Download Hydrogenic Rydberg atoms in strong magnetic fields: Theoretical

Atoms, Molecules
~~o and Ousters
Z. Phys. 0 - Atoms, Molecules and Clusters 5, 279-285 (1987)
o
Springer· Verlag 1987
Hydrogenic Rydberg Atoms in Strong Magnetic Fields:
Theoretical and Experimental Spectra in the Transition Region
from Regularity to Irregularity *
A. Holle, G. Wiebusch, J . Maio, and K.H. Welge
Fakultiit fUr Physik, Universi tat Bielefeld, Bielefeld, Federal Republic of Germany
G. Zeller, G. Wunner, T. Ertl. and H. Ruder
Lehrstubl fUr Theoretische ASlrophysik, Universita t Tiibingen,
Tiibingeo, Federal Republic of Gcrmany
Received December 5, 1.986
For deuterium Rydberg atoms in a magnetic field of ...... 6 T we compa re tbe complete
experimen tal spectrum io the range - 190 em - 1 to - 20 cm - I with the positions and
oscillator strengths of the correspond ing quantum theoretically calculated photoabsorption lines. The agreement is cxccllent. The range of energy covered extends from the
end of the I -mixing regime up to tbe regions where tbe approximate integrability of
the problem is completely lost, and the corresponding classical system undergoes a transition to chao!>.
PACS, 31.20.D; 31.50 ; 32.60.V
1. Introduction
The quanti tative description of the wealt h of spectral
structure observed in experimen ts wit h strongly magnetised Rydberg atoms both below and above the
field-free ionisa tion threshold has for many years presented a lasting challenge to theory. Classical and
semi-classical calcu lat ions were performed (e.g. [1- 3])
to account for the appearance and spacings of quasiLandau resonances observed in the spectrum [4], and
quantal calculations [ 5- lO] were able to explain at
least some general features of the spectra. It is, however, evident that the ultimate test for theory lies in
the quamitative compa rison between the theoretical
and experimenta l values for the positions and intensi ties of the hundreds of Rydberg lines observed in a
spectrum . It is tbe purpose of tltis note to carry o ut
such a comparison in an ex tended, nontrivial in terval
of energy ranging from the beginning of the II-mixing
0;
Rcse~rch
support ed by DeLllschc Forschungsgcmcinschaft (DFG)
regime way up to shortly below the ficld-free ionisation threshold. Thus il will be demonstrated that in
this essential part of the spectrum, whieh e.g. contains
the" build-up " (Delande and Gay [II]) of the quasiLandau resonances, this " principal rema ining problem in the elementary quantum mechanics of a oneelectron atom " (KJeppner et al. [1 2]), can be considered solved. We call this ra nge of energy the transition
region from regularity to irregularity since it is in
this range that, as a consequence of the nonintegrability ofthc underlying Ham ilto nian, in lhe corresponding classical system the fraction of regular orbits in
phase space is steadily decreased, and from a certain
energy on the classical system beha ves completely
chaotically. Therefore our in vestigations also allow
addressing the more genera l question of how the speetrum of a quantum system behaves in a range of energy where its classical counterpa rt undergoes a transition to cbaos. This question is of importance in the
context of current efforts to define the notion of
" quantum " stochasticity (see e.g. [1 3, 14]).
A. Holle 1:1 al.:
280
l -Iydrogeru~
Rydberg Atoms
(e)
n=24
, ,
[222'0 1816
Ii.
, ,
12 10
,
e
6
I
L
,
,
2
4
theory
bO
,
,
":25
0
0
I
0
0
,
,
,
I
II
I
experiment
j
1
!
0
_185
-190
-175
-180
(b)
n =26
theory
•
0
I
0
0
II I I
J
I
I
experiment
4
,
Fig. I a- f. Deuterium Rydberg atoms
in a magnetic field of 6.0 T :
comparison between the theoretical
2
1
1
o
- 170
oscillator strength spectrum and the
experimental photoabsorption
!
-165
transitions 10 m _ O, even-parity
Rydberg states, over lhe range of
energy - 190cm - J to -70cm - ' .
Osci1!ator st rengths are given in units
of 10- 6 , the experimental intensity
-155
-160
(e)
" : 28
n =27
spectrum for Llm = O Balmer
scale is in arbi trary units. The si~
successive energy intervals cover the
range from the onset of ,,-mixing (in
frame (aJ the n =24 and n .. 25
theory
•
,0
"
o
,
,
,
111
I
I
II I
I
I
o
-150
11
L
experiment
-145
-11,0
-135
-
energy (em 1)
man ifolds are still just sepa rated from
each other, while in frame (b) the
uppermost state of the 11 - 2~
mainfold lies already inside the 11 = 26
mainfold) to the regime where the
appro;o;imate integrability of the
problem is dest royed to an ever
increasing elltent (frame (f)). A total
of 177 lines contribute to tbe
spoctrum shown. The indices /I and k
labell ing the states arc the principal
quantum number a nd the separa tion
inde:t to which the corresponding
states can diabatieally be traced baek
in the limit B -> 0
281
A. HoJle CI al.: Hydrogenic Rydberg Aloms
f
n = 31
n= 30
n =29
F
Id)
theory
0
I
0
I
0
,I
I
I,
,
,I I I
experiment
3
2
.l
1
1
0
- 130
-125
-120
-115
~34
0=33
n= 32
energy (em-1)
Ie)
theory
0
0
,
II
I
II
I
I
I
1 I IL
I,
1
experiment
3
2
~
1
0
- 110
-105
I~ ! LL lLlu I~
n=
, 37
0=36
0=35
theory
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,
experiment
3
2
0
-9u
-85
-95
-TOO
I f)
A. Holle et at : Hydrogcnic Rydberg Atoms
'"
A quantitative comparison of experimental a nd
theoretical spectra has become possible since, on the
one hand, high resolution experiments with highly
excited one-eleclron systems (hydrogen and deuterium) have been performed in 4-6 T fields [15], and,
on lhe other, numerical procedures bave been developed which allow an almost routine calculation of
very accurate energy values. wave fu nctions, and,
thus, transition probabilities, of hydrogcnic systems
in uniform magnetic field s below the field-free iooisation limit. A preliminary comparison of experimental
a nd theoretical energy values, and, partly, some oscilla tor strengths, was recently performed in two small
sect ions of the energy scale [16]. We note that Delande and Gay [17] also compared sample results
for a few energy values of magnetised Rydberg states
with their own experimental caesium data. The objective of the present work is far mo re ambitious, namely
the quantitative comparison for -420 Rydberg lines
in the photoabsorption spectra of deuterium atoms
in magnetic fields .... 6 T from tbe point of view of
energy and intensity.
2. Experiment
The experimental method used to measu re the spectra
is the same as in Ref. 16, and has been described recently [ 15]. Here we give a brief summary of the substantial experimental fea tures. Rydberg states of even
parity were prepared by resonant two-photon absorption through Paschen-Back resolved substates of 2p.
The VUV laser light fo r the first excitation step (J.~
121.6 nm) was polarised parallel to the magnetic fi eld,
tuned to excite the 2p, m=O Paschen-Back state, and
kept at rued wavelength during the measurement.
The tunable UV laser light for the second excitation
step (~;::;364-366 nm) with a band width of
-0.07 cm - I was also polarised parallel to the magnetic field. [n this way pu re m= O final states with
even parity were excited. Both laser beams were directed perpendicular to each other, and perpendicular
to the well collimated beam of atomic deuterium,
which was directed parallel to the magnetic field. This
arrangement minimizes motional elect ric fie lds, reduces the effective Doppler width to well below the
band width of the lasers, and defines an effective excitation zone of ...... 0.5 mm diameter. The zone was located between fine-mesh grid electrodes, which allowed reducing electric stray fields orientated parallel
to tbe magnetic field to below 100 mV Icrn. After excitation the atoms drift ed, at thennal velocities, into
a constant parallel elect ric fie ld ( ...... 3.5 kV/ cm), wbich
lield~ i o nised them and accelerated the electrons to a
surface barrie r diode serving as the detector.
3. Theory
A diITerent numerical approach was taken to obtain
higbly accurate solutions of the - nonintegrable SchrOdinger equation descri bing an electron under
the simultaneous action of a fixed Coulomb potential
and a strong homogeneous magnetic field. For given
magnetic quantum number m we expanded the wave
functio ns in terms of spherical hannonics, "'... =
EhAr)· Y" ... (O, q,), and the radial functions in a com·
plete, o rthonormal basis wh ich is constructed from
generalized Lagucrre funct ions wit h fixed ex ponent.
We have used this approac h previously to explore
the properties of "circu lar " Rydberg states in strong
magnetic fields [ 18]. Matrix elements with respect to
this basis can be expressed in closed analytical form
and give rise to a banded Hamiltonian matrix which
can be diagonalized by efficient sta ndard algorithms.
Our choice of basis is similar to the Sturm ian basis
employed by Clark and Taylor [ 5-7] but avoids the
complications associated with the nonorthogonality
of the latter. As compared to the oscillator basis in
semiparabolic coordinates (used e.g. by Delande a nd
Gay [19], and Wintgen and Friedrich [8-10], where
the diagonalization produces eigenstatcs each of
which belongs to a diITerent value of the magnetic
field strength (solutions a re obtained in the E - B
pla ne along straight lines £/ 8 =const.), our basis has
the advantage that one diagonalization procedure
yields the spectrum a t fixed B. in accordance with
the experimen tal situation. In our calculations we
worked with basis sizes of up to 6400 for determining
both eigenvalues and eigenvectors of the first -600
states in given In·parity subspaees in the strong-field
regime. Convergence was established by varying the
size of the basis and the exponent of the Laguerre
functions. Accuracies of six significant ligures can be
guaranteed for energy values, and of two to three
sigoilicant figures for oscillator strengths. We note
that in the results to be presented below t he mass
scali ng laws valid in a magnetic fie ld [20, 21], which
relate the field st rength, energies, and oscillator
strengths pertaining to the infin ite·nuclear-mass hamiltonian to those belonging to fi nite core mass, are
taken into account.
4. Discussion
Figure 1 shows a comparison between theoretical and
experimental spectra for i! m= O Balmer transitions
to Rydberg states with m= O, even parity, and energies
between - l 90cm - 1 a nd -70cm - l • The magnetic
fi eld strength in tbe experiment was 6.0 T. The theoretical results were obtained fo r a value ofthe magnet-
A. Holle el al : Hydroge nic Rydberg Aloms
283
50
",
theory
( oj
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,
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L
I
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experiment
3
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A
.~A.lt
II
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o~~~~~~~~~Pll~~~W
,
I-"ig. 2a-c. Sa me as Fig. I but for a
magnetic field of 5.96 T and
energies of tbc Rydberg stales
between - 80em - 1 and - 20em - '.
Note that at the end of the energy
range (above _ - 25 em - I) the
corresponding classical system
becomes comptetely chaotic.
276 tines are shown in tnis spectrum
experiment
,
2
o
-35
-30
=
-25
=
ic field parameter fJ( B(Bo with Bo 2(m~c)21
(eh):::::4.70 1 x IO ~ T) of 1.275 x 10- 5 corresponding,
for deuterium, 10 B =5.997 T. A total of 177 line.s
are compared in Fig. 1. We encourage the reader to
go through the figure line by line. The agreement be·
tween the theoretical predictions a nd the experimen.
tal values for the positions and intensities of the lines
energy
(CIff1)
is evidently excellent. Moreover, theory can in fact
inform on where neighbouring lines were not resolved. Slight differences in the relative intensities of
a few lines are due to the fact that saturation effects
occurred in the experiment in strong lines, and, 31so,
to the uncertainty on the strength of the magnetic
field.
284
The range of energy considered in Fig. J extends
from the onset of the overlap of neighbouring n manifolds deeply into the n-mixing regime. The overall
structu re of the spectrum still seems relatively Ofdered, with the clusters of lincs belonging to the different n manifolds interpenetrating, at least at the beginning., without substantial mutual perturbation. This
behaviour, which was already pointed Qut in the earlier calculatio ns of Clark and Taylor [6, 7], and even
a llows labelling tbe states in terms of weak-field quantum numbers, is closely related to the existence of
an approximate constant of motion, and thus the approximate integrability of the pro blem (cf. [9]), in this
range of energy.
The comparison between experimental and theoretical spectra shown in Fig. 2 proceeds to regions
whcre this approximate integrability is rapidly destroyed, as witnessed by the disappearance of very narrow avoided crossings of the encrgy levels when
viewed as functions of the ficld (cf. Fig. 3 in Ref. 16).
Again we consider ..dm""O Balmer transitions to deuterium Rydberg states with m=O and even panty,
and the energies lie in the interval from -80cm - '
to - 20 em - I. I n this case the experimental data were
taken at a magnetic field of 5.96 T, and the theoretical
results were detcrmined for p= 1.2675 x 10- 5
(8=5.9619 T). A total of 276lincs cont ributes to the
theoretical spectrum. As onc moves up in cnergy in
Fig. 2, the line density is clearly growing, and the
structure of the spectrum becomes increasingly compl icated. Nevertheless it is also here that we find
practically complctc agrcement between theory and
experiment, with the experiment being limited, as regards the number of deteetable lines, by the finit e
rcsolution.
F igure 2 ean serve as a starting point fo r further
investigations. A Fourier analysis of the spectrum,
for example, reveals that in spite of the seeming irregularity of the speetrum the line sequence docs indeed
contain periodic features. From the Fourier transform
of the theoretical and experimental spectrum one can
recognise (Fig.3a and b, respectively) that at least
periodicities of - 1.7h wc and O.66hwc (w. is the electron cyclotron freq uency) arc present io the spectrum.
The peak marked by 1corresponds to the well known
quasi-Landau resonanccs which, at E=O, show a
spacing of 1.5/ICOe (4). The peak markcd by • corresponds to a new type of quasi-Landau resonances
discovered recently [15]. These resonances show a
spacing of 0.64 h w~ at E = O. The differences with respect to the above values are due to the energy dependence of the resonance spacings. It should be pointed
out tbat in the Fouricr transforms of the experimental
spectra which extend beyond the ionisation limit even
more resonances, with spacings smaller than
A. Holle et al.; liydrogeni!; Rydberg Atoms
''''
""
*
(a)
II"""
':;;" ""
,
'"
,
•
"
Ibl
oxperlment
:!" "
,
, .,
to
u
(21f!flwc)
1.S
"'i~ Ja a nd b. Square of the Fourier ulnsfonn of the theo retical
(a) and experimental (b) spectrum shown in Fig. 1 The unit of the
abscissa is chosen in such a way that a ~riodicily of },· /lw. in
the original spectrum is rcneetoo by a peak" at Ih in tbe Fourier
spectrum. Thus the feature around u ..... O.57, together with iu two
harmonics (dll$hed arrow5). corresponds 10 },- 1.7, while the more
prominent feature al II .... 1.5 corresponds to y-O.66
0.64 I!wv have bccn found , and have theoretically
been aceounted for by classical trajectory calculations
( 22). So far no exact quantum mechanical results arc
available at 6 T above the ionisation limit. The cxistence of the Fourier peaks in Fig. 3 (! and .) clearly
indicates the build-up of the corresponding quasiLandau resonances below the ionisation limit.
To investigate a possible connection to chaotic
behaviour in the corresponding classical system we
computed classical trajectories fo r 8 =6 T and m=O
as a function of energy. We find [23] that the first
irrcgular orbits appear around -- - 100 em - \ and
the last regions !illed with regular orbits vanish in
the Poincare ~urface of section at z= O around _
- 25 cm - I , which therefore is the critical classical energy of the onset of chaos for the set of parameters
chosen. Thus in Fig. 2 one act ually has the opportunity of watching the spectrum of a quantum system
evolve in a range of energy where its classical countcrpart undergoes a transition to irregularity. However,
a mere inspection of the spect rum does not exhibit
any perspicuous differences between thc structure of
the spect rum below and above the critical energy.
To find such differences it is necessary to look at
more basic properties of the spectrum, e.g. the statis-
A Holle el al : Hydrogenic Ryd berg Atoms
tics of the level sequences in different parts of the
spectrum, which display a transition from a Poissonlike to a Wigner-like distribution as one proceeds to
ranges of energy where classical motion becomes increasingly irregular [23- 25]. This underlines the fact
Ihat magnetised Rydberg atoms are indeed promising
objects in the current search for manifestations of stoehasticity in quantal systems.
To summarize, in this paper a complete experimental spectrum in the strong-field regime has successfully been accounted for, in quantitative terms,
by the positions and oscillator strengths of the corresponding quantum theoretically calculated (in total
426 different) photoabsorption lines. This certainly
can be considered essential progress in the quantitative spectroscopy of highly excited atoms in strong
magnetic fields.
Rdercnces
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(1982)
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12 Klcppner, D., Littman, M.G~ Z irnmerman, M.L: Ryd berg
atoms in strong fields. In : Rydberg Slates of atoms and molecules. Stcbbings, R.F., Dunning. F.B. (cds.), pp 73-11 6, Cambridge' Cam bridge Univers it y Press (1983)
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A. Holle
G. Wiebusch
1. Main
K.H. Welge
Pakultat mr Physik
UniversiUit Bielefeld
0-4800 Bielefeld
Federal Republic of Gerrnany
G. Zel ler
G. Wunncr
T. Erll
H. Ruder
Leh rst uhl fUr Theoret ische Astroph ysik
Un iversitiit Tiibingen
0-7400 Tiibingcn
Federal Repu btie of Germany