J. Phys.
B:
At. Mol. Opt. Phys.
21
(1988) 34993521. Printed in the
UK
Rydberg atoms
in
parallel magnetic and electric fields:
11.
Theoretical analysis of the Stark structure
of
the diamagnetic
manifold of hydrogen
P
Cacciani,
E
LucKoenig, J Pinard, C Thomas? and
S
Liberman
Laboratoire Aim6 Cotton$, Centre National de la Recherche Scientifique, 91405 Orsay
Cedex, France
Received 21 January 1988
Abstract. The semiclassical approach based on the secular perturbation method, primarily
introduced by Solov’ev to study the pure diamagnetism of hydrogen in the weakfield limit,
is extended to the case where the atom is subjected to electric and magnetic fields both
parallel to the
Oz
direction. The structure of the manifold of states with principal quantum
number
n
is governed by the existence of an approximate constant of motion and the states
can be classified into three groups. The semiclassical BohrSommerfeld quantisation rule
allows the energy
of
the states to be calculated. The main properties of the spectra and
their evolution when one of the fields increases are analysed. An original comparison
between the structure of the pure diamagnetic
n
multiplet with azimuthal quantum number
M
#
0
and the structure
of
the
n,
M
=
0
multiplet with parallel fields is proposed.
1.
Introduction
In the problem of highly excited states of the hydrogen atom perturbed by a weak
magnetic field directed along the
z
axis, recent theoretical works (Solov’ev 1981, 1982,
Gay and Delande 1983, Herrick 1982) brought into evidence an approximate constant
of the motion
A
=
4A2

5A: depending
on
the RungeLenz vector A
=
L
x
p
+
r/
r
and
describing the structure of a diamagnetic
n
manifold with a given total parity. The
sign of this constant is directly connected to the existence of two classes of states as
has been discussed in the previous paper (Cacciani
er
a1
1988a, hereafter referred to
as
I ).
The existence of these two different types of dynamics for the electron motion
has been confirmed experimentally in the analysis of the diamagnetic structure in odd
Rydberg states of lithium (Cacciani
er
a1
1986a,
I ).
The present paper aims to analyse the structure of an
n
manifold in the hydrogen
atom in the presence of parallel electric and magnetic fields. To understand this
behaviour we extend the semiclassical approach based
on
secular perturbation theory
developed by Solov’ev (1981, 1982).
In
this new situation the cylindrical symmetry is
conserved and the azimuthal quantum number remains an exact one. However the
electric field couples states with opposite total parity. The strengths of both fields are
supposed to be weak enough for the corresponding interactions to be considered as
t
Present address: Bureau International des Poids et Mesures, Pavillon de Breteuil, 92312 Shvres Cedex,
France.
f
The Laboratoire Aim6 Cotton is associated with the Universitt ParisSud.
09534075/88/213499
+
23$02.50
0
1988 IOP Publishing Ltd 3499
3500
P
Cacciani et
a1
perturbations compared with the Coulomb interaction. Then the principal quantum
number n related to the unperturbed energy of the manifold
E,(
n )
=

1/2n2 is still a
good quantum number. This ‘inter1 mixing’ regime appears when the maximum value
of the diamagnetic shift and the total energy extension of the linear Stark structure
are smaller than the energy difference
AE(
n, n
+
1) between two consecutive manifolds,
i.e. when yn7I2<<
1
and fn’<<
1
where
y =
BI B, with BC=2.35x 105T and
f
=F/Fc
with Fc =5.14x 109Vcm’.
By adding to
A
the constant of motion describing the linear Stark effect which is
proportional to
A,,
one obtains the constant of motion
A,
associated with the treatment
of the combined Stark and diamagnetic effects:
A,
=
4A2

5 Af
+
10PA,
where
p
(0
s
p
s
CO)
measures the relative strength of the linear Stark effect with respect
to the diamagnetic interaction
(§
2.1). Depending on
p
and
A,
the movement of the
RungeLenz vector is analysed, which leads to the definition of three different classes
of states
(0
2.2).
Related to the existence of the three independent constants
of
motion
n, M and
A,
the problem separates into three onedimensional motions, which are
described in terms of generalised coordinates. One of the coordinates is
0,
the angle
between the fields’ direction and the RungeLenz vector. Then by using a semiclassical
WKB
quantisation on this angle
0
and its conjugate momentum, the eigenvalues
A,
can be determined. Some analytical results have been obtained concerning the number
of states in each class
(§
2.3). The evolution of the structure
of
a diamagnetic manifold
( B
=
const) with increasing
F
field
or
of a Stark multiplet ( F
=
const) with increasing
B field is described
( 9
2.4).
By
analytically evaluating the derivative of the quantisation
integrals with respect to
A,,
the location of the crossings between states I and I1 is
determined, and the energies
of
the outermost states
of
the spectrum are calculated
(§
2.5). An original relation between the pure diamagnetic problem for the n manifold
with M
#
0 and the parallel fields problem with
M
=
0 and
PM
=
/Ml/n f i is discussed
in § 3.
2.
The hydrogen atom
in
weak parallel electric and magnetic fields
The Hamiltonian for a hydrogen atom subjected to electric and magnetic fields parallel
to the
Oz
direction can be written in atomic units:
P 2
1
Y Y 2
H
=
+
L,
+
p* +
f z.
2
r 2 8
H
is invariant under rotation around the
z
axis, M is an exact quantum number. In
the ‘interZ mixing’ regime each subspace n
=
const, M
=
const can be studied indepen
dently.
2.1.
Approximate constant
of
the motion in the lowjields regime
By using the secular perturbation method we calculate the values of
p2
and
z
averaged
over an unperturbed ellipse, the solution of the Kepler problem:
( p’ )
=
i n2( n2+ M2 + n2A)
( z ) =#n2 A,.
Hydrogen atom in the presence
of
parallel magnetic and electric$elds
3501
The energy is then equal to
E
=
Eo+Ep+&y2( p2) +f ( z)
where
Eo
is the unperturbed energy and E, the paramagnetic shift.
diamagnetic perturbations:
We introduce the parameter
p
which measures the ratio between the electric and
p
=yf/
y2n2.
(3)
The energy is expressed
E=Eo +E,+&y 2 n2 ( n2 +M2 +n2 Ap )
(4)
=
Eo+ E,+ E,
where the approximate constant
of
motion
A,
occurring in the diamagnetic plus linear
Stark problem is:
A,
=
4A2

5Ai
+
lOPA,
( 5 )
which satisfies:
d
(A,)
=
o+
o(
y4,f2).
dt
A,
is the third integral
of
motion specific to the problem of the hydrogen atom in
The
/3
=
0 limit is the pure diamagnetic problem with the
A
=
4A2

5A3
constant
Another expression for the energy is
weak parallel electric and magnetic fields.
of motion.
20
The
p
=
+CO
limiting case corresponds to the pure
with the constant
of
motion
A,/pl,+E
proportional
Stark problem.
electric perturbation associated
to
A,,
well known in the linear
During the slow evolution of the parameters of the elliptic trajectory, the extremity
of the RungeLenz vector
A
moves along a surface of equation
A,
=
4A2

5A:
+
10pA,
=
const.
This can be written in more convenient ways as:
A,
=4A2

5(AZ

p) ’ +
5p2
( 7a)
=4A:(A,5p)*+25P2
( 7 6 )
where
A,, A,
are respectively the components perpendicular and parallel to the
z
direction.
This surface is a hyperboloid
of
revolution centred at the point
(A,
=
0,
A,
=
5p).
The type of the hyperboloid depends on the sign of
Ap
25p2.
(i)
For
A,
 25p2<0
the hyperboloid has two sheets called
a
and
b
which are
symmetrical with respect to the plane
A,
=
5p.
Their respective tops are called
T,(O,
Taz)
and Tb(O, Tbz) with
T,,
=
5p
+
(25p’ A,)”’
Tbz
=
5p 
(25p’
Ap) ’/*
sheet
a
sheet
b.
(8)
(ii) For
Ap
25p2>
0
the hyperboloid has only one sheet called
c
(see figure
1).
3502
P Cacciani et a1
'\\
I/
\
/
hp<25P2
hp>25P2
Figure
1.
Secular variations of the RungeLenz vector
A
under the action of small magnetic
and electric fields parallel to the
z
direction. The extremity
of A
moves on the hyperboloid
of revolution
AB
=
constant (equation
( 7) ). /A/
is smaller than
1,
therefore
A
is bounded
by the sphere of unit radius centred at the origin, the double cone asymptotic to the
hyperboloid is shifted in the
z
direction by the quantity
Sp.
In the present case
p
=si,
and
there exist three sheets
a,
b
and
c.
To
and
Tb
denote the tops
of
the
a
and
b
sheets,
I,,
and
I,,
the
z
projections
of
the intersections between the sphere and the hyperboloid. The
part of the hyperboloid inside the sphere has been drawn in heavy lines.


,
asymptotic
cone with the top at
( A
=
0, A,
=
5 p)
and semiangle
Bo
with cot
Bo=
2.
The double cone asymptotic to these hyperboloids has a semiangle
Bo
with cot
Bo
=
2.
One can notice that the situation is quasisimilar to the pure diamagnetic one except
for the shift of
5p
in the
z
direction and for the
A
=
0
limiting condition changed into
A,
=
25p2.
The modulus of the RungeLenz vector is restricted to the range
A:+AI6
1

M2/n2.
A
just moves into the inner part of the sphere of radius one when
M
=
0.
The following
concerns the case
M
=
0
unless the contrary is specified.
2.2.
Variation range for
A,.
Definition
of
the classes
of
states
2.2.1.
Conditions for the existence
of
intersections between the sphere and the hyperboloid.
Under the joint actions of the electric and m.agnetic fields the extremity of
A
moves
on the surface
A,
=
const (equation
(7))
but remains included inside the sphere. These
two conditions can be simultaneously fulfilled when intersections between the sphere
and the hyperboloid exist.
The two conditions are summarised
4A:

(A,

5/3)'+
25p2

Ap
=
0
A:+A:S
1.
We deduce the inequalities satisfied by
A,
:
5Ai

IOPA,
+
A,
 4 6
0
IA,Is 1.
Hydrogen atom in the presence ofparallel magnetic and electricjelds 3503
A
general condition for the existence of a solution for inequality
( 9a)
is that the
discriminant is positive
A,
s5p2+4. (10)
The first term of inequality
( 9a)
vanishes for
I,=
( i
=
1,2)
represent the ordinates of the intersections between the sphere and the
hyperboloid only if
lItzl
s
1.
Table
1
shows that the number of intersections between the sphere and the hyper
boloid (or equivalently the number of solutions of the inequalities (9)) depends
on
p
and
A,
;
furthermore for a chosen
p
value, the energy
of
the states (i.e.
Ap)
is limited
from below and from above. This analysis points out the existence of the particular
value
p
=
1;
for
p
>
1,
I,, is necessarily greater than
1
and then there exists
no
mor:
than one intersection between the sphere and the hyperboloid. Let us remark that
when the sphere intersects the sheet c
(A,
>
25p2),
it
necessarily crosses this sheet
twice; for the sheet
a
there exists at most one intersection. For the sheet
b
one
0:’
two
intersections can be present.
Table
1.
Conditions
for
the existence of the intersections
I,,
and
I z,
(equations
(9),
(lo),
(11))
between the hyperboloid and the sphere as a function of
P
and 1,.
‘1,
lop

1
+lop

1
5p2+4
P < l
P ’ l
1
<
1,;
<
1

1
<
I,,
<
1
1
<I,,
<
I,;
<
1
2.2.2.
Motion of the A vector during its secular variations.
The portion of one sheet of
the hyperboloid swept by the tip of
A
during its slow motion can be of very different
types according to the
p
and
A,
values. Typical examples are presented in figure
2.
Various situations are to be considered according to the value of
p
with respect to the
limiting values
p
=
:
and
j3
=
1.
Indeed, when
p
=
:,
the top
of
the asymptotic cone is
located at the upper point of the sphere
( A,
=
0,
A,
=
1);
as a result, if
p
>
f
the sheet
a
necessarily lies outside the sphere (see figure
2).
Furthermore two intersections
between the sphere and the hyperboloid can exist only if
p
<
1
(see table
1);
if these
intersections take place
on
the same sheet, this sheet is necessarily either the
b
or the
c
one.
For
p
<:
the extremity of
A
can sweep one among the three sheets
a
or
b
if
A,
<
25p2
(figure
l ( a) )
or
c
if
A,
>
25p2
(figure
1(
b) ).
There exists only one intersection
between the sphere and the sheet
a
or
b,
but two intersections appear for the sheet
c.
As
a result the extrema1
A,
values are of a different nature according to the sheet
involved; they correspond either to the top T,, and Tbz or to the intersections I,, and
12z.
Consequently the sheets
a,
b
and
c
are connected to completely different spatial
localisation and symmetry for the
A
motion. Furthermore the variation range for
A,
associated with each sheet is determined
on
the one hand from the double cone and
on
the other hand from the
A,
values corresponding to the sheets
a,
b
or
c
externally
3504
P
Cacciani
et
a1
la1
1/5<
p<U6
Figure 2. Typical portions of space swept by the A vector during its secular motion.
 
,
asymptotic cone with semiangle
Bo
and top
(A,,
=
0,
A,
=
5p).
This top is internal to the
sphere for
p
<
f
and external for
p
>
f.
The portion of space swept by the A vector is
indicated by the stippled area. The conditions
for
the existence
of
the different types of
motion are indicated in table 2.
tangent to the sphere at the points with ordinates
A,
equal to To,= 1, Tbz=l and
I,,
=
I*,
=
P,
respectively (equations
(8),
(1
1)).
More precisely
To,
=
1
Tbz
=
1
11,
=
I 2r
=
P
if A,
=
1
+
lop
if A,
=
1
lop
if
A,
=5P2+4.
For
,6
<$,
the extension domain for A, and the extrema1 values of A,, when the
tip of A sweeps one of the three sheets, are specified in table
2.
Table
2.
Analysis of the A motion as a function of
p
and A,. The sheet swept by the tip
of A is specified, as well as the nature
of
the points limiting the
A,
variation.
a,
b,
c:
different sheets of the hyperboloid A,
=
const (equation
(20)); T,,T,,,
:
tops
of
the sheets
a
and
b
(equation
(8));
I,z121:
ordinates of the intersections between the sphere and the
hyperboloid (equation
(11));
Ap
=
25p2:
equation of the asymptotic cone, along which the
tip
of
A
can move if
p
S
I/&.
Boundaries for
Sheet the
A, motion Class
AD
extension
T,,
5
A,
s
I,,
I1
I11
I
I ZZSAz S
Tbr
I
I,,
s
A,
s
I,,
I z Z s A z s
I,,
I11
pz
I/&
111
p51/&
I,,
5
A,
5
I,,
I,,
5
A,
5
Tbr
111
I,, S A,
S
Tbr
I
1 +lop
G
A p
<25p2
110p5Ap<25pz
25p2
<
A,
5
5p2+4
1

lop
S
A,
5

1
+
lop
1
+lop
S
A,
55p2+4
1
+lop
S A,
<
25p2
25P2<A, G5p2+4
1

lop
5
AB
5

1
+
lop
Hydrogen atom in the presence ofparallel magnetic and electric fields 3505
For
4
<
p
<
1,
the tip of A can sweep either the sheet
b
or
the sheet
c;
furthermore
there can exist either one
or
two intersections between the sphere and the hyperboloid.
When only one intersection is present, it necessarily lies
on
the sheet
b
(figures
2(
a )
and
( c ) )
and the top
Tb,
is then located inside the sphere.
I n
this case
A,
varies from
I,,
to
Tb,.
The limiting A, values, for which such a motion exists, are deduced from
the extreme values
Tbz
=
1
and
$1.
When there exist two intersections between the sphere and the hyperboloid,
A,
varies from 12z to
I,,
and A, varies throughout the interval
[lop

1,
5p2+4] (table
1).
These intersections belong to the sheet
b
(respectively sheet
c)
if
A,
25p2 is
negative (positive). Consequently the
A
motion can take place along the sheet
c
only
if 25p2 is smaller than 5p2+4, that is if
/3
<
1/8;
in this case the sphere intersects
twice either the sheet
b
for
A,
E
[lop

1,
25p2]
or the sheet
c
for A,
E
[ 25p2,
5p2+4]
(figure
2 ( a )
sheets
b2
and
c).
Let us emphasise that the portion of space swept by the
A
vector is very similar for both sheets; therefore the asymptotic cone A,
=25p2
no
longer constitutes a discontinuity in the properties of the states for
+<
p
<
1/&.
For
p
=
l/&
the asymptotic cone is tangent to the sphere and for
1/8
<
p
<
1
the two
intersections between the sphere and the hyperboloid necessarily belong to the sheet
b
(figure
2(b)
sheet
b2).
In conclusion, for
1/8 <
p
<
1
two different motions exist for the
A
vector; the
different behaviours are not associated with the sheet of the hyperboloid involved but
they are tightly connected to the number of intersections between the sphere and the
hyperboloid or equivalently to the boundary conditions limiting the A, variation. The
extremal A, values, the sheet swept by the tip of
A
and the extension for A, are
summarised in table 2.
For
p
>
1,
necessarily there only exists one single intersection between the sphere
and the hyperboloid (table
1)
and it is unavoidably located
on
the sheet
b
(figure
2(c)
sheet
bl).
Then the
A
motion is limited by this intersection
12,
and the top
Tbz.
The
corresponding range of A, is easily deduced and is presented in table
2.
2.2.3. Definition
of
the classes
of
states.
In the evolution of the RungeLenz vector,
the nature of the extremal points limiting the variation range for A, permits us to
distinguish three types of behaviours associated with three classes of states called I,
I1 and
111:
I
where
Tb
is the top of the sheet
b
and
I2
=
b
n
s
I1
where
T,
is the top of the sheet
a
and
Il
=
a n
s
I11
where
I,
and
I,
are the two intersections of the
sphere with the sheets b
or
c
of the hyperboloid.
The classes are specified in table 2.
It must be pointed out that this definition
of
the classes of states for hydrogen in
the presence of parallel electric and magnetic fields generalises the definitions intro
duced in I for the problem of pure diamagnetism. In this latter case
p
=
0,
and the
hyperboloid is symmetrical with respect to the
z
=
0
plane. The sheets
a
and
b
are
swept by the tip of
A
when
1
s
A s
0
and the ordinates of their tops are equal to
122
S
A,
C
Tb,
T,,
s
A,
s
I,,
12z
SA,
S
I,,
T,
=
T,,
=

Tb,
=
a.
These sheets are associated with classes
I
and
I1
which, due to the symmetry with
respect to the
z
=
0
plane, correspond to degenerate states. The sheet
c
is related to
the class
111,
which has an extension
0
<
A
S
+
4. The
z
projections of the intersections
3506 P
Cacciani
et
a1
between the sphere and the hyperboloid exist
if 1
s
AS
+
4
and satisfy
It
is noteworthy that these three classes of states are not unambiguously determined
by the sheets
a,
b,
c
swept by the extremity of
A
when
p
>
i:
for example a motion
along the sheet
b
corresponds to either class I or
111;
furthermore class
I11
is associated
with sheets
b
or
c.
The three classes are exclusively defined by the nature of the
extrema1 positions of the
A
vector during its slow evolution. Other authors (Braun
and Solov’ev
1984)
have given an equivalent definition
of
the classes in a completely
different approach.
It
is to be noticed that the number of classes involved in the manifold depends on
the
p
values:
O<pS$
the three classes exist
;S p < l
p 2 1
only class I remains.
class I1 is no longer present
The range of variation of
A,
depends on the
p
value and on the studied class;
according to equation
(4)
or (6) the energy extension of the classes is a function of
p.
The results obtained for
M
=
0 states are summarised in table 3. Two situations are
simultaneously presented; one of the fields
y
(respectively
f )
is kept constant while
the other one
f
( y )
increases.
Table
3.
Range of variation of A, for the different classes of
M
=
0
states as a function
of
p
(equation
( 3) ).
The energy shift is related to
A,
by
E,=
(7’n4/16)(1+ilP)
=
&f n2[ (
1
+A,
)/PI.
p
range Class
I
Class I1
Class
111
f sps1
1
lop
S
A,
S
 1
+l op
 l o p s l + ‘ l p s +l op
 10s
( 1
+
A,)/p
s
10
lop
S
1
+A,
s
+
l op
 l oS( l +Ap)/ps
10
p 3 1
1
lop
S A, S  l +
106
 1
+
lop
s
AB
s
4+
5/32
l op
s
1
+A,
s
S ( l
+p2)
1OC
(l +Ap)/p
S
5(l/p
+ p )
2.2.4.
Transition
from
one class into another one, barrier and quasibarrier.
Figure 3
presents the domains in the
(p,
Ap)
plane where the different classes exist. The sheet
a, b
or
c
swept by the extremity of
A
is also specified.
For
p
<+,
the lower bounds
of
Ap
for the classes I and I1 vary linearly with
p
which reflects the linear Stark effect.
In
the same
p
range, the limits for the class
I11
increase quadratically with
p
according to the quadratic Stark effect. These distinct
behaviours for the classes I and
I1
on the one hand, and for the class I11
on
the other
hand, arise from completely different symmetry properties. Indeed for the states
belonging
to
classes
I
and
11,
one boundary in the
A
motion corresponds to a position
Hydrogen atom in the presence ofparallel magnetic and electricjelds
3507
0
1/5 1/{5
1
P
Figure
3.
Domains
(p,
.Ip)
where the three classes of states
I,
I1
and
111
for a given
M
=
0
hydrogenic
n
manifold occur. For a fixed magnetic field
y
and for an increasing electric
field strength
f
parallel to the magnetic field, this diagram presents the energy extension
of the different classes. The energy shift of a state is related to the approximate constant
of
motion A, by
E,,=
 1/2n2+( y2n4/16) ( 1+Ap).
For each class, the sheet
a,
b or
c
of
the hyperboloid swept by the extremity of A is also specified.
either parallel
(8
=
0)
or
antiparallel
(8
=
7 ~ )
to the fields’ direction;
on
the contrary,
for class I11 states the
A
vector can never be directed along
Oz;
as a result classes
I
and
I1
possess a librational symmetry while class
111
corresponds to a rotational one
(Delos et
a1
1983,
Waterland et
a1
1987).
For
0
<
p
<
i,
the line
A,
=
25p2
representing the asymptotic cone separates the
( p,
A,) diagram into two regions associated with states of different symmetry. This
curve generalises to the problem of parallel
B
and
F
fields the separatrix A
=
0
which
plays a very important role in the problem of pure diamagnetism, especially for the
transition from regular to chaotic character for classical motion (Delande and Gay
1986).
For parallel
B
and
F
fields, this separatrix
A,
=
25p2
with
0
s
p
si
is responsible
for the restructuring of the energy spectrum with increasing
p,
as is discussed below
(0
2.4);
this separatrix is called the ‘potential barrier’ by Braun and Solov’ev
(1984).
When
<
p
<
1,
the curve A,
=
lop

1
divides the diagram
( p,
A,) into two zones
where either librational (class
I)
or
rotational (class
111)
states appear; this A, value
corresponds to the particular case where the b sheet is internally tangent to the sphere
(
Tbz
=
+1).
This limit called the ‘quasipotential barrier’ is less crucial than the previous
one, and introduces less striking discontinuities in the evolution of the energy states.
If
p
>
1,
only librational states typical
of
the linear Stark effect subsist.
2.3. BohrSommerfeld quantisation of
L,(8)
2.3.1.
General expression f or the quantisation integral. The existence of the three
integrals of motion
(Lz,
Eo,
A,) allows a formulation of the BohrSommerfeld rules.
3508
P
Cacciani
et
a1
The origin of the frame is located at the nucleus, i.e. a focus of the ellipse. The position
r
of the electron is defined by the three angular coordinates
4,
0
and
9.
The
A
vector
is determined by the first two coordinates.
e
is the angle between
B
and
A.
4
gives the direction of the projection of
A
onto the plane perpendicular to the B
direction.
The
CC,
angle defines the position of the electron on the ellipse with respect to the
A
vector. The conjugate momentum
Zd
is equal to L, and the action
5
Z$
d$ to
27r/(
2Eo)1/2.
For these two coordinates the quantisation rules give the obvious results:
L,
=
M
Eo
=

1/2n2.
It is possible to demonstrate that the generalised momentum conjugated to the
third coordinate
8
is the component L,
of
the angular momentum perpendicular to
the plane defined by the
z
axis and the RungeLenz vector
A.
It is expressed in terms
of the angle
0
and the modulus of the RungeLenz vector
A:
L,(
e )
=
[
n2(
1

A2)

M2/sin2
e]1/2.
(12)
The modulus
A
satisfies the following equation, deduced from the expression of
Ap
(equation
(7))
(45
cos2
B)A2+10PA
COS
8 A,
= O.
(13a)
This quadratic equation can have two roots:
5p
COS
e
*
[25p2
cos2
e
+
h,(4

5
cos2
o)]1/2
45
cos2
0
A,
=

A,

5p
COS
e
*
[25p2
cos2
e
+
~,( 4

5
COS’
e ) p 2
which represent the modulus of the
A
vector only if
OsA,s
1.
(13b)
Equation
(13a)
defines the hyperboloid and the condition
(13b)
restricts this surface
to its part internal to the sphere of unit radius. Thus, the existence of the solutions
A,
is linked to the relative position of the sphere and the hyperboloid studied in
9
2.2:
our interest is not only to determine the range of
A,
value which defines the type of
motion but also to calculate the dependence of the modulus of
A
on the
0
angle at
fixed values of
(Ap,
p).
Let us remark that
A(
e )
is not necessarily a uniform function
of
0:
for some
0
values, two roots
A,
can exist. In relation to the equation
( 11),
the
slow motion of the extremity of
A
on one sheet of the hyperboloid defined by
( Ap,
p )
is unambiguously described by a trajectory C‘ in the phase space
(e,
15,).
The quantisation condition on the contour
C’
by restricting to the
L,zO
plane
and with an algebraic definition of the integral can be expressed:
where
O1
and
O2
are the
8
turning points where L, vanishes and dL,/de becomes infinite.
Hydrogen atom in the presence ofparallel magnetic and electricjelds
3509
For
M
=
0,
the
6
turning points correspond to
A2(
8 )
=
1: they are defined by the
intersections between the sphere and the hyperboloid (equation ( 1 1) ). Consequently
COS
e,
=
ii,
i
=
1
or
2.
(14)
In this case and for some trajectories
el
or
0,
can be equal to
0
or
7~
which
corresponds to the tops
T,
or
Tb
of the sheets
a
or b. These limiting
6
values are no
longer socalled
6
turning points
( L,
#
0)
but the topological nature of the trajectory
is such that we can keep the same quantisation condition.
For
M
=
0
the quantisation condition becomes
where the left integral part does not depend on the
n
value any more.
Let us remark that depending on the
p,
A,
values either the root
A+
or the root
A or both are to be introduced in the equation ( 15). For a chosen eigenstate,
8
varies
from
8,
to
e2
as the extremity of
A
sweeps progressively one sheet of the hyperboloid;
meanwhile
A,
varies in the range indicated in table
2.
However the
6
variation is not
necessarily a monotonic one. A nonmonotonic variation is observed when the cone
of semiangle
@T
centred at the origin is tangential to the hyperboloid, with the tangent
point lying inside the sphere of radius one. This
6T
angle defined by
lies outside the interval
[e,,
e,]
and in the algebraic integral (equation
( 15) )
0
varies
from
el
to with one root (ex:
A+)
and positive contribution, then from
67.
to
O2
with
the other root (ex: A ) with negative contribution.
To
illustrate the method, we present in figure
4
some typical examples of the
( 8,
L,(
e))
curves with their corresponding hyperboloids; in this example
p
=
0.16
and
six different values of
A,
are chosen (the three classes of states do exist).
0
x
e
l
Figure
4.
Typical motion
of
the
A
vector corresponding
to
p
=
0.16. Six demonstrative
values of A, have been retained and correspond
to
states belonging to the classes
I,
I1
or
111.
For each A, value, we show:
( a )
the trajectory in the phase space plane
(0,
L,(B)),
B
is the
( E,
A )
angle;
( b )
the part of the hyperboloid swept by the tip
of
the
A
vector:
class
I
states, curves
1,
2 or
3;
class
I1
states, curve
4;
class
I11
states, curves
5
or
6.
For
the trajectory
5,
the cone with the top located at the origin and with the semiangle
0,
is
tangential to the sheet c
of
the hyperboloid
( b).
Then
( l/n ) L,
is not a singlevalued
function
of
@( a ).
3510
P
Cacciani et
a1
The sense of motion
on
the closed contours is chosen to give positive integrals
corresponding to the area
S
in the
(e,
L,) plane. It is noteworthy that each type of
state I,
I1
or
111 is represented by a different kind of area. Then for each
p
value, the
A,
dependence of the areas
S,(p,
A,),
S,,(p,
A,)
and
SI,,(p,
AP)
are calculated
in
the
corresponding
A,
ranges (see table 2).
The semiclassical quantisation leads to solving the equations
SI@,
A,,k,)
=
(kI+t)T/H
kl
E
LO,
NI

11
kII
E
[O,
NI,

11
Sd P,
A,,k,J
=
(k1,+t).ir/n
(17)
SIII(P,
A,,,,,,)
=
(k11I+t)T/n
kill
E
[ O,
Niii

11
where
Ap,k,orkl l ork,,l
are the quantised values and
NI, NI,, NI,,
the number of states
of each type
( NI
+
NI,
+
N,,,
=
n).
The equations are solved by using an iterative process to determine the
AP,k
values.
For
p
s:
and some
n,
M,
p
values, the quantisation technique can break down
near the barrier
A,
=
25p2
and gives
n

lMl* 1 levels: a level can be created or
forgotten. However this feature appears very close to the barrier and in our drawings
to keep the right number of levels ( n

111.11)
we omit or add a level at the barrier. Such
a fact does not appear in a quantum mechanical treatment where all the states are
always obtained and must be continuously followed when
p
varies (Waterland
et a1
1987). It can be noticed that the semiclassical technique gives a significant partition
of the states between the three types by calculating the noninteger values
NI,
NI,
and
NITI,
but is unable to take into account quantum mechanical tunnelling between the
different types.
In
relation to the experimental results (Cacciani
et a1
1986b) the case
n
=
29, M
=
0
has been chosen and is illustrated in figures
5( a)
and
5 ( b ).
As
the critical values of
p
are 0, and 1, eight demonstrative
p
values have been retained. For each
p
value
the hyperboloids and the
(e,
L,) contours have been drawn for the 29 eigenstates.
2.3.2.
Explicit expression for the integrals S(A,) for
M
=
Ostates.
The explicit expression
of the quantisation integral
S
(equation (15)) depends
on
the class of states studied
and
on
the
p
and
A,
values. Details are given below.
2.3.2.(a) Class
I. When
O s
AP
6
25p2 and
p
s
4
or
OG
A,
s
lop

1 and
p
a:,
the
trajectory in the phase space
( 8,
L,)
is similar to the curve 3 in figure
4.
The quantisation
integral is:
S,(A,)
=
T

loo2
(1

de
with
A+(
e,)
=
1.
For the maximal
A,
value, independently of
p,
the integral
on
the righthand side
is minimal, therefore
SI(A,)
is maximal and consequently
k,
is maximal when
A,
is
as large as possible. For
p
>
1 and
A,
=
lop

1, the limiting trajectory reduces to the
point
( 8
=
0, L,
=
0) and the maximum value for the quantisation integral is equal to
T;
in this case the top Tb is at the highest point of the sphere
(
Tbz
=
+l).
When
 1

lop
6
A,
6
0 two situations are possible depending
on
the location, with respect
to the sphere of unit radius, of the circle defined by the cone of semiangle
OT
(equation
(16)).
Hydrogen atom
in
the presence ofparallel magnetic and electric$elds 351
1
Figure
5.
( a )
Hyperboloids representing the slow variations of the tip of the RungeLenz
vector, for all the states of the
n
=
29,
M
=
0,manifold of the hydrogen atom in the presence
of weak parallel electric and magnetic fields, with either relative strength (equation
( 3) )
chosen in the
/3
interval
[0,5].
( b )
Corresponding contours
C
in the section
[
0,
(l/n)Ll (
e ) ]
of
the phase space.
,
states belonging to class
I,
.
.
', states belonging to class
11;
_ _ 
,
states belonging to class
111.
For
p
=
0
a level belonging to class
111
exists
just
at the
barrier.
If
the circle is internal to the sphere the trajectory
(0,
L,)
is represented by the
curve
2
and
s,(A,)
=
J w
( 1
 A!) ~/~
do

(1
A:)'/'
do
*T
J*:
with
8 ~ <
e,<
7T
A+(&)
=A ( OT)
and
A+( 02) =1.
If
the circle is external to the sphere the trajectory is represented by the curve
1
and
s,( A~ )
=
J
71
(1

A?)"2
d0
81
with
A ( @,)
=
1.
For the smallest
AB
value
A,
=
1

lop,
the trajectory in the phase
space reduces to the point
( 0
=
n,
L,
=
0),
which corresponds to the top Tb located at
the lowest point
of
the sphere
(
Tbz
=
1).
Then
SI(l

lop)
=
0 and
k,
=
0
is associated
with the lowest
AD
value.
3512
P
Cacciani
et
a1
2.3.2(b) Class
II.
This class appears
for
p
S
5
and
lop

1
S
A,
6
25p2.
The trajectory
in the phase space is presented by the curve
4
and
Sll(Ap)
=
jOoi
(1
A?)’” d#
with
A(
e,)
=
1.
For
A@
=
lop

1,
the trajectory reduces to the point
( 0,
=
0, L,
=
0),
the top of the
hyperboloid
To
lying at the highest point of the sphere. Then
SI,
takes its smallest
value;
k,,
=
0
corresponds to the minimum of the quantised Ap value.
For
A,
=
25p2,
the integral is maximal and therefore the maximum value of
k,,
is
associated with the largest value of A,.
2.3.2(c)
Class
III.
This class exists when
2 5 p 2 S A,
s
5p2+4
for
,L?
S i
or when
lop

1
==A,
s
5 p 2 + 4
for
p
ai.
The trajectory is of the type
5
when the cone of semiangle
eT
is tangential to the
sheet
c
of the hyperboloid, the tangent circle being located inside the sphere. Then
with
eT
<
el
<
e,,
A+(
6,)
=
A(
0,)
and
A+(
0,)
=
A(
e,)
=
1.
When there is no circle
inside the sphere, the trajectory in the phase space is of the type
6
and
with
8,
<
O2
and
A+(
e,)
=
A+(
e,)
=
1.
For
the smallest A@ value, independently
of
the
p
value, the integral
SI,,
is maximal;
therefore the maximum
of
k,,,
corresponds to the minimum
of
the quantised A, value.
For
the largest A, value, the hyperboloid is exteriorly tangential to the sphere
8,
=
&=
and the trajectory in the phase space reduces
to
the point
(e,,
L,(
e,))
=
0.
The
value
k,,,
=
0
is associated with the maximum
of
the A, value.
2.3.3.
Number of
states
in
each
class.
As an immediate consequence of these expressions
for the quantisation integral, the number of states in each class can be calculated
analytically. Different situations are to be analysed successively depending on the
p
values.
2.3.3(a)
p
G;.
The value
A,
=
25p2
corresponds to the maximal values for the three
integrals
SI, SI,, SI,,.
It is associated with the cone
of
top ( A,
=
0,
A,
=
5p)
and
semiangle
Bo.
The analytical calculation of
S,(25p2), SI1(25p2),
and
S111(25p2)
leads
to the results:
~ ~ ( p )
=
~,( 2 5 p ~ ) n/~ = f n ( l
+&p) 
~,,,/2
~,,( p )
=
~ ~ ~ ( 2 5 ~ * ) n/‘ r r = f n ( l  J 3 p )

~,,,/2.
Let us remark that
NI@)
+
N,,(p)
+
NIII(P)
=
n. NI,,
decreases with increasing
p;
this
decrease is equally shared out amongst increasing contributions to
NI
and NI,.
(18)
Hydrogen atom in the presence ofparallel magnetic and electricjields
35 13
However,
N,,( p)
always decreases with increasing
p.
2.3.3(b)
4
S P
=s
1.
The class
I1
no longer exists; the limit value
A,
=
lop

1 is associ
ated with the maximal value for the integral
&(A,)
and
SI I I ( Ap).
For
A,
=
lop

1,
the sheet
b
of the hyperboloid is interiorly tangential to the sphere, its top
Tb
being
located at the highest point of the sphere
(
Tbz
=
+l ).
For
A,
>
lop

1 two intersections
exist corresponding to states of class 111. The analytical evaluation of
SI(
lop

1) and
SIII(lOp

1) leads to
n n
[
(
3 s p )fi pt an'()] 1 3
[
tan'
(

35p )~ptan1()] 1 3
NIll(p)
=Sl l I(l +l Op)=(l Ap)+
tan.'

(19)
7 r 2
ll
2P
2
NI(p)=Sl(11Op)=(l+fip)
n n
1 1 2
7r
2P
2
with
NI@)
+
N,,,( p)
=
n.
2.3.3(c)
p
2
1.
In such a case only states
of
the class
I
exist and
NI
=
n.
2.4.
Evolution with increasing
F of
the structure
of
the diamagnetic manifold. Evolution
with increasing
B
of
the structure
of
the Stark multiplet (BIIF)
The quantised values
Ap,k
have been calculated for all states
of
the n
=
30,
M
= O
manifold and for different
p
values. These results are valid in the weakfield limit,
where the diamagnetic plus Stark interaction can be treated to first order of the
perturbation theory coupling states belonging to the same n manifold.
All these values have been compared to those obtained by a diagonalisation process
using a spherical basis at vanishing fields. The agreement is quite good ( ~ 1 % ) but
quantal effects like the splitting of the low doublets when
p
=
0,
or the exact position
of states near the barrier A,
=
25p2,
cannot be explained with our semiclassical model.
Furthermore, as discussed above, there
is
sometimes a 'missing'
or
'added' state in the
semiclassical treatment.
2.4.1.
B
=constant;
F
increasing. The restructuring of the diamagnetic manifold
( n,
M, B
fixed) under the influence of an increasing electric field parallel to the magnetic
field is presented in figure
6.
The evolution of A, as a function of
p
is presented. In
figure
6 ( a )
the studied
p
range
( p
Si )
corresponds to the situation where the diamag
netic effect is predominant compared with the linear Stark effect. Figure
6( b)
concerns
higher
p
values. On these diagrams the critical values of
p0,
f
and lare reported
and the limiting curves describing the A, extension for the classes are drawn. These
figures summarise the different features mentioned above and previously discussed by
Braun and Solov'ev (1984).
(i)
The condition for the existence of the three classes of states and their energy
extension (table 2 ).
(ii) Twofold degeneracy of the lower part of the diamagnetic manifold
( p
=0)
giving rise to a linear Stark effect as soon as
p
#
0.
The degenerate parts of states split
up into class I and class I1 states with a negative and a positive slope, respectively.
(iii) Crossing between states belonging to the classes
I
and 11, for
p
si.
It is to
be emphasised that all the involved pairs of states undergo a crossing at the same
pK
values, these
p K
values being equally spaced. An analytical demonstration of this
remarkable feature is presented below
(§
2.5);
it is deduced from the explicit expression
of the semiclassical quantisation integrals.
3514
P
Cacciani et a1
115
\
P P
Figure
6.
Evolution
( Ap
against
p )
with increasing electric field
F
of
the structure
of
the
diamagnetic manifold
n
=
30,
M
=
0
and
B
=constant
for
the hydrogen atom
(BIIF).
The
energy shifts of the states is equal to (equations
(3)
and
(4))
E?,
=
(y2n4/16)(1
+Ap) with
p
=yf/n2y2.
y
is kept constant while
f
increases.
( a )
p
<&,
( b )
O <
p
<2.
The
three
classes
of
states
I,
I1
and
111
appear in the domains limited by the heavy curves defined
in table
3.
(iv) Progressive transformation, as
p
increases, of the states of class
I1
into states
of class
I.
When the
Ap
value for a class
I1
state reaches the potential barrier
Ap
=
25p2
this state reflects on it and becomes a class
I
state.
(v) Weak quadratic Stark effect for the nondegenerate class
I11
states in the upper
part of the diamagnetic manifold. When they reach the potential barrier for
p
S4
or
the potential quasibarrier for
<
p
<
1,
these class
I11
states transform one by one into
class
I
states.
(vi) For very large
p,
the
Ap
values become nearly equally spaced according to
the linear Stark effect.
2.4.2.
F=
constant;
B
increasing. The modifications appearing in the structure of the
Stark multiplet (n,
M,
F
fixed) under the influence of an increasing magnetic field
parallel to the electric field are reported in figure
7.
The variation of
(1
+
Ap)/p
as a
function of
1/p
is plotted; the
p
range
( p
>
4)
is presented in more detail in this figure.
For
1/p
=
0,
the equally spaced
(1
+
A p )/p values associated with
E,
varying in the
range
[
f
fn', +ffn'] are typical of the linear Stark effect. The three classes of states
can be observed. Depending on
1/p,
the conditions for the existence
of
the different
classes, and their corresponding energy extension, satisfy the relations reported in
table
3.
With increasing l/p, the transformation of the states from one class to another
one is manifest; lastly for
1/p
>
5
crossings between classes
I
and
I1
states are also
observable.
2.5.
Analytical analysis
of
dS/dAp
2.5.1.
Collective crossing between class
I
and class
11
states for
p < f.
States of the
classes
I
and
I1
coexist in the
(p,
Ap )
domain defined by
p
d
$
and
1

lop
d
Ap
d
25p2.
Hydrogen atom in the presence of parallel magnetic and electricjelds
35
15
Figure 7. Evolution
( ( 1
+Ap)/p
against
1/p)
with increasing magnetic field
B
of the
structure of the Stark multiplet
n
=
30,
M
=
0
and
F
=
constant for the hydrogen atom
(BIIF).
The energy shift of the states is equal to (equation
(6))
E,,, =&f nz( l +Ap)/p
with
l/p
=&y 2 n 2/j
f
is
kept constant while
y
increases. The three classes of states
I,
I1
and
I11 appear in the domains limited by the heavy curves defined in table
3.
Furthermore it has been shown that these states exhibit crossings which appear at the
same
/3
value (Braun and Solov'ev
1984).
To explain this phenomenon we demonstrate
below that in this domain, independently to the
p
value, the two integrals
SI
and
SI,
calculated for the same A, value verify
SI ( A~)  SI I ( A,)
=JsPr.
(20)
A P K n
=
K K
integer
(21)
Consequently for
P K
values such that
each state
k,,
of the class 11, associated with the
A,,
verifying the quantisation integral
(17),
can be associated with the state
k,= k,,+
K
of the class
I
with the same ApK,
which satisfies also the quantisation integral:
for each
kII
A6K.kii
=
'PK.ki
This demonstration is based on the analytical calculation of
dS,(A,)/dA,
and
dS,,(Ap)/aA,
in terms of complete elliptic integrals of the first kind.
As
a result we have proved that for the same A, value
Furthermore for A,
=25p2,
the two integrals
S,(25pz)
and
S,,(25p2)
differ by the
quantity
f i p n
(equation
(18)).
Consequently the relation
(20)
is valid independently
of the A, value.
In the range
p
si
there exist
n/f i
values
PK
with the equal spacing
l/n&,
for
which states of the classes
I
and
I1
are degenerate by pairs.
3516
P
Cacciani et
a1
2.5.2.
Approximate analytical expression f or the energy
of
the outermc:st states in each
class.
The analytical expressions of
aS(R,)/ah,
also allow us
to
calculate easily the
development of
S
around the extreme value of
A,.
For example
This leads to an approximate formula for the lowest states of group
I:
The lowest states of group
I1
are given by
On the other hand the upper states of group
TI1
are given by
These formulae
(23)
and
(25)
remain valid for
4s
p
s
1.
For
p
2
1
the
111
states
do not exist any longer and the upper states which belong to the group I are given by
4J5[(1p)(l5p)]”’
It is noteworthy that these formulae have already been obtained by Braun and
Solov’ev
(1984)
using a completely different method.
The limiting values
A,
=25p2
for
p=z+
and
A,
=+IOpl
for
4 s b s l
called
respectively barrier and quasibarrier by Braun and Solov’ev
(1984)
have the following
properties:
Thus when we use the semiclassical quantisation, the density
of
states in the discrete
spectrum equal to dk/dA,
=
(n/.rr)
dS/dA, becomes infinite near the barrier
or
the
quasibarrier.
Hydrogen atom in the presence ofparallel magnetic and electricjelds 3517
3.
Analogy in the structure
of
the pure diamagnetic multiplet
M
#
0,
p
=
0 and of the
multiplet M = O with parallel fields and
pM
=
IMI/&
For the case M
#
0,
it is possible to calculate in the same way the quantised values
Ap,k by using the expression (12) for
L,(
e )
fc(,pl
dB =2 ( k +f ) r
where
A
always satisfies the relation (13a). The probem is no longer dependent on
the single parameter ( k+$)/n, but depends also on lMl/n. The behaviour of these
states is qualitatively the same as for M
=
0.
Let us remark that for l Ml 3 n/f i at
p
=
0
only the class I11 exists and with
increasing
p
these class I11 states change into class I states; the class I1 states do not
exist whatever
p
is.
This last paragraph is a return to the problem of pure diamagnetism with M
#
0.
Its purpose is to compare the two situations:
( a ) pure diamagnetism
( p
=
0)
with M
#
0;
( b ) diamagnetism with M
=
0
and additional parallel electric field
p
#
0.
In the case ( a ) the limiting value is A
=
0
and the number of states in each class is:
This comparison begins with the calculation of the number of states in each class.
with sin
8,
=
/M//n and
0
s
8,
s 0,
for IM/
2
n/f i
NI,
=
NI l a
=
0
and
NI I l a
=
n IM12Nl I,
=
n  l Ml  2Nl a
In the ( b ) case when
p
S i
the limiting value is the 5pshifted cone defined by
Ap
=
25p'
and the number of states is given explicitly by the integrals S(25p2) (equation
( 18) ):
with sin(
eo

e,)
=
f i p
and
0
s
el
s
e,
with sin(
6,)
=
Jsp
and
0s
Ob
s
eo
Nib
=
N11b
+&pn
(see equation
(18))
3518
P
Cacciani
et
a1
It is clear that for the value
P M
=
I WJ S n
we have
ob
=
60
NI l b
=
Nl l a
N l I l b
=
n
 I M I
2NIIb
=
Nl I I a
=
Nl l b
+
Thus the number of states in the case
l M(
S
n/8 can be deduced from the formula (18)
)
/MI
2lMl

2n
NIIIa
=
2
(tan‘
lr
( n 2  5 ~ ” ” * n
(
n2

5 ~ ~ ) ” ~
NI,
=
n

IMI

NIIla.
Let us emphasise that this
p M
value is identical to
p K
corresponding to crossings
between states I and I1 (equation (21)).
This comparison can be extended to the analytical calculations of the integrals
dS,,,(A)/dA
and
dSl l l a ( h)/dA
in terms of elliptic integrals. By substituting
M
+
Ap n,
we can demonstrate that for
p
=
p M
The equation (27a) is valid for all typeI1 motion, that is for states with
0 s
(MI
S
n/8
in the case
( a )
and for
p
<
i
in the case (6).
The equation (27b) is only demonstrated for integrals corresponding to the motion
of A along the sheet c of the hyperboloid. Consequently in the case
( a),
this equation
concerns all the states
of
the class I11 with
OS
IMIS
n

1; in the case ( b) only the
ranges
p
s
1/A
and 25p2< A,
s
5 p 2 +4 are concerned (see table 2); as a result for
i <
p
s
1/&
only the more excited states of type I11 are
to
be considered. Then taking
into account the equalities
one obtains
The more striking result is that the quantisation integrals (equation (17)) lead to
constants
of
the motion linked by the following relation:
AM,k=
ApM.k25Pb
or
i l ~,k+5l M1~/n~=Ap,,k
Hydrogen atom in the presence ofparallel magnetic and electric$elds
35
19
where only the classes of states I1 and I11 are to be considered. In the pure diamagnetic
multiplet, states of the classes I and I1 are degenerate.
The extensions in energy for the classes I1 and I11 for the diamagnetic multiplet
with
M
#
0
are easily deduced from the corresponding ones in the multiplet
M
=
0
in
the presence of parallel fields with
/3
=
p M.
In case ( b ), the class I1 exists only when
p
< f,
therefore in case
( a )
it appears
only when
\MI
=
8 n p,
<
n/8.
In case ( b ), its extension is given by
1
 l o p
GAP
<25p2
which gives the variation of
AM,k=AP,,k 25Ph
for states of classes I and I1 in the
case
( a ):
(1
5pM) 2s A M
0
or
(1
 &l MI/n) *s
AM
e
0.
Similarly the extension of class I11 associated with a motion of the tip of A along the
c
sheet is given by:
25p2<APS4+5p2.
This transforms into
0
s
A M
42OpL
or
0
6
AM
c
4(
1

lMI’/n2).
The expansions of the A values for the states located at the bottom and at the top
of the manifold are deduced from
(24)
and
(25)
k=0,1,2,.
. .
I MI s n/8
AM,k s = 4
(
1  7
y2)
+
(

kln+;)
4
(
5  7
:’ )‘ I 2
+ l 2 ( Y )
i(
5n’M’)
k’+f * 3 15n2+M2
k’ =0, 1,2,.
,
. .
3520
P
Cacciani
et
a1
Disregarding the paramagnetic term, the energy shift is given for cases
( a )
and
( b )
respectively by:
y2n2
16
E,( M)
=

[
n'(
1
+
hM,k)
+
M ~ ]
16
It is quite remarkable that the spectrum of hydrogen for M
=
0 states, perturbed by
the diamagnetic interaction and the parallel electric field potential satisfying
p M
=
IMl/&n,
is directly related to the pure diamagnetic multiplet with M
#
0. Notice that
the dimensions of the subspaces are not the same:
n

IMI
in the case
( a ),
n
in the
case
( b).
For
p M
S
$
the lMl=
&np,
lowest states in the class I observed in case
( b )
correspond to states which are not degenerate with a state of class
11.
For
G
p M
6
l/&,
the IMl lowest states correspond to class I states or to class I11 states with the motion
of
A
along the sheet
b
of the hyperboloid; only the
( n

IM1)III states belonging to
the sheet
c
of the hyperboloid must be compared.
0
12 n/{5
n
1
M=&(3n
Figure 8.
Comparison between the constants of motion AD and AM for the two situations:
(i)
( n,
M
=
0)
manifold of hydrogen in the presence of weak parallel
B
and
F
fields
( p
#
0);
(i i )
( n,
M
ZO)
manifold of hydrogen in the presence
of
a weak magnetic field. The lines
present the evolution
of
the constant of motion for all the
n
states of the
n
=
30,
M
=
0
as a function of
p.
The limiting values
p
=
0,
f
and
1/&
are shown on this diagram.
For
each
pM
value such that
pu
=
l Ml/f i
with
( OS
lMI
n

l),
the
( n

IMI)
asterisks
present the values of
+
5 M2/n 2 )
which are related to the constants
of
the motion in
the
( n
=
30,
M )
pure diamagnetic manifold.
Hydrogen atom in the presence ofparallel magnetic and electric$elds
3521
As
an example we have plotted in figure
8
the diagram for
n
=
30,
M
=
0
as
a function of
p
in the range [ 0,1/8]; at each
PM
value defined by
p M
=
[ Ml/8 n
the
n
1M
stars represent the values of
AM,k+5M2/n2
of the pure diamagnetic
multiplet with azimuthal number
M.
4.
Conclusion
The problem of the hydrogen atom perturbed by weak parallel electric and magnetic
fields has been studied from the semiclassical point of view.
In
the ‘inter1 mixing’
regime, the structure of a given
(n,
M )
manifold is entirely determined by the approxi
mate constant of motion Ap, which is expressed in terms of the components of the
RungeLenz vector A. The analysis of the slow evolution of this vector permits us to
define unambiguously three classes of states according to the nature of the extrema1
positions of the
A
vector. The quantisation of the classical conjugate action variables
( 8,
L,)
allows one to determine the A, values, i.e. the energy of the states, for different
values of the parameter
p
representing the relative strength of the linear Stark effect
with respect
1:o
the diamagnetic interaction. As
p
increases, one observes the restructur
ing of the energy spectrum from pure diamagnetic structure to Stark linear structure.
The potential barrier (Ap
=
25p2
for
O<
p
<+)
and the quasibarrier (A,
=
1

lop
for
f <
p
<
1)
are shown to take a prominent part in this reorganisation. Analytical
computation of the integrals expressing the quantisation rules allow us to demonstrate
that each class
I1
state crosses a class
I
state at the same
p
value. Furthermore this
analytical analysis allows one to establish that there exists a tight connection between
the spectra observed in two different experimental conditions: the structure of the pure
( p
=
0)
diamagnetic manifold for
M
f
0
states can be obtained from the energies of
the
n

11111
more excited states or doublets observed in the presence of parallel
B
and
F
fields satisfying
PM
=
/Ml/n f i.
More thorough investigations are to be pursued in
order to explain more physically this striking result.
In the problem of parallel fields, the experimental evidence for the existence and
for the behaviour of the three classes of states is confirmed by a highresolution laser
spectroscopy experiment performed on an atomic beam of lithium. This experiment
is an extension of the previous one
( I )
dealing with odd Rydberg states of lithium
submitted to only an external magnetic field. The hydrogenic model we have developed
here gives the frame for the interpretation of the results which are presented in the
third paper of this series (Cacciani
et
a1
1988b).
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