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
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 Wieman4 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 Gilbert5 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
702
Am. J. Phys. 66 ~8!, August 1998
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 diodes7 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© 1998 American Association of Physics Teachers
702
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 910–
1020 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, 1100–1650 nm range, and
the lead salt laser diodes cover 10–33 mm. 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-kV thermistor to
monitor the temperature. We used a current controller also
manufactured by Light Control. Any simple low-noise ~10–
100 mA 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 m A in 3 h! 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 (dl/dt) cavity
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Am. J. Phys., Vol. 66, No. 8, August 1998
;0.06 nm/K and (dl/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 50–200 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 103 .
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.
Rao, Reddy, and Hecht
703
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 ap704
Am. J. Phys., Vol. 66, No. 8, August 1998
plications, 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, 1011/cm3 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 300–
800 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 INTERACTIONS12
A. Fine structure of atoms13,14
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 ( m s ),
m s5
2e\
s522 m Bs,
mc
~1!
where m B is the Bohr magneton.
Rao, Reddy, and Hecht
704
Because of the orbital motion of the electron in the electric
field of the nucleus, it experiences an apparent magnetic field
Bl proportional to the orbital angular momentum (l), and
thus the magnetic moment of the electron gives a term in the
Hamiltonian:
H52 m s •Bl .
~2!
The Hamiltonian due to the spin-orbit interaction after taking
relativistic effects into account is often written in the form
H5 z 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
~4!
J5L1S.
The total angular momentum J can have values
J5L1S,L1S21,...u L2S u .
~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–S5 ~ 21 !~ J2 2L2 2S2 ! ,
~7!
which is the well-known Landé interval rule.
The doublet energy splitting corresponding to J5l1 21 and
J5l2 21 for hydrogenlike atoms may be written as
DE 8 5
RE a 2 Z 4
,
n 3 l ~ l11 !
~8!
where RE is the Rydberg constant in energy units, and the
fine structure constant
1
e2
,
a5 '
\c 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 are14
DS50,
DL50,61,
L1L 8 >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.
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Am. J. Phys., Vol. 66, No. 8, August 1998
W5
(e (p
q pq e
,
u r p 2r e u
~10!
which may also be written as
W5
EE r
~ rn ! r ~ re ! 3
d r n d 3r e ,
u rn 2re u
~11!
where r (rn ) and r (re ) are the nuclear and electron charge
distributions, respectively. This is usually written in the
form15
W5
Er
n ~ r! F ~ r!
~12!
d 3 r,
where F~r! is the potential produced by the electrons at the
nucleus. The potential F~r! is a slowly varying function over
the nuclear volume and can be expanded in Taylor series15
~6!
where z is the fine-structure constant. Each J state splits into
(2S11) components if S<L or into (2L11) components if
S.L. The separation between the adjacent components of a
multiplet is
DE J 2DE J21 5 z nl J,
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 rp with an electron of charge q e at
position re ,
F ~ r! 5F ~ 0 ! 2r•E~ 0 ! 2
the shift in the energy of an electron may be written as
DE J 5 ~ 21 ! z nl @ J ~ J11 ! 2L ~ L11 ! 2S ~ S11 !# ,
B. Hyperfine interactions
3
1
6
(i (j ~ 3x i x j 2r2 d i j !
]E j
~ 0 ! 1¯ .
]xi
~13!
Introducing F~r! in Eq. ~12!, the total energy may be written
in the form15
W5qF ~ 0 ! 2p•E~ 0 ! 2
1
6
(i (j Q i j
]E j
~ 0 ! 1¯ .
]xi
~14!
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 F~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 i j ) 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
H5
(k 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 (k50) represents the Coulomb interaction between the electrons and the spherical part of the
Rao, Reddy, and Hecht
705
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 k51 term corresponds to the interaction between the
magnetic dipole moment of the nucleus m I and the magnetic
hyperfine field induced by the electrons at the nucleus
HJ (0). The k52 term is due to the interaction of the nuclear
electric quadrupole moment QI and the electric field gradients qJ (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 (k53) and
electric hexadecapole (k54) interactions are about 108
times smaller than their corresponding lower-order magnetic
dipole (k51) and electric quadrupole (k52) interaction and
require ultrahigh-precision techniques for measurements and
will not be discussed here.
m I 5 m n g I I,
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 8 may be written as
~16!
5
mI
I,
uIu
where m n is the nuclear magneton and g I is the nuclear g
factor:
DE5
2 m IH J~ 0 !
1
~ I–J! 5AI–J5 A ~ F2 2J2 2I2 ! .
u I uu J u
2
~19!
The magnetic dipole coupling constant A in frequency units
may be written as
A5
C. Isotope shift
d n A-A 8 5 d n A-A 8 FS1 d n A-A 8 NMS1 d n A-A 8 SMS,
The direction of HJ (0) is that given by the total angular
momentum of the atomic electrons J. The direction of HJ (0)
is opposite to the direction of J because the electrons have
negative charge. The nuclear magnetic moment can be written as
2 m IH J~ 0 !
.
h u I uu J u
~20!
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
~21!
F5I1J.
An atomic level of J is split into a number of sublevels with
all possible values of F such that
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,
d ^ r 2 & , between the isotopes.
If the nuclear spin is a half integer, say I5 , F will take
(J2 12 ) and (J1 21 ) values. If the nuclear spin is an integer,
say I51, F will take values (J21), J, and (J11) provided
J>1. If either I50 or J50, then there will be no magnetic
or higher order splittings. If I50, the nuclear dipole, electric
quadrupole and higher-order moments are zero. If J50, 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
D. Magnetic dipole interaction
where
The magnetic dipole moment m I of a nucleus with nonzero nuclear spin I interacts with the magnetic field HJ (0)
produced by the electrons at the nucleus. This corresponds to
the (k51) term in Eq. ~15!. The interaction Hamiltonian
may be written as a scalar product of nuclear and electronic
tensors, each of rank one (k51):
HM 1 52 m I •HJ ~ 0 ! .
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Am. J. Phys., Vol. 66, No. 8, August 1998
1
2
E M 15
hAk
2
~ I>1/2,J>1/2! ,
k5F ~ F11 ! 2I ~ I11 ! 2J ~ J11 ! .
~23!
~24!
The total angular momentum F can take values
~25!
F5I1J,I1J21,...,u I2J u .
The number of hyperfine components is 2J11 when I>J,
and 2I11 when I,J.
~17!
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 J5L
1S. This interaction results in a shift of the energy levels of
the atom by an amount
DE52 m I •HJ ~ 0 ! .
~22!
J2I<F<J1I.
~18!
E. Electric quadrupole interaction
The nuclear quadrupole moment QI interacts with the
electric field gradient qJ (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 QI and the second that of the electrons qJ (0) which
is written as
Rao, Reddy, and Hecht
706
HE2 5QI –qJ ~ 0 ! .
~26!
For diagonal matrix elements with respect to I and J Eq. ~26!
reduces to
H E2 5
hB @~ 3I•J ! 2 13/2~ I•J ! 2I 2 •J 2 #
,
2I ~ 2I21 ! J ~ 2J21 !
~27!
where the electric quadrupole coupling constant B is given
by
B5
e 2Q Iq J~ 0 !
.
h
~28!
The energy contribution due to the quadrupole interaction
may be written as
3
hB ~ 2 ! k ~ k11 ! 22I ~ I11 ! J ~ J11 !
,
E E2 5
4
I ~ 2I21 ! J ~ 2J21 !
I>1,J>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
formula16
E E2 5
3
hAk hB ~ 2 ! k ~ k11 ! 22I ~ I11 ! J ~ J11 !
1
.
2
4
I ~ 2I21 ! J ~ 2J21 !
~30!
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 structure17 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 eff
hfs between the
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 781.5 nm transition of Ho I recorded using a
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Am. J. Phys., Vol. 66, No. 8, August 1998
Fig. 2. The hyperfine structure spectrum recorded for the 781.5 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 700-865-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 0.008 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
Rao, Reddy, and Hecht
707
C. Intensities of the hyperfine transitions
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
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 781.5 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, 165Ho, has spin I
57/2, nuclear magnetic moment m I 514.173(27) m n , and
electric quadrupole moment Q512.716(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 20–55 GHz range. The hyperfine structure
components for the 781.5 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, 781.5 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:
F5I1J.
~21!
F can have values from u J2I u , J2I11,...,J1I21, J1I.
The selection rules for the electric dipole transitions are
DS50.
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Am. J. Phys., Vol. 66, No. 8, August 1998
DE5 ~ E F 8 5112E F511! 2 ~ E F 8 5102E F510!
5 ~ E F 8 5112E F 8 510! 2 ~ E F5112E F510! .
~31!
~32!
Let A, B and A 8 , B 8 be the hyperfine coupling constants
of the lower and the higher states, respectively, and let k 81
and k 82 correspond to F 8 510 and F 8 511 and k 1 , and k 2
correspond to F510 and F511, respectively. The energy
separation DE as defined by Eq. ~32! can be expressed in
frequency units as
F
D n 125 A 8
F
~ k 28 2k 18 !
2 A
2
13
B 8 @ k 28 ~ k 28 11 ! 2k 18 ~ k 18 11 !#
8 IJ 8 ~ 2I21 !~ 2J 8 21 !
G
G
B @ k 2 ~ k 2 11 ! 2k 1 ~ k 1 11 !#
k 2 2k 1
13
.
2
8
IJ ~ 2I21 !~ 2J21 !
~33!
We form four simultaneous equations in four unknown quantities A, B, A 8 , and B 8 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 8 , B 8 . The values obtained for A, B, and A 8 , B 8 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 by27
F~ x !5
DF50 or 61 ~no 0↔0!,
DJ50 or 61,
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
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 8 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→118 , 10→108 , 9→9 8 , 8→8 8 , and 7→7 8 !
could be easily identified. We measure the energy separation
(DE) between two hyperfine transitions, say F511→F 8
511 and F510→F 8 510 ~see Fig. 2!:
F
(n I n exp
2 ~ x2x n ! 2
0.36d x 2d
G
,
~34!
where I n is the intensity of the nth hfs component and d x d is
the half width of a Gaussian profile. The entire hyperfine
Rao, Reddy, and Hecht
708
Fig. 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.
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 8 and B 8 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 116s 2 .
For the presently investigated transition, the lower level corresponds to an energy of 18651.53 cm21 and its configuration as given by Wyart and Camus25 is @ Xe# 4 f 116s6 p. The
upper level at 31443.26 cm21 has a configuration
@ Xe# 4 f 116s7s which was also reported by Wyart and
Camus.25 Using the best values of A, B, and A 8 and B 8 , 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.
D. Hyperfine structure measurements in Rb using
absorption spectroscopy
Recently, Wieman and Preston28 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 attainTable I. Hyperfine structure constants A and B determined by laser optogalvanic spectroscopy for the 781.548 nm transition in Ho I.
hfs constants ~MHz!
Level
designation
Energy level
(cm21)
4 f 11ss 8 2
4 f 11s p1
31 443.26
18 651.53
709
This study
A
B
1045
870
21788
22560
Previous studies25
A8
B8
¯
864
Am. J. Phys., Vol. 66, No. 8, August 1998
¯
22574
able 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 (1025 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 ~m!
and the nuclear electric quadrupole moment (q) of the rubidium isotopes are available in the literature:20
Rao, Reddy, and Hecht
709
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.
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,
where m n is the nuclear magneton and b stands for barns
(1 b510224 cm2). 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 J5 21 . The 5 P 1/2
and 5 P 3/2 excited states are respectively at 794.76 and
780.023 nm.
E. Hyperfine structure calculations in
85
Rb
Ground state „5S1/2…: J5 21 , I5 25 , and A51011.91
MHz.18 Because of the hyperfine interactions, the ground
state splits into two states corresponding to F52 and F
53. There will be no quadrupole interaction because J,1.
The magnetic dipole interaction results in an energy separation of the F52 and F53 levels by 3036 MHz.
5P1/2 excited state: J5 21 , I5 25 , and A5120.72 MHz. 18
Because of the hyperfine interactions, this level splits into
two levels corresponding to F52 and F53. There will be
no quadrupole interaction. The magnetic dipole interaction
splits these levels by an energy equal to 362.16 MHz.
5P3/2 excited state: J5 23 , I5 52 , A525.01 MHz, and B
525.88 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.
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Am. J. Phys., Vol. 66, No. 8, August 1998
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 „5S1/2…: J5 21 , I5 23 , and A53417.34 MHz.
The ground state splits into two states corresponding to F
51 and 2. There will be no quadrupole interaction. The
magnetic interaction results in an energy separation of
6834.7 MHz.
5P1/2 excited state: J5 21 , I5 23 , and A5406.2 MHz. This
state splits into two states corresponding to F51 and F
52. There will be no quadrupole interaction. The magnetic
interaction corresponds to an energy separation of 812.4
MHz.
5P3/2 excited state: J5 23 , I5 23 , A584.8 MHz, and B
512.52 MHz. Now F can have values F50, 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 5 P 3/2 excited states at
;780 nm of both 85Rb and 87Rb are small compared to the
Doppler broadening (;550 MHz) of rubidium at room temperature and will not be resolved in Doppler limited specRao, Reddy, and Hecht
710
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.
APPENDIX: PARTS AND SUPPLIERS
Fig. 8. The hyperfine structure level scheme along with the expected hyperfine transitions for the 780-nm transition in 85Rb.
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 85Rb ground state hyperfine
splitting and two corresponding to 87Rb ground state hyperfine splitting ~Figs. 8 and 9!.
~1! Hollow Cathode Lamps: Holmium #14386 100Q,
$279.36; Rubidium #14 386 106N, $331.84; Fisher Scientific 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.
~2! 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 07860-0366, Phone: ~973! 579-7227.
~3! SI PIN Detector, Item #DET100, $81.00, 20 ns rise time,
13.7 sq. mm active area, range 350–1100 nm, THOR
LABS, 435 Route 206, P. O. Box 366, Newton,
NJ 07860-0366, Phone: ~973! 579-7227.
~4! Optical Isolators: Model I-80T-4 Single Stage, 4 mm
clear aperture, range 750–900 nm, $1,615.00, Isowave,
64 Harding Avenue, Dover, NJ 07801, Phone: ~201!
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 Laser Diode Mount, $645.00, Newport/Klinger, 18235
Baldy Circle, Fountain Valley, CA 92708, Phone: 800222-6440 ~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: ~303! 530-7785.
1
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Listings of commercially available diode lasers and their characteristics
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www.thorlabs.com
2
Fig. 9. The hyperfine structure level scheme along with the expected hyperfine transitions for the 780-nm transition in 87Rb.
711
Am. J. Phys., Vol. 66, No. 8, August 1998
Rao, Reddy, and Hecht
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22
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25
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27
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28
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