Laser Spectroscopy to Resolve Hyperfine Structure of Rubidium

Home , Alfred Kastler, Arthur Leonard Schawlow, Willis Lamb

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 spectrum and hyperfine structure of the two rubidium isotopes. 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 broadening 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 ω = ω (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 ω0 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 broadening 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 (ω0−ω) /2KT −ω tum. The total angular momentum, F, is then given by I(ω) = I0e 0 . (7)

F = I + J. (3) As seen above in Equation 7 the thermal doppler broadening 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 ˆ2 ˆ2 ˆ2 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 ∆ωfwhm = 2ω0 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. At the same time, the significantly stronger pump beam νrubidium ν1 = νc − νc (9) also went through the rubidium cell in the opposite di- c rection exciting the rubidium atoms. This is what caused the stimulated emissions that the PIN detector measured. ν In addition to getting an absorption spectrum for ru- ν = ν + rubidium ν (10) 2 c c c bidium, this experiment aimed to resolve the hyperfine structures of the two rubidium isotopes. To do this a The crossover frequency is exactly halfway between ν1 chopper and lock in amplifier where introduced to the and ν2 and the resulting equation for crossover frequency setup. The addition of these two elements allowed for is derived from the addition of Equation 9 and Equa- Phase Sensitive Detection; PSD in simple terms detects tion 10 and goes as frequency and then just throws out any additional noise that is not frequency dependent3. The introduction of ν1 + ν2 νc = . (11) the chopper and lock in amplifier were necessary to in- 2 crease the resolution of the setup enough to be able to The crossover frequency dips occur when two transitions resolve the hyperfine structures of rubidium 87. Unfor- differ in frequency less than the doppler line width. The tunately even with their increase of resolution, we were observed result is an increased rate of transmission at still unable to resolve the hyperfine structure of rubidium those crossover frequencies because the atoms are reso- 854. nant with ν1 at the pump and ν2 at the probe and also at ν2 at the pump and ν1 at the probe. IV. RESULTS

III. EXPERIMENTAL SETUP AND PROCEDURE A. Absorption Spectroscopy

The schematic diagram of the experimental setup is The ﬁrst section of this experiment employed the given by Figure 2 . The tunable diode laser used in this technique of absorption spectroscopy to create an absorption spectrum for rubidium. The data collected included measurements from thePIN detector giving us the absorption spectrum and also from the etalon which gave us the data necessary to calculate the frequency diﬀerences between the ground and excited states for both isotopes. Figure 3 shows the absorption spectrum for rubidium. The extra noise between the excited

FIG. 2. Schematic diagram showing the experimental setup of the laser spectroscopy experiment experiment provided a coherent light source at a wavelength of 780nm. The light from the laser traveled to a beam splitter that redirected some of the light towards the etalon while the rest remained headed to the rubidium sample. The etalon was a useful tool that helped us determine the frequency differences between states and the hyperfine splittings. the etalon did not tell us actual frequencies but it told use the changes in frequency which FIG. 3. Absorption Spectrum for Rubidium is what we really needed. The remaining beam was then directed towards another 90/10 beam splitter which di- states of rubidium 85 and rubidium 87 was attributed rected 90 percent of the beam to be what is called the to a cross over frequency. The difference in frequency 4

TABLE I. Table summarizing the frequency difference be- TABLE II. Table summarizing the frequency difference between states from the absorption spectrum seen in Figure 3. tween rubidium 87 hyperfine splitting Expected Value Calculated Value Expected Value Calculated Value Rb 87 6835 MHz 6950 MHz ± 90 MHz F=1 to F=2 157 MHz 160 MHz ± 10 MHz Rb 85 3036 MHz 3060 MHz ± 60 MHz F=2 to F=3 283 MHz 260 MHz ± 10 MHz between the isotope peaks was done by using the etalon frequency differences between the hyperfine splittings. data. Since the etalon peaks are each separated 300 Table II shows the hyperfine frequency differences be- MHz, I calculated the number of etalon peaks sepa- tween the F=1 to F=2 transition and F=2 to F=3 tran- rating the isotope peaks and multiplied by 300 MHz. sition. The uncertainty in the measurements once again This technique results in values that closely paralleled stems from the noise on the oscilloscope making it diffi- expected frequency values. The results are seen in cult to distinguish between noise and true etalon peaks Table I. The uncertainty of the measurements is the and also the averaging of five trials. Figure II matched result of noise from the oscilloscope making it difficult the theoretical quantum numbers that show that there to tell the difference between noise and etalon peaks. should be 6 hyperfine splittings for rubidium 87. The data was also taken five times and averaged which Figure 5 shows the hyperfine structure for rubidium 85. was also reflected in the uncertainty of the measurements. As seen in Figure 5 , the experimental setup did not have

Using Equation 8, the temperature of the emitting source was calculated using the doppler broadening of the absorption peaks. The temperature calculated was 310 K ± 40 K. This temperature value made sense because the experiment was performed at room temperature, which is generally around 300 K. The uncertainty introduced into the measurement was the result of the noise from the oscilloscope making measurement of the full width half max an approximation.

B. Saturation Absorption Spectrum

The second section of this experiment employed the technique of saturation absorption spectroscopy to try to resolve the hyperfine splitting for both rubidium isotopes. Figure 4 shows the hyperfine structure for rubid- FIG. 5. Hyperfine structure for rubidium 85 ium 87. Similarly to the technique used to determine the enough resolving power to resolve the hyperfine splitting of rubidium 85. According to the quantum numbers for rubidium 85, there are 10 hyperfine splittings that should have been visible.

V. CONCLUSION

The results of this experiment all fall within the range of the expected values when taking into account the uncertainty of the measurements. The doppler broadening of the absorption spectrum peaks allowed us to confirm that the temperature the experiment was in fact near room temperature. The experimental setup had enough resolving power to resolve the hyperfine splitting of rubidium 87 but it did not have enough resolving power to resolve the hyperfine splitting of rubidium 85. FIG. 4. Hyperfine structure for rubidium 87 Overall the experiment demonstrated the effectiveness of laser spectroscopy as a method of calculating an ab- frequency differences between the peaks on the absorp- sorption spectrum and hyperfine structures. This proved tion spectrum, the etalon data was used to calculate the that the techniques employed in this experiment are valid 5 and efficient techniques to determine the quantum num- partners, Will Weigand and Adam Egbert for helping bers and properties of a sample. take and analyze data.

1David J Griﬃths An Introduction to Quantum Mechanics Pearson VI. ACKNOWLEDGEMENTS Prentice Hall, India, (2004). 2P. Chantry Doppler broadening in beam experiments The Journal of Chemical Physics, 55, 2746-2759, (1971). I would like to thank Dr. Severn for his help with un- 3H. Lakhotia and C. Mitra Saturation Absorption Spectroscopy derstand the background theory and for providing the (2003). necessary equipment and assistance with the experimen- 4A.C. Melissinos and J. Napolitano Experiments in Modern tal setup. Additionally, I would like to thank my lab Physics 2nd Ed., Academic Press, (2003), p.270.