9.8 Diamagnetic Hydrogen
335
where
l β
is the sum over all repetitions of the βth orbit. In scaled coordinates, the
Green’s function (propagator) can be written
Im[[r| ˆ
G + ((, w)|r] =
β
l β
A β,l β (() sin
l β S β (()w −
π
2
n β
(9.167)
and involves the sum over all closed orbits. Some of the most important closed orbits
are shown in Fig. 9.7. The absorption spectrum obtained by summing over the 70
most important closed orbits (Du and Delos 1988a) gives the thin line in Fig. 9.6.
A more useful plot can be obtained by taking the Fourier transform of the w
dependence of the experimental data. The experimental data were obtained for 24
different scaled energies, with magnetic field varying between 6.0 T and 2.7 T. The
result is the plot of the absorption spectrum as a function of scaled energy and
scaled action shown in Fig. 9.8. One might think of Fig. 9.8 as a three-dimensional
plot with scaled energy and scaled action in the x − y plane and the peaks, showing
the Fourier amplitudes of the absorption cross section, along the z-axis. It is possible
to correlate the peaks in Fig. 9.8 to the various closed orbits in Fig. 9.7. We list some
of the main features below:
(a) The circled “1” indicates an almost vertical line, which is a plot of the scaled
energy versus scaled action of the periodic orbit that lies perpendicular to the
z-axis (perpendicular to the magnetic field). This was the orbit that Edmonds
showed gives the Garton–Tomkins oscillations. The three other almost parallel
vertical lines are plots of the scaled energy versus scaled action of the first three
repetitions of this periodic orbit. This orbit is not shown in Fig. 9.7 but is simply
a straight line along the ρ-direction. The peaks labeled with a circled “4” and
“5” are due to repetitions of this perpendicular periodic orbit. Those particular
repetitions have interference effects that reinforce them.
(b) The circled “2” indicates a curved line which is a plot of the scaled energy versus
scaled action of the periodic orbit which lies parallel to the z-axis (parallel to the
Fig. 9.7 Some of the
dominant unstable closed
orbits. Note that the two most
important orbits, a straight
line along the ρ-axis
(perpendicular to B) and a
straight line along the z-axis
(parallel to B), are not shown.
Reproduced from Main et al.
(1994) with permission of the
authors
335
where
l β
is the sum over all repetitions of the βth orbit. In scaled coordinates, the
Green’s function (propagator) can be written
Im[[r| ˆ
G + ((, w)|r] =
β
l β
A β,l β (() sin
l β S β (()w −
π
2
n β
(9.167)
and involves the sum over all closed orbits. Some of the most important closed orbits
are shown in Fig. 9.7. The absorption spectrum obtained by summing over the 70
most important closed orbits (Du and Delos 1988a) gives the thin line in Fig. 9.6.
A more useful plot can be obtained by taking the Fourier transform of the w
dependence of the experimental data. The experimental data were obtained for 24
different scaled energies, with magnetic field varying between 6.0 T and 2.7 T. The
result is the plot of the absorption spectrum as a function of scaled energy and
scaled action shown in Fig. 9.8. One might think of Fig. 9.8 as a three-dimensional
plot with scaled energy and scaled action in the x − y plane and the peaks, showing
the Fourier amplitudes of the absorption cross section, along the z-axis. It is possible
to correlate the peaks in Fig. 9.8 to the various closed orbits in Fig. 9.7. We list some
of the main features below:
(a) The circled “1” indicates an almost vertical line, which is a plot of the scaled
energy versus scaled action of the periodic orbit that lies perpendicular to the
z-axis (perpendicular to the magnetic field). This was the orbit that Edmonds
showed gives the Garton–Tomkins oscillations. The three other almost parallel
vertical lines are plots of the scaled energy versus scaled action of the first three
repetitions of this periodic orbit. This orbit is not shown in Fig. 9.7 but is simply
a straight line along the ρ-direction. The peaks labeled with a circled “4” and
“5” are due to repetitions of this perpendicular periodic orbit. Those particular
repetitions have interference effects that reinforce them.
(b) The circled “2” indicates a curved line which is a plot of the scaled energy versus
scaled action of the periodic orbit which lies parallel to the z-axis (parallel to the
Fig. 9.7 Some of the
dominant unstable closed
orbits. Note that the two most
important orbits, a straight
line along the ρ-axis
(perpendicular to B) and a
straight line along the z-axis
(parallel to B), are not shown.
Reproduced from Main et al.
(1994) with permission of the
authors