Atomic hyperfine structure studies using temperature/current tuning of diode lasers: An undergraduate experiment
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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 11530
~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 31 443.26 cm21 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. © 1998 American Association of Physics Teachers.
I. INTRODUCTION level splittings are larger than the Doppler broadening, and
therefore one can obtain well-resolved hyperfine spectra and
Lasers are playing an important role in the undergraduate reliable hyperfine structure coupling constants even in Dop-
physics laboratory curriculum for conducting a variety of pler limited spectroscopy. The experimental arrangement
interesting experiments in atomic physics and modern optics. does not involve any fabrication work and therefore no glass/
Details of some of these interesting experiments can be machine shop facilities are required. The entire setup can be
found in a report prepared by Bradenberger.1 During the last assembled in a couple of days with readily available com-
several years, semiconductor diode lasers have become mercial components. Since some colleges and universities do
popular for a variety of experiments to study atomic not have machine/glass shop facilities, the presently reported
structure.2,3 Because of their low cost, compact size and ease experiments are likely to be of special appeal to them.
of operation, they can be conveniently employed in an un- Most commercially available laser diodes7 can be em-
dergraduate instructional laboratory to carry out numerous ployed for the present studies. However, laser diodes operat-
interesting experiments in atomic physics. Recently, a num- ing in single frequency mode have significant advantages.
ber of undergraduate experiments have been proposed based One can use commercially available hollow cathode lamps
on diode lasers. Most of them use external cavity stabilized for optogalvanic spectroscopy work. Hollow cathode lamps
diode lasers with piezoelectric drives that require machine of most of the elements are commercially available as a stock
shop facilities for fabrication. item from a number of vendors and the typical cost is in the
MacAdam, Steinbach, and Wieman4 described the con- range ;$100– $230. If a diode laser setup is already avail-
struction of an external cavity narrow band tunable diode able, the hyperfine structure studies employing optogalvanic
laser system and a saturated absorption spectrometer for Cs spectroscopy can be carried out with a few hundred dollars.
and Rb. Wieman, Flowers, and Gilbert5 presented an inex-
pensive laser cooling and trapping experiment for under-
graduate laboratories. Libbrecht et al.6 reported the details of II. DIODE LASERS
the construction of stabilized lasers and lithium cells using a
670-nm diode laser to perform undergraduate atomic physics Compared to traditional ion and solid state lasers, diode
experiments. All of them involved the fabrication of a stabi- lasers are compact, reliable, easy to operate, amenable to
lized external cavity arrangement with a piezo drive control. high frequency electronic modulation and temperature tun-
They employed Doppler-free high-resolution saturation spec- ing, and are of low cost. The basic principles of diode laser
troscopy for the hyperfine structure studies. Here, we present operation were well documented in the literature ~see Ref. 3
a much simpler arrangement ~which can be assembled at and the literature cited therein!. The laser diode consists of a
minimal cost! to study the hyperfine structure of a number of double heterojunction surrounded by p-type and n-type clad-
atomic species using Doppler limited spectroscopy. We em- ding layers. When the laser diode is forward biased, elec-
ploy simple temperature/current tuning of the diode laser and trons and holes are injected into the active region and light is
optogalvanic spectroscopy technique for detection. This generated as a result of the recombination of the electron-
method can be used for almost all the rare earths, and a hole pairs. The electrons and holes confined to the active
number of other atomic species such as Nb, and Ta which region undergo a population inversion resulting in laser ac-
have large nuclear moments. For these atoms, the hyperfine tion. The wavelength of the emitted laser radiation is ap-
702 Am. J. Phys. 66 ~8!, August 1998 © 1998 American Association of Physics Teachers 702proximately equal to the band gap of the semiconductor ma- ;0.06 nm/K and (dl/dt) gain;0.25 nm/K. Because of this
terial. Compared to the ;100-nm tuning range of dye lasers mismatch of the temperature coefficients, as we change the
and the much larger tuning range of Ti-sapphire lasers, diode temperature, the wavelength output shows discontinuities.
lasers have a limited tuning range of ;10 nm. In general, the The first step of the tuning process is to set the laser output
continuously tunable range of diode lasers without mode wavelength close to the atomic/molecular transition of inter-
hops is considerably less and is of the order of 1 to 2 nm. est. If necessary, one can change both temperature and injec-
Therefore, one has to carefully choose an appropriate diode tion current to get the laser output at the desired wavelength.
laser which matches the atomic/molecular transition of inter- The stability of the laser output is important and it can be
est. Listings of commercially available diode lasers and their achieved by tweaking the temperature and current such that
characteristics are available on the Internet.7 We have tested reasonably stable output and tunability over a wider wave-
a number of lasers manufactured by Hitachi, Mitsubishi, length range is attained. The temperature tuning can be ac-
SDL, Sharp, and Toshiba for single frequency operation. We complished either by heating or cooling the diode laser. The
find that Hitachi, Mitsubishi, SDL, and Sharp lasers gave data were collected during the heating/cooling process and
good single frequency operation. GaAlAs diode lasers oper- the wavelength of the laser output was simultaneously moni-
ate in the 750- to 900-nm range and are useful for Rb and Cs tored. For current tuning, the temperature of the laser diode
atom traps. InGaAs diode lasers operate in the range 910– was kept constant at a particular value such that the laser
1020 nm, whereas AlGaInP diode lasers emit radiation vis- output was close to the atomic transition of interest and the
ible in the 630- to 700-nm range. InGaAsP laser diodes have diode laser injection current was modulated by a signal gen-
outputs in the far infra-red region, 1100–1650 nm range, and erated by a function generator. Typically, one uses a ramp
the lead salt laser diodes cover 10–33 mm. As stated earlier, signal of 50–200 mV ~peak to peak! at a modulation fre-
narrow linewidth single frequency diode lasers are optimally quency of 0.01 Hz to 15 kHz.
suitable for the present experiments. Diode lasers can be em-
ployed for hyperfine structure studies in atoms using simple
temperature/injection current tuning as described in this pa- III. OPTOGALVANIC SPECTROSCOPY OF
per or Doppler-free spectroscopy using the external cavity SPUTTERED ATOMS
arrangement as described in Refs. 4 and 6.
The first step is to choose an appropriate single frequency Optogalvanic spectroscopy ~OGS! is based on the ‘‘Opto-
diode laser which lases at a wavelength close to the atomic galvanic Effect’’ which is the change in the impedance of a
transition of interest. Any diode laser mount with thermo- gaseous discharge due to the resonant light absorption. It is a
electric cooler ~TEC! would be adequate for the present ex- simple and convenient detection technique for studying the
periments. We used a mount manufactured by Light Control spectroscopy of atoms, ions, molecules and radicals in elec-
which has a thermoelectric cooler and a 10-kV thermistor to trical, high frequency discharge plasmas and flame plasmas.
monitor the temperature. We used a current controller also Unlike emission or absorption spectroscopy which require
manufactured by Light Control. Any simple low-noise ~10– the use of optical detectors, OGS does not require an addi-
100 mA rms! current controller along with a temperature tional detector; instead the discharge plasma itself acts as a
stabilizer ~short term stability ;10 mK! will be adequate for sensitive nonoptical detector. No background filtering is
these experiments. A low-noise high-speed diode laser cur- needed in OGS and the signal-to-noise ratio is quite good
rent controller circuit ~which can be easily fabricated in an and is generally of the order of 103 .
undergraduate laboratory! capable of providing low noise
~total noise of ;45 nA rms in a 1-MHz bandwidth! and
A. Simple theory of optogalvanic effect
stable current ~current drift ,0.25 m A in 3 h! output was
reported by Libbrecht and Hall.8 There are two significantly different mechanisms for the
origin of the optogalvanic effect. In the first mechanism, the
A. Tuning of the diode lasers absorption of laser radiation in the discharge results in a
change in the steady-state population of bound atomic levels.
Diode lasers operate in general in single mode or multi- Different levels, in general, will have different ionization
mode. Often, a laser diode may give multimode output at probabilities. Hence, there is a net change in the ionization
lower currents and give single mode output at higher cur- balance of the discharge. A perturbation to the ionization
rents. Some diode lasers, even though stated to be of single balance leads to a change in the current through the dis-
frequency mode by the manufacturers, were found to be oth- charge or, equivalently, a change in the impedance of the
erwise. If you have access to a monochromator, it is a good discharge. In the second mechanism, the excitation of atoms
idea to first test the diode laser for single frequency output. A by the laser to higher electronic states perturbs the equilib-
diode laser can be tuned by temperature tuning, current tun- rium established between the electronic temperature and the
ing or external cavity tuning. Design, fabrication and char- atomic excitation temperature. But the superelastic collisions
acterization of diode lasers locked to an external cavity and between the electrons and the laser excited atoms in the dis-
their applications for a variety of atomic physics experiments charge tend to restore the equilibrium. In this process an
have been well documented.4–6 Here, we focus on tempera- excess amount of energy is released which often ends up in
ture and current tuning and their applications to hyperfine an increased electron temperature of the discharge. There-
structure measurements. fore, the laser excitation of atoms leads to an increase in the
The output wavelength of a free-running diode laser is conductivity or decrease in the impedance of the discharge.
determined by the temperature and injection current. The In fact, both mechanisms are expected to be present simul-
cavity tuning characteristics and the tuning characteristics of taneously in OGS. The relative importance of these two
the semiconductor medium gain have different wavelength to mechanisms depends on the discharge and excitation condi-
temperature coefficients and they have values (dl/dt) cavity tions.
703 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 703plications, sputtering techniques seem to offer some advan-
tages 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 em-
ployed 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 re-
Fig. 1. Schematic of the experimental arrangement for the diode laser ex- sulting in a distribution corresponding to a significantly
cited optogalvanic spectroscopy ~FPI–Fabry–Perot interferometer!. 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 steady-
B. Optogalvanic spectroscopy of sputtered atoms state densities of the order of, or greater than, 1011/cm3 of the
ground state atoms, metastable atoms and singly charged
The optogalvanic detection technique is well suited for the
ions. The number density is quite adequate for a variety of
spectroscopic study of sputtered atoms. The atomic spectros-
spectroscopy experiments, in particular high sensitivity and
copy of even refractory and nonvolatile elements can be car-
high selectivity techniques such as laser optogalvanic
ried out with ease using this method.
spectroscopy.10 The important points of interest are ~i! the
The hollow cathode discharge serves as a rich reservoir of
species get thermalized quickly, and the thermalized Doppler
sputtered atoms. Under the right conditions of gas pressure
broadening corresponds to temperatures in the range 300–
and bore diameter of the cathode, the negative glows from
800 K and high-resolution spectroscopy work is feasible; ~ii!
opposite walls of the inner surface of the hollow cathode
the sputtering yields, unlike the yields involved in the ther-
coalesce to produce neutral and excited atoms and ions in
mal methods, do not change drastically from element to ele-
high densities at the center of the hollow cathode. The hol-
ment and therefore laser optogalvanic spectroscopy ~LOGS!
low cathode discharge is highly self-sustaining and can
can be carried out on almost all the elements of the periodic
maintain high currents at small cathode-fall potential values.
table; ~iii! since many excited states are populated in the
Application of a potential difference of a few hundred
discharge, spectroscopy of the highly excited states such as
volts between the two electrodes of the hollow cathode lamp
Rydberg states can also be conducted; and ~iv! no light de-
~see Fig. 1! leads to breakdown of the rare gas buffer at low
tector is needed in this technique. The sputtering yields for
pressure and creation of a number of electron–ion pairs re-
refractory elements such as Zr and Nb are only about five
sulting in a discharge. The ions together with fast neutral
times lower than the fast sputtering elements such as Cu and
atoms produced by resonant charge exchange are accelerated
Ag.
in the high field of the cathode dark space and bombard a
The OGS technique is applicable for the study of both the
cathode which is made of, or coated with, the material of
ground and the excited states of atoms.11 In fact, optogal-
interest. The highly energetic ions and fast neutral atoms
vanic spectroscopy technique offers greater sensitivity for
impart sufficient energy to the crystal lattice of the cathode
the study of the highly excited states of atoms than do optical
material to dislodge and eject the atoms from the lattice sites.
detection methods.
The sputtered species, predominantly single ground state
neutral atoms, which initially possess high kinetic energies,
rapidly lose their kinetic energy by elastic collisions with IV. HYPERFINE INTERACTIONS12
rare gas atoms and attain thermal equilibrium. As the sput-
tered atoms diffuse from the cathode surface into the nega- A. Fine structure of atoms13,14
tive glow, some of them are excited or ionized by electron
impact or by collisions with metastable atoms or ions present The development of high resolving power spectroscopic
in the discharge. In this way, a reasonably high steady-state instruments at the end of nineteenth century led to the dis-
density of atoms, metastable atoms and singly ionized ions covery of many finer details of the atomic structure. Michel-
can be maintained in the negative-glow region of the dis- son, Fabry, Perot, Lummer, and Gehrcke noted that many
charge which is suitable for carrying out optogalvanic spec- spectral lines consist of not only fine structure, but in fact
troscopy. each fine structure line consisted of many closely spaced
lines ~hyperfine structure!.
The fine structure of atomic states is a result of the orbital
C. Sputtering process motion of the electrons with intrinsic spins through the elec-
tric field caused by the nuclear charge. The spin angular
In atomic spectroscopy experiments, preparation of the
momentum ~s! of an electron gives rise to its magnetic mo-
sample often demands a major effort and its importance need
not be overemphasized. Discharge, arc, and spark sources ment ( m s ),
were commonly employed in traditional optical spectroscopy 2e\
studies.9 For studies on atomic structure using lasers, optical m s5 s522 m Bs, ~1!
mc
cells maintained at high temperatures, atomic beams, heat
pipe ovens, and sputtering cells are popular. For specific ap- where m B is the Bohr magneton.
704 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 704Because of the orbital motion of the electron in the electric B. Hyperfine interactions
field of the nucleus, it experiences an apparent magnetic field
Bl proportional to the orbital angular momentum (l), and We are interested in the electrostatic interaction between
thus the magnetic moment of the electron gives a term in the the atomic nucleus and the surrounding electrons. In the fol-
Hamiltonian: lowing, we consider the nucleus interacting with the external
fields produced by the electrons. The total interaction energy
H52 m s •Bl . ~2! may be written as a sum of the interactions of each proton
The Hamiltonian due to the spin-orbit interaction after taking charge q p at position rp with an electron of charge q e at
relativistic effects into account is often written in the form position re ,
H5 z nl L–S. ~3! q pq e
W5 (e (p u r p 2r e u
, ~10!
The spin-orbit interaction depends not only on the magni-
tudes of the L and S but also on their orientation and is which may also be written as
EE r
proportional to L–S. The total angular momentum J of the
~ rn ! r ~ re ! 3
atom is W5 d r n d 3r e , ~11!
u rn 2re u
J5L1S. ~4!
where r (rn ) and r (re ) are the nuclear and electron charge
The total angular momentum J can have values distributions, respectively. This is usually written in the
J5L1S,L1S21,...u L2S u . ~5! form15
Equation ~3! is to be corrected for the interaction of the mag-
netic moments of different electrons, interaction of the or-
W5 Er n ~ r! F ~ r! d 3 r, ~12!
bital motion of one electron and the spins of the other elec- where F~r! is the potential produced by the electrons at the
trons, etc. Since nucleus. The potential F~r! is a slowly varying function over
the nuclear volume and can be expanded in Taylor series15
L–S5 ~ 21 !~ J2 2L2 2S2 ! ,
1
the shift in the energy of an electron may be written as F ~ r! 5F ~ 0 ! 2r•E~ 0 ! 2
6 (i (j ~ 3x i x j 2r2 d i j !
DE J 5 ~ 21 ! z nl @ J ~ J11 ! 2L ~ L11 ! 2S ~ S11 !# , ~6!
]E j
3 ~ 0 ! 1¯ . ~13!
where z is the fine-structure constant. Each J state splits into ]xi
(2S11) components if Snuclear charge distribution. This produces the same shift for The direction of HJ (0) is that given by the total angular
all the levels of a configuration. For different isotopes of an momentum of the atomic electrons J. The direction of HJ (0)
element the shifts vary, leading to the so-called isotope shift. is opposite to the direction of J because the electrons have
The k51 term corresponds to the interaction between the negative charge. The nuclear magnetic moment can be writ-
magnetic dipole moment of the nucleus m I and the magnetic ten as
hyperfine field induced by the electrons at the nucleus
mI
HJ (0). The k52 term is due to the interaction of the nuclear m I 5 m n g I I, 5 I,
electric quadrupole moment QI and the electric field gradi- uIu
ents qJ (0) produced at the nuclear site due to charges exter- where m n is the nuclear magneton and g I is the nuclear g
nal to the nucleus. The higher order terms are usually negli- factor:
gibly small. For example, the magnetic octupole (k53) and
2 m IH J~ 0 ! 1
electric hexadecapole (k54) interactions are about 108 DE5 ~ I–J! 5AI–J5 A ~ F2 2J2 2I2 ! .
times smaller than their corresponding lower-order magnetic u I uu J u 2
dipole (k51) and electric quadrupole (k52) interaction and ~19!
require ultrahigh-precision techniques for measurements and The magnetic dipole coupling constant A in frequency units
will not be discussed here. may be written as
2 m IH J~ 0 !
A5 . ~20!
C. Isotope shift h u I uu J u
The monopole interaction results in a small energy shift in An atomic level with the total angular momentum value J
the nuclear and electron levels. The relative shift of the elec- will split according to the possible values (I–J) which are
tron levels of a given configuration for two isotopes of an quantized. In this case analogous to the spin-orbit interaction
element is known as the isotope shift. The energy shifts as- giving rise to (L–S) term in the fine structure of atoms, J and
sociated with the nuclear levels can be measured by employ- I couple resulting in the total angular momentum, which is
ing a variety of techniques such as Mössbauer spectroscopy. designated by F, such that
The energy shifts of an atomic transition corresponding to F5I1J. ~21!
two isotopes of mass numbers A and A 8 may be written as
An atomic level of J is split into a number of sublevels with
d n A-A 8 5 d n A-A 8 FS1 d n A-A 8 NMS1 d n A-A 8 SMS, ~16! all possible values of F such that
where the first term on the right-hand side corresponds to the
J2I1/2,J>1/2! , ~23!
2
D. Magnetic dipole interaction where
k5F ~ F11 ! 2I ~ I11 ! 2J ~ J11 ! . ~24!
The magnetic dipole moment m I of a nucleus with non-
zero nuclear spin I interacts with the magnetic field HJ (0) The total angular momentum F can take values
produced by the electrons at the nucleus. This corresponds to F5I1J,I1J21,...,u I2J u . ~25!
the (k51) term in Eq. ~15!. The interaction Hamiltonian
may be written as a scalar product of nuclear and electronic The number of hyperfine components is 2J11 when I>J,
tensors, each of rank one (k51): and 2I11 when I,J.
HM 1 52 m I •HJ ~ 0 ! . ~17!
The magnetic field at the nucleus is produced by the orbital E. Electric quadrupole interaction
motion and the spin dipole moments of the electrons. From
The nuclear quadrupole moment QI interacts with the
symmetry considerations, this field is linearly related to the
electric field gradient qJ (0) produced by the electrons at the
total angular momentum of the electrons J such that J5L
nuclear site. The interaction Hamiltonian is a scalar product
1S. This interaction results in a shift of the energy levels of of two second-order tensors, one corresponding to the
the atom by an amount
nucleus QI and the second that of the electrons qJ (0) which
DE52 m I •HJ ~ 0 ! . ~18! is written as
706 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 706HE2 5QI –qJ ~ 0 ! . ~26!
For diagonal matrix elements with respect to I and J Eq. ~26!
reduces to
hB @~ 3I•J ! 2 13/2~ I•J ! 2I 2 •J 2 #
H E2 5 , ~27!
2I ~ 2I21 ! J ~ 2J21 !
where the electric quadrupole coupling constant B is given
by
e 2Q Iq J~ 0 !
B5 . ~28!
h
The energy contribution due to the quadrupole interaction
may be written as
hB ~ 2 ! k ~ k11 ! 22I ~ I11 ! J ~ J11 !
3
E E2 5 , I>1,J>1.
4 I ~ 2I21 ! J ~ 2J21 !
~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
formula16
hAk hB ~ 2 ! k ~ k11 ! 22I ~ I11 ! J ~ J11 !
3
E E2 5 1 . ~30!
2 4 I ~ 2I21 ! J ~ 2J21 !
Clearly, the hyperfine interactions depend on both the
nuclear and atomic properties of an atom. In fact, the mea-
sured energy shifts are products of them. Precision hyperfine Fig. 2. The hyperfine structure spectrum recorded for the 781.5 nm transi-
structure measurements have provided a wealth of informa- tion in Ho I using temperature tuning of the diode laser and optogalvanic
spectroscopy.
tion on nuclear structure17 and electron wavefunctions.18 The
nuclear information includes the nuclear charge radii, nuclear
magnetic dipole moments, electric quadrupole and octupole holmium hollow cathode lamp with neon buffer gas is given
moments, Sternheimer shielding and antishielding effects, in Fig. 2. Using the intense neon lines for calibration, we
nuclear hyperfine anomaly, etc. could identify a number of holmium lines in the spectra re-
The hyperfine structure spectra not only allow us to obtain corded in the 700-865-nm range. The optogalvanic detection
the magnetic and the quadrupole hyperfine coupling con- using CW laser excitation involves chopping the laser beam,
stants, but also permit unambiguous assignment of the J val- and phase sensitive detection as shown in Fig. 1. In our
ues of the atomic levels involved. The hyperfine coupling experiment the beam was chopped at 2.2 kHz. Phase-
constants depend strongly on the electronic wavefunctions in sensitive lock-in detection improves the signal to noise ratio.
the vicinity of the nucleus. Since the relativistic corrections A Fabry–Perot interferometer with a free spectral range
are important, one has to use the relativistic Dirac wavefunc- ~FSR! of 300 MHz provided the frequency markers for the
tions. The true Hamiltonian H hfs corresponding to LS calibration of the observed hyperfine spectra. For the cases
coupled relativistic eigenfunctions can be expressed as ma- studied, since the hyperfine splittings are large, one can cali-
trix elements of an effective Hamiltonian H eff hfs between the brate the spectrum with a low resolution Fabry–Perot inter-
nonrelativistic LS-coupled states.19 The effective operator ferometer with a free spectral range ~FSR! of about 2 GHz as
not only accounts for the relativistic effects, but also for the well. In fact, one can use even a monochromator for calibra-
configuration interactions and polarization effects. The ex- tion purposes. For example, a Spex monochromator ~model
pressions for the electronic tensor operators contain the ra- 1000M! has a resolution of 0.008 nm, whereas the separation
dial integrals. Since it is difficult to calculate the radial inte- between the extreme hyperfine lines in the present measure-
grals, they are often represented as free single-electron hfs ments is ;25 GHz, which corresponds to ;0.05 nm for
parameters, which can be determined by a fit procedure to 780-nm radiation.
the experimental hfs data. When an adequate number of hfs
constants A and B are determined experimentally, the single A. Hyperfine spectrum of holmium
electron hfs parameters can be determined from a least-
squares fit of the parametrized single electron parameters to In general, the atomic hyperfine splittings are quite small,
the experimental values. often demanding Doppler-free techniques. However, Dop-
pler limited spectroscopy techniques can be applied if the
magnetic and or quadrupole interactions are strong and the
V. EXPERIMENTAL resulting hyperfine splittings are large compared to the Dop-
pler broadening, as in the case of holmium which is investi-
The experimental arrangement for continuous wave ~CW! gated in this paper.
diode laser excited OGS is shown in Fig. 1. The hyperfine We employed a LTO27MD Sharp laser diode operated at
spectrum of the 781.5 nm transition of Ho I recorded using a 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 707C. Intensities of the hyperfine transitions
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
White21 and by Kopfermann22 for values up to J5 132 and I
5 27 . However, for the transition presently studied, I5 27 and
J5J 8 5 152 . The relative intensities of the different hyperfine
transitions were calculated by us using the formulae given by
Candler.23
For the transition presently studied, I5 27 and J5J 8 5 152
~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
Fig. 3. The hyperfine structure level scheme for the 781.5 nm transition in
Ho I. The expected ~calculated! intensities of the hyperfine transitions are
observed intensities somewhat differ from the theoretical
given at the bottom. 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
employing temperature scanning of the diode laser during hyperfine components by assigning F and F 8 values to the
the heating or cooling cycle. The temperature tuning of peak positions according to their intensity pattern. The diag-
the LTO27MD diode laser was measured to be onal components are much stronger than the off-diagonal
;0.06 nm/°C~;29.6 GHz/°C! at 781 nm. We could typically components. For example, in our observed spectrum ~Fig. 2!,
scan ;1.5 nm without mode hops in this region. The tuning the intense hyperfine peaks corresponding to diagonal com-
range is quite adequate to cover the entire hyperfine structure ponents ~11→118 , 10→108 , 9→9 8 , 8→8 8 , and 7→7 8 !
of the 781.5 nm transition in Ho I which is ;25 GHz. The could be easily identified. We measure the energy separation
hyperfine spectra were calibrated using the markers obtained (DE) between two hyperfine transitions, say F511→F 8
from a 300 MHz FSR Fabry–Perot interferometer. As stated 511 and F510→F 8 510 ~see Fig. 2!:
earlier, a high resolution Fabry–Perot interferometer is not
DE5 ~ E F 8 5112E F511! 2 ~ E F 8 5102E F510!
necessary for the present hyperfine structure measurements.
We used it because it is readily available in our laboratory. 5 ~ E F 8 5112E F 8 510! 2 ~ E F5112E F510! . ~32!
The only stable isotope of holmium, 165Ho, has spin I
57/2, nuclear magnetic moment m I 514.173(27) m n , and Let A, B and A 8 , B 8 be the hyperfine coupling constants
electric quadrupole moment Q512.716(9) b. 20 Because of of the lower and the higher states, respectively, and let k 81
the large nuclear moments and also the hyperfine coupling and k 82 correspond to F 8 510 and F 8 511 and k 1 , and k 2
constants, the hyperfine structure of Ho I transitions is usu- correspond to F510 and F511, respectively. The energy
ally spread over a 20–55 GHz range. The hyperfine structure separation DE as defined by Eq. ~32! can be expressed in
components for the 781.5 nm transition in Ho I ~see Fig. 2! frequency units as
F G
are well resolved even in the Doppler limited spectra because
the energy separations of the hyperfine components are ~ k 28 2k 18 ! B 8 @ k 28 ~ k 28 11 ! 2k 18 ~ k 18 11 !#
D n 125 A 8 13
larger than the Doppler broadening. The hfs level-scheme of 2 8 IJ 8 ~ 2I21 !~ 2J 8 21 !
F G
Ho I, 781.5 nm, transition along with the expected theoreti-
cal intensities is given in Fig. 3. The Doppler broadening, k 2 2k 1 B @ k 2 ~ k 2 11 ! 2k 1 ~ k 1 11 !#
2 A 13 .
which is the dominant contributor to the broadening of the 2 8 IJ ~ 2I21 !~ 2J21 !
spectral lines, was estimated to be ;750 MHz at the hollow ~33!
cathode lamp operating current of 13 mA.
We form four simultaneous equations in four unknown quan-
B. Hyperfine transitions tities A, B, A 8 , and B 8 corresponding to the observed energy
seperations of the diagonal components. The four simulta-
The hyperfine interaction couples the electron angular mo- neous equations are solved to obtain the values of A, B, and
mentum J and the nuclear angular momentum I to form the A 8 , B 8 . The values obtained for A, B, and A 8 , B 8 serve as
total angular momentum F: initial guess values for the lower and the upper levels, re-
F5I1J. ~21! spectively, which are used as free parameters to fit the entire
spectrum. The complete hyperfine spectrum was fitted to a
F can have values from u J2I u , J2I11,...,J1I21, J1I. sum of Gaussian functions given by27
F G
The selection rules for the electric dipole transitions are
2 ~ x2x n ! 2
DF50 or 61 ~no 0↔0!, F~ x !5 (n I n exp 0.36d x 2d
, ~34!
DJ50 or 61, ~31!
where I n is the intensity of the nth hfs component and d x d is
DS50. the half width of a Gaussian profile. The entire hyperfine
708 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 708Fig. 5. Schematic of the experimental arrangement for diode laser based
absorption spectroscopy of rubidium.
Fig. 4. Computer generated hyperfine spectrum of the 781.5-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.
able in the presently proposed Doppler limited spectroscopy
technique is significantly lower than the saturation spectros-
spectrum is fitted with the normalized intensities of the indi- copy, it is much simpler and quite adequate to investigate the
vidual hfs components and the fitted spectrum is shown in hyperfine structure of the ground state of rubidium in an
Fig. 4. The observed hyperfine transition intensities in opto- undergraduate laboratory.
galvanic spectroscopy deviate slightly from the expected the- For rubidium hyperfine structure measurements one can
oretical intensities because of saturation effects.26 These use a hollow cathode lamp and optogalvanic spectroscopy
saturation effects are accounted for by introducing a single technique as demonstrated above or a rubidium cell and
optical saturation parameter into the intensity formulae. The simple absorption spectroscopy. For the present measure-
best values of A, B and A 8 and B 8 obtained are tabulated in ments, we employed a rubidium cell and simple absorption
Table I along with the values available in the literature. The spectroscopy. Rubidium cells can be easily fabricated if
vacuum and sealing facilities are available. Rubidium cells
ground state electronic configuration of Ho is @ Xe# 4 f 116s 2 .
are also available commercially at a cost of about $250. This
For the presently investigated transition, the lower level cor-
should be compared with ;$150 which is the cost of the
responds to an energy of 18651.53 cm21 and its configura-
hollow cathode lamp. A Pyrex glass tube about 5 cm long
tion as given by Wyart and Camus25 is @ Xe# 4 f 116s6 p. The and 2-cm diameter was fitted with optical windows and
upper level at 31443.26 cm21 has a configuration evacuated to high vacuum (1025 Torr), degassed a couple of
@ Xe# 4 f 116s7s which was also reported by Wyart and times, and a small quantity of rubidium was introduced and
Camus.25 Using the best values of A, B, and A 8 and B 8 , we the tube was sealed. It should be mentioned that ultra-high
generate the expected spectrum using Eq. ~34! which is vacuum is not critical for this experiment. Rubidium can be
shown in Fig. 4. To the best of our knowledge, the hyperfine introduced into the cell by distillation. The details on the
structure constants of the upper level are reported for the first fabrication of the rubidium cells were presented in detail by
time. McAdam et al.4 Since rubidium vapor pressure at room tem-
perature is high, one would have adequate density of ru-
D. Hyperfine structure measurements in Rb using bidium atoms in the vapor state to carry out hyperfine
absorption spectroscopy structure/absorption spectroscopy measurements. The experi-
mental arrangement for the study of the absorption spectros-
Recently, Wieman and Preston28 presented a detailed copy of rubidium employing the temperature/injection cur-
writeup on Doppler-free spectroscopy of rubidium atoms for rent tuning of a diode laser is given in Fig. 5. The Doppler
undergraduate laboratory. They used an external cavity tun- limited hyperfine structure spectrum of rubidium recorded
able diode laser for saturation spectroscopy experiments. using the temperature tuning of the laser diode is shown in
However, the ground state hyperfine structure of rubidium Fig. 6. A Sharp LTO27MD laser diode lasing at 780 nm was
can be investigated by studying the absorption spectra of used for the measurements. The hyperfine structure spectrum
rubidium employing simple temperature/injection current of Rb vapor obtained by modulating the injection current by
tuning of the diode laser. Even though the resolution attain- 4 mA is shown in Fig. 7. Figure 7~a! was obtained by mea-
suring the absorption as a function of laser frequency with no
lock-in detection. Figure 7~b! was obtained with lock-in de-
Table I. Hyperfine structure constants A and B determined by laser opto- tection which considerably improves the signal-to-noise ratio
galvanic spectroscopy for the 781.548 nm transition in Ho I. and also minimizes the constant sloping background. If a
hfs constants ~MHz!
lock-in amplifier is not available, simple absorption spectros-
copy can be employed for ground state hyperfine structure
Level Energy level This study Previous studies25 measurements of rubidium.
designation (cm21) A B A8 B8 Rubidium has two stable isotopes. The measured values of
4 f 11ss 8 2 31 443.26 1045 21788 ¯ ¯ the nuclear spin (I), the nuclear magnetic dipole moment ~m!
4 f 11s p1 18 651.53 870 22560 864 22574 and the nuclear electric quadrupole moment (q) of the ru-
bidium isotopes are available in the literature:20
709 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 709Fig. 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.
Rb~ 72.15% ! ,
85
I5 25 , m 511.353m n ,
and q510.273 b,
Rb~ 27.85% ! ,
87
I5 23 , m 512.751m n ,
and q510.132 b, 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
where m n is the nuclear magneton and b stands for barns recorded with no lock-in detection whereas spectrum ~b! was obtained with
(1 b510224 cm2). The Doppler broadening for Rb is lock-in detection. Note the improvement in the signal-to-noise ratio and
;550 MHz at room temperature. This is an alkali atom with reduction in the background level with lock-in detection.
the ground state configuration @ Kr# 5s 1 , and J5 21 . The 5 P 1/2
and 5 P 3/2 excited states are respectively at 794.76 and
780.023 nm. 87
F. Hyperfine structure calculations in Rb
85 Ground state „5S1/2…: J5 21 , I5 23 , and A53417.34 MHz.
E. Hyperfine structure calculations in Rb
The ground state splits into two states corresponding to F
Ground state „5S1/2…: J5 21 , I5 25 , and A51011.91 51 and 2. There will be no quadrupole interaction. The
MHz.18 Because of the hyperfine interactions, the ground magnetic interaction results in an energy separation of
state splits into two states corresponding to F52 and F 6834.7 MHz.
53. There will be no quadrupole interaction because J,1. 5P1/2 excited state: J5 21 , I5 23 , and A5406.2 MHz. This
The magnetic dipole interaction results in an energy separa- state splits into two states corresponding to F51 and F
tion of the F52 and F53 levels by 3036 MHz. 52. There will be no quadrupole interaction. The magnetic
5P1/2 excited state: J5 21 , I5 25 , and A5120.72 MHz. 18 interaction corresponds to an energy separation of 812.4
Because of the hyperfine interactions, this level splits into MHz.
two levels corresponding to F52 and F53. There will be 5P3/2 excited state: J5 23 , I5 23 , A584.8 MHz, and B
no quadrupole interaction. The magnetic dipole interaction 512.52 MHz. Now F can have values F50, 1, 2, and 3. We
splits these levels by an energy equal to 362.16 MHz. calculated the total energy splittings corresponding to both
5P3/2 excited state: J5 23 , I5 52 , A525.01 MHz, and B magnetic and quadrupole interactions using the Casimir for-
525.88 MHz. The F values correspond to 1, 2, 3 and 4. We mula @Eq. ~30!#. The level separations are shown in Fig. 9.
calculated the total energy splittings corresponding to both The hyperfine splittings of the 5 P 3/2 excited states at
the magnetic and the quadrupole interactions using the Ca- ;780 nm of both 85Rb and 87Rb are small compared to the
simir formula @Eq. ~30!#. The level separations are shown in Doppler broadening (;550 MHz) of rubidium at room tem-
Fig. 8. perature and will not be resolved in Doppler limited spec-
710 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 710VI. CONCLUSIONS
A simple experimental arrangement employing tem-
perature/current tuning of diode lasers and optogalvanic
spectroscopy can be effectively employed to measure the hy-
perfine interactions ~magnetic dipole and electric quadrupole
interactions! of a number of atomic species. All the compo-
nents ~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 as-
sembled easily in an undergraduate instructional laboratory
and requires no fabrication work involving machine/glass
shop facilities.
APPENDIX: PARTS AND SUPPLIERS
~1! Hollow Cathode Lamps: Holmium #14386 100Q,
$279.36; Rubidium #14 386 106N, $331.84; Fisher Sci-
entific Company, 52 Fadem Rd., Springfield, NJ 07081,
Phone: 800-766-7000.
Hollow Cathode Lamps: Holmium #062829-04,
$168.00; Rubidium #062824-04, $175.00, Scientific
Measurement Systems, Inc., 606 Foresight Circle East,
Grand Junction, CO 81505, Phone: 800-229-4087.
Fig. 8. The hyperfine structure level scheme along with the expected hyper- ~2! Diode Lasers: Sharp #LTO27MD, 780 nm, 10 mW,
fine transitions for the 780-nm transition in 85Rb. $45.00; Sharp #LTO30MD, 750 nm, 5 mW, $69.10,
THOR LABS, 435 Route 206, P. O. Box 366, Newton,
NJ 07860-0366, Phone: ~973! 579-7227.
~3! SI PIN Detector, Item #DET100, $81.00, 20 ns rise time,
troscopy. However, the ground state splittings are much 13.7 sq. mm active area, range 350–1100 nm, THOR
larger than the Doppler broadening and will be well resolved LABS, 435 Route 206, P. O. Box 366, Newton,
even in Doppler limited spectroscopy. Therefore, in the case NJ 07860-0366, Phone: ~973! 579-7227.
of Doppler limited spectroscopy, we expect a total of four ~4! Optical Isolators: Model I-80T-4 Single Stage, 4 mm
peaks, two corresponding to 85Rb ground state hyperfine clear aperture, range 750–900 nm, $1,615.00, Isowave,
splitting and two corresponding to 87Rb ground state hyper- 64 Harding Avenue, Dover, NJ 07801, Phone: ~201!
fine splitting ~Figs. 8 and 9!. 328-7000.
~5! Chopper: $995.00, Stanford Research Systems, Inc.,
1290D Reamwood Avenue, Sunnyvale, CA 94089.
~6! Temperature Controller Model 320, $975.00; Model 502
Laser Diode Driver, $895.00; Model 700-10, 9 mm La-
ser Diode Mount, $645.00, Newport/Klinger, 18235
Baldy Circle, Fountain Valley, CA 92708, Phone: 800-
222-6440 ~Newport/Klinger acquired Light Control In-
struments, Inc.!.
~7! Rubidium Cell: Rubidium vapor cell, 7.5 cm long and
2.5 cm diameter, $250.0, delivery—4 weeks, Environ-
mental Optical Sensors, Inc., 6395 Gunpark Drive, Boul-
der, CO 80301, Phone: ~303! 530-7785.
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fine transitions for the 780-nm transition in 87Rb. www.thorlabs.com
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~1977!. 28
Carl Wieman, private communication, preprint.
712 Am. J. Phys., Vol. 66, No. 8, August 1998 Rao, Reddy, and Hecht 712You can also read
