Interaction of resonance radiation with atomic beams
by Steven Charles Seitel
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Physics
Montana State University
© Copyright by Steven Charles Seitel (1972)
Abstract:
A dual atomic beam device for investigating the resonance transitions in the rare gases is described.
The device is useful wherever low beam densities or windowless light paths are desired. A simple
model is developed for the frequency distribution of a resonance line excited by electron bombardment
in an atomic beam light source. The model is used to interpret the observed absorption and scattering of
the 1048 å (3Pi) and 1067. å (3Pi) argon resonance lines. The ratio of the oscillator strengths of these
lines is measured by a new method with the result f(1048)/f(1067) = 3.99+0.55 in agreement with
values given in the literature, The electron impact excitation functions for these lines are observed for
the first time and compared to a recent theoretical treatment. INTERACTION OF RESONANCE RADIATION WITH ATOMIC BEAMS
by
STEVEN CHARLES SEITEL
A thesis submitted to the Graduate Faculty in partial
fulfillment of the requirements, for the degree
of
DOCTOR OF PHILOSOPHY
in
Physics
Approved:
Head, M^gof Department
'Cf /I J dto.d&i’
//J/Xscntf
Chairman, Examining Committee
MONTANA STATE UNIVERSITY
Bozeman, Montana
March, 1972
iii
ACKNOWLEDGEMENTS
I wish to thank Professor David K. Anderson for
suggesting this problem, and for his advice and encourage­
ment.
I am indebted to Mr. Cecil Badgley and to Mr. Fred
Blankenburg for valuable technical assistance.
Finally, I
wish to thank my charming and capable wife, Delores, for
typing this manuscript.
TABLE OF CONTENTS
Page
LIST OF TABLES ...................... ..
LIST OF FIGURES
vi
............................... ..
vii
Chapter
I.
II.
III.
INTRODUCTION ....................
I
THEORY . . . . . . . . . . . . . . . . . . . .
5
EXPERIMENTAL APPARATUS
LIGHT SOURCE
. . . . . ' ..........
.
.................................
12
...............................
15
MONOCHROMATOR
SCATTERING BEAM
..................
15
VACUUM S Y S T E M ..........
PRESSURE GAUGES
PROCEDURE
.........
.............
. .
. . . . . . . . . . . . . . . . . .
PRELIMINARY EXPERIMENTS
V.
16
. . . . . . . . .
OPERATING CHARACTERISTICS
IV.
12
16
17
18
. . . . . . . . . .
18
EXCITATION FUNCTIONS . . . . . . . . . . . .
18
LIGHT SOURCE INTENSITY . . . . . . . . . . .
21
ABSORBED INTENSITY . . . . . . . . . . . . .
25
SCATTERED INTENSITY
........................
27
DISCUSSION . . . . . . . . . . . . . . . . . . .
30
SPECTRAL SCANS
......................
EXCITATION FUNCTIONS ........................
30
30
V
LIGHT SOURCE INTENSITY ......................
32
ABLS LINESHAPE ...........
. . . . . . . . .
37
ABSORBED INTENSITY ..........................
37
SCATTERED INTENSITY
. . . . . .
40
. . . . . . . .
40
A P P E N D I X ................................................
43
...........
CONCLUSION . ...............
A.
ABSORPTION PROFILE OF
AN ATOMIC BEAM . . .
B.
RECOIL-BROADENED EMISSION PROFILE
C.
COMPUTER PROGRAMS
REFERENCES
. . . .............
. . . .
. .......................
44
48
SI
ES
vi
LIST OF TABLES
Table
I.
,
'
ABLS Intensity Parameters . .............
page
35
vii
LIST OF FIGURES
Figure
I.
Page
Partial Term Diagram.for A r g o n , Showing
the First Resonance Lines .................
4
2.
Recoil-Broadened Beam Emission Profile
...
9
3.
Equivalent Gaussian Width y vs.
Convolution Parameter F .r .................
11
4.
(a) Schematic Diagram of the Apparatus
...
(b) Geometry of the Light Source
...........
13
13
5.
Spectrum From an Argon Beam, Showing the
Ar I and Ar II First Resonance Lines
...
19
6.
Spectrum Showing the Argon Resonance Series .
20
7.
Electron Impact Excitation Functions for
the Argon First Resonance Lines . . . . . .
o
ABLS Intensity (1048 A) vs. Source Oven.
Pressure
. . . . . .
...............
...
O
ABLS Intensity (1067 A) vs. Source Oven
Pressure
. . . . •......... .................
8.
9.
10.
11.
12.
Absorption of Argon Resonance Radiation
(1048 A) vs. Scattering Oven Pressure .
.
22
23
24
26
O
Scattered Intensity (1048 A) vs.
Scattering Oven Pressure
. . . . . . . . .
28
ABLS Lineshape
38
. .................... ..
viii
ABSTRACT
A dual atomic beam device for investigating the
resonance transitions in the rare gases is described.
The
device is useful wherever low beam densities or windowless
light paths are desired.
A simple model is developed for
the frequency distribution of a resonance line excited by
electron bombardment in an atomic beam light source.
The
model is used to interpret the observed absorption and
scattering of the 1048 A (1P") and 1067. A (3P") argon
resonance lines.
The ratio of the oscillator strengths of
these lines is measured by a new method with the result
f (1048)/ f (1067) = 3.99+0.55 in agreement with values given
in the literature.
The electron impact excitation functions
for these lines are observed for the first time and compared
to a. recent theoretical treatment.
I.
INTRODUCTION
Atomic first resonance transitions are of interest
for the information they provide about the lowest-lying
excited states.
The oscillator strengths are intimately
related to the lifetimes
I
and to the small-angle inelastic
electron-scattering cross sections.
In the rare gases, the
resonant transitions occur in the vacuum ultraviolet spectral
region
3
where special optical techiques are required.
Two types of vacuum ultraviolet light sources have
recently been described in the literature: Verkhovtseva,
4
e t . a l ., employ an ultrasonic gas jet excited by a high5
energy electron beam; Govertsen and Anderson use a
collimated atomic beam excited by electron bombardment to
produce narrow spectral lines.
Because of the low atom
densities and the ability to operate near threshold energies,
the latter source is uniquely suited to an investigation of
resonant transitions.
A feature common to sources of this type is the
presence of "background" atoms in the region where the
beam is excited.
If these atoms are of the same species as
the beam atoms and have large velocity components in the
direction of observation, the background emission takes the
form of a broad spectral line superimposed upon the narrow ■
2
beam line.6
The presence of stray atoms is particularly
troublesome in the case of resonant transitions; long path
lengths and the extremely large resonant cross sections1
result in substantial self-absorption, even if background
densities are reduced by efficient pumping techniques.
The process of excitation by electron bombardment
involves a transfer of momentum from the electron to the
excited atom.
The spectral line emitted by the beam is
broadened as a result.
These "recoil-broadening" effects
are not well understood.
An early analysis (for
by
7
Mack and Barkofsky predicts a broadening mechanism which
is essentially non-Gaussian in character.
A more recent
O
calculation by Korolyov and Odintsov has yielded theoret­
ical widths for several lines in the singlet spectrum of
helium which agree with the observed widths.
Details of the
broadening mechanism unfortunately are not presented.
g
Larson and Stanley have observed substantial broadening in
He II; they comment that the source profiles were "mainly"
Gaussian in character.
Recoil-broadening effects should be
less important with heavier atoms.
In this w o r k , a simple model for the frequency
distribution of a resonance line excited by electron
bombardment in an atomic beam light source is developed.
3
The model is used to interpret the observed absorption and
scattering .of the 1048 A
(1P 0 ) and 1067 A
(3P 0) fine-
structure components of the first resonance transitions in
argon
(figure I) .
A new method for. measuring relative
oscillator strengths is used to determine the ratio
O
O
f (1048 A ) / f (1067 A).
The electron impact excitation
functions for these lines are observed and compared to a
recent theoretical calculation.
10
4
ARGON RESONANCE LINES
3 P 54 S —*-3P6
9 3 1 4 3 .8 0 0
CM
1048A
I067A
FIG. I
II.
THEORY
An optical transition of an atom between an
excited state and the ground state is called a resonant
transition; the radiation emitted or absorbed in the
process is called resonance radiation.
If resonance
radiation corresponding to a particular transition propa­
gates in the ^-direction from the point of emission x=0
through an absorbing vapor of density n (x), the total
intensity' reaching a point x=L is
(I)
I (L) oc n (O) / dyE (w) exp{-/dxn (x) a (x, co) >
The emission profile E (m )is the frequency distribution of
the radiation emitted; a (x,co) is the cross section for
resonant absorption, or absorption profile, near x*
is determined by the distribution of the
the velocities
11
E (w)
components
of
of the excited-state atoms near x=0 , and
Cr(XirIo) is determined by a similar distribution for groundstate atoms near x, both according to the Doppler relation
12)
(U-U0 )
6
Here W 0 is the separation in angular frequency of the
ground and excited states, and c is the speed of light.
The distribution of atomic velocities throughout a
gas in thermal equilibrium is Maxwellian.
The correspond­
ing absorption profile is independent of x:
a (w) a
exp{- (a^ t0a) 2> ,
(3)
Y
W 2. / 2kT
c /
M
The absolute temperature T of the gas and the mass M of an
individual atom determine the absorption width y .
mann's constant is denoted by k.
Boltz­
The oscillator strength f
of the resonant transition appears because of the normal­
ization requirement^
(4)
/ dwc (w) =
I
V
f
me2
— 00
where m is the mass and e the charge of an electron.
It is shown in Appendix A that an atomic beam
produced by effusion through a small aperture and collimated
7
with a circular opening downstream exhibits an absorption
profile of the form
(3), provided T is interpreted as an
effective beam temperature 0. This effective temperature is
a measure of the geometric collimation of the beam and is
in general less than the temperature of the gas in the
source.
The absorption width y is correspondingly reduced.
The velocity distribution in an atomic beam under-
going excitation by electron bombardment is altered by ■
momentum transfer from electron to atom.
The beam emission
profile differs in form from the absorption cross section
as a result.
The distribution of atomic recoil momenta
along a direction perpendicular to the axis of a perfectly
collimated atomic beam
(0°K effective temperature)
can be
calculated from the differential, cross section for inelastic
electron scattering.
imation in Appendix B.
This is done in the first Born approx­
The corresponding emission profile
is
15)
sin"1 Is|
8
The angular frequency Wm -Co0 corresponds to the maximum
recoil velocity commensurate with energy-momentum conser­
vation.
The function is strictly zero for values
|c|>l
(recoil cutoff).
If the beam is not perfectly collimated, the
emission profile is a convolution of forms
(3) and
(5);
c t n (9/2)
(6 )
E(w) can be evaluated numerically with the aid of the first
program in Appendix C.
The results for several 'F are
shown in figure 2 .
The parameter T depends through y and
upon the
atomic properties, the beam temperature, and the incident
electron energy E :
(7)
9
RECOIL-BROADENED BEAM
EMISSION PROFILE
r = 0.0459
0.0
FIG. 2
1.0
'
C
2.0
10
For values r>0.4, the emission profile can be represented
by a simple Gaussian whose characteristic width Yr depends
upon F , as shown in.figure 3.
A least-squares comparison
between approximate and "exact" curves was made to select
the Yr corresponding to each T.
11
EQUIVALENT GAUSSIAN WIDTH Tr
L IM IT
FIG.3
r
III.
EXPERIMENTAL APPARATUS
The general features are illustrated in figure 4(a).
Resonance radiation from an atomic beam light source
analyzed to wavelength by a vacuum monochromator
scattered by an atomic beam
(C).
(A) is
(B) and
Intensities transmitted
and scattered at 90° are detected at
(D).
LIGHT SOURCE
The atomic beam light source
described by Govertsen and Anderson.^
parallels their description.
(ABLS) has been
The following closely
The source consists of a
focused and collimated atomic beam excited by electron
bombardment.
The beam originates in an oven equipped with
a I imn x 10 mm multichannel aperture of the type described
by Larson and Stanley.
stacked copper foils
9 12
'
The aperture consists of 40
(16 nun x 3.5 mm x 0.025 mm) into each
of which has been etched 40 channels of rectangular cross
section
(0.25 mm x 0.012 mm).
The channels are "aimed" at
a geometric focal point in the excitation region 30 mm from
the aperture.
Such mechanical focusing increases the beam
density achievable in the excitation region for a given oven
pressure.
The transmittance^^ of the aperture is =45%.
Stable beam density is achieved by regulating the rate of
13
14
flow into the oven with a Granville-Phillips series 203
variable leak valve.
A I mm x 8 mm slit 10 mm downstream
in the wall between the oven and excitation chambers colli­
mates the beam.
The I mm slit widths are in the direction
of the monochromator.
The electron gun consists of a tungsten filament
and a U-shaped copper anode.
The excitation region is
within the anode structure and radiation emerges through a
small slot in the surface facing the monochromator.
The
assembly is mounted between the pole pieces of a permanent
magnet which provides a field of =600 G to confine the
electrons to a narrow sheet.
The relationship between
atomic beam, electron sheet, and emitted radiation is shown
in figure 4 (b).
This geometry is selected to allow viewing
of the radiation in a direction perpendicular to the major
components of the atomic velocities.
A tungsten filament
can be used to heat the anode before operation; this impedes
anode contamination.^
Stable emission current density is achieved with a
feedback-type regulator which adjusts filament voltage in
response to anode current fluctuations.
Anode voltage and
emission current are separately adjustable.
The electron
gun and collimating magnet assembly is mounted on a trans­
15
fer table which allows precise positioning of the anode in
the plane perpendicular to the beam axis.
Optimum align­
ment is achieved from outside the vacuum chamber by means
of a mechanical feedthrough.■
MONOCHROMATOR
The radiant output of the light source is analyzed
with a McPherson model 235 half-meter vacuum U.V. scanning
monochromator of the Seya-Namioka type.
The MgF -coated
2
grating is ruled 600 L/mm and has a first-order dispersion
O
of 34 A/mm and a resolving power of -25,000.
SCATTERING BEAM
The scattering beam originates in an oven with a
2 mm x 10 mm multichannel aperture constructed by stacking
80 of the foils described above.
An adjustable slit set at
2 mm x 8 mm in the wall between the oven and interaction
chambers
(8 mm downstream)
collimates the beam.
Radiation
from the monochromator is absorbed in the direction of the
2 mm slit w i dth.
The intensity transmitted and that scat­
tered at 90° to the incident and beam directions are
observed with Bendix model 4503C channeltrons.
Masks are
inserted between the detectors and beam to insure that only
16
photons from the interaction region are counted.
VACUUM SYSTEM
Individually-forepumped NEC oil diffusion pumps
using Dow-Corning DC-704 fluid are employed throughout.
The chambers containing source- and scattering-beam ovens
are jointly evacuated through a tee by a 6-inch pump
equipped with a cryogenic baffle.
A 2-inch pump directly
evacuates the scattering chamber, and the monochromator is
equipped with a 4-inch pump and Granville-Phillips cryosorb
cold trap.
PRESSURE GAUGES
Ion gauges of the inverted Bayard-Alpert type are
used to monitor background pressures in the vacuum chambers.
Source- and scattering-oven pressures in the range 0-1 Torr
are measured with a type 77 MKS Baratron pressure metter
with type 77H-1 capacitance sensor.
Source oven pressures
above I Torr are inferred from the background pressure in
the excitation chamber, which is observed to be a linear
function of source pressure below 10 Torr.
17
OPERATING CHARACTERISTICS
For source oven pressures in the range I to 10 T o r r ,
the corresponding range of beam, densities in the excitation
region is estimated to be IO12- I O 13 per c m 3.
The cross
section of the electron sheet is approximately 0.1 mm x
IO mm for anode voltages of 0-300 V; these dimensions were
obtained by observing the blackening of the anode surface
struck by the electrons.
The maximum emission current
obtainable at any voltage is determined by space-charge
limitation; at 150 V a stable current of 10 mA can be main­
tained.
The maximum spread in electron energies is esti­
mated to be ±10 ev.
Scattering oven pressures in the range 0-1 Torr
correspond■to estimated beam densities in the interaction
region of IO11- I O 12 per cm 3.
The detectors have an electron
gain of 10 7 and a dark noise <5 counts/minute when operated
at. 3000 V.
° IF
The detection efficiency is =10% at 1000 A.
IV.
PROCEDURE
The experimental investigation was conducted in the
following five stages.
PRELIMINARY EXPERIMENTS
Preliminary experiments were performed to select
resonant transitions suitable for detailed investigation
and to determine optimum operating conditions.
The spectrum
from an argon beam excited in the ABLS was examined in the ■
O
range 700-1100 A for several source oven pressures, anode
voltages and currents, and monochromator slit widths.
figures 5 and 6 .
See
The reduced efficiency of the monochromO
ator grating at wavelengths below 800 A limited observed
O
intensities.
The Ar I resonance lines at 1048 A
(1P 0) and
O
1067 A
(3P°) were well-resolved with 300 y slits.
The
intensities of these lines were compared to the intensities
O
of the. ion
O
(Ar II) lines at 920 A and 932 A for several
source oven pressures.
The Ar I lines were found to be
significantly self-reversed at source oven pressures above
0.I Torr.
EXCITATION FUNCTIONS
The electron impact excitation functions for the
ARGON
FIG.
IO
ro
•o
ARGON
SOURCE PRESSURE 0.4 TORR
ELECTRON ENERGY-IOOeV(AV=ISO)
SLITS SOyu
CD CO
tD LO
CVi CM
CO CO
FIG. 6
WAVELENGTH (A)
21
Ar I resonance lines were studied by observing the intensity
of the light source as a function of anode voltage
electron energy)
(hence
for a source oven pressure of 0.1 Torr and
an emission current of I mA.
This current was the largest
that could be maintained over the entire voltage range.
low oven pressure was chosen to avoid self-reversal.
results are presented in figure 7.
The
The
The error bars corre­
spond to 10% fluctuations in the anode current caused by
poor filament regulation.
The observed threshold was at an
anode voltage of 63 V, corresponding to a space-charge
barrier in the electron gun of about 51 ev.
was evident for voltages below threshold.
17
Some emission
From t h i s , the
spread in electron energies was estimated to be ±10 ev.
LIGHT SOURCE INTENSITY
The intensity of the Ar I resonance lines was
observed as a function of source oven pressure at an anode
voltage of 150 V and an emission current of 10 mA.
Three
separate measurements were made of the intensity at each
pressure.
The mean values are plotted in figures 8 and 9.
The standard deviation in each mean is in all cases less
than 3% of the mean value, averaging less than 1%.
ARGON EXCITATION FUNCTIONS
m 0.6
-
THEORY
0.2
—
EXPERIMENT 9 1 0 4 8 A
20
40
60
80
P
O 1067 A
IOO
120
ELECTRON ENERGY IN EV ABOVE THRESHOLD
23
LIGHT SOURCE INTENSITY
IN T E N S IT Y
(ARBITRARY)
(1 0 4 8 A)
FIG. 8
•
EXPERIMENT
—
BEST FIT
SOURCE OVEN PR ESSU R E (TORR)
24
LIGHT SOURCE IN T E N S IT Y
(1 0 6 7 A )
EXPERIMENT
BEST FIT
SOURCE OVEN PRESSURE (TORR)
25
ABSORBED INTENSITY
Argon resonance radiation from the monochromator was
partially absorbed by a beam of argon atoms in the scatter­
ing chamber.
The intensity of radiation transmitted was
measured as a function of scattering oven pressure for
several source oven pressures,
were 150 V and 10 mA.
more times.
The electron beam parameters
Each measurement was made three or
The standard deviation in each mean was always
less than 13% of the mean v a l u e , averaging less than 3%.
The absorption A, defined as"^
(8)
a _ i_■ Transmitted Intensity
Incident Intensity
O
is shown in figure 10 for the 1048 A line.
The ratio of the initial slopes of the transmitted
O
O
intensity curves for the 1048 A and 1067 A lines was
measured as a function of source oven pressure and extraprolated to zero pressure.
From this ratio, the ratio of the
absorption coefficients and hence of the oscillator
strengths
was determined to be f (1048)/ f (1067) = 3.99±0.55.
The uncertainty quoted is three times the rms deviation as
A B S O R P T IO N
26
ABSORPTION
( 1 0 4 8 A)
SCATTERING
0 « EXPT
SOURCE PRESS.
THEORY
n,
0.09 T
0.17
0.55
1.04
0.08
0.25
0.90
2
OVEN PRESSURE (TORR)
27
determined by a least squares fit.
with the value obtained by Lawrence
lifetime measurements, Geiger
loss spectroscopy, and Lewis
broadening studies.
gives 4.08.
Geiger
19
20
This is in agreement
18
(3.86) from direct
(3.96) from electron energy
(4.4) from resonance-
The Hartree-Fock calculation of Knox
19
21
reports that H. Schmoranzer has
measured the ratio directly and found it to be 4.01.
SCATTERED INTENSITY
The fluorescent intensity was observed at 90° to
the direction of incident radiation as a function of
0
scattering oven pressure. The 1048 A results for several
source oven pressures are presented in figure 11.
The
intensity scattered to the detector with no beam present in
the chamber is due to background gas and is normalized to
unity.
Typical error bars, corresponding to the /N
statistical uncertainty, are indicated at the peak of the
lowest source pressure curve.
Because observed scattered intensities were low
O
(-30 photons/minute for the 1048 A line), long counting
periods were required.
An attempt was made to achieve
maximum stability of the light source.
The anode was
preheated at pressures below 5x10“ 7 Torr before the light
0 .0
fig .
Il
0.1
0 .2
0 .3
SCATTERING OVEN PRESSURE (TORR)
29
source was operated.
Stable beam pressures were achieved
15 minutes after adjusting the leak valves.
The filament
regulator maintained a constant emission current.
The
maximum drift in radiant output under these conditions was
10% per hour.
This was compensated for by monitoring the
light source intensity with no scattering beam present at
frequent intervals.
V.
DISCUSSION
SPECTRAL SCANS
The spin-orbit interaction of the argon excited
states is sufficiently large to establish intermediate
coupling.
22
Consequently all J=I levels of the p 5s and p 5d
excited state configurations will contain a 1P" component
and will make dipole transitions to the 1S
o
ground state.
There are two such levels in each p 5s configuration and
three in each p sd.
This shows clearly in figures 5 and 6 .
O
O
The lines at 894 A (3d) and 843 A (4d) are mostly triplet
and are not strongly excited at 150 V.
EXCITATION FUNCTIONS
The electron impact excitation functions for the
Ar I resonance lines apparently have not previously been
studied experimentally.
A theoretical calculation has
recently been made by Sawada, Purcell, and G r e e n . T h e y
employed a phenomenologically-determined independent
particle model of the atom and a distorted-wave approxima­
tion to the resultant scattering potential.
The formalism
included the effects of electron exhange in the singlet
excitation but made t h e .simplifying assumption of purely
LS-coupled atomic wave functions.
31
In figure 7 , the singlet curve is normalized to the
data at 138 ev.
The theory correctly predicts that the
O
1067 A excitation function will decay with increasing energy
O
more rapidly than the 1048 A function.
Beyond this, there
is little agreement.
The discrepancy occurs in part because
the argon atom is not LS coupled.
O
The relative intensity of 1067 A radiation above
50 ev due to singlet-triplet mixing can be estimated as
follows.
The wave function is written as the sum
of pure.
Russell-Saunders triplet and singlet contributions:
(9)
ip(10 67 A) = aip(3P°) + IBiM2P 0) .
i
i
The intermediate coupling constants a and 8 are determined
from the ratio of the lifetimes,
hence of the oscillator
strengths ,^ with the result a = 0 ,893, 8=0.451.
ratio of intensities at high energy is
observed ratio is =2,
The expected
(a/8)2-4,
The
O
Preferential population of the 1067 A
state by cascading from higher excited states may account
for the remaining emission.
The exchange contribution to the singlet curve
32
aPPears as a spike 2-3 ev wide at low energy. The large
O
initial slope of the 1048 A function is consistent with
such a spike.
LIGHT SOURCE INTENSITY
Interpreting the beam-pressure dependence of the
ABLS intensity
(figures 8 and 9) in terms of expression
(I)
requires a detailed knowledge of the density and velocity
distributions throughout the excitation chamber.
A treat­
able model consists of a sharply-defined beam of uniform
density superimposed upon a fully-thermal!zed background
gas =
The predicted intensity of resonance radiation
originating in such a light source is
(10)
I (n ) =C n / d w [E
I
Here n
Ij
(w) +~E
I
P
(w) Jexp{-n
2
is proportional to beam density
Ia (w) +§a
1 1
P
(to) J }
2
(hence source
pressure), 3 is the ratio of beam density to background
density, and a is the ratio of background path length to
beam path length.
E^ (to) and a ^ (to) are profiles representing emission
33
and absorption, respectively, by"beam atoms; E
(to) and
2
(w) are similar background profiles.
The absorption profile ct (w) of the atomic beam in
the case of a J=I-KT=O resonant transition is
(cu) = j [ct (m - m h )+a (co)+a (ti+tuH )]
(11 )
where My is the Zeeman splitting of the excited state in the
field of the collimating magnet.
The functions
ct (m
- m h ),
et c . , are calculated for the reduced beam temperature from
(3),
in
It is convenient to regard the parameter y appearing
(3) as the "width" y j of
(m ) .
Since most of the back­
ground absorption occurs away from the magnetic field, O^(M)
is a simple Gaussian of width y '.
2
The geometry of the system is such that the axis of
the collimating field is contained in the plane defined by
the normal to the monochromator grating and the optical axis
of the system.
Consequently, the ir-polarized (AMj=O)
component is substantially attenuated at the grating.
Zeeman
This
effect is accounted for in the theory by writing the beam
emission profile as
34
(12)
(w) = .^Q IE (M-Uh )+GE(W)+E (m + o)h )]
where the "grating factor" G is less than I.
The component
functions E (to-w^) , etc., have in general the form
(6).
For
the experimental conditions of .figures 8 and 9, the convo­
lution parameter has the value T=O.46, and the components
can be approximated by equivalent Gaussians as discussed in
Chapter II.
The equivalent width of a component is the
"width" Y 1 °f E
(to) .
The background emission profile
(to)
is similar in form to E^ (to) except that the functions
E((o-(oH ) , etc., axe calculated for the actual temperature of
the background gas.
The intensity curve
(10) was evaluated numerically
for the argon resonance lines with parameters estimated from
the experimental geometry and operating conditions.
parameters appear under "best guess" in Table I .
These
Agreement
between the calculated curves and the experimental data was
poor.
The model as formulated does not adequately describe
the relevant velocity and density distributions.
The shape of
(10) is sensitive to the various line-
35
TABLE I
Parameter
Best
Guess
Best
Fit
3
Density Ratio
16
2
a
Path Length Ratio
64.4
G
Grating Factor
. §
V ^ D
V
yD
y
;/ y d
Y / YD
y
V yd
+
0.55
0.55 +
(1048 A)
0.251
0.251+
(1067 A)
0.320
0.320+
Beam Emission Width
0.14
0.2
Beam Absorption Width
0.1
0.1-0.16*
*
Background Emission Width
I
I
Background Absorption Width
I
0 .6
.Zeeman Splitting
+These parameters are non-adjustable.
§
64.4
Y q = Room Temperature Doppler Width
2 .OSSx IO 10 radians/second.
*See text.
(HWl/e) =
36
widths and to the density ratio 6 .
A computer search was
made to find a set of parameters which characterize the
observed behavior.
These "best fit" parameters appear in
Table I 7 and the corresponding curves are plotted with the
experimental data in figures 8 and 9.
The normalization is
chosen to make the low-pressure slopes agree.
Functions of
this form can be generated with the aid of the second
program in Appendix C.
With the "best fit" parameters, the intensity curve
is not sensitive to values of y' within the indicated limits.
i
Taken at the center of its range, the value Y 1=O. 13y _. (Yn-'is
the argon room-temperature Doppler width)
indicates a beam
absorption width about 30% greater than that estimated from
the geometric collimation of the beam; this indicates partial
decollimation due to the anode structure.
The beam emission
width Y 1 is greater than the absorption width by about 15%.
This is in agreement with the predicted recoil broadening.
The values Y 1=O-G y t. and 8=2 are reasonable in that the
2
D
background gas is not fully thermalized but is partially
directed along the axis of the beam, at least near the
excitation region.
Substantial absorption width and density
variations exist throughout the chamber as a result.
quently these parameters represent averages
Conse­
Iin the sense of
37
expression
(I)J over the entire chamber.
ABLS LINESHAPE
The predicted distribution in frequency of a
resonance line excited in the model ABLS is given by the
integrand of expression (10).
This distribution is plotted
O
for the 1048 A line for several source oven pressures in
figure 12.
For pressures <0.1 T o r r , beam atom contributions
to the total emitted intensity are dominant.
The relative
contribution due to background emission increases with
pressure, causing the distribution to "widen" in the sense
that the intensity is increasingly concentrated in the wings
For pressures >2 To r r , beam atom contributions are neglig­
ible .
ABSORBED INTENSITY
If I(n^,w) is the intensity per unit frequency of
resonance radiation entering the scattering chamber, then
the intensity I (n^ ,nz ,to) reaching the detector isI
(13)
I (n ,n ,w) = I (n ,w)exp{-n a (a>) } .
I
2
I
.23
ABLS LINESHAPE
K u ))
2 TORR (SOURCE
OVEN PRESSURE)
FIG. 12
U)/%
39
Here
is proportional to the pressure in the scattering
The absorption profile a (w) is a Gaussian
beam oven.
3
whose width y ^ is regarded as an average over the entire
chamber.
The total transmitted intensity is found by
integrating
(13) over all frequencies.
The absorption A is found from equation
(8 ) and is
plotted with the experimental data in figure 10.
source pressure curve
The lowest
(Uj= .08)corresponds to Y g= O .65 and is
normalized to the data at a point near the knee. ; The
remaining curves are generated by varying n .
The param­
eters n^ increase at about the same rate as the correspond-,
ing source pressures, except for the extreme high-pressure ■
curve
(n^ = 8) ., Deviations at high pressures are to be
expected, since at a source pressure -4 Torr the mean free
path of an argon atom in the oven becomes comparable to the
narrow dimension of a collimator channel.
Under these
conditions, simple molecular effusion does not occur but
rather a turbulent hydrodynamic flow, with the result that
the "effective" source area increases.^
Thus the beam
collimation varies with pressure, and the assumption of
constant path ratio a and density ratio 8 must be abandoned.
At high scattering beam pressures, the experimental
absorption saturates at a value which depends upon the
40
source pressure.
This result is correctly predicted by the
theory and corresponds to the broadening of the ABLS lineshape discussed above; as the intensity shifts away from the
central frequency, it has a smaller probability of being
absorbed in the scattering chamber.
SCATTERED INTENSITY
Because of the large experimental uncertainties, a
detailed extension of the theory to the case of resonant
scattering has not been made.
The general features of
figure 11 are consistent with the model.
The intensity of
scattered radiation peaks as a function of scattering oven
pressure.
The decrease at higher pressure is due to the
attenuation of the scattered radiation in the background
gas before reaching the detector.
Resonant excitation is
most efficient for low source pressures.
to a favorable overlap between the
This corresponds
(narrow) ABLS lineshape
and the scattering beam absorption profile.
As the line-
shape widens with source pressure, the overlap decreases
and so also does the excitation efficiency.
CONCLUSION
The atomic beam is a useful tool for examining the
41
resonance transitions in the rare gases; it meets the
requirements of low density and windowless light paths.
An
atomic beam light source of the type described can be
represented by a simple two-component
model.
(beam and background)
The model parameters can be used to calculate
roughly the effects of recoil broadening and to predict the
frequency distribution of a resonance line excited in the
light source.
This capability is important when direct
interferometric examination of the lineshape is not possible
(as in the vacuum ultraviolet).
The recoil-broadening analysis suggests an important
experiment.
Momentum transfer effects should be greater
g
with lighter atoms.
Korolyov and Odintsov predict that the
widths of all lines arising from resonant levels will vary
approximately as the inverse of the electron velocity.
They
observed a strong energy dependence in the width of the
O
5016 A line (S1P ->21S ) in helium but were not equipped to
O
investigate the 584 A resonance line directly.
A systematic
study of the recoil broadening of this line
25
as a function
of energy is indicated.
2g
Anderson
has observed anomalous behavior in the
Hanle effect in xenon which he attributes to the Gaussian
profile of the resonance lamps.
The scattered intensity
42
measurements suggest the possibility of observing the Hanle
effect for the case of narrow linewidth excitation.
Such an
experiment would be most efficient at low beam densities and
would require sensitive low-noise detection systems.
APPENDIX
A.
ABSORPTION PROFILE OF- AN ATOMIC BEAM
Consider an atomic beam formed by effusion through
an ideal aperture in the wall of an enclosure or oven
containing a gas of atoms in thermal equilibrium.
The
aperture is sufficiently small that the Maxwellian distri­
bution of atomic velocities within the oven is not altered
by the effusion process.
spherical coordinates
The polar axis. 2 of a set of
(r,9 ,([>) with origin at the aperture
is defined by the outward normal to the aperture.
The
probability that an atom will be effused into a velocity
range dv about a particular velocity v is, by a simple
statistical mechanical argument,
p (v) dv
£( M ) 2vcos 0exp {-^srJdv
TT v2kT
2kT
(Al)'
dv = v 2sin0dvd0d<f> .
Here M is the mass of an atom, T the absolute temperature
of the gas in the o v e n , and k is Boltzmann's constant.
The
projection vx of v along a particular direction & perpendic­
ular to the polar axis is vsin0cos<j> if the azimuth cf> is
measured from £.
This constraint may be used to write
p (v)dv in terms of v^, v, and 0 :
45
(A2)
P(»)d» - Il5Hr)
2
v 2sin6cos6
✓
exp{-$ T ^ dvx dvd6
sin2 6- (vx/v)
The probability P ^vx^ dvx
an effused atom will have a
velocity component v^ in the range dvx is found by integrat­
ing
(A2) over all allowable v and 9.
Suppose the atomic beam is collimated by a plane
circular opening normal to and centered on the polar axis at
some distance downstream from the oven.
The maximum angle
that an effused atom can make with the polar axis and still
pass through the opening is 60 .
Then in the region on the
downstream side of the opening.
fl(^)
re.
■!rt
p tv Xld v X
Jc
^ecose,
sin26-(vx/v) 2
x
(A3)
c = vx/sin8 o , d = sin- 1 (vx/v)
The lower limit on the 6-integration is required by the
definition of v / and the lower limit on the v-integration
46
is prescribed by system geometry.
(A4)
P tV
dvX =
After some work.
sin2 00- l (---- * L _ ) 2exp{
/? 2kTsin26 0
-------}dv
2kTsin260
.
x
The factor sin20o appearing in the normalization is the
probability that an effused atom will pass through the
collimating aperture.
The "effective" beam temperature 0
is defined by Tsin2Q0', and is related to the oven temperature
T by
(AS)
c = / I = n&r •
C is called the geometric collimation of the beam.
For
sufficiently small Q 0 , C-l/Q0 .
The beam absorption profile obtained from (A4)
according to
(2 ) is
47
cr(co)
^ e x p {Y
0.) z ) ,
Y
CA6)
Y
_ Wo. /2kTsin280.i
c
v
M
'
which is a Gaussian with a characteristic width y , called
the "width" of the absorption profile.
\
B.
RECOIL-BROADENED EMISSION PROFILE
Consider a beam of atoms moving in the 2-direction
(only!) which undergoes bombardment by electrons of mass m
incident along
After an inelastic collision causing an
excitation to an optical state of energy hw, the atom will
in general exhibit a momentum component along the S - axis.
The distribution of atomic recoil momenta along x is clearly
the same as the distribution of electronic recoil momenta
along - S .
The angular distribution p (6 ,<j>) of inelastically
scattered electrons is given in the first Born approximation
to the differential cross section:
(BI)
27
p(8,*)d0 = do = --- — ---si n 2 (8/ 2)
where the polar angle 8 is measured from f , and the azimuth
(p from S.
If the change in kinetic energy of the atom is
neglected, the energy balance gives
(B2)
^ ( P 2
t- P 2
1 ) = hw.
49
where p Q and
are the constant magnitudes of the initial
and final electronic momenta, respectively.
Thus the dis­
tribution of scattered momenta p is
(B3)
p (p)dp « 6 (p-p^ )p (8 ,<f))dpdft = 6 (p-p^ )
dpd6d(f) .
The constraint px = psinBcosej) can be used to eliminate <J> in
favor of px with the result
(B4)
p(p)dp cc 6 (p-p
P-
—
d p d 6dpx .
izSin2G- (PxZp) 2
The desired distribution p(px )dpx is achieved by integrating
over all allowable p , 6 .
The p-integration is immediate; the
6-integration is not:
'Tr-g
p(Px )dpx « dp JdB(B5)
: Q t n (B/2)
. 1S l n 2 6 - (px / p
g = sin- 1 Ip x Z p 1 I
)2
50
The limits are occasioned by the fact that only angles
6 1$ 6<Tr-0 1 can contribute to a particular p^, where
p'=p sin6 1.
X
I
The absolute value signs are added to insure
symmetry of the function about p^=0.
The corresponding
emission profile is given by equation
(5).
C:
COMPUTER. P R O GRAMS
I REM: BA SI C PROGRAM CALCULATES RECOIL-BROADENED
2 REM: BEAM E M IS SI O N P R O F I L E ( R B S H P E )
10G = .03
I IN =• 0228/G
12PRIN T"G ='UG
I S P R I N T nX 1S " P C X ) " •
3 0 FORX= I • 0 2 T O I • 0 6 STEP* 0 2
40P=0
5 0 F O R Y = - • 9 9 TO ®9 9 STEP* 0 2
60A =(X -Y )/G 7 0 F l =EXPC - A*A )
77 Z= AB SC Y)
8 0L=ATNC Z / S Q R C 1 - Y * Y ) )
9 0 U = 3 . I Al 6 - L
10 0 D=C U - L ) / 1 0 0
1201=0
I 3 0 F O R T = L + DTOU-DSTEPD
1 4 0 F 2 = S I N C T)
I 5 0 F 3 = I-COSC T)
I 6 0 I = I + D * C F 2 / F 3 ) / SQRCF 2 * F 2 - Y * Y )
I 70NEXTT
180P = P + .02*F 1*I
190NEXTY
200PRIN TX/P
210NEXTX
999END
1
52
I REM: BASIC PROGRAM CALCULATES ABLS I MTEMSI TT ( L S I 'MT)
10DIMEC 3 ) » FC 3 ) j>GC 3 )
20A 1=64.4
30B1.= 2
40G= o 5 5
50G1= • 2
6062=1
70G3=eI
8 0G4=» 6
90 S 1 =» 1 7 9 5 / 3 . 4
1 0 0 1 0 = 1 /1 .36806E-02
I I 0 FORPI = « 0 5 T 0 I • 4 S T E P . 0 5
I 20N1=S1*P1
I 30J=0
I 40FORX=0TO2STEPl/350
150E 4=EX P(-C X /G 4)t2)/G 4
I 6 0 FORI = I T03
170Y = X -.502+ .251*i
180ECI ) = E X P ( - ( Y / G l ) t 2 ) / G l
190FC I ) = EXPC-C Y / G 2 ) t 2 ) / G 2
20.0GC D = EXPC -C Y / G 3 ) f 2 ) / G 3
SI 0MEXTI
220E 1 = CEC I ) + G*EC 2 ) + EC 3 ) ) / C 2+G)
2 3 0 E 2 = ( FC I ) + G*FC 2 ) + FC 3 ) ) / C 2+G)
240E3= C GC I ) + GC 2 ) +GC 3) ) / 3
2 5 0 J = J + I 0 * ( i M 1 / 3 5 0 >*C E1 + E 2 / B 1 )*EXPC - M D C E3+C A l / B l ) * E 4 ) )
260MEXTX
270PRINTP1> J
280MEXTP1
290EMD
53
I REMJ BASIC PROGRAM CALCULATES ABLS L I MESHAPE ( L S H P E )
10DIMEC 3)> FC 3>»G< 3 )
20A1 = 64» 4
3 0 B 1 =2
4 0 G = . 55
50G1 = o 2
60G2=I
70G3=. I
8 0 GA=o6
90S 1=«1795/3® 4
100 10 = 1 / 1 . 3 6 8 0 6 E -0 2
I 1 0 P l = o 55
120N1=S1*P1
I 30J=0
I 4 0 FORX = 2* I TO 2 • 5 S T E P . I
I 50E4=EXP<-CX/G4)»2 ) /G 4
I 60FORI = I TO 3
I 70Y = X ~ .5 0 2 + o 251*1
18 0EC I )=E XP ( -C Y /G 1 ) t 2 > / G l
190FC I ) =E X P ( -C Y / G 2 ) r 2 ) / G 2
2 0 0 G C I ) = EXPC-C Y / G 3 ) » 2 ) / G 3
2 10NEXTI
22 0E 1 = CEC I ) + G*E( 2 ) + E( 3 ) ) / ( 2+G)
2 3 0 E 2 = ( F( 1) + G*F( 2 ) + FC 3 ) ) / ( 2+G)
240E3= C GC l.)+GC 2 ) + GC 3 ) ) / 3
2 5 0 J = J + I 0 * ( N 1 / 3 5 0 ) * C E 1 + E 2 / B 1 ) * E X P C ~M1*CE3+C A l / B l ) * E 4 ) )
260PRIMTX.» J
270NEXTX
290END
54
IREM: BASIC PROGRAM CALCULATES TRANSMITTED
2REM: INTENSITY AND ABSORPTION < TRIMT)
10DIMEC 3), FC 3)>G< 3)
20A1=64.4
30B1 -2
40G= o55
50GI= o2
60G2=1
70G3=•I
80G4=o 6
8 5G5 = . 65
90S!=.1795/3»4
9552=7/.19
10010=1/1»36806E”02
I I0P1 = •55
I20NI= Sl*P1
I27F0RP2=.ITOl STEP®I
I29N2=S2*P2
I30J=0
I35Jl = 0
I40FORX=0TO2STEPl/350
I50E4=EXP(-CX/G4)t 2)/G4
I 60FORI = IT03
I70Y=X-» 502+» 251*1
180EC D = E X P C -CY/G1) t 2>/Gl
190FC I )= EXPC-C Y/G2)t 2>/G2
200GC D = E X P C -C Y/G3) t 2)/G3
210NEXTI
220E1 = C EC I )+G*EC 2) + EC 3) )/C 2+G)
230E2=C FC I )+G*FC 2) + FC 3) )/C 2+G,)
240E3=CGC I)+ GC2)+GC3))/3
245J2=10*CNI/350)*CE1 + E2/B1)*EXPC-N1*CE3+C Al/Bl)*E4))
250J=J+J2
260E5=EXPC-CX/G5)t 2)/G5
280J1=J1+J2*EXPC -N2*ES)
290NEXTX
300PRINTP2,JT,1-J1/J
3 10NEXTP2
320END
REFERENCES
REFERENCES
1.
A. C. G. Mitchell and M. W. Zemansky, Resonance
Radiation and Excited Atoms (Cambridge Univerity
Press, London, 1961).
2.
J. Geiger, Z. Physik 177, 138
3.
R. L. Kelly, Atomic Emission Lines Below 2000 AngstromsHydrogen through Argon (Naval Research Laboratory,
Washington, D. C . , 1968).
4.
E. T. Verkhovtseva, V.. P. Strelnikov, V. .N. Sokolov, and
B. N. Popov, Zh. Prikl. Spectrosk. 1_, 859 (1967) .
5.
G. A. Govertsen and D. K. Anderson, J. Opt. Soc. Am. 60,
600 (1970).
6.
V. I . Odintsov, Opt. Spectry. 3j0, 202
7.
J. E. Mack and E. C. Barkofsky, Rev. Mod. P h y s . 14, 82
(1942) .
~
8.
F. A. Korolyov and V. I. Odintsov, Opt. Spectry. 8 , 547
(1965) .
9.
H . P. Larson and R. W. Stanley, J. Opt. Soc. Am. 12,
1439 (1967).
10.
(1956).
(1961) .
T. Sawada, J. E. Purcell, and A. E. S. Green, Phys. Rev.
A. 4, 193 (1971).
11. ,In this and what follows, the natural linewidth is
assumed to be negligibly small.
12.
R. W. Stanley, J. Opt. Soc. Am. 56_, 350
(1966).
13.
Ratio of open area to total area.
14.
W. E. La m b , Jr. and R. C. Retherford, P h y s . Rev. 79, 41
(1950).
15.
E. S . Fry and W. L. Williams
16.
Bendix Corporation specification sheet.
(to be published).
57
17.
The threshold potentials for the 1048 A and 1067 A
lines are at 11.8 ev and 11.6 ev respectively.
18.
G. M . Lawrence, Phy s . Rev. 1 7 5 , 40
19.
J. Geiger, Phys. Lett. 33A, 351
20.
E. L. Lewis, Proc. Phy s . S o c . 9^2, 817
21.
R. S. Knox, Ph y s . Rev. 110, 375
22.
E . U. Condon and G. H. Shortley, Theory of Atomic
Spectra (Cambridge University Press, Lond o n , 1964)
p. 304.
23.
A. Lurio, M. M a ndel , and R. Novick, Phys. Rev. 126,
1758 (1962).
24.
N. F. Ramsey, Molecular Beams (Oxford University Pre s s ,
London, 196371
O
At 584 A a grazing incidence monochromator may be
required.
25.
(1968).
(1970).
(1967).
(1958).
26.
D. K. Anderson, Phys . Rev. 1 3 7 , A21
27.
N. F. Mott and H. S . W. Massey, Theory of Atomic
Coliisons (Oxford University Press, London, 1965).
U<
at
(1965).
'»»*«»Slw “'"asm’UWMtte
3 1762 IOO11221
t
Seitel, Steven C
Interaction of
resonance radiation
with atomic beams
WAm H ANP A D O W g W
5c. jZ S