Z. Phys. D 39, 127—137 (1997)
Interference effects in Stark spectra of weakly autoionising
5dnf states of barium
G.J. Kuik, W. Vassen, C.T.W. Lahaije, W. Hogervorst
Laser Centre Vrije Universiteit, Department of Physics and Astronomy, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
Received: 1 October 1996
Abstract. Weakly autoionising 5d nf Rydberg states of
3@2
barium around n"60 have been studied in the presence
of a static electric field. The experiment has been carried
out in a CW laser-atomic-beam setup. In between the
overlapping n"60 and 61 angular momentum manifolds
broad 5d63d resonances interact with the manifold states
resulting in pronounced interferences. These interferences
(anti-crossings) have been analysed by a direct diagonalisation procedure neglecting interactions with the continuum, and by a Multichannel Quantum Defect Theory
(MQDT) analysis including continuum interactions.
PACS: 32.60.#i; 32.80.Rm; 32.80.Dz
1 Introduction
In recent years several studies of Rydberg states of alkali
and alkaline-earth atoms in the presence of electric fields
have been performed [1—4]. Most of these studies concerned the behaviour of bound Rydberg states in external
fields. Recently we reported on electric field effects in
autoionising series [5].
In the electric field case both the quadratic and the
linear Stark effect have been studied in detail. The quadratic Stark effect manifests itself particularly in excited
states of nonhydrogenic atoms with low angular momenta, i.e. in states which exhibit large quantum defects. Rydberg states with a high orbital quantum number l have
small quantum defects and are nearly hydrogenlike. In
these states the linear Stark effect may be observed, i.e.
angular momentum manifolds appear that are fanning out
approximately proportional to the strength of the static
electric field applied during the excitation. In several recent papers [3—5] results on scaled-energy spectroscopy
experiments are reported as well.
The presence of a continuum adds interesting features
to the investigation of excited atoms in the presence of an
electric field. Drastic changes in shapes and widths of
autoionising resonances as well as electric-field induced
interferences have been observed in experiments with
pulsed dye lasers [e.g. 6, 7].
Here we present a study of the autoionising 5d nf
3@2
Rydberg series of barium applying CW laser spectroscopy
in an atomic beam. The 5d nf Rydberg series autoionises
3@2
into the 6s
continuum through quadrupole coupling.
1@2
As this quadrupole coupling is weak slow autoionisation
rates are observed, justifying the use of narrowband CW
lasers in the experiment. In the presence of an electric field
the 5d nf states are coupled with broad 5d nl states
3@2
3@2
with low l-value and peculiar interference effects are observed. The experimental data are analysed by direct diagonalisation of the energy matrix (neglecting continuum
interactions) and also within the framework of multichannel quantum defect theory (MQDT). Our paper is organised as follows. In Sect. 2 the theory of the linear Stark
effect and the MQDT model are briefly summarised. The
experimental setup is presented in Sect. 3. Section 4 contains the results and discussion. Some conclusions are
given in Sect. 5.
2 Theory
Throughout the discussion in this paper atomic units will
be used, i.e. m"+"e"1. The electric field strength F in
0
atomic units is 5.142]109 V/cm.
2.1 Linear Stark effect in hydrogen
The Hamiltonian of a hydrogen atom placed in a homogeneous static electric field F directed along the z-axis is
[8]:
p2 1
H" ! #Fz
2
r
(1)
From first-order perturbation theory follows that the
change in the energy eigenvalues by the electric field is
given by:
DE(1)"F · St DzDt T
n
n
n
(2)
128
Here t are eigenstates of the unperturbed Hamiltonian in
n
zero electric field. In non-hydrogenic atoms the first-order
shift is zero because eigenstates have definite parity and
different parity states are nondegenerate. So the matrix
element in eq. (2) is zero. In hydrogen states of different
parity are degenerate. First order perturbation theory
then requires a zero-order basis adjusted to the perturbation: the parabolic basis. In this basis the degeneracy is
lifted by the perturbation and a linear Stark splitting is
obtained [12]. To describe the linear Stark effect in hydrogen it is therefore convenient to decouple the Schrödinger
equation into one-dimensional differential equations
using a coordinate transformation to a parabolic basis
(m, g, /), where
Fig. 1. The potential »(m)t A and »(g) B of (4). The dashed lines are
the pure Stark potentials 1Fm and !1Fg
4
4
m"r#z,
with A given by the following 3j-symbol:
kl
g"r!z,
/"/.
(3)
The decoupled one-dimensional differential equations in
m and g become (the equation for / remains unchanged):
C
C
D
d2
m2!1 b F
E
!
# ! m# f (m)"0 ,
dm2
4m2
m 4
2
(4a)
D
d2
m2!1 1!b F
E
!
#
# g# g(g)"0 .
dg2
4g2
g
4
2
(4b)
b is a charge separation constant reflecting the effective
Coulomb charge for the two differential equations. E
is the energy and m is the conserved quantum number
corresponding to the z-component of the angular
momentum. The uphill equation (Eq. 4(a)) has bound
solutions characterised by the number of nodes of f (m) for
m'0 (quantum number n ). The downhill equation (Eq.
1
4(b)) has a potential barrier with a maximum at the
saddle-point (the classical field-ionisation limit) allowing
an electron to escape to gPR (see Fig. 1). Its eigenstates
are characterised by the parabolic quantum number n .
2
The charge separation constant b is obtained through the
Bohr-Sommerfeld quantisation rule for m [9, 16]:
C
D
A
B
1@2
mb E
mb E m2 b F 1@2
dm": ! # ! m
dm
: !»(m)
2
2 4m2 m 4
ma
ma
"(n #1/2)n .
(5)
1
In Eq. (5) a factor 1/4m2 is omitted. This factor would
induce a breakdown of the WKB approximation near
m&0 (Langer correction) [9]. The principal quantum
number n in the Coulomb field is related to the parabolic
quantum numbers (n , n ) by the relation n #
1 2
1
n #DmD#1"n. Parabolic quantum states are usually
2
represented by DnkmT with k"n !n . For a fixed n and
1
2
m, k ranges from n!DmD!1, n!DmD!3, 2 to
!n#DmD#1. A parabolic Stark state DnkmT can be expressed as a linear superposition of spherical Coulomb
state DnlmT [10, 11]:
DnkmT"+ A DnlmT
kl
l
(6)
A
B
(n!1)/2 (n!1)/2 l
A "(!1)mJ2l#1
.
kl
(m#k)/2 (m!k)/2 m
(7)
The energy eigenvalue of the parabolic Stark state DnkmT
up to first order is given by [12]:
1
3nkF
E"! #
.
2n2
2
(8)
The k quantum number physically represents a measure of
the projection of the charge distribution on the field axis
when Stark states are considered as permanent dipoles.
The resulting energy shift is proportional to k. At zero
electric field all k states are degenerate. With increasing
electric field F and the k states fan out resulting in the
so-called angular momentum manifold.
2.2 Linear Stark effect in alkali atoms
The eigenstates t of the unperturbed zero-field Hamiln
tonian in alkali atoms are no longer degenerate. Therefore, the expectation values of the operator z (Eq. (2)),
which has odd parity, are zero and only in second-order
perturbation theory a field-dependent energy shift occurs
proportional to F2 (quadratic Stark effect). However,
when the eigenvalues of the eigenstates t are nearly
n
degenerate and the differences are small compared to the
energy shift contributions of the external electric field
(i.e. when 3n2F<d n~3, d is the l-dependent quantum
2
l
l
defect), linear Stark effects again can be observed.
However, a striking difference for angular momentum
manifolds of alkali atoms compared to hydrogen exists.
In the case of hydrogen angular momentum manifolds
belonging to different principal quantum numbers n
simply cross with increasing field strength. For alkali
atoms, whenever the electric field results in the mixing of
opposite parity states into the original wavefunctions,
second-order perturbation theory predicts a repulsion
between these states and the result will be an anti-crossing. The minimum energy separation observed between
anti-crossing states is a measure of the coupling of the
states.
129
2.3 Linear Stark effect in barium
In the case of barium, a two-electron alkaline-earth atom,
linear Stark effects can be observed under the same conditions as in the alkali atoms. The Stark matrix elements for
LS-coupled states are given by [13]:
Sn l n l ¸SJMDzDn@ l@ n@ l@ ¸@S@J@M@T"d @ $ d @ d @
l2l2 1 M M S S
11 22
11 22
g(g)"C
g
·(!)2L`S`J@`J~M~l1`l2 J(2J#1) (2J@#1)
gN (g)"C
g
·J(2¸#1) (2¸@#1)
·
A
BG
J
1 J@
!M 0 M
¸
J@
HG
J S
¸@ 1
l
¸ l
2
1
¸@ l@ 1
2
· (!1)(l2~l2@ `1)@2 Jl · F · Rn@2l@2 .
n2l2
2.!9
Here
H
(9)
=
(10)
Rn2@ l2@ " : R rR @ @ r2dr .
n2l2
n2l2 n2l2
0
is
R (r) is the radial wave function of the nl-electron. l
nl
2.!9
the largest value of l@ and l . In other coupling schemes
2
2
the matrix elements follow from a transformation to LS
wavefunctions using standard angular momentum theory
[13].
2.4 Stark effect of autoionising states near the saddle-point
(alkali model)
To describe the Stark effect in Rydberg spectra of bound
and autoionising series Harmin [14, 15] has developed
a quantum defect theory (QDT) model for the electric field
case for one-electron systems. This model is identical to
conventional QDT at short range. At large distances the
Stark effect is taken into account.
Short-range effects, limited to the core region (r(r ),
0
may be considered independent of the external electric
field as the Coulomb force near the core is of the order of
1 a.u. compared to laboratory strengths of the external
field of less than 10~6 a.u. The long-range behaviour is
governed by the combined Coulomb and Stark potential
(see Eq. (1)). The appropriate wavefunctions in this region
are solutions of the decoupled one-dimensional differential equations (4). Within the region r )r;r , where r is
0
S
S
the radius at the classical field ionisation limit (saddlepoint, Fig. 1B) of the Stark potential, the Schrödinger
equation is separable in both spherical and parabolic
coordinates. The short-range and long-range solutions of
the Schrödinger equation are connected in this region
through the corresponding frame transformation.
2.4.1 Long-range and short-range wavefunctions. As the
Schrödinger equation in a Coulomb plus electric field is
separable in parabolic coordinates the long-range
wavefunctions may be expressed as [16]:
t(FEbm)"(2nmg)~1@2 f (m)g(g)eim” ,
tM (FEbm)"(2nmg)~1@2 f (m)gN (g)eim” ,
where g(g) is the regular and g6 (g) the irregular solution of
Eq. (4b) at g"0. As the electron can only escape for
gPR (see Fig. 1) we only have to consider the downhill
equation (Eq. (4b)) and the long-range wavefunction g in
a parabolic basis. Using a WKB approximation, the solutions for g and gN are [9, 16]:
(11)
AS
AS
S
S
B
B
k
1
sinD · ¼ (g)#
cosD · ¼ (g) ,
b
1
b 2
2
2k
k
1
cosD · ¼ (g)!
sinD · ¼ (g) .
b 1
b 2
2
2k
(12)
C and k are constants, D is a phase integral given by [17]:
g
b
gb E m2 1!b Fg
D ":
! #
# dg .
(13)
b
2 4g2
g
4
ga
g and g are the classical turning points (see Fig. 1). ¼
a
b
1
and ¼ are parabolic cylinder functions exponentially
2
decreasing and increasing with g respectively [16, 17].
The short-range wavefunctions in a spherical basis are
the regular (g) and irregular (g6 ) Coulomb functions [8] in
zero-field (denoted by ‘0’):
S
t0(Elm)"g (r) ½ (h, /),
El
lm
0(Elm)"g
tM
6 (r) ½ (h, /),
El
lm
where ½ are spherical harmonics.
lm
(14)
2.4.2 Frame transformation. For r;r the parabolic
S
wavefunctions may still be considered to be nearly independent of the applied electric field. In that case it is
appropriate to expand the parabolic wavefunctions in
spherical harmonics:
t(FEbm)"+ º t0(Elm) ,
bl
l
(15)
tM 0(FEbm)"+ Ut tM (Elm) .
bl
l
The frame transformation matrix elements º (FEm) can
bl
be expressed as a product of normalised real ClebschGordan coefficients scaled with the effective quantum
number l"n!d , where d again is the quantum defect
l
l
[18]. Ut is the transposed matrix of U.
2.4.3 QDT for quasi-bound Stark states. Standard quantum defect theory (QDT), following the R-matrix formalism [19], may now be applied to calculate the quasibound Stark spectra of alkalis. The long-range electron
wavefunction, a linear combination of t(FEbm) and
tM (FEbm), in matrix notation is given by:
G"w#wN R (r'r ) .
(16)
0
To obtain the total wavefunction G is multiplied by the
ion-core wavefunction. The R-matrix is determined
by connecting G to the solution for r(r at the core
0
boundary. In the intermediate range (r (r(r ) the
0
S
wavefunction may also be expressed in spherical
Coulomb functions as:
G0"w0#wN 0R0 (r (r(r ) .
0
S
(17)
130
The R0-matrix is the usual R-matrix from quantum defect
theory. The R-matrix now follows from:
R"UR0Ut .
(18)
where U is the matrix defined in eq. (15). For alkali atoms
the R0-matrix reduces to [20]:
R0 "tan(nk )d .
(19)
ij
l ij
k is the reduced quantum defect. The functions G are the
l
appropriate functions to describe the electron motion.
The boundary condition for a quasi-bound state around
the saddle-point for one-electron systems requires that for
any linear combination of functions G the terms that
exponentially grow with g will vanish within the region
before the potential barrier is reached. This condition
yields [16]:
K
K
det cot D d @ !+ º @ tan(nk )ºt "0 .
bl
l lb
b bb
l
(20)
2.5 Stark effect in autoionising states of barium
In the case of barium, a two-electron system, a closed
(quasi-) bound channel and an open (continuum) channel
has to be taken into account. Sakimoto [16, 17] extended
Harmin’s single-channel QDT model [9, 14, 15] to
a multi-channel QDT model including ion cores of any
state and to incorporate autoionisation. In the case of
autoionisation (energies above the first ionisation threshold) open as well as closed channels are available. The
motion of the free electron in an open channel may be
considered independent of the applied electric field, and is
described by Eq. (14) as well. In the description of closed
channels the electric field has to be included. To describe
dielectronic recombination and autoionisation Sakimoto
used the S-matrix formalism introducing incoming and
outgoing waves [17, 19]:
1
w " (wN #iw) ,
` 2
1
w " (wN !iw) .
~ 2
(21)
In the multichannel case the electronic part of the
wavefunction G has to be multiplied with all possible
ion-core state wavefunctions ½ to obtain the total
c
wavefunction t (E); it can be written as (using Eq. 16, 21):
j
G"w !w v
(22)
~
`
with the matrix v related to R by:
v"(1#iR) (1!iR)~1 .
(23)
To describe Stark effects on autoionising states between
the first (6s)- and second (5d )- ionisation limit the
3@2
matrix v is divided into submatrices representing openopen (v )-, open-closed (v )- and closed-closed (v ) chan00
0#
##
nel parts, where the open channels are labelled with (c, l, m)
and the closed channels with (c, b, m). Introducing the
matrix D (diagonal with respect to (c, b, m), with matrix
b
elements deduced from Eq. (13)) a scattering matrix S can
be derived imposing boundary conditions in the region
g ;g;g :
b
c
S"v !v [v #exp(!2iD )]~1v .
(24)
00
0# ##
b
#0
The absorption spectrum in the autoionisation region
between the first and second ionisation limit is calculated
from the cross sections for the bound(a)Pfree( j)
transitions as a function of energy [8]:
d
2
f (E)" (E!E )DSt (E)DdD t TD2 .
ja
a
j
a
dE
3
(25)
t is in our case the 5d2 1G wavefunction; t and t are
a
4
a
j
normalized functions. In the case of t , the wavefunction
j
may be expanded in terms of the ion-core wavefunction
and a wavefunction describing the electron motion. The
electron wavefunction satisfies the boundary condition
that for large electron-core separation it is a linear combination of spherical Coulomb functions independent of the
electric field.
The dipole matrix element of Eq. (25) may be separated into a resonance and a non-resonance part, i.e. a part
where the electron motion is still quasi-bound and depending on the electric field, and a part where there is
a free motion. Only the resonance part contributes to the
oscillator strength distribution and needs to be considered. This resonance part is given by [17, 21]:
St (E)DdDt T"! + + + (v )
j
a
0# c0lm, c@c b@m@
cc@ b@m@ ccbm l@
](v #exp(!2iD ))~1
##
b c@cb{m@ccbm
]exp(!iD ) (sin D · Y @ · A @ #cosD · Y~1@ · B @ )
ccl m
b
b bl ccl m
b bl
(26)
with Y the eigenfunctions of the core. The dipole
c
transition moments A @ and B @ are defined by:
ccl m
ccl m
A @ "SY t0(El@m)DdD t T ,
cc
ccl m
a
(27)
B @ "SY tM 0(El@m)DdD t T ,
cc
a
ccl m
The matrix elements A @ and B @ (Eq. (27)) can be
ccl m
ccl m
deduced from zero-field data. The only other data
required to calculate the oscillator strength distribution
(Eq. (25)) are the R-matrix elements. These elements are
obtained from the R0-matrix elements using Eq. (18). For
the two-channel case with one open and one closed channel R0-matrix elements are obtained from zero-field experimental data [20]:
R0 "0,
11
JnCl3
R0 "
,
12
cos nk
(28)
R0 "tan nk .
22
The index 1 refers to the continuum channel and index
2 to the discrete channel; k is the reduced quantum defect
and Cl3 the reduced half width of the autoionising
Rydberg series (in a.u.).
131
3 Experimental setup
5d nf series of barium were excited from the metastable
3@2
5d2 1G level using a frequency-stabilized Stilbene 3 ring
4
dye laser (Spectra Physics 380D). The laser light perpendicularly intersected a collimated atomic beam, resulting
in a residual Doppler linewidth in the absence of an
electric field of about 10 MHz. The beam of metastable
atoms was produced by radiatively heating a tantalum
oven filled with barium and running a low-voltage DC
discharge between the tungsten heating filament and the
oven. The interaction region consisted of a well-shielded
chamber with two capacitor plates separated by 6 mm.
The lower capacitor plate was set to a negative potential
to apply a static electric field. The upper capacitor plate
contained a grounded fine wire mesh. Electrons produced
in the autoionisation process were accelerated towards the
mesh and counted by a channeltron electron multiplier
mounted behind this mesh. The electron signal and the
transmission peaks of a 750 MHz confocal etalon for
calibration purposes were simultaneously stored on
a computer which also controlled the frequency scan of
the ring dye laser. More detailed information on the
experimental setup can be found in [5]. Spectra were
taken for n+60. The polarisation of the laser light was
perpendicular to the electric field axis (p-excitation). Stark
spectra originating from the n"60 and n"61 states
could be recorded simultaneously in one scan using the
ring laser’s maximum scan range of 30 GHz. The discussion which follows focuses on these n"60, 61 spectra.
4 Results and discussion
4.1 Low-field experiments
From the metastable 5d2 1G state in zero electric field
4
one 5d nf series with J"5 (strongly) and two with J"4
3@2
(weakly) are excited. In low electric fields first the quadratic Stark effect for the J"5 series is observed. For
various n-values the six different DMD components could
easily be resolved. As the quantum defect of the 5d nf
3@2
J"5 series is as low as 0.074 the f-electron can be considered to be nearly hydrogenic and decoupled from the
5d-electron. In Fig. 2 the recorded spectra for n"60 for
low fields are shown. At low field strengths the appropriate quantum number is M, the z-component of the total
angular momentum J. At a field strength F&0.4 V/cm
the Stark splitting is of the same order of magnitude as the
fine structure splitting causing mixing between equal Mstates. At high field strengths the fine structure of 5dnf can
be neglected and the appropriate quantum number is m,
the magnetic quantum number of the Rydberg electron (at
most three in excitation from 5d). The spectra are a superposition of DmD"0, 1, 2 and 3 contributions. Apart from
the J"4 components at field of F"0.42 V/cm the J"5
DMD"0 and 1 components are still resolved. The DMD"2
and 3 components already merged into a linear Stark
manifold on the high energy wing of the J"5 DMD"1
component. Two interesting features can be observed for
F"0.42 V/cm. Firstly, the onset of the n"60 manifold,
superposed on a broad autoionising resonance. The broad
Fig. 2. Low field autoionising n"60 Stark manifolds in barium
originating from the zero-field transition 5d2 1G P5d 60 f
4
3@2
resonance may be attributed to the 5d64s J"2 state with
a quantum defect of 4.154 [22] and which has a high
autoionisation rate. Secondly, on the high frequency wing
of the manifold an interference pattern is observed, composed of a series of Fano profiles related to the excitation
of discrete manifold Stark states interacting with the continuum. This effect is less clearly observed for the spectra
recorded at higher field strengths. For F"0.83 V/cm only
the lowest M component of the J"4 state has not yet
merged into the manifold. The manifold now consists of
a superposition of different M components. Decomposed
into one-electron wavefunctions and taking into account
the excitation probabilities of the f-electron with DmD"0,
1, 2, 3 results in a manifold as observed for the higher field
strengths. What can clearly be seen in the spectrum for
F"1.23 V/cm is that the central manifold resonances are
narrower than those at the low and high frequency wings.
The resonances at the low frequency wing are broadened
due to the fact that different m components are not degenerate but too close in energy to be resolved individually.
All resonances in the manifold are broadened compared
to the zero field resonances as the Stark states are more
sensitive to field inhomogeneities (experimentally determined to be of the order of 10 mV/cm). The additional
broadening at the high frequency wing is mainly due to
these field inhomogeneities as the high-k Stark states have
the largest induced dipole moments and are therefore
most sensitive to field variations. A second cause for
broadening is the coupling of the Stark states with the
underlying 6s continuum. However, it may be concluded
that this latter effect is relatively unimportant given the
narrow Stark resonances observed. This is not surprising
given the fact that for the 5d 60f state a Doppler limited
3@2
linewidth of 10 MHz is observed. Therefore l-values larger
132
than 3 will even have smaller linewidths. Only the low-l
values give rise to high autoionisation rates. At the field
strengths used these are not yet mixed into the manifold.
4.2 High-field experiments
4.2.1 Experimental observations. When the field is increased to about 2 V/cm fully resolved linear Stark manifolds at n"60, 61 are observed. In Fig. 3 the n"60 and
n"61 manifolds are shown at a field of 2 V/cm. Again
clearly resolved is the broadening of the manifold resonances at the low frequency wing for n"61. For n"60 the
high frequency wing is not so clearly visible due to the low
oscillator strengths within the range of 4 to 10 GHz (frequencies are relative to the zero-field 5d 60f state). How3@2
ever, also in this region the broadening of the Stark
resonances can still be observed. This relatively strong
decrease in oscillator strength at the high frequency wing
of the n"60 manifold is due to mixing with the 5d64p
states, located at 3 GHz in zero-field. Because of their
large quantum defects 5d64p states (d"3.71 for J"3)
hardly shift in a field of 2 V/cm. The mixing of the discrete
manifold states and broad 5d64p states, which are strongly couple to the 6s continuum, results in a Fano-type
1@2
interference profile in the envelope of the manifold.
Interesting features are observed within the frequency
range of 10 to 17 GHz. In fact, within this range several
anti-crossings are observed between Stark manifold states
and the 5d63d states for small field changes. The 5d63d
states are located halfway between the n"60 and 61
manifold (d"2.543 for J"4). In Fig. 4 and 5 several
observed anti-crossings are shown in more detail. Experimental data are collected in Table 1. All broad resonances
are due to 5d63d states with different J and M values. The
narrow Stark manifold states interact with broad 5d63d
states. As a result, the manifold states gain intensity and
a clear narrowing of the 5d63d state involved in the
anti-crossing is observed. From Fig. 5 at F"2.13 V/cm it
can be seen that Stark manifold states in fact anti-cross
with at least two different 5d63d M states. The spectrum
recorded at F"2.08 V/cm in Fig. 5 shows three distinct
peaks where the manifold anti-crosses with one or more
5d63d M states.
Fig. 3. Spectrum of the frequency region between n"60 and 61 for
F"2 V/cm. The frequency is relative to 5d 60 f J"5 in zero-field.
3@2
The 5d 61 f J"5 state is located at 29.6 GHz in zero-field
3@2
Fig. 4. Anti-crossing between Stark manifold states and 5d 63d
3@2
states
Fig. 5. Stark manifold states anti-crossing with 5d 63d states.
3@2
The upper part shows three resonances at the anti-crossing, the
lower part Stark manifold states anti-crossing with two 5d 63d
3@2
states
133
Table 1. Measured splitting of level
crossing between 5d 63d state and Stark
3@2
manifold states with m"2 and
experimental widths (FWHM) of the
states that anti-cross
F (V/cm)
DE%91 (MHz)
1
2.08
2.13
2.15
2.16
2.18
2.19
2.20
2.22
2.24
2.25
2.26
2.27
2.30
2.32
2.34
2.35
2.37
2.39
103 (5)
87 (5)
126 (5)
135 (5)
120 (5)
141 (5)
130 (5)
137 (5)
139 (5)
176 (5)
174 (5)
174 (5)
172 (5)
184 (5)
198 (5)
208 (5)
210 (5)
214 (5)
4.2.2 Diagonalisation of the energy matrix. To understand
the features observed between 10 and 17 GHz in Figs. 3—5
we first consider a model neglecting continuum interactions. The total energy matrix in the 5d nl LSJ basis
3@2
(n"59—62) is a symmetric band matrix with on the diagonal zero-field energies of 5dns, 5dnp up to 5dnl("n!1)
fine structure states and off-diagonal terms proportional
to F for l@"l$1 (Eq. (9)). It is a good starting point for
the analysis as it could, in principle, provide correct energy positions for the anti-crossings. However, this requires (as does MQDT) zero-field input parameters which
unfortunately are not fully available in the present case.
For example, the fine structure multiplet of the 5d63d state
is not completely known. The quantum defects for low-n
5d nd states for all J-values are known [23—26]. How3@2
ever, around n"60 the 5d nd-series is perturbed due to
3@2
interactions with the 5d nd-series [25], so scaling laws
5@2
can not be applied. For the 5d np states data are only
3@2
available for J"1 and 3 [27, 28]. For the 5d ns-series
3@2
quantum defects are available for J"0—3 [23, 24, 26].
For the 5d nf-series quantum defects are available for
3@2
J"1, 3—5 [27—31], and for the 5d ng-series for J"2—4
3@2
[26]. It is therefore not possible to completely diagonalise
the matrix. This also hampers identification of the broad
5dnd resonances shown in Fig. 3 and in particular does
not allow an M-assignment. To analyse the experimental
data we will consider an alkali-like problem, i.e. we neglect
the fine structure of 5dnl for all l-values. On the diagonal
we use a single 5dnl state, i.e. the ones expected to be most
intense in the electric field (5dnf J"5, 5dnd J"4, 5dnp
J"3, and 5dns J"2). For l'3 quantum defects can be
set to zero. The relevant reduced quantum defects are then
given by k "0.074, k "!0.457, k "!0.290 and
3
2
1
k "0.154. The Stark matrix elements (Eq. (9)) simplify to
0
[12]:
SnlDzD n@l@T"d @ $ d @ d @ ·
ll 1 MM SS
S
(l#1)2!m2
· F · Rnl@ @.
nl
(2l#3) (2l#1)
(29)
DE%91 (MHz)
2
66 (5)
82 (5)
109 (5)
133 (5)
C (MHz)
1
C (MHz)
2
C (MHz)
3
47 (3)
43 (3)
—
63 (3)
86 (3)
73 (3)
74 (3)
80 (3)
76 (3)
112(3)
107 (3)
94 (3)
103 (3)
93 (3)
107 (3)
107 (3)
115 (3)
97 (3)
53 (3)
42 (3)
50 (3)
95 (3)
68 (3)
66 (3)
72 (3)
76 (3)
66 (3)
90(3)
93 (3)
85 (3)
95 (3)
91 (3)
92 (3)
119 (3)
106 (3)
114 (3)
33 (3)
—
73 (3)
76 (3)
Fig. 6. Calculated sublevel energies as a function of electric field
strength for m"0 by diagonalisation of the energy matrix. The
energy is scaled to the 5d 60f J"5 state in zero electric field
3@2
Even with this simplification the problem remains complex as we have to consider all allowed m-values (l*0 for
m"0, l*1 for m"1, l*2 for m"2 and l*3 for
m"3).
For the diagonalisation of the energy matrix four nvalues (n"59—62) have been taken into account. The
diagonalisation is carried out for all allowed m-values
separately. In Fig. 6 part of the angular momentum manifold fanout for m"0 is shown, calculated for field
strengths ranging from 0 to 2.4 V/cm. The upper part of
the n"60 manifold and the lower part of the n"61
manifold are shown. The frequency is again scaled relative
to n"60. At a field of about 1 V/cm the 64p state starts to
anti-cross with the n"60 manifold. The 63d state starts to
anti-cross with the n"60 manifold at about 2 V/cm. At
slightly higher field strengths the n"60 and 61 manifolds
134
begin to merge as well. The 65s state anti-crosses with the
n"61 manifold at about 0.7 V/cm (only for m"0). The
61f state at about 28 GHz in zero field is merging with the
linear Stark manifold of n"61 at about 0.6 V/cm. Figure
7 shows a detailed enlargement of the angular momentum
manifold for field strengths between 1.7 and 2.4 V/cm for
m"2. Interestingly, both from the diagonalisation and
from the MQDT analysis (see following section) it follows
that the m"2 components are excited most strongly. The
discussion may therefore concentrate on m"2, simplyfiing the analysis considerably. The spectra for the first and
second anti-crossing, derived from this diagonalisation
procedure, are shown in Fig. 8. From Fig. 7 we conclude
that the anti-crossing is most pronounced and narrowest
for the first (n"60, k"57) manifold level. As the intensity of the anti-crossing levels primarily comes from the
63d state, manifold peaks are only observed due to their
admixture with the 63d state as is clearly visible in Fig.
8 and in the experiment. Close to the minimum distance
the two levels are mixed strongly resulting in two equally
strong lines. The first anti-crossing in this model occurs at
2.03 V/cm with an energy splitting of 100 MHz. The second occurs at 2.10 V/cm (183 MHz). The merging of the
63d level into the manifold nicely manifests itself in the
distribution of oscillator strength over more manifold
states at increasing field strength. Comparison with experimental results, shown in Figs. 4 and 5 and Table 1,
shows clearly the limitations of the model. Firstly, the
anti-crossings do not occur exactly at the calculated field
strengths but at slightly higher values. Secondly, the width
of the anti-crossing is much narrower than calculated
although the intensities behave as expected. Thirdly, additional anti-crossings occur in the experiment (Fig. 5) that
can not be reproduced within our model as their positions
and widths do not fit separations of neighbouring mani-
Fig. 7. A detailed enlargement of the calculated sublevel energies as
a function of field strength for m"2 showing the first anti-crossing
of the Stark angular momentum manifold states with 5d 63d. At
3@2
higher field strengths the anti-crossings due to n-mixing between
n"60 and 61 are visible. The energy is scaled to the 5d 60f J"5
3@2
state in zero electric field
fold states. Especially the doublet observed at 2.13 V/cm
in Fig. 5 is intriguing. It may be explained assuming that
they are induced by two different 5d 63d states (but with
3@2
the same m"2).
4.2.3 MQDT analysis. The references quoted in Sect. 4.2.2
regarding quantum defects include only information on
autoionisation widths of a limited number of states. However, the widths of all states involved in the electric field
case are necessary input parameters for an MQDT analysis. With the same approximations as used for the diagonalisation of the energy matrix, an MQDT analysis was
nevertheless performed. The missing input parameter was
the scaled autoionisation width of the 5dnp states. However, as confirmed by the diagonalisation procedure as
well as by the MQDT analysis, m"2 states give the
dominant contribution to the overall spectra. Furthermore, as shown in Fig. 6, the 5d64p state which anticrosses with the n"60 manifold states at a field strength
of about 1 V/cm does not have a significant influence on
manifold states within the region of interest at a field of
about 2 V/cm. The scaled autoionisation width (HWHM)
for the 5dnp states was tentatively fixed at 2.34]10~2 a.u.
The other non-zero scaled autoionisation widths
(HWHM) are 6.78]10~5 (5dnf), 6.04]10~3 (5dnd) and
2.51]10~2 (5dns). In Fig. 9 the result of the m"2 MQDT
analysis for the first and second anti-crossing between the
manifold states and the 5d 63d state is shown, allowing
3@2
for a comparison with the diagonalisation method of Fig.
8. The outcome of these calculations for the energy positions are identical to those obtained from the diagonalisation procedure. However, in the MQDT analysis also the
interaction with the underlying 6s continuum is taken
1@2
into account, which in particular shows in the widths of
the resonances. An interesting feature is the change in the
Fano profile, corresponding to an inversion of the Fano
q-parameter, for manifold states before and after the anticrossing (compare e.g. the spectra with field strengths of
1.99 V/cm and 2.06 V/cm). Noteworthy is also the intensity maximum and linewidth minimum behind the anticrossing at F"2.09 V/cm. These features have not been
observed experimentally due to field inhomogeneities that
tend to smear out narrow resonances as was discussed in
Sect. 4.1.
One feature of Fig. 5 for F"2.08 V/cm could not be
explained in both models: the occurrence of three resonances at an anti-crossing. Two explanations are possible.
The first would be to assume that a third resonance relates
to a manifold state with mO2. In this case a narrow
manifold state, not interacting with the other states would
accidentally shift in. However, we reject this explanation
as the intensity of such a resonance would be nearly zero,
unless it also anti-crosses with a 5d63d state with the
same m-value. Such an anti-crossing would then also
occur at the other field strengths, which is not the case.
The other explanation is to assume that a second 5d63d
state shifts the energy of two particular manifold states at
the field strength of 2.08 V/cm such that two manifold
states anti-cross with one of the 5d63d states at the same
time.
The assignment of other broad resonances observed in
the region between 10 and 17 GHz is not clear (Figs. 3—5).
135
Fig. 8. Calculated spectra for m"2 showing the
first anti-crossing between Stark angular
momentum states (labeled by Dn, kT) and the 63d
state using a diagonalisation procedure neglecting
continuum interaction. The energy is scaled to the
5d 60f J"5 state in zero electric field
3@2
They do not seem to interact with the manifolds. Probably
they are due to excitations of 5d63d states not containing
a large m"2 fraction. Manifold states with m-values
matching those of 5d63d should also anti-cross. However,
due to their low oscillator strengths these anti-crossings
are not observed experimentally. The low oscillator
strengths for m-values other than m"2 were confirmed in
our MQDT analysis.
5 Conclusions
We have demonstrated that the linear Stark effect can be
observed for autoionising states in a carefully selected
system such as the 5d nf series of barium. It requires that
3@2
the interaction with the underlying continua must be
small. A qualitative analysis of such Stark manifold is
feasible neglecting the influence of the continuum. It involves a direct diagonalisation of the energy matrix to
determine the position of the manifold states. This procedure does not result in an oscillator strength distribution to
be compared with experiment. Using an MQDT model
which incorporates the Stark potential for bound states to
describe the long-range effects, oscillator strength distributions can be calculated as well. This procedure provides
extra information on the widths of resonances. New features such as q-reversal and unexpected line narrowing of
manifold states are calculated. However, in the present
study a complete MQDT analysis is hampered by lack of
zero-field experimental data on relevant autoionising
states. Still, with the simplifications made, reducing the
problem basically to that of an autoionising alkali atom,
the observed anti-crossings qualitatively can be understood.
136
Fig. 9. Calculated spectra for m"2 showing the
first anti-crossing between Stark angular
momentum states (labeled by Dn, kT) and the 63d
state using the MQDT model incorporating
continuum interactions. The energy is scaled to the
5d 60f J"5 state in zero electric field
3@2
The authors are indebted to Jacques Bouma for his technical assistance. We would like to thank Dr. K. Sakimoto for valuable suggestions and making available the compute code for the MQDT treatment. Financial support from the Foundation for Fundamental
Research on Matter (FOM) and the Netherlands Organization for
the Advancement of Research (NWO) is gratefully acknowledged.
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