lance) (Dated: March 26, 2011) A gas of natural Rubidium, which consists of Rubidium-87 and Rubidium-85 was optically pumped in order to determine the nuclear spin number (I) for each isotope experimentally, as well as to confirm the Breit-Rabi equation for large magnetic fields. The g-factor was determined to be 0.47 for Rb-87 and 0.33 for Rb-85, and the results of the large field calculations can be found in Table (II). I. INTRODUCTION In 1950, French physicist Alfred Kastler introduced the concept of modern optical pumping, for which he was awarded the Nobel Prize in 1966. The process of optical pumping involves using photons of a desired wavelength and polarization to change the distribution of a gas of atoms from thermal equilibrium. Today, optical pumping is a critical tool in modern Physics experiments, and is the basis of lasers. This process utilizes quantum mechanical ideas, and can therefore be used to gain a better understanding of quantum and atomic theory. In this experiment optical pumping processes were performed on natural Rubidium gas, which is a mixture of Rb-87 and Rb-85. Because Rubidium only has a single valence electron, it can be approximated as a singleelectron atom (i.e. Hydrogen). Optically pumping the Rubidium sample achieved two main goals, which was the determination of the nuclear spin angular momenta (I) for each isotope, as well as the confirmation of Zeeman splitting due to a magnetic field according to the Breit-Rabi equation. In Section (II), the theory behind the optical pumping is discussed. Section (III) explores the experimental setup and procedure. The results and their analyzations are in Section (IV), and the conclusions can be found in Section (V). II. A. THEORY Atomic Structure In order to understand the concept of optical pumping, it is first necessary to understand a few basic principles about atomic structure. Because Rubidium is an alkali atom, it only has a single valence electron. That is to say that all the electrons are paired except for one. It follows that it can be approximated as a hydrogen-like atom, with a single free electron. This electron has three important quantities associated with it: the orbital angular momentum L, the spin angular momentum S, and the total electron angular momentum J. These numbers are quantized, are all in values of h̄, and are related by J = L + S. (1) Each electron is described in spectroscopic notation by 2S+1 LJ ; the valence electron in Rubidium is therefore designated by 2 S1/2 . This corresponds to J = 1/2, L = 0, and S = 1/2. It is also important to consider the angular momentum of the nucleus. The nuclear spin I combines with J to form the total angular momentum number, F. This coupling has a few important properties. First of all, because of the interaction of the nucleus and the electron, the quantized energy levels of the atom are split into 2S + 1 sublevels for each energy state of the electron (known as Hyperfine Splitting). Second, this splitting is degenerate, meaning that atoms with slightly different quantum numbers will occupy the same Hyperfine split states, which are designated by F. It should be noted that the difference in energy between the F = 1 and F = 2 states is insignificant compared to the difference between the energies of the ground state and first excited state. B. Zeeman Splitting The degeneracy in the Hyperfine splitting described earlier becomes important when a magnetic field is introduced to the atom in question. When this occurs, the F energy states are split further into 2F + 1 sublevels, designated by the new quantum number M. At very small magnetic field levels, the difference between the Zeeman states is approximately equal for each value of F, and as the magnetic field is increased the difference between the states also increases linearly. As is the case when comparing the energy levels of the Hyperfine splitting and the electron energy states, the Zeeman splitting is significantly smaller than the Hyperfine splitting. However, as the magnetic field is further increased beyond the low-field levels, the Zeeman split energy states begin to change in a non-linear fashion. These changes are designated by the Breit-Rabi equation, 1/2 ∆W µI ∆W 4M 2 W (F, M ) = − − BM ± 1+ x+x , 2(2I + 1) I 2 2I + 1 (2) 2 FIG. 2: Allowed transitions due to photon absorption during the experiment. Note that while electrons in the M=2 state cannot absorb photons, excited electrons can relax back into this state. D. FIG. 1: Plot of the Breit-Rabi equation. The horizontal axis shows the magnetic field strength as the dimensionless unit x, and the vertical axis shows the energy as the dimensionless unit W/∆W. Individual lines are designated by their quantum numbers as |F M > where W is the interaction energy and ∆W is the Hyperfine energy splitting. Furthermore, µ0 B , ∆W µI gI = − . Iµ0 x = (gJ − gI ) (3) (4) A plot of the Breit-Rabi equation, using the dimensionless axis W/∆W (energy) versus x (magnetic field strength) can be seen in Figure (1). C. Photon Absorption As photons are absorbed by the atoms, the electrons are excited to higher energy states designated by the energy level of the particular photon absorbed. When the electrons relax, they re-emit a photon corresponding to the drop in energy they undergo. It is important to understand that the re-emitted photon does not have to be of the same energy of the absorbed photon. Furthermore, because the possible energy levels of the electron are quantized, and because of the interaction of the quantum numbers, there are certain selection rules that designate how an atom can absorb and emit a photon. The rules for an atom in a magnetic field are ∆S=0, ∆J=0, ±1, ∆L=0, ±1 (but not L=0 to L=0), ∆F=0, ±1, and ∆M=0, ±1. Optical Pumping In this experiment the light introduced is of a very particular wavelength and therefore energy. The light introduced to the sample of natural Rubidium will only induce changes between the ground state and the first excited state. Because this energy is so much larger than the Hyperfine and magnetic splitting energies, the selection rules for F and M still apply (and in fact are the only important selection rules as other transitions are not possible). The light introduced will also be circularly polarized, which means that it will have a very particular angular momentum. It follows that for the atoms absorbing these photons, only transitions of ∆M=+1 are possible (however the normal selection rules for M still apply when the electrons relax and re-emit photons). The allowed transitions are shown in Figure (2). Because the incident photons are circularly polarized, only transitions of ∆M = +1 are possible, and therefore any electron in the M=2 state will be unable to absorb a photon. Any electron in this state will be unable to leave. However, the excited electrons do not have this constraint when they re-emit a photon during their relaxation process, which means that transitions into the M=2 state are allowed. It follows that eventually all the electrons will occupy the M=2 state, and the incident light will not be absorbed. In a sense, the sample of natural Rubidium gas becomes “invisible” to the incident light! This is process is referred to as optical pumping. The optical pumping will break down in certain situations. First of all, if the Rubidium atoms collide with the walls of the container, it is possible to force electrons from the M=2 state (more on this later). Second, if the magnetic field applied to the sample is just strong enough to cancel out the magnetic field of the Earth, then there will be no net magnetic field at the sample and the pumping will also break down (if there is no magnetic field then there is no Zeeman splitting). Because the strength of the magnetic field applied in this experiment is actually swept over a wide range, this breakdown is observed and is referred to as the “zero-field transition.” 3 Transmitted Light Intensity bidium gas, two perpendicular pairs of Helmholtz coils, a set of RF pulse coils, a Silicon photodiode detector, and several lenses and optical filters. These components were all arranged along a straight track. Additionally, there was a base unit for controlling the apparatus (the strength of the magnetic fields, the temperature of the Rubidium lamp, etc.), an oscilloscope for reading the photodetector output, a computed connected to the oscilloscope capable of printing the oscilloscope data, and a wave function generator. The experimental setup is shown in Figure (4). Magnetic Field 1. FIG. 3: Observed resonances during the low field optical pumping trials. The large dip to the left represents the zero-field transition, and the two dips to the right represent the Rb-87 and Rb-85 Zeeman resonances, respectively. Note that the magnetic field strength increases to the right and the transmitted light intensity increases upward There is another way in which the optical pumping can be broken down, and that is by introducing an RF signal perpendicular to the path of the incident light. By doing this at the resonant frequency for the M=2 to M=1 transition, it is possible to induce transitions from the M=2 state and thus to temporarily break the optical pumping. This has a few important implications. Because the frequency needed to cause this transition is proportional to the applied magnetic field with relation to the particular atomic structure of the atom in question, it is possible to experimentally determine the quantum numbers of the atom. And because the atomic structure of Rb-87 and Rb-85 are different, the energy gap corresponding to the M=2 to M=1 transition is different. It is therefore possible to observe both dips corresponding to each isotope’s resonant breakdown of the optical pumping, as shown in Figure (3). Furthermore, when a large magnetic field is applied to the sample, it is possible to break down the pumping even further and elicit transitions between all the split Zeeman states. In this way the full nature of the Zeeman splitting can be observed (however it should be noted that the transitions that occur from ∆F = ±1 take place at several GHz and will not be observed in this experiment). The lamp was filled with natural Rubidium gas and a small amount of gaseous Xenon as a buffer. This was done so that the collisions between the Rubidium atoms and the walls of the container were minimized. The gases in the lamp were heated and thus excited, and as they relaxed photons relating to the relaxation energy were released. It should be noted that this results in a broad spectrum of emitted light, but this was filtered by the optical setup described later. The natural Rubidium in the sample also contained a buffer gas, which in this case was Neon. Again, the buffer gas was used to minimize the collisions with the walls of the container. If this were not done, collisions with the container would result in photon emission, which would destroy the optical pumping. 2. Optics The optical setup consisted of two plano-convex lenses, an interference filter, two linear polarizers, and a quarter wavelength plate. As the Rubidium lamp was heated, the gas emitted two main lines at 780nm and 795 nm (due to the two isotopes found in natural Rubidium). The interference filter removed the 780nm line, and the remaining light was linearly polarized. Next, the quarter wavelength plate turned the linearly polarized light into circularly polarized light before it was introduced to the natural Rubidium sample in the cell. This was all done to ensure that the electrons were excited by photons of a precise energy and angular momentum. The two planoconvex lenses were used to focus the light. 3. III. Natural Rubidium Helmholtz Coils EXPERIMENT AND APPARATUS A. Apparatus The TeachSpin optical pumping apparatus was used for all experiments performed. The apparatus consisted of a Rubidium discharge lamp, a sample of natural Ru- There were three main sets of magnetic fields applied to the Rubidium sample. These were calibrated to counteract the effect of the Earth’s magnetic field along the x, y, and z axis of the sample. The entire apparatus (i.e. the track with the sample, lamp, etc.) was alined along one of the Earth’s magnetic axis. The vertical 4 FIG. 4: Overhead view of the experimental setup. Helmholtz coil was adjusted until it produced a magnetic field just strong enough to counteract the vertical field of the Earth. Both of these processes were done by making adjustments until the observed zero-field dip was as deep and narrow as possible. It should be noted that these adjustments did not have any effect on the numerical values of the data, but rather they simply made the dips easier to find and read. The other fields generated were the main horizontal field and the horizontal sweep field. Both fields were perpendicular to the vertical field axis and parallel to the track (and thus both the main and sweep fields were coaxial). The sweep field started from a predetermined value and increased linearly for a predetermined range (the maximum current possible through the coils was 1A). The time the sweep coil took to complete a full sweep was also adjustable. Therefore, by controlling the value of the start field, the range over which the field was swept, and the time it took to complete a full sweep, it was possible to observe many of the characteristics of the optically pumped Rubidium. The main field was simply a more powerful horizontal field coaxial with the sweep field. This was applied in order to observe the quadratic Zeeman splitting at higher fields, which the sweep field was not able to find on its own. Both horizontal fields had a sense resistor on the base unit, across which it was possible to read the voltage difference and thus current through the coils by use of V = IR. The sense resistor for the main field was 0.5 Ω, and the voltage in the sweep coils was read across a 1 Ω resistor. 4. B. 1. Low Field Resonance Determination of I First, the value of the residual magnetic field was determined. Because the orientation of the apparatus and the strength of the vertical field only counteracted two axial components of the Earth’s magnetic field, the zero-field transition was actually not observed at B= 0. The voltage across the sense resistor of the sweep coil (and thus the current through the sweep coils) was measured as the field was slowly swept through the zero field transition. The strength of the sweep field was roughly determined by B = 8.991 × 10−3 IN/R̄, (5) where B is the magnetic field strength in gauss, I is the current through the coils in Amperes, N is the number of turns of coil on each side (11 for this setup), and R̄ is the mean radius of the coils (which was 0.1639 m). The value determined to be the strength of the residual magnetic field was then subtracted from all future magnetic field values determined by Equation (5). Next, an RF signal was introduced in order to generate transitions from the M= 2 to M= 1 state for each isotope. Because the two isotopes have different g-factors (and thus different I values), this resulted in two separate observed dips. The RF signal was applied at six separate frequencies ranging from 40 KHz to 65 KHz; and for each frequency the current through the sweep coils was measured at the two observed dips. From this data, plots were created of transition frequency versus the magnetic field strength, and the data was fit to two separate straight lines. ν = gF µ0 B/h RF generator The RF generator was applied perpendicular to the horizontal and vertical magnetic fields. It was also connected to the wave function generator. Therefore it was possible to apply an oscillating RF signal of a known frequency and amplitude. This was done to allow transitions from the M= 2 state, which made it possible to observe the transitions between the Zeeman states. Procedure (6) was used to determine the Lande g-factor for each isotope, where ν is the frequency in MHz, µ0 is the Bohr Magneton, h is Plank’s constant, and B is the magnetic field in gauss. Once the g-factors were determined, gF = gJ F (F + 1) + J(J + 1) − I(I + 1) 2F (F + 1) (7) was used to determine I, as gJ is known for both isotopes. Transmitted Light Intensity Transmitted Light Intensity 5 Magnetic Field Magnetic Field (a) The first 5 dips (b) The second 5 dips FIG. 5: Observed resonance at high magnetic fields. The magnetic field strength increases to the right, and the transmitted light intensity increases upward. Note that there are only 6 total dips, and therefore a 4 dip overlap between figures. 2. Quadratic Zeeman effect For the next experiment the accepted g-factor values were used, along with the data already taken, to generate a more precise equation for the magnetic fields with respect to current. The resonance equation (Equation (6) from above) was used on all the low field data to create a plot of the magnetic field strength versus the current through the sweep coils. Next, the main horizontal coils were turned on in a direction opposite the sweep coils. The main field was then increased slightly, and the sweep coil was also increased in the opposite direction so as to center on the zero-field transition. Because the strength of the magnetic sweep field with respect to the current through the sweep coils was known, the main coils were therefore generating a magnetic field of equal (though opposite) strength. By repeating this procedure over a few points, it was possible to generate another plot for the main magnetic field strength versus the current through the main coils. By doing this, two equations were created (one for the sweep field and one for the main field) that gave a precise measurement of the magnetic field strength. Next, the sweep field was centered on the Rubidium87 dip, and the main field was turned to zero. By slowly increasing both the main field and RF pulse frequency, it was possible to “follow” the dip out to a high field region. In this region transitions between the 2F+1 sublevels were observed as six separate dips. The strength of the combined magnetic fields was determined at each dip, and the results compared to the Breit-Rabi equation. The resonant dips can be observed in Figure (5). IV. RESULTS AND ANALYSIS A. Low Field Determination of I The zero field dip was observed to occur at 314 mV, which corresponds to a current through the coils of 314 mA and a magnetic field of .189 gauss. This value was then subtracted from the values of the magnetic field calculated from Equation (5); the final calculated values can be found in Table (I). The data in the fourth and fifth column were plotted against the frequency at which they were observed to form the straight lines described earlier. From this, the Lande g-factor of Rb-87 was determined to be 0.48, and the g-factor of Rb-85 was determined to be 0.33. The accepted values are 1/2 and 1/3, which is in good agreement with the data. Equation (7) yielded nuclear spin values of 3/2 for Rb-87 and 5/2 for Rb-85. B. RF Spectroscopy The sweep field was calibrated as described in Section (III), which resulted in the calibrated equation B = −0.187 + 0.595I. Note that the residual magnetic field from the Earth is taken into account when I = 0. Likewise, the calibrated equation for the main field was B = 6.829I. The quadratic splitting was observed at a frequency of 4.064 MHZ, and a main field voltage of 506.8 mV. This corresponded to a 6.921 gauss magnetic field. The data from the spectroscopy measurements can be found in Table (II). The calculated values are actually not in good agreement with those predicted by the Breit-Rabi equation, which were all around 5.8 gauss. 6 TABLE I: Low field data used to determine the g-factor for each isotope. There is an uncertainty of ±0.5 mA for each current measurement, which corresponds to a ±3 × 10−4 uncertainty in the magnetic field calculations. Frequency (KHz) 40 45 50 55 60 65 Current at Rb-87 dip (mA) 408 422 434 446 459 469 Current at Rb-85 dip (mA) 455 476 496 511 529 546 TABLE II: High field resonant dips. All data taken at a main field strength of 6.921 gauss. The Breit-Rabi equation was used for the values in the last column. Sweep Coil Current (A) Sweep Field (gauss) Total Field (gauss) 0.319 0.358 0.397 0.435 0.472 0.509 0.003 0.027 0.050 0.072 0.094 0.116 6.918 6.895 6.871 6.849 6.827 6.805 Predicted Value (gauss) 5.805 5.813 5.820 5.813 5.805 5.798 [1] Melissinos, A, & Napolitano, J. (2003). Experiments in modern physics. San Diego: Academic press. [2] Optical pumping of Rubidium: Instructor’s manual. Buffalo: TeachSpin Magnetic field at Rb-87 dip (gauss) 0.057 0.066 0.073 0.080 0.088 0.094 V. Magnetic Field at Rb-85 dip (gauss) 0.086 0.098 0.110 0.119 0.130 0.140 CONCLUSIONS It appears that there may be something wrong with the values predicted by the Briet-Rabi equation. For instance, the fact that there is repetition in four of the values is unsettling, as six dips were clearly observed. That is not to say that the measurements were perfect, as the calibration equations could be wrong. If this experiment were to be performed again, more data would be taken for the calibration measurements to ensure a more accurate line fitting. However, it should be noted that the theory behind the experiment was shown to be sound. The sample of Rubidium was observed to have been Optically Pumped, and the values obtained for the nuclear spin numbers were very close to the accepted values. Furthermore, the dips observed confirm that Zeeman splitting of the nuclear energy states did in fact take place, and that the optical pumping was disrupted by the applied RF signal. The process of optical pumping remains a valuable tool for any Physicist. Quantum mechanical theories were explored and confirmed throughout the experiment. [3] Wolff-Reichert, B. (2009). Conceptual tour of optical pumping. TeachSpin Newsletters, Retrieved from http://teachspin.com/newsletters/OP% 20ConceptualIntroduction.pdf.