Laser spectroscopy to resolve hyperfine structure of rubidium Hannah Saddler, Adam Egbert, and Will Weigand (Dated: 12 November 2015) This experiment had two main goals: to create an absorption spectrum for rubidium using the technique of absorption spectroscopy and to resolve the hyperfine structures for the two rubidium isotopes using saturation absorption spectroscopy. The absorption spectrum was used to determine the frequency difference between the ground state and first excited state for both isotopes. The calculated frequency difference was 6950 MHz ± 90 MHz for rubidium 87 and 3060 MHz ± 60 MHz for rubidium 85. Both values agree with the literature values. The hyperfine structure for rubidium 87 was able to be resolved using this experimental setup. The energy differences were determined to be 260 MHz ± 10 MHz and 150 MHz ± 10 Mhz MHz. The hyperfine structure for rubidium 85 was unable to be resolved using this experimental setup. Additionally the theory of doppler broadening was used to make measurements of the full width half maximum. These values were used to calculate a temperature of 310K ± 40 K which makes sense because the experiments were performed at room temperature. I. INTRODUCTION in the theory section and how they were manipulated and used to derive the results from the recorded data. Addi- tionally there is an explanation of experimental error and The era of modern spectroscopy began with the in- uncertainty associated the results. Section V is a conclu- vention of the laser. The word laser was originally an sion that ties the results of the experiment we performed acronym that stood for light amplification by stimulated to the usefulness of the technique of laser spectroscopy. emission of radiation. The first functioning laser was built in 1960 by Theodore H. Maiman. Lasers differ from other light sources because they emit light coher- II. THEORY ently; this means that the light is spatially coherent al- lowing it to be tightly focused with minimal scattering. Even though lasers were not successfully built until 1960, The two techniques of laser spectroscopy used in this the theoretical foundations were laid out by Albert Ein- experiment aim to understand and measure the absorp- stein in 1917, Rudolf Landenburg in 1928, Willis Lamb tion spectrum for rubidium and the hyperfine splitting and R.C. Retherford in 1947, and Alfred Kastler in 1950. of the two rubidium isotopes. To understand the pro- The Nobel Prize in Physics in 1981 was jointly received cedure, results, and observations it is important to first by Nicolaas Bloembergen and Arthur Leonard Schawlow look at the classical and quantum physics behind the for their contribution to the development of laser spec- structure of rubidium. With a firm understanding of the troscopy and Kai M Siegbahn for his contribution to basics, it is then important to understand the physics the development of high-resolution electron spectroscopy. behind two more phenomena experienced in this experi- The type of laser used in this experiment is known as a ment: doppler broadening and crossover frequencies. tunable diode laser. This means that its wavelength can be altered by a controller to be tuned over a specific range of wavelengths. A. Basic Physics Laser spectroscopy is a useful technique to learn about the structure and energy levels of atoms and their con- Atoms typically contain a central nucleus with elec- stituent electrons. Spectroscopy has played a critical role trons that orbit the nucleus. The electrons fill into these in furthering quantum mechanics, atomic and nuclear different orbitals based on the Pauli Exclusion Principle. physics, and many other fields. Lasers and laser spec- The Pauli Exclusion Principle states that no two elec- troscopy have even found their ways to cosmology and trons in an atom can have the same four electronic quan- medicine. The two types of spectroscopy used in this ex- tum numbers. The orbitals have different energy states. periment were absorption spectroscopy to observe the ab- Electrons that are in the outermost shell, meaning fur- sorption spectrum of rubidium and saturation absorption thest away from the nucleus, can be excited into higher spectroscopy to try to resolve the hyperfine structures of energy states. They jump into these states excited states both rubidium isotopes. by absorbing photons with energies equal to the energy difference between the two states. This is shown by The second section of this paper will cover the theory necessary to understand the procedure, results, and ob- ∆E = E2 − E1 = hf; (1) servations from the laser spectroscopy experiment. Sec- tion III is an overview of the experimental design and ap- where f is the frequency of light that is required to make paratus used. Section IV further details the experimental the transition between the states. Figure 1 is an energy procedure and the results of the experiment. This section level diagram that shows the energy levels for rubidium includes a discussion of the various equations presented 85 and rubidium 87. This energy level diagram show the 2 energy difference between the ground and first excited = 3,2,1,0 for rubidium 87 and F = 4,3,2,1 for rubidium states. The diagram also shows the hyperfine splitting 85. Using Equation 5 and the quantum numbers, the caused by non-time dependent energy perturbations that energy differences between the hyperfine splittings can remove the degeneracy of the states. From the figure it be determined1. is obvious that a frequency of 384x106 MHz is required To understand the quantum addition of angular mo- to excite both of the rubidium isotopes from the ground mentum from a more classical perspective is may be use- state to the second excited state. ful to think of the electron as a bar magnet. The magnet can either be aligned or antialigned. The aligned state is a lower energy state that refers to the subtraction of J from I. The antialigned state is a higher energy state that refers to the addition of J and I. B. Doppler Broadening In the laser spectroscopy experiment , the tunable diode laser is directed through a cell of rubidium gas to allows us to make observation of the absorption spec- trum and hyperfine structure of the two rubidium iso- topes. The rubidium gas cell does not contain a station- ary sample to be measured. As a result of the thermal motion of the rubidium atoms in the gas sample, doppler broadening occurs. Doppler broadening is the broaden- ing of spectral lines due to the doppler effect caused by a FIG. 1. Energy diagram for rubidium 87 and rubidium 85. distribution of velocities of the atoms. The different ve- The diagram shows the difference between the ground state locities of the rubidium particles result in doppler shifts and the excited state. The diagram also shows the hyperfine based upon the direction and speed of the particle rela- splitting of both states for both isotopes. tive to the incoming light waves. The expected frequency with the doppler shift is given by To understand the hyperfine splitting of the energy v ! = ! (1 − ); (6) states we must turn to quantum mechanics. The mul- 0 c tiple substates that both energy levels split into are the result of energy perturbations that are explained by the where !0 is the resonant frequency of light the rubid- perturbing hamiltonian operator. The perturbing hamil- ium atom would experience if it was not moving relative tonian operator is given by to the incoming light beam. Therefore doppler broaden- ing stems from the fact that instead of receiving a single Hhfs = A(J • I); (2) spectral line at the resonant frequency, there is a spread of frequencies measured. The spread of frequencies is a where I is the nuclear angular momentum and J is the gaussian distribution given by sum of the electronic orbital and spin angular momen- 2 2 2 −m0e (!0−!) =2KT −! tum. The total angular momentum, F, is then given by I(!) = I0e 0 : (7) F = I + J: (3) As seen above in Equation 7 the thermal doppler broad- ening is dependent of frequency of spectral line, mass By combining Equation 2 and Equation 3, the Hamilto- of emitting particle, and the temperature. Therefore the nian can be rewritten as doppler broadening of the absorption spectrum peaks can be used to determine the temperature of the emitting A ^2 ^2 ^2 body. The equation to do this is known as the equation Hhfs = [F − I − J ]; (4) 2 for full width half maximum and is given by which yields p 2 ∆!fwhm = 2!0 2ln(2KT=m0)=c : (8) A 2 Hhfs = ~ [F (F + 1) − I(I + 1) − J(J + 1)]: (5) 2 C. Crossover Frequencies The ground state of rubidium 85 has I = 5/2 and J = 1/2 which leaves us with F = 2 and F = 3. The ground state When analyzing the results of this experiment it is im- of rubidium 87 have I = 3/2 and J = 1/2 which leaves us portant to know what crossover frequencies are, what with F = 2 and F = 1. For the excited states there are they represent, and what they look like. Crossover fre- four allowed J values (J = ±3=2; ±1=2). This yields F quencies, νc, occur when the laser frequency, is such that 3 the moving rubidium atoms see the pump beam red- pump beam and the remaining 10 percent to be the probe shifted to a frequency of ν1 and at the same time the beam. The probe beam was then directed through the moving rubidium atoms see the probe beam blueshifted rubidium sample and into a PIN detector. The PIN de- to a frequency of ν2. tector was then used to make absorption measurements.