Z. Phys. B 98, 371-376 (1995)
ZEITSCHRIFT
FORPHYSIKB
9 Springer-Verlag 1995
Pressure shift of atomic resonance lines in liquid and solid helium
S. Kanorsky*, A. Weis, M. Arndt, R. Dziewior, T.W. H~insch
Max-Planck Institut fi.ir Quantenoptik, Hans Kopfermannstrasse 1, D-85748 Garching, Germany
(Tel: + 49-89-32905 292, Fax: + 49-89-32905 200)
Abstract. We show that for atoms trapped in a liquid He
matrix the excitation and emission spectra can be quantitatively described by the bubble model which treats the
matrix as an incompressible continuous medium. Comparison with experimental results shows that the model
also holds for a solid matrix. The model is only valid in the
case where the attractive H e - H e interaction dominates
the attractive part of the foreign atom-helium interaction.
Its failure in the case of implanted Na atoms is discussed.
PACS:
67.80; 32.80.; P 32.70.J
Introduction
Recent demonstrations of optical pumping of alkali
atoms in a matrix of liquid [-1] and solid [-2] helium open
a number of possibilities to use such atoms for the purposes of high precision spin physics [-2]. For these studies
the effects of the trapping site on the atomic properties,
such as shifts of the atomic energy levels, modifications
of the electron wave functions and corresponding changes in the constants of the fine and hyperfine structure,
are of primary importance. Theoretical approaches to
these problems are based on the so called bubble model,
which was first introduced for the description of the
structure of the point defect formed by a single electron
in liquid helium [3]. Several questions arise when this
model is to be used for the description of the trapping
sites for neutral atoms that form bubbles with radii
comparable to the interatomic separation in the helium
bulk:
- Can the helium matrix around the point defect still be
treated as a continuous medium with a smooth density
distribution function?
*Permanent address: Lebedev Physical Institute, Moscow, Russia
- Can macroscopic concepts such as pressure volume
work and surface tension still be used for cavities of radii
as small as 6 A?
- In case of an affirmative answer to the previous question, which value for the surface tension coefficient should
be used for pressurized liquid and solid helium ?
- What is the trapping site for atoms implanted in solid
helium ?
- Can the bubble model developed for the liquid helium
matrix be used for solid helium as well ?
In order to examine these questions we have performed
systematic studies of the broadening and shift of the excitation and emission lines of the 6s 2 1So ~ 6s6p 1Pt singlet
transition of Ba atoms implanted in liquid and solid
helium at 1.5 K. The advantage of the chosen transition is
the absence of the fine structure of the line, which simplifies the quantitative analysis of the observed spectra. We
also compare the predictions of the bubble model with the
experimental findings for the D1 line of Cs atoms implanted in solid helium, and finally we address a case
where the bubble model in the form presented in this
paper fails, namely, the absence of fluorescence from light
alkalis (Li, Na, K) in liquid helium.
Formulation of the model
The description of the bubble model for neutral foreign
atoms can be found in [-4,5]. In brief, due to a strong
repulsion of helium atoms at short distances the impurity
atoms form cavities (bubbles) of diameter of db ~ 10 A or
larger. Liquid helium outside this cavity is described as
a continuous incompressible medium with a smooth density profile:
p(R; ~Ro, ~) =
0;
R < 9to
po{1 - [1 + c~.(R - ~Ro)]-exp[ - cc(R - 9~o)3}; R _> 9l o
(la)
372
where
91o(0)=Ro.( l + f l
3c~ 0 - 2
1)
(lb)
The helium density vanishes inside the cavity (R < 9to)
and reaches its bulk value Po as R ~ Go. The parameter
determines the width of the bubble interface. Following
the symmetry considerations we assume that the equilibrium shape of the bubble reflects the symmetry of the
electronic wave function of the foreign atom. For the
excited state of Ba (~P1) the deviation from the spherical
symmetry of the trapping site is taken into account by
adding a quadrupolar dependence ocflY~(O, (p) to the
parameter 910 (0). Here O is the azimuthal angle in the
reference frame in which the P-state electronic wave function has the form ~P(r) = R(r) "Y~(O, cp). In this reference
frame the equilibrium bubble shape should be invariant
with respect to rotations about the z-axis, for this reason
we have discarded Ym~:o(,-9,
z
(p) terms in the quadrupolar
deformation. For the spherical states So, S~/2, and P1/2 the
bubble is spherically symmetric and hence fi = 0
The distribution of liquid helium around the point
defect is thus determined by three parameters Ro, c~,and ft.
These are to be found by minimizing the total energy of
the point defect, which is the sum of the foreign atomhelium interaction energy and the energy needed to form
a cavity of given shape: Etot= Eint + Ec. Following
Hiroike et al. [3] the cavity contribution is commonly
written as a sum of the surface energy, pressure volume
work for the bubble formation, and volume kinetic
energy:
h2
f (Vj~
93
(2a)
regions with helium atoms by making the bubble interface
very thin. The volume kinetic energy term which is proportional to the square of the gradient of the helium
density prevents such a collapse of the density profile and
assures the stability of the energy minimization procedure.
Therefore we keep the volume kinetic energy term in its
form (2a) for all pressures at the expense of having to
admit that the value of the surface tension coefficient is
not well defined for our bubble (see discussion in the next
section).
In the present paper we calculate the foreign atom
- helium interaction energy by summing over all pair
interactions. This approach is equivalent to the treatment
of this interaction in lowest order perturbation theory. Its
validity is justified by the smallness of the corresponding
shift of the atomic energy level. This condition is fulfilled
for Ba and Cs, but not for Na as will be discussed below.
Optical transitions studied in this paper are the
transitions between 2s+ 1Sj and 2s+ 1p j, states. In this case
the interaction energy can be found as
Eint = 4~z~R2 dR'p(R)
V~,nl/z (R)
(3a)
for the spherically symmetric So, $1/2, and P1/2 states, and
as
E~nt = ~ d3R'p (R) (V~ (R) cos20 + Vn (R)sin20);
for ~
state. For barium we have used the numerical
values of the adiabatic pair potentials V~(R), VrI(R) of
Czuhaj [7], and for alkalis those Pascale [8].
Once the equilibrium helium distribution p(R) is found
by minimizing the total energy of the point defect, the
optical spectrum is calculated according to the standard
statistical line broadening theory in its static limit [9]. The
spectrum is found as the Fourier transform of the
transition dipole autocorrelation function:
where the cavity radius Rb is defined as the center of
gravity of the bubble interface:
I (co) = S dq:' e~'~ C (z)
Rb
with
P(r)rzdr = ~ [Po -- P(r)]rzdr
0
(2b)
Rb
The last term in (2a) (volume-kinetic energy) deserves
a special discussion. For typical values of the shape parameters (91 ~ 10 ao, ~ v 1 ao 1) the contribution of this term
is approximately 10% of the total energy of the bubble.
For a spherical bubble and the adopted form of the
helium density profile (Eq. la) this term can be expressed
as E~.k.(91o,c~)=~z/2 him Po (1.411c~ 1 + 2 . 0 5 4 9 1 o +
1.454e91oZ).
The leading term oc c~91o
2 is proportional to the surface
of the bubble. For this reason it has been suggested in [6]
that a part of this term is already included in the definition
of the surface tension coefficient, so that only its increase
with respect to the value at zero pressure should be considered. We have found that in this case the minimization
of the total energy of the point defect at low helium
pressure yields non physically large values of c~ on the
order of 100. This is due to the fact that pair potentials of
the foreign atom
helium interaction have attractive
minima at intermediate and large distances. The minimization routine then tends to completely fill these attractive
(3b)
(4a)
--O9
C (v) = exp{ - 5 d3R[1 - i kv (R) ~)] p (R)}
(4b)
where Av(R) is the shift of ~ e transition frequency by
a helium atom at the position R with respect to the foreign
atom.
This line broadening theory was originally developed
for dense gases. In [6] it was used to describe the optical
spectra of transitions between excited states of helium
atoms in liquid helium. Several conditions should be fulfilled to justify its application to foreign atoms in a liquid
helium matrix.
First, the theory assumes that the energy shifts of the
atomic levels due to various perturbers are additive. This
assumption is already used in the present model where the
total energy of the foreign atom-helium interaction is
calculated as the sum over all pair interactions. This
approximation can be partially justified by considering
the foreign atom - helium interaction within first order
perturbation theory, but finally only the general success or
failure of the model can substantiate its validity.
Second, the theory assumes that on the time scale
determined by the autocorrelation time of the atomic
373
polarization the various perturbers do not move, that
their positions are uncorrelated and their density fluctuations are similar to those of a dense gas. In our case this
autocorrelation time is estimated from the measured
spectra (Figs. 3,4) to be on the order of 10 -~3 s. This is
about one order of magnitude shorter than the inverse of
the roton frequency which can be considered as the time
needed for the correlation between the atoms in liquid
helium to take place. With the thermal velocities on the
order of 104 cm/s the helium atoms will move by only
about 0.1 A during this autocorrelation time, which is
negligible compared to the radius of the bubble. Thus the
applicability of the statistical line broadening theory for
atoms in liquid helium is justified by the shortness of the
corresponding autocorrelation time of the atomic polarization. And finally, the present model does not take into
account possible deviations from the equilibrium shape of
the bubble which can lift the 3-fold degeneracy of the
P-states (static Jahn-Teller effect). Therefore it is not able
to reproduce the corresponding splitting of the spectral
lines.
555
Free atom
transition
wavelength
"~
0.50 F
.....
g"
E
~
' .........
0.25
550
0.00 ~'
0.5
i
0.6
,
i
0.7
,.~
0.8
545
0.65
~
o 540
= 0.7
e~
,r-.
•
c~ = 1 . 3 - 1 . 7 ( c o m p l e t e m o d e l )
liquid
5350_
i
i
i
~ 1;
i
_,
i
i 201
~
i
solid
i ~ t 301
r
i
He p r e s s u r e (bar)
Fig. 1. Pressure shift of the excitation line barycenter for the
Comparison with the experiment (Ba and Cs)
1S0 __, 1p~ transition of Ba atoms implanted in liquid helium. The
insert shows the region in a - c~ parameter space for which the
present theoretical model reproduces the experimental results
The theoretical excitation spectra of the singlet ts 0 ~ ~Pt
transition of Ba atoms implanted in liquid and solid
helium at different pressures have been compared with
experimental ones. The detailed description of the experimental procedure is given in [10]. Figure 1 shows the
measured pressure shift of the excitation line barycenter.
At the saturated vapour pressure (SVP) at 1.5 K the line is
blue shifted by 8.1(3)nm with respect to the free atomic
transition and the increase of the pressure shifts the
line further to shorter wavelengths at a rate of
- 0.13(2) nm/bar. The bubble model in its standard form
as described above overestimates both the SVP line shift
and the pressure shift rate (solid line 5 in Fig. 1), although
it should be pointed out that the discrepancy - on the
order of 3 nm - is well within the width of the excitation
line ( ~ 8nm). There are several possible explanations for
this discrepancy, none of which can be excluded a priori.
Among them, inaccuracy of the adiabatic Ba-He pair
potentials, the neglect of many body effects in the calculation of the barium-helium interaction energy, and inaccuracy of the functional describing the helium matrix
around the point defect (2a, b). This form of the bubble
energy functional was introduced for electron bubbles,
which have a diameter Db of approximately 65 ao and
a number of helium atoms on the surface of the bubble
N~u~f~ 300. The energy minimization procedure gives for
the atomic bubble Db~20 ao which corresponds to
Nsurf~30.
In this situation the applicability of the phenomenological functional (2a, b) becomes questionable. It
is also not clear which value of the surface tension coefficient should be used for a bubble of such a small radius.
For this reason we have also looked for a density distribution function which is consistent with the experimental
results. The shape parameter ~ and the surface tension
coefficient a were considered to be pressure independent.
At every pressure point the equilibrium bubble radius was
found by minimizing the total energy with respect to the
parameter Ro only. The values of c~ and a which are
consistent with the experimental pressure dependence of
the excitation line barycenter were found using a Za criterion with a significance level (false-reject probability)
of 10%. The result is shown as insert in Fig. 1 The best
value of Z2 of 1.2 per degree of freedom is achieved for
= 0.6, and a = 0.32 erg/cm 2. The value of the surface
tension coefficient is compatible with the value
0.34erg/cm 2 obtained in [ l l l from the analysis of the
infrared spectra of electron bubbles in liquid helium. The
corresponding pressure shift of the excitation line of Ba is
shown in Fig. 1 as the dashed line, also shown as dotdashed lines are the two extreme cases still consistent on
the adopted significance level with the experimental findings: {~ = 0.34 erg/cm 2, ~ = 0.55} and {a = 0.34 erg/
cm 2, ~ = 0.75}. The helium distribution function determined in such a way shows a smoother behaviour in the
bubble interracial region than the one found by the energy
minimization with respect to both bubble shape parameters (Fig. 2). Since the present simplified model involves
a number of other approximations, for example, the use of
molecular pair potentials in an essentially many body
problem, we do not want to emphasise the physical significance of this result. Further theoretical and experimental
studies are needed to clarify this question. The comparison of the theoretical and experimental excitation line
shapes for different pressures is presented in Fig. 3. The
theoretical line shapes were calculated for the "optimum"
values of the parameters e = 0.6, and ~ = 0.32 erg/cm 2.
A remarkable result is that the excitation spectrum recorded in solid helium can be still satisfactorily reproduced by the bubble model developed for the liquid helium matrix. This indicates that the trapping site for the
implanted atom is not significantly changed in the liquidto-solid phase transition.
374
1.0
1.0
9
/"
complete model
0.5
] /
0
0.8
0.6
"optimum"
y profile
0.4
0.0
I
I
I
r
I
,
,
,
J/I
5
0
,
,
t
10
,
I
t
i
15
r
,
20
0.2
R (a.u.)
Fig. 2. Density profile of liquid helium in the interfacial region of
the bubble for the complete model (solid line) and for the "optimum"
value of the parameter e (dashed line)
0"020
830
840
850
860
870
880
890
900
Wavelength (nm)
Fig. 4. Comparison between theory and experiment for the excitation and emission lines of the 6S1/2 --, 6PI/2 transition of Cs atoms
implanted in solid helium at 30 bar
1.0
a P= 1.1 bar
0.8
Limitations of the model (Na)
0.6
0.4
0.2
x~
9
1.0
0.8
b P = 20 bar
0.6
o
ov.,~
4..a
0.4
0.2
1.0
C P = 32bar
0.8
0.6
0.4
0.2
530
535
540
545
550
555
Excitation wavelength (nm)
Fig. 3. Comparison of the theoretical and experimental excitation
line shapes for the 1So ~ 1P1 transition in Ba at different helium
pressures
Pressure shifts of the Cs D1 line in liquid helium are
reported elsewhere in this volume [12]. We have measured the excitation and emission spectra of Cs atoms in
solid at 30(2) bar. The comparison with the model calculations is shown in Fig. 4. With the surface tension coefficient of 0.34 erg/cm z, the best agreement of the line centers
is achieved for c~ = 0.4. Keeping in mind all crude model
assumptions discussed above the agreement with the experiment may be regarded as satisfactory.
An attractive feature of the version of the bubble model
described above is its simplicity. In the demonstrated
cases the excitation and emission spectra of the foreign
atoms could be readily reproduced by the model calculations without significant computational efforts. Nevertheless, there are situations where this simple model can
not be applied. One example of such a situation is provided by the emission spectra of light alkali atoms in
liquid helium. It is a well know fact, that no fluorescence
from light alkalis (Li, Na, K) has ever been observed in
experiment [13]. However, the bubble model in the form
presented above predicts the existence of emission lines for
the resonance transitions. For example, in the case of Na,
the emission on the resonance 3P ~ 3S transition should
be shifted according to the model calculations to the
region of 620 nm. This controversy indicates that for light
alkalis the present form of the bubble model fails.
The main reason for this failure is the adopted form of
the model density distribution function, namely the implicit assumption that helium is an incompressible liquid:
according to (la) the helium density around the point
defect can never exceed the liquid helium bulk density. Ba
and Cs atoms in both the ground and the excited states
show no strong attraction with He and the above assumption seems to be reasonable. This is definitely not the case
for light alkalis. For example, Fig. 5 shows a three-dimensional plot of the adiabatic pair potential of Na*(3P)- He
interaction [8]. In Fig. 5 it is assumed that the angular
part of the N a electronic wave function has the form
Y~o(O, ~o) with the quantization axis along the z-axis. Clear
seen are the two deep potential wells in the regions around
the points y = + 4 a.u., z = 0. The depth of these wells is
about 650 K. The pair potential has a rotational symmetry around the z-axis. It is thus clear that a finite
number of helium atoms will be localized within a ring of
radius 2-3 A in the nodal plane of the electronic wave
function, thus forming a Na*(3P)He, exciplex. The density
375
-~. 80.0
C~
,.~ 77.5
75.0
lO
~(a.O.)
.~ .1-~..,.~-_10 -o "~~, .....
Fig. 5. Adiabatic Na-He pair potential [8] as a function of the
position of the He atom in the z = 0 plane
of helium in such an exciplex can exceed by far the liquid
helium bulk density, thus giving rise to much larger energy
shifts of the Na atomic energy levels than those expected
on the basis of the simple model described above. Therefore we have assumed the following structure of the point
defect for a sodium atom in the excited 3P state: the
exciplex consisting of the Na atom and several helium
atoms trapped in a ring in the nodal plane of the Na
electronic wave function is located in a bubble. This structure is illustrated in Fig. 6. It is clear that the main perturbation of the Na atomic levels is due to these close
atoms and in the following discussion we neglect the effect
of the rest of the helium around the point defect. We have
estimated the number of helium atoms in the exciplex by
a variational minimization of the total energy of the
Na*(3P) He, complex. The Hamiltonian of the system was
written as
/)
h2
--
2m
A -[- U H e _ N a "Ji- U H e - H e
(5)
The energies of H e - N a and H e - H e interaction Une-Na
and Une-ne were found by summing over all pair interactions. In cylindrical coordinates (z, p, ~) the Une-Na part
of the potential energy was approximated as a sum of
a harmonic potential in the z-coordinate and a Morse
potential in the p-coordinate:
mfO 2Zz2
UHe-Na : ~
"t- De [1 -- Exp[ - b (p - pe)]) 2 - 1],
(6)
where co~ = 2.11013s -~,pe = 4.2 a.u., b : 0.93, and
D e = 519.2cm -1. For the H e - H e interaction we have
used the analytical H F D - I D form of the adiabatic pair
potential of Aziz and Slaman 1-14]. For this form of the
Hamiltonian the variables in the Schr6dinger equation
can be separated, so that the trial ~-function of helium
atom can be chosen in the following form:
zx~
(7)
where qS(z) and Z(P) are solutions of the Schr6dinger
equations for the motion in the harmonic potential (zdirection) and Morse potential (p-direction), n is the
number of helium atoms in the cluster, and the parameter
Aq) determines the helium atom localization in the q0
coordinate. This choice of the trial ~-function uses the
Fig. 6. The model for the structure of the point defect formed in
liquid helium by a Na atom (see discussion in the text). The toroidal
distribution of He in the nodal plane has a radius of approximately
4.5 a.u.
symmetry of the problem by assuming that the equilibrium positions of He atoms form a symmetric polygon in
the z = 0 plane. Minimization of the total energy of the
system with respect to both free parameters n and A~0
g i v e s Emin = - 1 8 1 0 c m
-1 for n = 5. When 5 helium
atoms are located in such a close proximity of the Na core
the 3S and 3P levels are practically degenerate and a radiationless quenching of the 3P state is quite possible. It
should be pointed out that strictly speaking the use of pair
potentials in a situation where the atomic levels are so
strongly perturbed is not justified. Therefore, the question
of the existence of a level crossing still remains open.
However, even if such a crossing does not occur, the
radiation of the 3P ~ 3S transition should be shifted into
the ~tm region and be spread over a spectral interval of
several microns, which makes the detection of this radiation a very difficult problem.
This result is in exact agreement with that obtained by
D u p o n t - R o c [15] within the bubble model using a different form of the functional for the description of the liquid
helium around the sodium atom. The difference between
the two approaches to the problem is that while in [15]
liquid helium is treated as a continuous compressible medium, in our approach we assume a molecular-like "discrete" structure of the trapping site for the implanted
atom. The agreement of the results of the two complimentary approaches shows that even in this extreme situation
the bubble model can still be used for the description of
the structure of the point defect, although a more elaborate form of the energy-density functional is required.
Conclusions
We have shown that the bubble model in its simplest form
in combination with the quasistatic line broadening theory is able to explain quantitevely the main features of the
376
excitation and emission spectra of atoms in a matrix of
liquid helium. However, the applicability of the model
which treats liquid helium as an incompressible medium is
limited to the case where foreign atoms do not exhibit
strong attraction on helium atoms, i,e. where the potential
well in the foreign a t o m - helium interaction does not
exceed that of the H e - H e interaction. A remarkable result
is that the bubble model can be applied to atoms implanted
into a solid helium matrix as well, which indicates that the
trapping site for such atoms does not undergo a significant
change under the liquid-to-solid phase transition.
This work has been supported in part by DFG (Deutsche Forschungsgemeinschaft).
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