Atomic hyperfine structure studies using temperature/current tuning of diode lasers: An undergraduate experiment

Transcription

1 Atomic hyperfine structure studies using temperature/current tuning of diode lasers: An undergraduate experiment G. N. Rao, M. N. Reddy, and E. Hecht Department of Physics, Adelphi University, Garden City, New York Received 27 May 1997; accepted 28 January 1998 We present a simple and inexpensive experimental arrangement for hyperfine structure studies in atoms using commercially available laser diodes and hollow cathode lamps. The experiment is highly suitable for the undergraduate laboratory. This technique can be employed to investigate the hyperfine structure of rare earth and other elements such as Ta and Nb which have large nuclear magnetic and or quadrupole moments. In this paper, we report well-resolved hyperfine structure spectra recorded for holmium employing optogalvanic spectroscopy. We also report Doppler limited hyperfine structure measurements on the ground state of rubidium using injection current/ temperature tuning of the diode laser. This involves a simple experimental arrangement suitable for undergraduate laboratories. The hyperfine coupling constants for the level at cm 1 in Ho I are reported for the first time. Details of the data analysis to obtain accurate hyperfine structure coupling constants from the observed spectra are presented. A number of commercially available diode lasers in the visible and the near infrared regions and simple in-house developed or commercially available low cost current and temperature controllers can be employed for the present studies. We employ simple cooling/heating or current modulation for tuning the output wavelength of the diode laser. The presently proposed experimental arrangement can be assembled easily and requires no machine/glass shop facilities American Association of Physics Teachers. I. INTRODUCTION Lasers are playing an important role in the undergraduate physics laboratory curriculum for conducting a variety of interesting experiments in atomic physics and modern optics. Details of some of these interesting experiments can be found in a report prepared by Bradenberger. 1 During the last several years, semiconductor diode lasers have become popular for a variety of experiments to study atomic structure. 2,3 Because of their low cost, compact size and ease of operation, they can be conveniently employed in an undergraduate instructional laboratory to carry out numerous interesting experiments in atomic physics. Recently, a number of undergraduate experiments have been proposed based on diode lasers. Most of them use external cavity stabilized diode lasers with piezoelectric drives that require machine shop facilities for fabrication. MacAdam, Steinbach, and Wieman 4 described the construction of an external cavity narrow band tunable diode laser system and a saturated absorption spectrometer for Cs and Rb. Wieman, Flowers, and Gilbert 5 presented an inexpensive laser cooling and trapping experiment for undergraduate laboratories. Libbrecht et al. 6 reported the details of the construction of stabilized lasers and lithium cells using a 670-nm diode laser to perform undergraduate atomic physics experiments. All of them involved the fabrication of a stabilized external cavity arrangement with a piezo drive control. They employed Doppler-free high-resolution saturation spectroscopy for the hyperfine structure studies. Here, we present a much simpler arrangement which can be assembled at minimal cost to study the hyperfine structure of a number of atomic species using Doppler limited spectroscopy. We employ simple temperature/current tuning of the diode laser and optogalvanic spectroscopy technique for detection. This method can be used for almost all the rare earths, and a number of other atomic species such as Nb, and Ta which have large nuclear moments. For these atoms, the hyperfine level splittings are larger than the Doppler broadening, and therefore one can obtain well-resolved hyperfine spectra and reliable hyperfine structure coupling constants even in Doppler limited spectroscopy. The experimental arrangement does not involve any fabrication work and therefore no glass/ machine shop facilities are required. The entire setup can be assembled in a couple of days with readily available commercial components. Since some colleges and universities do not have machine/glass shop facilities, the presently reported experiments are likely to be of special appeal to them. Most commercially available laser diodes 7 can be employed for the present studies. However, laser diodes operating in single frequency mode have significant advantages. One can use commercially available hollow cathode lamps for optogalvanic spectroscopy work. Hollow cathode lamps of most of the elements are commercially available as a stock item from a number of vendors and the typical cost is in the range $100 $230. If a diode laser setup is already available, the hyperfine structure studies employing optogalvanic spectroscopy can be carried out with a few hundred dollars. II. DIODE LASERS Compared to traditional ion and solid state lasers, diode lasers are compact, reliable, easy to operate, amenable to high frequency electronic modulation and temperature tuning, and are of low cost. The basic principles of diode laser operation were well documented in the literature see Ref. 3 and the literature cited therein. The laser diode consists of a double heterojunction surrounded by p-type and n-type cladding layers. When the laser diode is forward biased, electrons and holes are injected into the active region and light is generated as a result of the recombination of the electronhole pairs. The electrons and holes confined to the active region undergo a population inversion resulting in laser action. The wavelength of the emitted laser radiation is ap- 702 Am. J. Phys. 66 8, August American Association of Physics Teachers 702

2 proximately equal to the band gap of the semiconductor material. Compared to the 100-nm tuning range of dye lasers and the much larger tuning range of Ti-sapphire lasers, diode lasers have a limited tuning range of 10 nm. In general, the continuously tunable range of diode lasers without mode hops is considerably less and is of the order of 1 to 2 nm. Therefore, one has to carefully choose an appropriate diode laser which matches the atomic/molecular transition of interest. Listings of commercially available diode lasers and their characteristics are available on the Internet. 7 We have tested a number of lasers manufactured by Hitachi, Mitsubishi, SDL, Sharp, and Toshiba for single frequency operation. We find that Hitachi, Mitsubishi, SDL, and Sharp lasers gave good single frequency operation. GaAlAs diode lasers operate in the 750- to 900-nm range and are useful for Rb and Cs atom traps. InGaAs diode lasers operate in the range nm, whereas AlGaInP diode lasers emit radiation visible in the 630- to 700-nm range. InGaAsP laser diodes have outputs in the far infra-red region, nm range, and the lead salt laser diodes cover m. As stated earlier, narrow linewidth single frequency diode lasers are optimally suitable for the present experiments. Diode lasers can be employed for hyperfine structure studies in atoms using simple temperature/injection current tuning as described in this paper or Doppler-free spectroscopy using the external cavity arrangement as described in Refs. 4 and 6. The first step is to choose an appropriate single frequency diode laser which lases at a wavelength close to the atomic transition of interest. Any diode laser mount with thermoelectric cooler TEC would be adequate for the present experiments. We used a mount manufactured by Light Control which has a thermoelectric cooler and a 10-k thermistor to monitor the temperature. We used a current controller also manufactured by Light Control. Any simple low-noise A rms current controller along with a temperature stabilizer short term stability 10 mk will be adequate for these experiments. A low-noise high-speed diode laser current controller circuit which can be easily fabricated in an undergraduate laboratory capable of providing low noise total noise of 45 na rms in a 1-MHz bandwidth and stable current current drift 0.25 A in3h output was reported by Libbrecht and Hall. 8 A. Tuning of the diode lasers Diode lasers operate in general in single mode or multimode. Often, a laser diode may give multimode output at lower currents and give single mode output at higher currents. Some diode lasers, even though stated to be of single frequency mode by the manufacturers, were found to be otherwise. If you have access to a monochromator, it is a good idea to first test the diode laser for single frequency output. A diode laser can be tuned by temperature tuning, current tuning or external cavity tuning. Design, fabrication and characterization of diode lasers locked to an external cavity and their applications for a variety of atomic physics experiments have been well documented. 4 6 Here, we focus on temperature and current tuning and their applications to hyperfine structure measurements. The output wavelength of a free-running diode laser is determined by the temperature and injection current. The cavity tuning characteristics and the tuning characteristics of the semiconductor medium gain have different wavelength to temperature coefficients and they have values (d /dt) cavity 0.06 nm/k and (d /dt) gain 0.25 nm/k. Because of this mismatch of the temperature coefficients, as we change the temperature, the wavelength output shows discontinuities. The first step of the tuning process is to set the laser output wavelength close to the atomic/molecular transition of interest. If necessary, one can change both temperature and injection current to get the laser output at the desired wavelength. The stability of the laser output is important and it can be achieved by tweaking the temperature and current such that reasonably stable output and tunability over a wider wavelength range is attained. The temperature tuning can be accomplished either by heating or cooling the diode laser. The data were collected during the heating/cooling process and the wavelength of the laser output was simultaneously monitored. For current tuning, the temperature of the laser diode was kept constant at a particular value such that the laser output was close to the atomic transition of interest and the diode laser injection current was modulated by a signal generated by a function generator. Typically, one uses a ramp signal of mv peak to peak at a modulation frequency of 0.01 Hz to 15 khz. III. OPTOGALVANIC SPECTROSCOPY OF SPUTTERED ATOMS Optogalvanic spectroscopy OGS is based on the Optogalvanic Effect which is the change in the impedance of a gaseous discharge due to the resonant light absorption. It is a simple and convenient detection technique for studying the spectroscopy of atoms, ions, molecules and radicals in electrical, high frequency discharge plasmas and flame plasmas. Unlike emission or absorption spectroscopy which require the use of optical detectors, OGS does not require an additional detector; instead the discharge plasma itself acts as a sensitive nonoptical detector. No background filtering is needed in OGS and the signal-to-noise ratio is quite good and is generally of the order of A. Simple theory of optogalvanic effect There are two significantly different mechanisms for the origin of the optogalvanic effect. In the first mechanism, the absorption of laser radiation in the discharge results in a change in the steady-state population of bound atomic levels. Different levels, in general, will have different ionization probabilities. Hence, there is a net change in the ionization balance of the discharge. A perturbation to the ionization balance leads to a change in the current through the discharge or, equivalently, a change in the impedance of the discharge. In the second mechanism, the excitation of atoms by the laser to higher electronic states perturbs the equilibrium established between the electronic temperature and the atomic excitation temperature. But the superelastic collisions between the electrons and the laser excited atoms in the discharge tend to restore the equilibrium. In this process an excess amount of energy is released which often ends up in an increased electron temperature of the discharge. Therefore, the laser excitation of atoms leads to an increase in the conductivity or decrease in the impedance of the discharge. In fact, both mechanisms are expected to be present simultaneously in OGS. The relative importance of these two mechanisms depends on the discharge and excitation conditions. 703 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 703

3 Fig. 1. Schematic of the experimental arrangement for the diode laser excited optogalvanic spectroscopy FPI Fabry Perot interferometer. B. Optogalvanic spectroscopy of sputtered atoms The optogalvanic detection technique is well suited for the spectroscopic study of sputtered atoms. The atomic spectroscopy of even refractory and nonvolatile elements can be carried out with ease using this method. The hollow cathode discharge serves as a rich reservoir of sputtered atoms. Under the right conditions of gas pressure and bore diameter of the cathode, the negative glows from opposite walls of the inner surface of the hollow cathode coalesce to produce neutral and excited atoms and ions in high densities at the center of the hollow cathode. The hollow cathode discharge is highly self-sustaining and can maintain high currents at small cathode-fall potential values. Application of a potential difference of a few hundred volts between the two electrodes of the hollow cathode lamp see Fig. 1 leads to breakdown of the rare gas buffer at low pressure and creation of a number of electron ion pairs resulting in a discharge. The ions together with fast neutral atoms produced by resonant charge exchange are accelerated in the high field of the cathode dark space and bombard a cathode which is made of, or coated with, the material of interest. The highly energetic ions and fast neutral atoms impart sufficient energy to the crystal lattice of the cathode material to dislodge and eject the atoms from the lattice sites. The sputtered species, predominantly single ground state neutral atoms, which initially possess high kinetic energies, rapidly lose their kinetic energy by elastic collisions with rare gas atoms and attain thermal equilibrium. As the sputtered atoms diffuse from the cathode surface into the negative glow, some of them are excited or ionized by electron impact or by collisions with metastable atoms or ions present in the discharge. In this way, a reasonably high steady-state density of atoms, metastable atoms and singly ionized ions can be maintained in the negative-glow region of the discharge which is suitable for carrying out optogalvanic spectroscopy. C. Sputtering process In atomic spectroscopy experiments, preparation of the sample often demands a major effort and its importance need not be overemphasized. Discharge, arc, and spark sources were commonly employed in traditional optical spectroscopy studies. 9 For studies on atomic structure using lasers, optical cells maintained at high temperatures, atomic beams, heat pipe ovens, and sputtering cells are popular. For specific applications, sputtering techniques seem to offer some advantages over the other methods. The technique is simple and can be applied to most of the elements of the periodic table including the refractory materials. This method can be employed in hollow cathode lamps which are commonly used for chemical analysis of samples using atomic spectroscopy. Commercially available hollow cathode lamps can be readily employed for the present experiments. A number of vendors keep hollow cathode lamps of most elements in stock. The ejected species consist of predominantly ground state neutral atoms, and a small fraction of excited atoms, ions, and clusters of atoms. Even though the species are released with a range of energies of the order of up to 10 ev or so, they experience direct collisions with the rare gas atoms resulting in a distribution corresponding to a significantly lower temperature. The emitted atoms may be further excited or ionized due to collisions with energetic electrons and ions of the discharge. Using this technique, one can obtain steadystate densities of the order of, or greater than, /cm 3 of the ground state atoms, metastable atoms and singly charged ions. The number density is quite adequate for a variety of spectroscopy experiments, in particular high sensitivity and high selectivity techniques such as laser optogalvanic spectroscopy. 10 The important points of interest are i the species get thermalized quickly, and the thermalized Doppler broadening corresponds to temperatures in the range K and high-resolution spectroscopy work is feasible; ii the sputtering yields, unlike the yields involved in the thermal methods, do not change drastically from element to element and therefore laser optogalvanic spectroscopy LOGS can be carried out on almost all the elements of the periodic table; iii since many excited states are populated in the discharge, spectroscopy of the highly excited states such as Rydberg states can also be conducted; and iv no light detector is needed in this technique. The sputtering yields for refractory elements such as Zr and Nb are only about five times lower than the fast sputtering elements such as Cu and Ag. The OGS technique is applicable for the study of both the ground and the excited states of atoms. 11 In fact, optogalvanic spectroscopy technique offers greater sensitivity for the study of the highly excited states of atoms than do optical detection methods. IV. HYPERFINE INTERACTIONS 12 A. Fine structure of atoms 13,14 The development of high resolving power spectroscopic instruments at the end of nineteenth century led to the discovery of many finer details of the atomic structure. Michelson, Fabry, Perot, Lummer, and Gehrcke noted that many spectral lines consist of not only fine structure, but in fact each fine structure line consisted of many closely spaced lines hyperfine structure. The fine structure of atomic states is a result of the orbital motion of the electrons with intrinsic spins through the electric field caused by the nuclear charge. The spin angular momentum s of an electron gives rise to its magnetic moment ( s ), s e mc s 2 Bs, 1 where B is the Bohr magneton. 704 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 704

4 Because of the orbital motion of the electron in the electric field of the nucleus, it experiences an apparent magnetic field B l proportional to the orbital angular momentum (l), and thus the magnetic moment of the electron gives a term in the Hamiltonian: H s B l. 2 The Hamiltonian due to the spin-orbit interaction after taking relativistic effects into account is often written in the form H nl L S. 3 The spin-orbit interaction depends not only on the magnitudes of the L and S but also on their orientation and is proportional to L S. The total angular momentum J of the atom is J L S. 4 The total angular momentum J can have values J L S,L S 1,... L S. 5 Equation 3 is to be corrected for the interaction of the magnetic moments of different electrons, interaction of the orbital motion of one electron and the spins of the other electrons, etc. Since L S 1 2 J 2 L 2 S 2, the shift in the energy of an electron may be written as E J 1 2 nl J J 1 L L 1 S S 1, where is the fine-structure constant. Each J state splits into (2S 1) components if S L or into (2L 1) components if S L. The separation between the adjacent components of a multiplet is E J E J 1 nl J, 7 which is the well-known Landé interval rule. The doublet energy splitting corresponding to J l 1 2 and J l 1 2 for hydrogenlike atoms may be written as E R E 2 Z 4 n 3 l l 1, 8 where R E is the Rydberg constant in energy units, and the fine structure constant e2 c 1 137, Z is the atomic number, n is the principal quantum number, and l is the orbital quantum number. The selection rules for electric dipole transitions are 14 S 0, L 0, 1, L L 1. 9 The transitions are allowed between terms of one multiplet only. The fine structure typically has splittings in the range from a fraction of a wave number to several hundred wave numbers. The largest contribution is usually from the spin-orbit interaction. Normally, higher J values have a higher energy than the lower J value. However, the ordering may get reversed if other contributions such as core polarization are dominant. 6 B. Hyperfine interactions We are interested in the electrostatic interaction between the atomic nucleus and the surrounding electrons. In the following, we consider the nucleus interacting with the external fields produced by the electrons. The total interaction energy may be written as a sum of the interactions of each proton charge q p at position r p with an electron of charge q e at position r e, q W e p q e p r p r e, 10 which may also be written as W r n r e r n r e d 3 r n d 3 r e, 11 where (r n ) and (r e ) are the nuclear and electron charge distributions, respectively. This is usually written in the form 15 W n r r d 3 r, 12 where r is the potential produced by the electrons at the nucleus. The potential r is a slowly varying function over the nuclear volume and can be expanded in Taylor series 15 r 0 r E i j 3x i x j r 2 ij E j x i Introducing r in Eq. 12, the total energy may be written in the form 15 W q 0 p E i j Q ij E j x i This shows that the total energy of interaction may be expressed in terms of interactions of various multipoles with the external field. The monopole interaction corresponds to the nuclear charge (q) interacting with the potential 0 at the nucleus, the dipole interaction corresponds to the interaction of the dipole moment p with the electric field E 0 produced at the nucleus, and the quadrupole interaction is given by the quadrupole moment (Q ij ) interacting with the field gradients ( E j / x i ) produced by the electrons at the nuclear site. In general, the electrostatic interaction Hamiltonian of the nucleus and its electrons is usually expressed in a compact form as a multipole expansion H T n k T e k. 15 The expansion represents the product of a nuclear-multipole tensor operator of rank T(n) k and electronic charge distribution tensor of rank k, T(e) k. By symmetry considerations of parity and time-reversal, the nonvanishing terms with even k values represent electric, and those with odd k values, magnetic interaction. The monopole term (k 0) represents the Coulomb interaction between the electrons and the spherical part of the 705 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 705

5 nuclear charge distribution. This produces the same shift for all the levels of a configuration. For different isotopes of an element the shifts vary, leading to the so-called isotope shift. The k 1 term corresponds to the interaction between the magnetic dipole moment of the nucleus I and the magnetic hyperfine field induced by the electrons at the nucleus H J (0). The k 2 term is due to the interaction of the nuclear electric quadrupole moment Q I and the electric field gradients q J (0) produced at the nuclear site due to charges external to the nucleus. The higher order terms are usually negligibly small. For example, the magnetic octupole (k 3) and electric hexadecapole (k 4) interactions are about 10 8 times smaller than their corresponding lower-order magnetic dipole (k 1) and electric quadrupole (k 2) interaction and require ultrahigh-precision techniques for measurements and will not be discussed here. C. Isotope shift The monopole interaction results in a small energy shift in the nuclear and electron levels. The relative shift of the electron levels of a given configuration for two isotopes of an element is known as the isotope shift. The energy shifts associated with the nuclear levels can be measured by employing a variety of techniques such as Mössbauer spectroscopy. The energy shifts of an atomic transition corresponding to two isotopes of mass numbers A and A may be written as A-A A-A FS A-A NMS A-A SMS, 16 where the first term on the right-hand side corresponds to the field shift FS, the second term to the normal mass shift NMS and the third term to the specific mass shift SMS. The field shift is a combination of the shifts resulting from the changes in the nuclear volume and the nuclear charge distribution. The normal mass shift arises from the change in the nuclear mass when the correlations in the momenta of the electrons are neglected, whereas the specific mass shift is the contribution from the correlations of the moments of the electrons. The nuclei of the isotopes differ in their radii and nonsphericity, resulting in different charge distributions inside the nuclei. Isotope shift measurements enable us to obtain information on the changes in the nuclear charge radii, r 2, between the isotopes. D. Magnetic dipole interaction The magnetic dipole moment I of a nucleus with nonzero nuclear spin I interacts with the magnetic field H J (0) produced by the electrons at the nucleus. This corresponds to the (k 1) term in Eq. 15. The interaction Hamiltonian may be written as a scalar product of nuclear and electronic tensors, each of rank one (k 1): H M1 I H J The magnetic field at the nucleus is produced by the orbital motion and the spin dipole moments of the electrons. From symmetry considerations, this field is linearly related to the total angular momentum of the electrons J such that J L S. This interaction results in a shift of the energy levels of the atom by an amount E I H J The direction of H J (0) is that given by the total angular momentum of the atomic electrons J. The direction of H J (0) is opposite to the direction of J because the electrons have negative charge. The nuclear magnetic moment can be written as I n g I I, I I I, where n is the nuclear magneton and g I is the nuclear g factor: E IH J 0 I J AI J 1 I J 2 A F2 J 2 I The magnetic dipole coupling constant A in frequency units may be written as A IH J h I J An atomic level with the total angular momentum value J will split according to the possible values (I J) which are quantized. In this case analogous to the spin-orbit interaction giving rise to (L S) term in the fine structure of atoms, J and I couple resulting in the total angular momentum, which is designated by F, such that F I J. 21 An atomic level of J is split into a number of sublevels with all possible values of F such that J I F J I. 22 If the nuclear spin is a half integer, say I 2, 1 F will take (J 1 2) and (J 2) 1 values. If the nuclear spin is an integer, say I 1, F will take values (J 1), J, and (J 1) provided J 1. If either I 0 orj 0, then there will be no magnetic or higher order splittings. If I 0, the nuclear dipole, electric quadrupole and higher-order moments are zero. If J 0, the magnetic field induced by the electrons and also the electric field gradients and higher order terms induced at the nuclear site by the atomic electrons are zero. The energy contribution due to the magnetic dipole interaction is E M1 hak I 1/2,J 1/2, 23 2 where k F F 1 I I 1 J J The total angular momentum F can take values F I J,I J 1,..., I J. 25 The number of hyperfine components is 2J 1 when I J, and 2I 1 when I J. E. Electric quadrupole interaction The nuclear quadrupole moment Q I interacts with the electric field gradient q J (0) produced by the electrons at the nuclear site. The interaction Hamiltonian is a scalar product of two second-order tensors, one corresponding to the nucleus Q I and the second that of the electrons q J (0) which is written as 706 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 706

6 H E2 Q I q J For diagonal matrix elements with respect to I and J Eq. 26 reduces to H E2 hb 3I J 2 3/2 I J I 2 J 2, 27 2I 2I 1 J 2J 1 where the electric quadrupole coupling constant B is given by B e2 Q I q J h The energy contribution due to the quadrupole interaction may be written as E E2 hb 2 k k 1 2I I 1 J J 1 3, I 1,J 1. 4 I 2I 1 J 2J 1 29 The total hyperfine energy of a free atom is the sum of the magnetic dipole Eq. 23 and the electric quadrupole Eq. 29 interactions, resulting in the well-known Casimir formula 16 E E2 hak 2 hb 2 k k 1 2I I 1 J J I 2I 1 J 2J 1 Clearly, the hyperfine interactions depend on both the nuclear and atomic properties of an atom. In fact, the measured energy shifts are products of them. Precision hyperfine structure measurements have provided a wealth of information on nuclear structure 17 and electron wavefunctions. 18 The nuclear information includes the nuclear charge radii, nuclear magnetic dipole moments, electric quadrupole and octupole moments, Sternheimer shielding and antishielding effects, nuclear hyperfine anomaly, etc. The hyperfine structure spectra not only allow us to obtain the magnetic and the quadrupole hyperfine coupling constants, but also permit unambiguous assignment of the J values of the atomic levels involved. The hyperfine coupling constants depend strongly on the electronic wavefunctions in the vicinity of the nucleus. Since the relativistic corrections are important, one has to use the relativistic Dirac wavefunctions. The true Hamiltonian H hfs corresponding to LS coupled relativistic eigenfunctions can be expressed as matrix elements of an effective Hamiltonian H hfs between the eff nonrelativistic LS-coupled states. 19 The effective operator not only accounts for the relativistic effects, but also for the configuration interactions and polarization effects. The expressions for the electronic tensor operators contain the radial integrals. Since it is difficult to calculate the radial integrals, they are often represented as free single-electron hfs parameters, which can be determined by a fit procedure to the experimental hfs data. When an adequate number of hfs constants A and B are determined experimentally, the single electron hfs parameters can be determined from a leastsquares fit of the parametrized single electron parameters to the experimental values. V. EXPERIMENTAL The experimental arrangement for continuous wave CW diode laser excited OGS is shown in Fig. 1. The hyperfine spectrum of the nm transition of Ho I recorded using a Fig. 2. The hyperfine structure spectrum recorded for the nm transition in Ho I using temperature tuning of the diode laser and optogalvanic spectroscopy. holmium hollow cathode lamp with neon buffer gas is given in Fig. 2. Using the intense neon lines for calibration, we could identify a number of holmium lines in the spectra recorded in the nm range. The optogalvanic detection using CW laser excitation involves chopping the laser beam, and phase sensitive detection as shown in Fig. 1. In our experiment the beam was chopped at 2.2 khz. Phasesensitive lock-in detection improves the signal to noise ratio. A Fabry Perot interferometer with a free spectral range FSR of 300 MHz provided the frequency markers for the calibration of the observed hyperfine spectra. For the cases studied, since the hyperfine splittings are large, one can calibrate the spectrum with a low resolution Fabry Perot interferometer with a free spectral range FSR of about 2 GHz as well. In fact, one can use even a monochromator for calibration purposes. For example, a Spex monochromator model 1000M has a resolution of nm, whereas the separation between the extreme hyperfine lines in the present measurements is 25 GHz, which corresponds to 0.05 nm for 780-nm radiation. A. Hyperfine spectrum of holmium In general, the atomic hyperfine splittings are quite small, often demanding Doppler-free techniques. However, Doppler limited spectroscopy techniques can be applied if the magnetic and or quadrupole interactions are strong and the resulting hyperfine splittings are large compared to the Doppler broadening, as in the case of holmium which is investigated in this paper. We employed a LTO27MD Sharp laser diode operated at 56 ma and 19 C. The hyperfine spectrum could be recorded 707 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 707

7 C. Intensities of the hyperfine transitions Fig. 3. The hyperfine structure level scheme for the nm transition in Ho I. The expected calculated intensities of the hyperfine transitions are given at the bottom. employing temperature scanning of the diode laser during the heating or cooling cycle. The temperature tuning of the LTO27MD diode laser was measured to be 0.06 nm/ C 29.6 GHz/ C at 781 nm. We could typically scan 1.5 nm without mode hops in this region. The tuning range is quite adequate to cover the entire hyperfine structure of the nm transition in Ho I which is 25 GHz. The hyperfine spectra were calibrated using the markers obtained from a 300 MHz FSR Fabry Perot interferometer. As stated earlier, a high resolution Fabry Perot interferometer is not necessary for the present hyperfine structure measurements. We used it because it is readily available in our laboratory. The only stable isotope of holmium, 165 Ho, has spin I 7/2, nuclear magnetic moment I 4.173(27) n, and electric quadrupole moment Q 2.716(9) b. 20 Because of the large nuclear moments and also the hyperfine coupling constants, the hyperfine structure of Ho I transitions is usually spread over a GHz range. The hyperfine structure components for the nm transition in Ho I see Fig. 2 are well resolved even in the Doppler limited spectra because the energy separations of the hyperfine components are larger than the Doppler broadening. The hfs level-scheme of Ho I, nm, transition along with the expected theoretical intensities is given in Fig. 3. The Doppler broadening, which is the dominant contributor to the broadening of the spectral lines, was estimated to be 750 MHz at the hollow cathode lamp operating current of 13 ma. B. Hyperfine transitions The hyperfine interaction couples the electron angular momentum J and the nuclear angular momentum I to form the total angular momentum F: F I J. 21 F can have values from J I, J I 1,...,J I 1, J I. The selection rules for the electric dipole transitions are F 0 or 1 no 0 0, J 0 or 1, 31 S 0. The intensities of the hyperfine transitions correspond to the multiplet intensity formulas. The relative intensities of the transitions between hfs multiplets have been tabulated by White 21 and by Kopfermann 22 for values up to J 13 2 and I 2. 7 However, for the transition presently studied, I 7 2 and J J The relative intensities of the different hyperfine transitions were calculated by us using the formulae given by Candler. 23 For the transition presently studied, I 7 2 and J J 15 2 Refs. 24 and 25. Therefore, F takes values from 4 to 11. Out of a total of 22 hyperfine structure components see Fig. 3 expected, 14 separate lines were resolved out of which 12 of them were single transitions. Even though the contribution from the quadrupole interactions in holmium is significant, it is easy to identify the strong diagonal hfs components. Due to the saturation effects and possibly interatomic fields, the observed intensities somewhat differ from the theoretical values. 26 The preliminary estimates of the hyperfine coupling constants for both lower and upper levels can be obtained with the help of measured spacings between a set of selected hyperfine components by assigning F and F values to the peak positions according to their intensity pattern. The diagonal components are much stronger than the off-diagonal components. For example, in our observed spectrum Fig. 2, the intense hyperfine peaks corresponding to diagonal components 11 11, 10 10, 9 9, 8 8, and 7 7 could be easily identified. We measure the energy separation ( E) between two hyperfine transitions, say F 11 F 11 and F 10 F 10 see Fig. 2 : E E F 11 E F 11 E F 10 E F 10 E F 11 E F 10 E F 11 E F Let A, B and A, B be the hyperfine coupling constants of the lower and the higher states, respectively, and let k 1 and k 2 correspond to F 10 and F 11 and k 1, and k 2 correspond to F 10 and F 11, respectively. The energy separation E as defined by Eq. 32 can be expressed in frequency units as 12 A k 2 k 1 3 B k 2 k 2 1 k 1 k IJ 2I 1 2J 1 A k 2 k 1 3 B 2 8 k 2 k 2 1 k 1 k 1 1. IJ 2I 1 2J 1 33 We form four simultaneous equations in four unknown quantities A, B, A, and B corresponding to the observed energy seperations of the diagonal components. The four simultaneous equations are solved to obtain the values of A, B, and A, B. The values obtained for A, B, and A, B serve as initial guess values for the lower and the upper levels, respectively, which are used as free parameters to fit the entire spectrum. The complete hyperfine spectrum was fitted to a sum of Gaussian functions given by 27 F x n I n exp x x n x, 34 d where I n is the intensity of the nth hfs component and x d is the half width of a Gaussian profile. The entire hyperfine 708 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 708

8 Fig. 5. Schematic of the experimental arrangement for diode laser based absorption spectroscopy of rubidium. Fig. 4. Computer generated hyperfine spectrum of the nm transition in Ho I using the fitted hyperfine structure coupling constants A and B. This should be compared with the observed spectrum given in Fig. 2. spectrum is fitted with the normalized intensities of the individual hfs components and the fitted spectrum is shown in Fig. 4. The observed hyperfine transition intensities in optogalvanic spectroscopy deviate slightly from the expected theoretical intensities because of saturation effects. 26 These saturation effects are accounted for by introducing a single optical saturation parameter into the intensity formulae. The best values of A, B and A and B obtained are tabulated in Table I along with the values available in the literature. The ground state electronic configuration of Ho is Xe 4 f 11 6s 2. For the presently investigated transition, the lower level corresponds to an energy of cm 1 and its configuration as given by Wyart and Camus 25 is Xe 4 f 11 6s6p. The upper level at cm 1 has a configuration Xe 4 f 11 6s7s which was also reported by Wyart and Camus. 25 Using the best values of A, B, and A and B, we generate the expected spectrum using Eq. 34 which is shown in Fig. 4. To the best of our knowledge, the hyperfine structure constants of the upper level are reported for the first time. Table I. Hyperfine structure constants A and B determined by laser optogalvanic spectroscopy for the nm transition in Ho I. Level designation Energy level (cm 1 ) hfs constants MHz This study Previous studies 25 A B A B 4f 11 ss f 11 sp D. Hyperfine structure measurements in Rb using absorption spectroscopy Recently, Wieman and Preston 28 presented a detailed writeup on Doppler-free spectroscopy of rubidium atoms for undergraduate laboratory. They used an external cavity tunable diode laser for saturation spectroscopy experiments. However, the ground state hyperfine structure of rubidium can be investigated by studying the absorption spectra of rubidium employing simple temperature/injection current tuning of the diode laser. Even though the resolution attainable in the presently proposed Doppler limited spectroscopy technique is significantly lower than the saturation spectroscopy, it is much simpler and quite adequate to investigate the hyperfine structure of the ground state of rubidium in an undergraduate laboratory. For rubidium hyperfine structure measurements one can use a hollow cathode lamp and optogalvanic spectroscopy technique as demonstrated above or a rubidium cell and simple absorption spectroscopy. For the present measurements, we employed a rubidium cell and simple absorption spectroscopy. Rubidium cells can be easily fabricated if vacuum and sealing facilities are available. Rubidium cells are also available commercially at a cost of about $250. This should be compared with $150 which is the cost of the hollow cathode lamp. A Pyrex glass tube about 5 cm long and 2-cm diameter was fitted with optical windows and evacuated to high vacuum (10 5 Torr), degassed a couple of times, and a small quantity of rubidium was introduced and the tube was sealed. It should be mentioned that ultra-high vacuum is not critical for this experiment. Rubidium can be introduced into the cell by distillation. The details on the fabrication of the rubidium cells were presented in detail by McAdam et al. 4 Since rubidium vapor pressure at room temperature is high, one would have adequate density of rubidium atoms in the vapor state to carry out hyperfine structure/absorption spectroscopy measurements. The experimental arrangement for the study of the absorption spectroscopy of rubidium employing the temperature/injection current tuning of a diode laser is given in Fig. 5. The Doppler limited hyperfine structure spectrum of rubidium recorded using the temperature tuning of the laser diode is shown in Fig. 6. A Sharp LTO27MD laser diode lasing at 780 nm was used for the measurements. The hyperfine structure spectrum of Rb vapor obtained by modulating the injection current by 4 ma is shown in Fig. 7. Figure 7 a was obtained by measuring the absorption as a function of laser frequency with no lock-in detection. Figure 7 b was obtained with lock-in detection which considerably improves the signal-to-noise ratio and also minimizes the constant sloping background. If a lock-in amplifier is not available, simple absorption spectroscopy can be employed for ground state hyperfine structure measurements of rubidium. Rubidium has two stable isotopes. The measured values of the nuclear spin (I), the nuclear magnetic dipole moment and the nuclear electric quadrupole moment (q) of the rubidium isotopes are available in the literature: Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 709

9 Fig. 6. Doppler limited hyperfine spectrum of rubidium in a vapor cell obtained by the temperature tuning of the diode laser. The spectrum on the left was obtained during the natural heating cycle and the spectrum on the right was obtained during the Peltier cooling cycle. Note the different laser detuning scales for heating and cooling cycles. 85 Rb 72.15%, I 5 2, n, and q b, 87 Rb 27.85%, I 3 2, n, and q b, where n is the nuclear magneton and b stands for barns (1 b cm 2 ). The Doppler broadening for Rb is 550 MHz at room temperature. This is an alkali atom with the ground state configuration Kr 5s 1, and J 2. 1 The 5P 1/2 and 5P 3/2 excited states are respectively at and nm. E. Hyperfine structure calculations in 85 Rb Ground state 5S 1/2 : J 1 2, I 5 2, and A MHz. 18 Because of the hyperfine interactions, the ground state splits into two states corresponding to F 2 and F 3. There will be no quadrupole interaction because J 1. The magnetic dipole interaction results in an energy separation of the F 2 and F 3 levels by 3036 MHz. 5P 1/2 excited state: J 1 2, I 5 2, and A MHz. 18 Because of the hyperfine interactions, this level splits into two levels corresponding to F 2 and F 3. There will be no quadrupole interaction. The magnetic dipole interaction splits these levels by an energy equal to MHz. 5P 3/2 excited state: J 3 2, I 5 2, A MHz, and B MHz. The F values correspond to 1, 2, 3 and 4. We calculated the total energy splittings corresponding to both the magnetic and the quadrupole interactions using the Casimir formula Eq. 30. The level separations are shown in Fig. 8. Fig. 7. Doppler limited hyperfine spectrum of rubidium in a vapor cell recorded by the injection current tuning of the diode laser. Spectrum a was recorded with no lock-in detection whereas spectrum b was obtained with lock-in detection. Note the improvement in the signal-to-noise ratio and reduction in the background level with lock-in detection. F. Hyperfine structure calculations in 87 Rb Ground state 5S 1/2 : J 2, 1 I 2, 3 and A MHz. The ground state splits into two states corresponding to F 1 and 2. There will be no quadrupole interaction. The magnetic interaction results in an energy separation of MHz. 5P 1/2 excited state: J 2, 1 I 2, 3 and A MHz. This state splits into two states corresponding to F 1 and F 2. There will be no quadrupole interaction. The magnetic interaction corresponds to an energy separation of MHz. 5P 3/2 excited state: J 2, 3 I 2, 3 A 84.8 MHz, and B MHz. Now F can have values F 0, 1, 2, and 3. We calculated the total energy splittings corresponding to both magnetic and quadrupole interactions using the Casimir formula Eq. 30. The level separations are shown in Fig. 9. The hyperfine splittings of the 5P 3/2 excited states at 780 nm of both 85 Rb and 87 Rb are small compared to the Doppler broadening ( 550 MHz) of rubidium at room temperature and will not be resolved in Doppler limited spec- 710 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 710

10 VI. CONCLUSIONS A simple experimental arrangement employing temperature/current tuning of diode lasers and optogalvanic spectroscopy can be effectively employed to measure the hyperfine interactions magnetic dipole and electric quadrupole interactions of a number of atomic species. All the components diode lasers, hollow cathode lamps, etc. needed for the experimental setup are readily available from commercial sources at low cost. The experimental setup can be assembled easily in an undergraduate instructional laboratory and requires no fabrication work involving machine/glass shop facilities. Fig. 8. The hyperfine structure level scheme along with the expected hyperfine transitions for the 780-nm transition in 85 Rb. troscopy. However, the ground state splittings are much larger than the Doppler broadening and will be well resolved even in Doppler limited spectroscopy. Therefore, in the case of Doppler limited spectroscopy, we expect a total of four peaks, two corresponding to 85 Rb ground state hyperfine splitting and two corresponding to 87 Rb ground state hyperfine splitting Figs. 8 and 9. APPENDIX: PARTS AND SUPPLIERS 1 Hollow Cathode Lamps: Holmium # Q, $279.36; Rubidium # N, $331.84; Fisher Scientific Company, 52 Fadem Rd., Springfield, NJ 07081, Phone: Hollow Cathode Lamps: Holmium # , $168.00; Rubidium # , $175.00, Scientific Measurement Systems, Inc., 606 Foresight Circle East, Grand Junction, CO 81505, Phone: Diode Lasers: Sharp #LTO27MD, 780 nm, 10 mw, $45.00; Sharp #LTO30MD, 750 nm, 5 mw, $69.10, THOR LABS, 435 Route 206, P. O. Box 366, Newton, NJ , Phone: SI PIN Detector, Item #DET100, $81.00, 20 ns rise time, 13.7 sq. mm active area, range nm, THOR LABS, 435 Route 206, P. O. Box 366, Newton, NJ , Phone: Optical Isolators: Model I-80T-4 Single Stage, 4 mm clear aperture, range nm, $1,615.00, Isowave, 64 Harding Avenue, Dover, NJ 07801, Phone: Chopper: $995.00, Stanford Research Systems, Inc., 1290D Reamwood Avenue, Sunnyvale, CA Temperature Controller Model 320, $975.00; Model 502 Laser Diode Driver, $895.00; Model , 9 mm Laser Diode Mount, $645.00, Newport/Klinger, Baldy Circle, Fountain Valley, CA 92708, Phone: Newport/Klinger acquired Light Control Instruments, Inc.. 7 Rubidium Cell: Rubidium vapor cell, 7.5 cm long and 2.5 cm diameter, $250.0, delivery 4 weeks, Environmental Optical Sensors, Inc., 6395 Gunpark Drive, Boulder, CO 80301, Phone: Fig. 9. The hyperfine structure level scheme along with the expected hyperfine transitions for the 780-nm transition in 87 Rb. 1 John R. Brandenberger, Lasers and modern optics in undergraduate physics Lawrence University Report unpublished J. C. Camparo, The diode laser in atomic physics, Contemp. Phys. 26, C. Wieman and L. Hollberg, Using diode lasers for atomic physics, Rev. Sci. Instrum. 62, K. B. MacAdam, A. Steinbach, and C. Wieman, A narrow-band tunable diode laser system with grating feedback, and a saturated absorption spectrometer for Cs and Rb, Am. J. Phys. 60, C. Wieman, G. Flowers, and S. Gilbert, Inexpensive laser cooling and trapping experiment for undergraduate laboratories, Am. J. Phys. 63, K. G. Libbrecht, R. A. Boyd, P. A. Willems, T. L. Gustavson, and D. K. Kim, Teaching physics with 670 nm diode lasers-construction of stabilized lasers and lithium cells, Am. J. Phys. 63, Listings of commercially available diode lasers and their characteristics can be downloaded on the internet: Thor s guide to laser diodes, Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 711

11 8 K. G. Libbrecht and J. L. Hall, A low-noise high-speed diode laser current controller, Rev. Sci. Instrum. 64, R. Gupta, Resource Letter LS-1 Laser Spectroscopy, Am. J. Phys. 59, G. N. Rao, J. Govindarajan, and M. N. Reddy, Optogalvanic spectroscopy of sputtered atoms, Hyperfine Interact. 38, B. Barbieri, N. Beverini, and A. Sasso, Optogalvanic Spectroscopy, Rev. Mod. Phys. 62, L. Amstrong, Theory of Hyperfine Structure Wiley, New York, W. R. Hindmarsh, Atomic Spectra Pergamon, London, 1967, pp I. Sobelman, Atomic Spectra and Radiative Transitions Springer-Verlag, Berlin, 1991, pp J. D. Jackson, Classical Electrodynamics Wiley, New York, 1975, p H. B. G. Casimir, On the interaction between atomic nuclei and electrons Freeman, San Francisco, K. Heilig, Nuclear charge radii from optical spectroscopy, Hyperfine Interactions 38, E. Arimondo, M. Inguscio, and P. Violino, Experimental determinations of the hyperfine structure in the alkali atoms, Rev. Mod. Phys. 49, P. G. H. Sandars and J. Beck, Relativistic effects in many electron hyperfine structure, I Theory, Proc. R. Soc. London Ser. A 289, Pramila Raghavan, Table of Nuclear Moments, At. Data Nucl. Data Tables 42, H. E. White, Introduction to Atomic Spectra McGraw Hill, New York, 1934, p H. Kopfermann, Nuclear Moments Academic, New York, 1958, p C. Candler, Atomic Spectra and the Vector Model Van Nostrand, New York, 1937, p W. C. Martin, R. Zalubas, and L. Hagan, eds., Atomic Energy Levels The Rare Earth Elements U.S. Government Printing Office, Washington DC, J. F. Wyart and P. Camus, Etude du spectre de l holmium atomique II, Physica C93, R. Engleman, Jr., R. A. Keller, and C. M. Miller, Effect of optical saturation on hyperfine intensities in optogalvanic spectroscopy, J. Opt. Soc. Am. B 2, M. N. Reddy, S. A. Ahmad, and G. N. Rao, Laser optogalvanic spectroscopy of holmium, J. Opt. Soc. Am. B 9, Carl Wieman, private communication, preprint. 712 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 712