Hyperfine Energy Levels of 85Rb & 87Rb Chance J. Haycock, Fernando H. Jabaloyes, and Joe R. Gibbs University of Warwick, UK (Dated: February 13, 2019) The technique of optical pumping was used to study the hyperfine energy level splitting of Rubidium-85 and Rubidium-87. The low field and quadratic Zeeman effects were demonstrated in order to verify the theoretical values given by the Breit-Rabi equation. The atomic Lande g- 85 87 factor gF was obtained for the isotopes Rb and Rb and were found to be 0:3323 ± 0:0006 and 0:5008±0:0014 respectively. These values agree with the predictions for both isotopes, giving a good agreement of the expected low field Zeeman effect. The instantaneous dynamics of a system under- going a transition from a pumped to depumped state were also investigated, however, the analysis was limited by the sources of error. I. INTRODUCTION defined as F = I + J (1) French physicist Alfred Kastler is credited with the development of optical pumping techniques in the 1950's where J = L+S is the electron's total angular momentum [1][2]. The process of optical pumping uses a focused and I is the nucleus' spin angular momentum. For both beam of coherent, high frequency light with particular isotopes of Rubidium, quantum mechanical properties. These imposed proper- 1 1 ties allow us to manipulate the distribution of electrons L = 0; S = ; J = ; 2 2 amongst energy levels and is particularly useful for the construction of lasers; creating a population inversion of but their nuclear spin values, I, are distinct. In the ab- the system [3]. In this case, the system exists in a state sence of an external magnetic field, the hyperfine levels in which more electrons occupy higher energy states than can not be well resolved. In the presence of an exter- lower ones. The de-excitation of these electrons, and the nal magnetic field, each F level splits into 2F + 1 (where subsequent emission of photons, then produce a coherent F = j F j) Zeeman levels. Then, the Hamiltonian is light source. The technique for measuring the hyper- µ µ H = A(I · J) − J (J · B) − I (I · B) (2) fine energy level transitions of atoms will be explored in J I depth throughout the paper. In particular, the splitting of the 5s energy level of Rubidium-85 and Rubidium-87 where B is the external magnetic field, A is the hyperfine subject to an external magnetic field will be investigated. splitting frequency for B = 0, J =j J j, I =j I j and Two main regimes are focused on, namely the Low Field µJ , µI are the electron and nuclear magnetic moments Zeeman Effect and the Quadratic Zeeman Effect [4]. respectively. In the weak field regime, the Hamiltonian of equation (2) can be resolved for the energy eigenvalues: E = mF gF µBB (3) −24 −1 II. THEORY where µB = 9:27 × 10 JT [5] is the Bohr Magneton and mF = −F; −F + 1;:::;F − 1;F . We also obtain In an atom, the magnetic dipole moment of the J(J + 1) + S(S + 1) − L(L + 1) g = g nucleus interacts with the magnetic field generated by F J 2J(J + 1) the orbiting electrons. The alignment of spins in the atom cause the electron energy levels to split. This is F (F + 1) + J(J + 1) − L(L + 1) known as hyperfine splitting. In order to quantitatively gJ = 1 + investigate this splitting, the role and interaction of the 2F (F + 1) electron's angular momentum and intrinsic spin angular Equation (3) can be recast as momentum must be discussed. g µ B ν = F B [6] (4) Rubidium has an electron configuration given by h [Kr]5s1. This configuration consists of filled inner elec- and is known as the weak-field Zeeman equation. For tron orbitals, which are stable and spatially symmetric, 1 stronger fields and the special case of J = 2 , the eigen- and one valence electron. Due to screening, we can ig- values of (2) are given by the Breit-Rabi Equation nore the effects of these inner electrons and the atom can −∆E µ ∆E r 4m x be modelled as hydrogen-like. Taking into account all mF hf I hf F 2 EF =I± 1 = − BmF ± 1 + + x contributions to angular momentum from the outer elec- 2 2(2I + 1) I 2 2I + 1 tron and the nucleus, the total angular momentum, F, is (5) 2 where, (gJ − gI )µBB −µI x := gI := ∆Ehf µBI and ∆Ehf is the hyperfine splitting energy at zero mag- netic field. This theoretical result shows an approxi- mately linear relationship for weak fields, and a non- linear relation for larger fields. Lastly, in a transitioning system from a pumped to depumped state or vice-versa in a very short period of time, we expect the populations to overshoot giving rise to ringing. This is referred to as having transient effects. It is predicted that in these cases, the frequency of the ringing will be proportional to the RF Amplitude applied [7]. III. EXPERIMENTS 1. Experimental Setup FIG. 2. Energy level schemes of 87Rb (left) and 85Rb (right) showing fine structure, hyperfine structure and Zeeman split- The presented setup, as shown in Figure 1, uses a Ru- ting. For a given energy level, it is shown how the application bidium discharge lamp in series with an interference fil- of a magnetic field B increases the number of allowed tran- ter, a linear polariser and a quarter-wave plate to achieve sitions via the division of states of a single F into 2F + 1 the conditions required for this experiment. From the states. theory and Figure 2 it can be seen that each F level splits into 2F + 1 Zeeman levels. In order to pump the system, the selection rules that are required are as follows: One of the major difficulties with measuring transitions between hyperfine level transitions in a magnetic field is ∆F = 0 ∆mF = +1 (6) that there is nearly no distinction between the popula- This will allow electrons to be excited into higher energy tion size of electrons in two adjacent energy levels. The states except those which are trapped by a lack of higher ratio of the populations of two adjacent levels with an energy levels available to transition to for which selection energy difference ∆E, is determined by a Boltzmann dis- −∆E rules (6) hold. tribution, exp . Taking for example, a splitting kBT of 1MHz at room temperature (about 300K), this gives a ratio of ≈ 1 − 2 × 10−7 ≈ 1. 2. Calibrations of Fields The predicted value for the Earth's magnetic field at the location of the laboratory was ≈ 49µT [8]. How- FIG. 1. Apparatus used. From left to right: 1 is the Ru- ever, the setting in which the experiment took place had, bidium discharge lamp; 2 is an interference filter; 3 is a linear itself, a resultant magnetic field measured to vary in ori- polariser; 4 is a quarter wave plate; 5 is the Rubidium absorp- entation throughout the laboratory. Thus, the resultant tion cell kept at a constant 323K; 6 is the optical detector. background magnetic field was different from simply that of the Earth's magnetic field. The coils used satisfied For this pumping to occur, the combined use of the the Helmholtz condition, and hence the residual field at IN linear polariser, quarter wave plate and the interference the coils can be calculated from B = k Gauss [7] filter is vital. The first pair will achieve right hand cir- R cularly polarised light, and the interference filter is used where k = 8:991 × 10−3 Gauss m, N is the number of to ensure that the transmitted light is of approximate turns on each side and R is the mean radius of the coils. wavelength λ = 795nm. These are the conditions that The magnitude of the Earth's magnetic field, (though it are necessary in order to stimulate the D1 transition, is more precisely the resultant background field; it will visualised in Figure 2. still be referred to as BEarth) at the coils was measured to be 3 BEarth = 0:2137 ± 0:00005 Gauss (7) As the known gF values are either integer or half- integer, the approximated value for Bcoil is sufficient for the estimation of gF values. For later experiments, a greater accuracy than that obtained from the geometry of the coils was required. For the calibration of the sweep coils, the knowledge of the gF values for both isotopes is assumed and the mea- surements of the value of current at which the resonances occurred are used to arrive at the relationship Bsweep = 0:621Isweep − 0:222 Similarly, for the main coils B = 7:1628I (8) main main FIG. 3. Low Field Zeeman Effect at 70kHz (top graphic) and 120kHz (bottom graphic). Largest dip is the zero resultant B The condition Bmain(0) = 0 was imposed. These calibra- field and the following 2 on either side correspond to the 85Rb tions were used throughout the experiment. and 87Rb. Note: As the frequency increases the symmetry is maintained but the isotope peaks are further away from the central peak. 3. Low Field Zeeman Effect The current at which resonances occurred was mea- 5. Transient Effects sured for each isotope at frequencies in the range 70- 150kHz. Following on from the calculation of the Earth's Lastly, the behaviour of electrons for a system in tran- field strength (see result (7)), the geometry of the coils sition from a pumped to depumped state was investigated.