Energy levels of Pr3+ in Y(OH)3 determined from absorption and fluorescence spectra
by James Keith Boger
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in
Physics
Montana State University
© Copyright by James Keith Boger (1987)
Abstract:
In this thesis, the electronic energy levels in the 4f subshell of a trivalent lanthanide in an optically
transparent crystal are found. The particular crystal studied was trivalent praseodymium in yttrium
hydroxide. Determination of the levels was accomplished by comparing experimentally found energies
to the eigenvalues of a Hamiltonian whose form had been given. Experimental energies were found by
studying emission and absorption spectra from the sample. This thesis begins by outlining the theory
behind the project. Details of the experimental arrangement, data analysis and results from the
calculations are then presented. As final results, the calculated energy levels as well as their
experimental counterparts are presented. Also presented are the magnitudes found for the parameters in
the Hamiltonian of the sample. ENERGY LEVELS OF Pr3 * IN Y(OH)3 DETERMINED FROM
ABSORPTION AND FLUORESCENCE SPECTRA
by
James Keith Boger
t
A thesis submitted in partial fulfillment
of the requirements for the degree
of
Master of Science
in
Physics
MONTANA STATE UNIVERSITY
Bozeman, Montana
July 1987
MAIN UB-
M37g
ii
APPROVAL
of a thesis submitted by
James Keith Boger
This thesis
has been read by each member of the thesis
committee and has been found to be satisfactory regarding
content,
English
usage,
format, citations, bibliographic
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to the
College of Graduate Studies.
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Date
Approved for the Major Department
n
Date
(f?i
Head,Major Department
Approved for the College of Graduate Studies
Date
Graduate Dean
iii
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TABLE OF CONTENTS
LIST .OF TABLES
..........................................
v
LIST OF F I G U R E S ..................................
vi
A B S T R A C T ......................................... ..
. . vi i i
1. INTRODUCTION .........................................
1
2. T H E O R Y ............................
6
Spectroscopy .......................................
Praseodymium as a Free I o n ........................
Praseodymium in a Crystal Field
.................
7
12
17
3. EXPERIMENTAL ARRANGEMENT . . .
27
Absorption Experiment
............................
Fluorescence Experiment
. ............... . . . .
27
39
4. DATA A N A L Y S I S .......................
5. COMPUTER FITTING THE HAMILTONIAN .............
51
...
58
The Fitting P r o g r a m ..............................
Fitting to Data from Pr3*: L a C l s .............
Fitting to Data from Pr3*: Y( OH) 3 . . ..............
59
51
66
6. RESULTS AND C O N C L U S I O N S ..................... : . .
71
REFERENCES CITED
79
A P P E N D I X ...............................................
81
Experimental Plots of Spectra
.. . .
81
V
LIST OF TABLES
Table
Page
1.
States of the 3Fs term........ ...............
24
2.
Selection rules for a crystal having Ca h
symmetry.......................................
26
Example list of possible transitions
responsible forspectral lines recorded . . .
55
Numerical results from fitting program using
data from Pr3+ : LaCla..................... .. .
63
Numerical results from fitting program using
data from Pr3 + : Y(OH) a ..........
70
Absorption lines recorded and levels
determined....................................
72
Fluorescence lines recorded and levels
determined.............................. •......
74
3.
4.
5.
6.
7.
8.
Final calculated energy levels for
Pr3 + : Y(OH) 3 ..................................... 76
9. ' Final parameters for the Hamiltonian of
Pr3 + : Y( OH) a
78
vi
LIST OF FIGURES
Figure
Page
1.
Thesis outline...............................
3
■2.
Plots of the harmonic oscillator...........
10
3.
Estimated energy levels for Pr 3 + : YC OH) 3 . .
16
4.
3Fj submatrix for the crystal field
Hamiltonian .......................
23
Pr3*: YC OH) 3 sample and relevant
polarization................................
26
Top view of the Spex 14018 spectrometer
used in e x p e r i m e n t s.......................
31
7.
Optical arrangement used in experiment.
. .
33
8.
Cross sectional view of the glass dewar . .
35
9.
Block diagram of the signal processing used
in absorption experiment...................
3 7,
5.
6.
10.
Schematic of system used in the fluorescence
experiment including the Hansch type dye
laser a r rangement..........................
41
11.
Block diagram for the signal processing used
in the fluorescenceexperiment..............
46
12.
Absorption with sigma polarization at
22400 ......................................
82
13.
Absorption with pi polarization at 22400 .
83
14.
Absorption with pi polarization at 21400
15.
.
Absorption with sigma polarization at
2 1 1 0 0 .......... •...........................
16.
84
Absorption with pi polarization at 2 1 0 2 0
85
.
86
vi i ■
LIST OF FIGURES-confcinued
Figure
Page
17.
Absorption with sigma polarization at
2 0 4 4 5 ....................................... -g7
18.
Fluorescence with no polarization at
1 8849 .......................................
88
Fluorescence with no polarization at
1 6841 . '........................... ..
89
19.
20.
Absorption withpi polarization
21.
Absorption with sigma polarization at
1 6790 .
gi
Absorption with sigma polarization at
1 6490 .......................................
92
Fluorescence with pi polarization at
1 6 4 8 0 ......................
93
Fluorescence with si.gma polarization at
1 6 4 8 0 .......................................
94
Fluorescence with sigma polarization at
1 6290 ..................................... . .
95
Fluorescence with
pipolarization at
1 6290 ...........• •................. ........
96
Fluorescence with
nopolarization at
1 2400 .......................................
97
Fluorescence with no polarization at
11500 .......................
98
22.
23.
24.
25.
26.
27.
28.
at16900
.
90
viii
ABSTRACT
In this thesis, the electronic energy levels in the 4f
subshell
of a trivalent
lanthanide
in
an optically
transparent
crystal
are found.
The particular crystal
studied was trivalent
praseodymium in yttrium hydroxide.
Determination of the levels
was accomplished by comparing
experimentally found energies
to the eigenvalues
of a
Hamiltonian
whose form had been given.
Experimental
energies were found by studying emission and absorption
spectra from the sample.
This thesis begins by outlining
the theory behind the project.. Details of the experimental
arrangement,
data
analysis
and
results
from
the
calculations
are then presented.
As final
results, the
calculated
energy levels
as
well
as their experimental
counterparts
are presented.
Also presented
are the
magnitudes found for the
parameters in the Hamiltonian of
the sample.
I
CHAPTER 1
INTRODUCTION
This
thesis
concentrated
absorption spectra of
a
lattice.
one
In
general
trivalent lanthanides.
these
elements
are
on
trivalent
the
emission
lanthanide
in
a
scientific
interesting
point
because
they
of view
have many
extremely narrow emission and absorption linewidths
spectrum. 111
can
incorporated . into
be
materials.
This is
Furthermore,
various
or transition
free
ions
small,
when
the
oscillator
ion
very
is
lanthanide ions may be
state.
of
trivalent lanthanides
to the
fact that the
probabilities,
of the
while they can be significant
trapped . in
strengths
in the
optically transparent
convenient due
oscillator strengths,
are
a host
might ask what is special about
From
visible
and
suggest
optically
a
host
that
a
pumped
lattice.
large
into
Large
number
of
an excited
These properties lead to the practical applications
trivalent
materials.
lanthanides
The YAG
laser,
in
host
lattices
as
with tri valent neodymium,
laser
is a
graphic example of the practical role these materials have.
Typically,
be envisioned,
before any applications for a material can
it is necessary to establish some
2
fundamental properties .of
that
transparent
crystals
glasses
rare earth,
and
the energy levels are
information
established.
was
3,
the
among the
the
crystal of
motivation
information could then
be
For optically
doped with a trivalent
Finding
trivalent praseodymium in a
Pr3 + : Y(OH)
material.
used
energy levels of
yttrium hydroxide,
for
in
first bits of
this thesis.
carrying
This
out further
experiments on the sample such as optical hole burning. (2)
A brief
outline of
the procedure used in seeking the
energy levels of the sample is facilitated by following the
flow chart
given in
Figure 1.
The first box in this flow
chart labeled ' Experiment’ , indicates
in
the
project
transitions in
was
the
to
experimentally
sample
recording absorption
as
transitions made up
conducted.
Of
course just
does not mean
that the states involved in each transition are
necessary to
the
The
,data.
indicates
a
second
general
interpret the
data!
make
of
a
list
responsible for
all
each
develop a
box
outline
In general,
the
known.
It
method for analyzing
entitled
of
as many
As indicated,
spectral lines may be recorded,
is therefore
first step
observe
possible.
and fluorescence
the two types of experiments
because many
that the
' Data
Analysis' ,
scheme designed to
the method,
used was to
possible transitions which might be
spectral
making a choice from each list,
line
recorded.
Then by
a set of experimentally
3
EXPERIMENT
Review
experiment
n
DATA ANALYSIS
Select
Make a
from table
table of
transitions -----» transitions
causing
responsible
spectra
for lines
Construct
final
table
of energy
levels
n
If data
are not
clear
1
I
Check
analysis
COMPUTER FITTING HAMILTONIAN TO FIT DATA
Enter best
experimental
energy
levels
for fitting
program
Run program
to calculate
energy levels
and find
Hamiltonian
parameters
Figure 1. Thesis outline
Enter
remaining
data
and rerun
program
When no
fit is
f ound
4
determined energy
impossible to
in the
levels could be constructed.
experimentally determine
sample, so
But it was
every energy level
it was necessary to turn to theoretical
aspects of the energy levels.
Theoretically,
calculated
if
all
the
of
the
energy
Hamiltonian
of
levels
the sample was known.
While the form of the Hamiltonian is known,
of
the
various
necessary
to
experimentally
these
parameters
take
to
are
the
parameters.
not.
energy
empirically
This
could be
the magnitudes
It
is therefore
levels
determined
determine the magnitude of
was
done
by
entering
the
experimental data into a computer program which modeled the
Hamilton!an by
comparing
solving
the
results
the
to
eigenvalue
the
problem
entered
data.
Hamiltonian is found having eigenvalues which
the experimental
values,
and then
If
are close to
then all of the energy levels can
be considered known.
The typical, situation however
find
between
contradictions
values.
Contradictions
selections
made
selections
often
during
proved
a
made
the
to
experimental
it
and calculated
necessary
data
be
was to
analysis
ambiguous,
to
check the
step.
forcing
These
the
experiment itself to be repeated.
This thesis
reports on the method used in seeking the
energy levels of Pr3 + : Y(OH) 3 and the results obtained.
discussion
begins
in
Chapter
2
with
the
The
fundamental
5
theoretical ideas necessary to understand the goals of this
work.
After investigating
discussion
closely
explaining
each
Chapter
3
follows
section
explains
the
in
the
theoretical
flow
So,
arrangement
and
Chapter 4
investigates the exact procedure used to analyze
the
spectra
detail.
used
from
recording
the
chart in Figure 1,
considerable
experimental
aspects,
procedure
the data
in
some
experiments.
from
Chapter
the
5
is
sample.
then a
discussion of the method and tools used to find the correct
Hamiltonian
Pr3 + : Y(OH)
3
and
.
thereby
Finally
the
Chapter
correct
6
energy
presents
obtained and conclusions made from this work.
levels for
the
results
6
CHAPTER 2
THEORY
The main thrust of this thesis concerned recording and
studying spectra.
discussion of
some
It is
therefore
the theory
fundamentals
directed towards
of
natural
to
begin the
behind the project by explaining
spectroscopy.
the specific
Attention
sample studied.
is then
The sample
which this project focused on was trivalent praseodymium in
a crystal
of yttrium
hydroxide (Pr3 + : Y(OH) 3 ).
several aspects
of the
sample which
before studying
the experimental spectra.
There are
should be understood
Specifically we
need to know what possible energy levels to expect and what
the selection
data
are
used
rules are.
After studying the spectrum,
to
construct
a
diagonalized should
yield the
energy eigenvalues.
any modeling can be done,
needs to be introduced.
Hamiltonian,
the full form
crystal field Hamiltonian.
which when
Before
of the Hamiltonian
To accomplish this,
is broken into two parts:
the
the discussion
the free ion Hamiltonian and the
7
Spectroscopy
Spectroscopy
concerns
spectrum for signals
being studied.
scanning
emitted
Generally
or
here wavenumbers,
are
The
electromagnetic
absorbed
frequency is
spectrum but
used.
the
by
the sample
used to index the
which have
units of cm"1 ,
relationship between wavenumbers and the
energy states of the electron is
simply given
by the Bohr
postulate for radiating atoms.
c ( E f - Bi)
h
Here
Ef
aind
_
"
Ei
1_
A
^
refer
to the initial and final energies
respectively and % is used for wavelength.
One
may ask at
this point what good this equation does since there are two
unknowns in energy.
It is
in determining
that spectroscopy finds its greatest use.
these energies
The experimental
section specifically addresses this question.
For now let
it suffice to say that by obtaining many spectral lines one
can logically deduce what the initial and final states are.
The catalyst
for this
deduction lies
in finding spectral
absorption first where the initial energy can be assumed to
be zero.
Bohr's postulate
simply
a
interesting
statement
to
for the
of
wavenumber of the photon is
energy
investigate
conservation.
the
origins
It
of
is
this
8
electromagnetic
radiation
because
it
into the quantum mechanics of the atom.
source of
gives some insight
The most efficient
electromagnetic radiation is the electric dipole
oscillator.
The
power
which
is
radiated
by
such an
oscillator is. given b y (31
P ,
Idl1 '
2
3 ■
Here d
is the
wavevector.
dipole moment and k is the magnitude of the
From this it can be seen that the link between
radiated electromagnetic
energy and the states of the atom
lies in the nonzero value for the dipole moment.
To establish this link it is helpful to
electric
dipole
moment
starting
quantum states of the atom.
calculate the
with the time dependent
The strength
of a
dipole is
given as
d = e|<i|r|f>|.
In this expression,
3
<i|r|f> refers to the expectation value
for the electron’s position when it is between
final
states
and
e
is
its
charge.
To calculate the
expectation value it is necessary to know the
atom.
initial and
state of the
In a one electron atom the state may be expressed in
ket form as |nlsj mj >.
making a
thought of
involved.
During the time that the electron is
transition from
as
being
Upon
in
one state
a
calculating
to another,
superposition
the
of
expectation
it can be
the states
value
an
9
oscillation in time is found as shown by
<iIrIf>
cos(wt).
4
Here w is the angular frequency ( Ei - E f)/h, and translates
as
the
angular
frequency
oscillating
expectation
oscillating
and
of
the
value
therefore
P
emitted
means
in
photon.
the
equation
The
dipole
is
2 is nonzero,
proving the atom radiates.
Figure
2
oscillating
model.
The
a
dipole
graphical
using
the
representation
harmonic
of
the
oscillator
as a
first two graphs in Figure 1 represent the wave
functions
of
harmonic
is
the
ground
oscillator.
and
The
first excited state of the
third
graph
represents
the
superposition state that the electron must be in during the
transition.
If the
probability
distribution
shown in
clearly
superposition
the fourth
shows
evolution
of
oscillating as
the
graph.
the
the
of
nonzero
states
suggested by
also
transition
noticed
were
requirement for
of
squared,
the
electron is obtained as
dipole.
shows
that
the double
the
different
a nonzero
fact that the quantum
arbitrary.
that
is
This asymmetric distribution
the fourth plot and by equation 4.
have
state
Following the time
this
dipole
is
headed arrow over
The c a r e f u l . reader may
states
involved
parit y . 4 ’
dipole moment
numbers involved
This
in
the
is
a
and leads to the
are not completely
1O
(a) Ground State
(b) First excited state
(c) Superposition of states
«6------ >
(d) Probability distribution
Figure 2.
Plots of the harmonic oscillator.
The
restrictions
referred to
as the
placed
on
the quantum numbers are
selection rules.
For
a one electron
atom emitting electric dipole radiation the selection rules
are as follows:
An
unrestricted
Al = ± 1
Aj
= ±1,0
A mj
= ±1,0
These selection rules may be found
mechanics
such
as
Liboff. <3>
in any
Of
course
violations of these rules resulting from
are not dipole in nature.
a good example
nondipole
of
However,
is
very
ignored in this project.
large enough
to worry
there
can
be
transitions which
Electric quadrupole radiation is
this.
transition
book on quantum
If
weak
the
about,
the
intensity
of a
and could safely be
quadrupole
moment were
a new set of selection rules
would have to be introduced.
While the transitions
dipole in
nature,
the
out to be invalid.
we consider
studied
The most blatant
that all
seemingly forbidden
interacts
different parity
of the
project were
violation occurs when
transitions studied occurred
Al equal
transitions occur
with
the
with the
the transitions. t6) When
detail,
this
one electron selection rules turned
within the 4f subshell suggesting
field
in
free
ion,
4f states
considering
to zero.
The
because the crystal
mixing
states
of
and thereby allowing
the
sample
new pertinent selection rules are presented.
in more
These
selection rules turn out
to be
important when determining
energy levels.
Praseodymium as a Free Ion
Praseodymium is a rare earth element in the lanthanide
series.
Like all
unfilled 4f
other lanthanides,
subshell which is spatially located within the
5s, 5p and 6s
subshells.
electrons being
This
shielded.from
in turn gives rise to sharp
atoms
are
praseodymium has an
fixed
in
results
in
the valence
outside perturbations which
spectral lines,
a host lattice.
even when the
Another interesting
property of the lanthanides is the fact that there are many
visible
spectral
lines
resulting from transitions within
the 4f subshell.
The electronic configuration of • the praseodymium atom
is given by
The
[Xe]
4f3 6s2.
atom
which
was
studied was triply ionized.
xenon is a noble element with a relatively
energy,
Because
high ionization
the.three electrons will be removed from the 4f and
6s subshells.
The electronic configuration of Pr3* is then
given by
[ Xe]
Typically,
4f2.
the
states
spectroscopic notation.
s and
j quantum
of
the
system
In this notation,
numbers for
all of
are
expressed in
the sum of the I
the electrons in the
valence subshell are specified.
are
two
contributing
In the case
electrons,
orbital angular momentum of 3.
This
terms
possible
follow!ng
each
for
of Pr3+ there
having
gives
total
a maximum
rise
orbital
to the
angular
momentum:
P
1
S
L = O
D
2
F
3
G H I
4 5 6.
AdditionalIy each electron has
one
half,
suggesting
that
either singlet or triplet.
a spin
each
The
angular momentum of
term given above may be
next goal
is to determine
what order these terms must be in.
In finding what order the term notation must be in, it
is only natural to begin
This
is
easily
Doing this,
with
finding
accomplished
it is found that
by
the
ground state.
applying
Hund's rules.
the ground
state of
Pr3* is
given in term notation as 3H a .
To
find
necessary
in
Hamiltonian.
the
principal
Dieke
accounting of
to
of
all
other
diagonalize
terms,
the
it is
appropriate
The Hamiltonian for the lanthanide series of
free ions has been
Judd,
ordering
and
worked out
Wybourne.
the free
by several
workers such as
The
reader may find a full
ion theory
in Wybourne' s book. (7 1
Most of the discussion on the Hamiltonian presented here is
based on an excellent review of the subject by Hiifner. t6> .
The only atomic Hamiltonian for which
equation
may
be
solved
the Schrodinger
exactly is that for the hydrogen
atom.
This
means
that
for
theory must
be used.
This
Hamiltonian,
as in
equation
representing
the
hydrogenic
the
Pr3+
ion,
perturbation
theory allows us to split the
5,
into
a
system
unperturbed term
and a perturbed term
representing complicating factors such as electron-electron
interactions.
H = Hh s + Hp
The
idea
5
behind
perturbation
theory
is
to
solve
the
Schrodinger equation with the unperturbed Hamiltonian first
and
thereby
states,
a
obtain
matrix
constructed and
a
of
set
the
of
basis states.
perturbation
diagonalized for
energies.
With these
term
may
be
The energy of
the system may then be expressed by
E=E
h s
+ E p.
6
So it is seen that calculating the perturbation energies Ep
is
the
primary
goal
since
the
energy derived from the
hydrogenic term Eh s , simply adds a constant to the total
energies.
With the ideas of perturbation theory in hand,
it is possible to
focus on
the pertinent
Hamiltonian for
Pr3 + .
In light of perturbation theory,
perturbation terms to a
considered.
For
central field
it is clear that only
Hamiltonian need be
trivalent praseodymium the perturbation
Hamiltonian has been given by W. T. Carnall efc al. 1® 1 as
Hf
= S F kFk + ^A50 + ccL( L + 1 ) + 6G( G 2 ) + y F( R 2 )
+ Z M kmk + S P kPk.
The first
term in
this expression
interaction between the
adjustable
parameter.
two
The adjustable
electrons
The
orbit interaction between
represents the. coulomb
with • F k
being an
second term considers the spin
the
parameter for
electron
and
the nucleus.
spin orbit is £.
Spin orbit
corrections appear even for one electron systems,
not
surprising
that
These additional
accounted for
by the
account
relatively
interactions
and
interactions
are
Finally,
cc, B and
the
it is
two electron systems have additional
spin orbit corrections.
for
so
Pk parameters. (9 '
small
the
The Mk parameters
effects
spin-other-orbit
between
corrections are
two
of
spin-spin
interaction.
These
valence electrons./4 ’
y parameters consider corrections due
to higher configuration interactions. <10)
At this point,
if the magnitude of each parameter were
■
known,
then the energy eigenvalues could be calculated.
However,
Pr3+
is usually
The exception to this
the spectra
values,
of Pr 3 +
studied in a crystal or a glass.
was work
done by
J. Sugar studying
.in the vapor state.
Using Sugar's
and studying the values obtained from
of Pr3+:LaCls
and Pr3+:LaFs, ‘8 ’
ion energy levels was made.
Figure 3.
the spectra
an estimation of the free
The estimated levels appear in
Energy
(cm"')
3 P2
22000
'I*
3 Pi
20000
I 8000
1D2
1 6000
1 4000
12000
1 0000
8000
6000
4000
2000
0
Figure 3.
Estimated energy levels for Pr3*: Y(OH) 3.
Praseodymium in a Crystal Field
When the
Pr3 * ion
is in a crystal field,
2j+1 degeneracy of the free ion is partially
is due
to the
fact that
adding the
Hamiltonian removes the spherical
environment.
Naturally it
are
removed.
This
crystal field to the
symmetry from
is necessary
before
some of the
possible
states
spectra.
To determine the these states,
the ion's
to know what the
studying
the
experimental
it is necessary to
know the form of the crystal field part of the Hamiltonian.
To facilitate
convenient
the explanation of finding the states,
to
Hamiltonian
consider
alone.
the
By
crystal
doing
field
this,
with
the
of the
we are effectively
diagonalizing a submatrix of the Hamiltonian.
concludes
part
it is
selection . rules
This section
pertinent
for
Pr3*: Y(OH) 3.
The crystal field results
occupying a
by a
site in
The
Pr3 *
sample studied
ions.
of the
The
primarily by the (OH):
ion.
praseodymium ion
was Y(OH): doped
This means that the crystal was grown from a.
solution such that I %
with
the
the crystal lattice normally occupied
yttrium atom.
with I % Pr3*.
from
The potential
yttrium atoms
crystal
field
molecules which
is
were replaced
then produced
surround each Pr3*
can be expressed in terms of a series
of spherical harmonics.
This means that the crystal
field
Hamiltonian can be expressed by
He f — 2 I m B ^m C *m
Here
the
B' s
are
8
parameters
while
renormalized spherical harmonics.
the
The
C 1m
Y' s
represent
and
Cs
are
related by
The
C
4ir
(2m+1)
reader
may
9
Y 'm
find
a good explanation of crystal field
expressions in Weissbluth' s book. (12 1 So at this point,, Hcf
is expressed
in an
infinite series in the orbital angular
momentum sum which must be truncated.
The triangle relation
for angular momenta immediately limits the series to values
less than 21. ‘12 1
sum
can
only
For two f electrons this means
run
as
high
as
six.
Also,
lattice
with
a
occupies a point in
the
symmetry
allows
and
this
that the
the Pr3+ ion
relatively high
us to further limit the series
representing the crystal field Hamiltonian.
The point group symmetry for the Pr3+ ions is Cah..
the
language
of
group
properties are found in
theory,
Ca
the crystal
means that identical
for each
rotation of
(2/3) ir in a plane perpendicular to the crystal axis.
in
this
notation
properties
are
horizontal plane.
crystal with
refers
found
to
upon
the
a
fact
shown.
that
reflection
Figure 5 on page 26 is a
the axis
In
The h
identical
through
the
drawing of the
By definition of a symmetry
operator,
ion
it must commute with the
Hamiltonian
is
spherically
necessary only to consider
In .other
words,
Hamiltonian.
any
symmetric,
the crystal
The free
so
it
is
field Hamiltonian.
terms in the expansion in spherical
harmonics which do not commute with
have coefficients of zero.
the C 3 h
operator must
This leads to the crystal field
Hamiltonian given by Carnall et al . (8)
Hcf = B20C20 + B40C40 + B60C60 + B66CC66 + C6 - 61
Combining equation 10 with
total perturbation
equation
7
Hamiltonian for Pr2*:Y(OH) 3 .
a total of 18 parameters which must be
parameters,
we
experimental energies.
compared
Chapter
us the
There are
determined in order
to calculate all of the energy eigenvalues.
for the
gives
10
To find values
calculated
energies to
5 explores the method for
determining the values of these parameters in more detail.
No
comparison
energies can
between
calculated
and
experimental
occur until we have experimentally determined
several levels.
the experiment
The methods for determining the levels for
are discussed in the data analysis section.
Here it is only important to note that determination, of the
levels is
and what
dependent on
knowing exactly
selection rules • govern dipole
what levels exist
transitions.
The
number of levels is easily obtained by examining the matrix
elements
of
the
selection rules
crystal
field
Hamiltonian
while
the
can be obtained by knowing the point group
20
symmetry of the Pr3+ ion.
Finding
the
new
states
diagonalizing the submatrices
each of
the 2j +1
is
which
accomplished
are
constructed from
degenerate free ion states and Hc f.
basis states used are
those for
by
the hydrogenic
atom
The
and
are given in the uncoupled representation as
|LSJMj > = S a( mi , ms ) Ym i |sms > .
Here
the
a(m,,m,)
coefficients.
1I
represent .
the
Clebsch-Gordan
Recall that the crystal field Hamiltonian is
constructed from a set of renormalized spherical harmonics.
Because of
this,
a
general outline
of the matrix element
calculation can be presented by examining
one term.
over all
However,
terms
each element
in
Hc f.
the results from
in the
Each
term
matrix is a sum
in
the
matrix is
calculated using the inner product.
< I' s' j' m' j jLMi IIsj IT ij >
Because
we
removed
from each term in the free ion,
possible
to
counterparts.
seek
set
to
12
,
find
I,s
out
and
Writing out
j
how
the mj degeneracy is
equal
the inner
it
to
is immediately
their
product in
primed
terms of
equation 11 results in expression 13.
/ E a* ( m' ,, m' „) Ym ' i< sm' « |t Y m l S a( m 1 ,m$ ) Ym i |sm, > dfi 13
The T which appears
before
the
spherical
harmonic
is a
21
constant relating
B parameter.
confusing,
the Y' s to the C s
This
expression
at
and also includes the
first
so it is helpful to note that the Y m l ' s from the
crystal field Hamiltonian depend only on
The
kets
appears somewhat
representing
spin
exist
spatial variables
in spin space and are
therefore unaffected by the Hamiltonian.
us
to
combine
the
spin
variables
This fact allows
and thereby obtain a
relation between m' « and m, .
<Sm' 5 |sms> = 6 nsa's
14
This simply dictates that for a nonzero element,
m' , and m«
must be equal.
Rewriting
mathematical
equation
problem
13,
lies
it
in
is
clear that the basic
evaluating
the
integral
containing the product of three spherical harmonics.
2 S 2 I ra*a ( Y" ’ ,) * Y m l Yn , dQ
Evaluation of
by using the
this integral
Gaunt
formula.
15
is easily accomplished simply
The
primary
objective in
evaluating the inner product is. to determine which elements
in
the
matrix
are
nonzero:
Therefore
the
following
property derived from the Gaunt formula is important.
-m' i + M + mi = 0
Using this
IG
condition and
we can immediately see
1.
noting the values of M from Hc f,
a relationship
between the orbital
22
azimuthal quantum numbers.
mi = ±6,
This
O
17
information
immediately
allows
us
to
construct
matrices from which it is possible to determine what states
are non-degenerate.
The next problem which must be addressed at
concerns labeling
the states.
this time
To explain the labeling of
the new set of
non-degenerate
states, it
construct
submatrix
the 3Fa term as an example.
the
Figure 4 shows the
the 3Fs
for
nonzero elements
is
in the
term using ' the conditions
helpful to
submatrix for
given in 17.
The dark
outlines in Figure 4 suggest that the matrix can be thought
of as two submatrices.
which
diagonalizes
submatrix,
states.
being
However,
degeneracy)
the
to
The
lower
appears to give five possible
state differing only by a sign in the mi's
The
labeling
of
these
states is
a scheme given by Hellwege and discussed
in Hufner's book. (6 ’
For
C sh this
scheme
defines
a new
p, which is defined by
p = mj ( mod 6) .
1.
two states.
due to Kramer's degeneracy (time reversal
accomplished with
The states
give
diagonal,
are still degenerate.
quantum number,
The upper submatrix is a two by two
18
resulting from the 3 F3 term are listed in Table
Notice that the states from the two by two matrices are
23
3
-3
3
X
X
-3
X
X
mj
=
2
1
0
-I
-2
Figure 4.
2
I
0
-I
-2
X
X
X
X
X
3Fi submatrix for the crystal field
Hamiltonian. Nonzero elements are
indicated.
24
labeled with
the same
p quantum
are clearly different so a
prime
numbers.
The two states
is
to distinguish
used
between them.
3 F 3 ( 0) ,
' 3 Fa(I),
3 Fa( 2 ),
3 FaO),
3 Fa( 3 ' ) ,
Table 1.
Figure
3
P
W
W
P
P
States of the 3Fa term.
lists
all
Pr3+:Y(OH))a.
Energy level
estimated
comparing
by
Pr3+: LaCla,
The
explored
state
state
state
2
3 state
3' state
0
1
=
,=
=
=
=
of
the
possible
positioning on
the
states
in
this Figure is
energy levels determined for
established by Sarup and Crozier . (13’
last
theoretical
before
analyzing
consideration
the
which
experiment
possible transitions an electron can make.
basic transitions
must
concerns
There
which may occur in a crystal:
be
the
are two
radiative
and nonradiative.
Nonradiative transitions may occur via crystal lattice
phonons.
If phonons
away from the excited
are
possible.
are easily
able to carry the energy
ion,
nonradiative transitions
This
event becomes more probable when it
takes one or two phonons to
Y(OH) a,
phonons
exist
then
carry
with
away
energies
the
energy.
corresponding
In
to
wavenumbers on the order of 3600 cm"1 and 900 cm 1. (2 * With
phonons in this energy range, it is relatively easy to have
25
nonradiative relaxations occur.
A
good
example
of this
type of transition occurs between the 3Po and 1 D 2 states in
Pr 3 + : Y(OH) 3 .
This transition
energy separation
readily
between these
occurs
because the
states is on the order of
3600 cm " 1 as seen by inspection of Figure 3.
Radiative transitions are primarily electric dipole in
nature.
These transitions are governed by selection rules
which are given
represent
absorbed
the
crystal
perpendicular
Table
two
photon
polarization,
the
in
possible
may
then
In
this
If
the
its polarization
the
table
ir
and a
polarizations the emitted or
have.
axis.
to
2.
Naturally,
crystal,
axis.
photon . has
a
ir
vector is parallel to
a
polarization
is
This is graphically
represented in Figure 5.
This completes the discussion of the basic theoretical
aspects of the project.
The ideas presented here help make
the experimental arrangement and analysis more plausible.
26
u=
O
-2
-1
Table 2.
3
Selection rules for a crystal
having Cj» symmetry.
4
I polarized wave
Figure 5.
2
crystal axis
a polarized wave
Pr3 *: Y( OH) 3 sample and relevant polarization.
27-
CHAPTER 3
EXPERIMENTAL ARRANGEMENT
To
experimentally
Pr 3 + : YC OH) 3 ,
it
was
determine
necessary
fluorescence spectra.
the
to
energy
study
levels
of
absorption and
Many,of the experimental arrangement
details in the absorption and fluorescence experiments were
similar.
This
similarities
by
section
takes
describing
one
advantage. / of
experiment
in
these
full and
referring back to the details during the explanation of the
second experiment.
and the
The absorption experiment was the first
easiest experiment.
Therefore this section will
begin with a full description of
the absorption experiment
followed by a description of the fluorescence experiment.
Absorption Experiment
One way
of finding the transitions of the electron is
by studying the absorption spectrum.
This was accomplished
by passing
crystal and observing
white light
the resulting
continuous
spectrum.
spectrum
indicate that
through the
so
White
the
they receive
light
of
course
has a
recording instruments simply
a signal
for all wavenumbers.
Since the white light passes through the crystal,
it is
r
28
possible
This
for
some
absorption
parts
shows
of the spectrum to be absorbed.
up
as
a
dip
in
the
otherwise
continuous spectrum.(See absorption plots in the Appendix).
Absorption occurs when the
energy
of
the
incident light
exactly matches the difference in energy between two states
of the ion!
may not
It should be noted
necessarily be
ground state of the
that the
from electrons
ion.
Instead
transitions seen
originating in the
the
electron
may be
starting from
a state with slightly higher energy than the
ground state.
This results from the thermal
excitation in
the crystal.
Recall that Boltzman' s result tells us the probability
that an ion will be in the excited state.
P(E)
= P( 0) e" (E z kT ’
19
Here E is used for energy and k is Boltzman' s
constant.
insure that most of the ions are in the ground state,
sufficient to require that
state with
Plugging
wavenumbers gives
have to
a value
state
corresponding
be
above
to
at
into
a
of about
the
1 Ocm " 1
this
ground
or
more,
temperature
of
assumption was based on studies of
trivalent
praseodymium
in
other
in a
This suggests
relation
in
1.44K/cm"1.
state
it is
of being
energy E be at or below P(O)B"1.
that kT = E.
the first
the probability
To
has
terms of
Assuming
an energy
then the crystal would
at
the
least
energy
14K.
levels
crystals. (13>14>
This
of
The
29
relatively
high
temperature
suggests
submerged in liquid helium (T <
of the
electrons would
be in
4K)
would
the
energy
lines recorded.
It
insure that all
it was
is
also
advantageous
to
compile
on
maximum' amount
of
data
experiments were made at liquid helium,
temperatures were
However,
not used
study the
This can yield information
In an
temperatures.
With the
possible to
about the levels near the ground state.
room
crystal
levels directly from the absorption
crystal at higher temperatures.
a
a
the ground state.
crystal at these very low temperatures,
determine
that
attempt to
the
crystal,
liquid nitrogen and
signals recorded at higher
in the
final analysis because
they were too weak.
With these
ideas in
mind,
we
can move on to examine
how the experiment was designed and run.
point of
view,
the
From a practical
goal of the experiment was to maximize
the sensitivity of the equipment measuring the.signal while
minimizing the
noise.
Therefore,
the following paragraphs
concentrate on the logistics of the
the optical
arrange me nt constitutes
experiment.
However,
is helpful
used.
experiment.
Naturally
the foundation of the
in order to understand the design,
it
to. discuss several aspects of the monochromator
In addition
also consider
to the
optical
arrangement,
we must
the arrangement for signal processing,
is the last aspect discussed.
which
At the heart of the absorbtion experiment was the . 85meter
This
Czerny-Turner
instrument
was
consequently is
of the paper.
Double
produced
referred to
In keeping
best signal,
it was
Monochromator-Spectrometer.
by
Spex
Industries
and
as the Spex for the remainder
with the
goal of
recording the
necessary to design the optics of the
experiment in such a way as to take
full advantage
of the
Spex.
The
Spex
separates
the
many
components
of
the
electromagnetic spectrum by diffracting the signal off of a
holographic grating.
Figure
of the instrument . ( 131
possible
signal
at
6
shows a top view schematic
In order
the
to receive
the strongest
photomultiplier
tube,
it
necessary to fill the diffraction grating with light.
situation also
wasThis
had the additional benefit of improving the
resolution of the signal.
Recall that the
resolving power
of a grating is given by
R=
where R
A / AA
= mN
is the
diffracted light,
of slits exposed.
20
resolving power, Xis the wavelength of the
m is the order and N is the
By inspection of equation 20 it is clear
that the resolution improves
grating as
possible.
the edges of the grating,
Fortunately
the
total number
optimum
upon
using
as
much
of the
Of course if there is light beyond
it can show up as unwanted noise.
situation was realized by simply
31
mirrors
center
slit
grating
grating
mirror
exit
slit
mirror
entrance
slit
photomultiplier
tube
Figure 6 . Top view of the Spex 14018 spectrometer
used in experiments
32
making sure that the
mirror.
signal.
completely filled
the first
This is illustrated by the rays drawn in Figure 6 .
Designing
the
optics
to
send
the
signal from, the
crystal to the spectrometer begins with the first mirror in
the Spex.
This
was easily
accomplished due
to the fact
that the speed of the mirror was given as. f/7. 8 .
that the
speed of
aperture size,
a mirror
relates the
Recalling
focal length and
it can be seen that the relation for finding
the angle between the peripheral rays is given by
0 = 2 arctan(%a/f)
where f
is the
21
focal length and a is the apenature of the
optic.
Figure 7 shows the actual optical
G is
arrangement.
Since
dependent only on the speed of the mirror or lens,
is clear that the
must be
f/7. 8 .
effective speed
By effective
of lens
speed it
3 in
it
Figure 7
is meant that the
actual diameter of the signal traveling from lens 2 to lens
3 is
used to
of lens
3.
combination
find the speed,
This
of
situation
lenses
used.
instead of the physical size
was
a
consequence
of the
The parallel rays running
between lens 2 and lens 3 were achieved by positioning lens
2 with
its focal
point on
the.crystal.
had the advantage of maximizing the signal.
noted that
In addition this
It may also be
the positioning of these lenses was achieved by
focusing a telescope on infinity and
then sighting through
33
f/7. 8 mirror in
Spex
slit of Spex
A/4 waveplate
fi = 2 0 0 mm
lens 3
polarizer
lens 2
75 mm
cryostat
lens I
quartz
halogen bulb
Figure 7.
shielded
crystal y
sample holder
Optical arrangement used in experiment.
34
the lenses focusing on the crystal and then the Spex slits.
The next phase of the optical arrangement includes the.
crystal,
cryostat and light source.
The light source was a
quartz halogen bulb focused onto the crystal by
as seen
in Figure
7.
In order
lens three
to shield unwanted white
light which does not pass through the crystal,
was positioned
over a
from a piece of
slit of
shim stock.
the crystal
comparable dimensions made
This
is illustrated
in the
inset of Figure 7 along with the design of the brass sample
holder.
Finally the cryostat was necessary to submerge the
crystal in liquid helium.
The design
cryostat,
with
with
referred
three
viewed.
of the cryostat is shown in Figure 8 .
to as
windows
a dewar,
through
The sample space of
liquid
helium.
was made
which
the
the dewar
If
out of glass
sample
could be
was usually filled
the liquid helium was not well
isolated from the room it would quickly boil away.
time
was
not
a
factor,
the
several
other
Even if
bubbling would scatter the
signal making it very hard to see.
space,
The
spaces
To isolate
were
the sample
designed
into
the
de war. (2) The first space" adjacent to the sample space is a
vacuum space
filled with
used to
liquid . nitrogen
temperature gradually.
vacuum
insulate.
The
in
The next space is usually
order
last space
step
elaborate insulating system,
to room
is under a common
with the first vacuum insulating space.
this rather
up
Even with
the liquid helium
35
valve to
vacuum pump
Insulating
space
liquid
nitrogen
space
sample
space
Figure 8 .
windows
Cross sectional view of the glass dewar.
36
would still boil.
the dewar
pumped.
could
To eliminate
light scattering bubbles,
was designed such that the sample space could be
By pumping away helium
take
place
superfluid.
thereby
vapor,
changing
evaporative cooling
the
Pumping with a mechanical
helium
into
a
pump to
38 microns
signal,
it is also
eliminates the bubbling.
In
crucial
addition
to
to
optimizing
discriminate
Originally,
the
between
polarization
inserting a polarizing sheet
when reviewing
the
various
was
in the
contradictory data,
polarizations.
determined simply by
signal beam.
Later,
it was found that this
method of discrimination was
unsatisfactory.
for
from the poor response of the
this
inadequacy
stems
Spex in certain regions
polarizations.
of
the
The response
spectrum
under different
of the Spex has been studied
and found to respond poorly with a signal
having
vertical
polarization. M 61
balance
polarized.
the
response,
the
below 13000 cm " 1
The
horizontally polarized signals was good,
to
response
a quarter
to
therefore in order
signal
was
Circularly polarized light was
by inserting
The reason
circularly
easily achieved
waveplate into the signal as shown
in Figure 7.
The
second
experiment is
part
in
the signal
the
of
the absorbtion
processing which starts with the
photomultiplier tube and ends
Figure 9 . shows a
design
with
the
storing
of data.
block diagram for the signal processing.
37
SPEX 14018 DOUBLE
MONOCHROMETER
EXTERNAL
DRIVE
CONTROL
EMI 9558QB
PHOTOMULTIPLIER
TUBE
KEITHLY 416
HIGH SPEED
PICOAMMETER
NORTHERN SCIENTIFIC
575
MULTICHANNEL
ANALYZER
PDP-11 COMPUTER
DISK STORAGE
Figure 9.
Block diagram of signal processing used in
absorption experiment.
38
This diagram begins
with
the
EMI
9558QB photomultiplier
tube which initially sees the signal from the spectrometer.
This signal is
then
sent
to
a
Keithly
416
high speed
pi coammeter which in turn registers a current.
the
tube,
appropriate
and
voltage
applied
to
By choosing
the photomultiplier
the appropriate scale on the pi coammeter,
possible
to
obtain
picoammeter's
panel
a
full
scale
meter.
deflection
it was
on
the.
An absorption transition is
found when the signal disappears momentarily while scanning
the spectrum.
The pi coammeter then passes whatever signal
it receives to
the
analyzer
which
digitally.
the
Northern
is
Scientific
capable
of
The multi channel
stepping
motor
in
575 multichannel
storing
the
analyzer was
the
Spe x.
information
interfaced with
This
allowed
instrument to scan along with the spectrometer.
this
synchronization,
it
was
possible
Because of
to
relate
particular address to the appropriate wavenumber.
step in
This
the process
was
easily
the
a
The last
was to store the data on floppy disk.
accomplished
because
the.
multichannel
analyzer was interfaced to the PDP-11 computer.
The procedure used in the absorption experiment was to
simply scan the spectrum for absorption
lines.
absorption
and
line
was
found
Pr 3 + >Y(OH) 3 , the f irst line
After
finding
■began
by
the
scanning
by
to
error.
found was • 3 Po at
first line,
3Po
trial
The first
For
20438 cm-1.
all subsequent experiments
insure
that
the
system was
39
actually
recording
signals
With all systems set,
recording began.
16000 cm"1.
the tedious
process of
scanning and
The region scanned was from 23500 cm " 1 to
Most of the absorption experiment
liquid helium
at 77
originating from the crystal.
K.
temperatures,
Generally
but
however,
was done at
some lines where recorded
the
data
at
the higher
temperatures was too noisy to interpret due to the nitrogen
boiling.
Fluorescence Experiment
In
brief,
measuring
and
the
fluorescence
recording
ions in the crystal.
excite the
ions.
experiment
is
experiment.
view,
the
experiment
signals
emitted by excited Pr 3 *
This means that the first
After exciting the ions,
similar
in
principle
to
step is to
the rest of the
the
absorption
In other words, from an experimental point of
goal is
to find
as many
fluorescence lines as
possible with the least amount of noise.
goal,
consists of
In obtaining this
it was found that the biggest difference
absorption and
between the
fluorescence experiments came in the signal
processing techniques.
Before the fluorescence experiment could begin,
necessary to
corresponds
examining the
it was
tune the exciting laser to a wave number which
to
a
fluorescence
energy level
state
diagram for
Argonne National Laboratories report ( 81
in
the
ion.
By
Pr3+ given in the
it was
found that
40
three states
pump)
the
fluoresce:
ions
wavenumber of
a
1 D2 ,
tunable
the level
3Po and
dye
3 Pi. . To excite
laser
whose energy
contain
a
brief
set
to
the
was determined from
results of the absorption experiment.
follow
was
(or
The paragraphs which
description of the laser used to
excite the ions.
One
of
the
spectroscopy was
most
significant
the advent
was this laser which was at
experiment.
The lasing
organic dye
tunable
dissolved
over
a
the heart
the dye
ethanol.
Since
in this
range containing the
This selection can be made by examining
dye
manufacture's
data.
The
project were • coumarin and rhodamine at
To
to
frequency,
the
dye is
it was first
20400 cm " 1 and 16500 cm " 1 respectively.
tune
It
laser is an
each
band of frequencies,
the tuning curves from the
dyes used
laser
of the fluorescence
necessary to choose a dye. which has a
desired frequency.
in
of the tunable dye laser.
material of
in
finite
advances
laser
to
the
desired
understand how
it
is
instructive to investigate the structure of the laser.
All of the lasers used
University
students.
by
Dr.
R. L.
were
Cone
built
and
at
Montana State
his previous graduate
The design of the dye laser was based on a paper
written by T. W. Hansch in 1 972. *1 71
the laser are shown in Figure 10.
The basic components of
The system begins with a
repetitively pulsed nitrogen gas laser.
41
sample
fluorescence
signal to
Spex
focusing
lens
<
partially
reflecting
mirror
nitrogen gas
laser
mirror
photodiode
telescope
beam
expander
variable
angle
grating
Figure 10.
partially
reflecting
mi rror
oscillator
dye cell
amplifier
dye cell
mirror
Schematic of system used in fluorescence
experiment including the Hansch type dye laser
arrangement.
42
The.nitrogen
the dye laser.
tight
four
laser was used as the pumping source for
The laser
foot
long
is basically
cavity
output end and a plane mirror
gas is
excited transversely
running the length of
the cavity
made up
with a glass window on the
on the
back.
with two
the cavity.
The nitrogen
pairs of electrodes
To
operate the laser,
is first pumped out by a mechanical vacuum pump
and then nitrogen gas is allowed to flow in.
constant
of an air
pressure,
the
chamber
is slowly pumped while a
fresh supply of gas continued to flow in.
typically held
To maintain a
The
at 48 mmHg during operation.
flow of nitrogen gas in the chamber,
pulsed across
pressure is
With a stable
approximately 25 kV is
the electrodes at a repetitive rate of 6 Hz.
This high voltage
excites
nitrogen molecule.
the
electronic
states
of the
The lifetime of these excited states is
quite short (on the order of I 0 nsec) , and consequently the
nitrogen laser
emits bursts
10 nanoseconds
long.
shaped and
in the
mirrors on a
The
of radiation typically
resultant beam
violet region
Newport
Research
about
is spatially fan
of the
spectrum.
Corporation
Using
4'x 8 * optical
bench,
the fan shaped beam is split up, part being directed
to an
oscillator ' dye
cell
and
part
being
sent
to an
amplifier dye cell.
The discussion
begin
with,
amplifier
the
cell
may now
difference
should
be
focus on
between
clarified.
the dye lasers.
the
To
oscillator and
Simply
put,
the
43
oscillator cell is within the oscillating
amplifier is
outside the
cavity are made up of
cavity.
a
632
cavity while the
The boundaries of this
lines
per
inch diffraction
grating at one end and a partially reflecting mirror on the
other end.
Changing the angle
changes the
because
grating
particular frequency
section,
allows
for ■ feedback
in
This is
only
one
giving rise to a preferential gain at
frequency.
As
pointed
out
in
the theory
the best resolution from a diffraction grating can
be obtained by
This is
diffraction grating
frequency at which the dye will lase.
the
the desired
on the
illuminating
accomplished by
as
many
lines
as possible.
inserting a telescope arrangement
between the dye cell and the grating which acted as
expander.
a beam
The dye solution itself was contained in a cell
known as a cuvette-type
design which
has a
dye cell.
This
is a rectangular
magnetic stirrer in the bottom to help
keep the dye solution homogeneous during
operation.
the beam
it passes through
leaves the
an amplifier cell.
beam.
Since
laser,
oscillating cavity,
The amplifier then simply
the
After
enhances the
experiment was dependent on tuning the
it is productive to outline the tuning procedure.
To set the dye laser,
the wavenumber.
To
there must be
measure this,
a way
the beam is directed to
the slit of the
spectrometer which . is covered
paper
The
card.
card
is
in
to measure
place
spectrometer slits by diffusing the light.
to
by a white
protect
The
the
signal is
44
then
seen
by
the
Spex
photomultiplier tube.
From
and
consequently
here the
by
the
signal goes directly
to an oscilloscope where it can be viewed.
To see the signal on the screen it is necessary to set
the time constant on the scope.
the
RC
signal.
time
constant
of
This
was done
the circuit which delivers the
The resistance of the circuit can be determined by
simply knowing
the input
resistance of the scope.
experiment a Tektronix 485 oscilloscope was
50
ohm
by finding
or
one
megohm
input
used which has
resistance.
The value for
capacitance can be determined by knowing the length
coaxial cable used to transmit the signal.
these cables is given as 30 picofarad per
constant
generally
turned
out
to
of the
Capacitance for
foot.
be
In the
The time
approximately 0. 3
milliseconds using the 1 megohm input.
With the
set, the
beam hitting
wavenumber of
with the Spex.
the Spex
and the oscilloscope
the laser can be found by scanning
Once the
wavenumber is
found,
it
can be
changed by adjusting a micrometer attached to the dye laser
grating.
Setting
conceptually
practice it
a
the
very
seems
to
wavenumber
simple
take
thing
at
of
to
least
the ' laser
do.
twice
is
However in
as
long as
initially planned.
After tuning
the crystal.
using a
The
the laser,
beam
convergent lens.
was
the beam
positioned
was directed on to
onto
the crystal
A logical question, at this point
45
is:
how do you know when the beam
is hitting
the crystal?
This could be determined by actually looking at the crystal
and watching for
adjusted.
fluorescence
as
the
beam
This turned out to be a tedious process indeed.
The
optical
arrangement
for
the
experiment was the same as that used for
that the
exciting beam
was incident
fluorescence
absorption except
on the crystal at 90
degrees from the optical axis in Figure 10.
looking for
fluorescence lines
absorption lines.
states in
the ion
However,
the
A block
lifetime
of
the excited
was very short (approximately 150 nsec)
diagram of
processing system.
the signal processing system used
in the fluorescence experiment is given
in the absorption experiment,
and ends as data on a disk.
part of the experiment
used for
was
in Figure
11.
As
the signal starts at the Spex
The first
photomultiplier tube
cooled tube
In principle,
was similar to looking for
and this led to a more elaborate signal
in the
position was
used.
the
difference is found
The tube used for this
RCA
weak signals.
C31034A
which
is a
The second and major
difference in the signal processing was the introduction of
the signal
averager.
was
as
known
produced by
the
To be more precise,
boxcar
averager,
Princeton Applied
this instrument
gated
Research Company (PARC).
gate has the effect of
allowing
data while
is present. (gate open)
data in
the signal
between signals
integrator,
the
instruments
(gate closed).
This
A
to take
and not take
leads to a
46
SPEX 14018 DOUBLE
MONOCHROMETER
EXTERNAL
DRIVE
CONTROL
RCA C3034A
PHOTOMULTIPLIER
TUBE
Trigger
DELAY LINE
SCOPE
PARC MODEL 162
BOXCAR AVERAGER
Gate
PARC MODEL 1 64
GATED INTEGRATOR
Trigger
NORTHERN SCIENTIFIC
575
MULTICHANNEL
ANALYZER
PDP-11 COMPUTER
DISK STORAGE
Figure 1 I.
Block diagram for the signal processing used in
fluorescence experiment.
47
significant reduction
of noise.
. The boxcar was introduced
because it provided a gate and could be used for very short
signals.
The
boxcar
had
the
the
pulse
advantage of
integrating
and
repetitions.
To understand how the boxcar was adjusted and
used,
averaging
additional
over
several
it is helpful to trace the path of the signal.
The signal began with, the laser
the crystal.
The
rise time
exciting the
ions in
of the signal corresponds to
the amount of time the laser was on which was about 5 to 10
nanoseconds.
After
spontaneously emit
This time
ions
photons over
signals
a finite
which
nanoseconds.
During
emitted,
photomultiplier
the
oscilloscope.
So,
been
excited
they
amount of time.
This time should be compared to the amount of
between
to adjust
have
was observed on the oscilloscope to be about 150
nanoseconds.
time
the
the
is
time
about
that
sends
0.17
photons
a
billion
are being
signal
to
The signal is sent to the scope first,
the size
and delay
of the
the
only
gate on the boxcar.
one channel of the scope shows the signal and the other
channel shows the gate.
Both the scope and the boxcar were
triggered by a photodiode which picked up stray
light from
the dye laser (see Figure 10).
At this
point it
is necessary to consider the amount
of time it takes for the photodiode to
and consequently
response time
for
for the
the
boxcar to
PARC
1 62
trigger the boxcar,
respond.
The nominal
is , 75- nanoseconds. (18)
48
Because
of
this
the response of the gate,
the
signal was arriving before the gate could be initiated.
To
rectify this
delay
in
situation,
a
50 ohm delay cabIe was cut to a
specific length in order to induce
in the
signal.
a 100
This allowed.both the signal and the gate
to appear on the screen of the scope.
duration
and
delay
time
where
been made,
removed
and
the
scope
Finally the aperture
set to match the signal.
After this adjustment has
from
nanosecond delay
put
the signal
on
line was
the PARC 164 gated
integrator.
The PARC I 64 gated integrator operates by sampling the
input
signal
a
number
exponential average.
storing the
of
times
Physically,
and
then computing an
this
can
be
voltage from each input signal on a capacitor.
Unfortunately a capacitor can leak off some of
particularly when
to the
aperture
experiment.
the time
To
solve
allows
one
to
its charge,
between gates is long compared
duration.
This
the
digital storage option is
option
done by
was
leaky
the
case
capacitor
available on
take
the
in the
problem,
the boxcar.
a
This
analog input signal and
convert it to digital format where it can be stored without
loss.
After
averaging,
the signal is converted back to
analog form and sent to the multichannel analyzer.
Before leaving the boxcar averager,
it is important to
understand the time it takes the instrument to respond to a
signal,
for this directly effects' the
rate
at
which the
49
Spex can scan.
is
known
The time it takes for the boxcar to respond
as
the
observed
time
constant
(OTC)
and is
calculated b y <1S)
OTC =
Here AD
( AD)
TC
( REP)
refers to
22
the aperture
duration time,
time constant selected on the instrument
experiment)
Hz).
and
For
psec
(1
fluorescence,
seconds.
So,
a signal was approximately
experiment,
the
allows
the
2 cm-1,
,
output
OTC
was
since the wavenumber spread of
then the
maximum safe
scan speed for the Spex was 0.5 cm " 1 per second.
speed
for this
REP is the repetition rate of the laser ( 6
the
typically 1.6
TC is the
signal
to
reach
This scan
a signifcant
fraction of its maximum gain.
After the
signal leaves 'the boxcar,
is essentially
Plots
for
the same
both
absorption
given in the Appendix.
two types
as in
of plots
and
signal processing
the absorption experiment.
fluorescence spectra are
The primary difference
is the
baseline.
between the
In the fluorescence
experiment the baseline corresponded to zero signal and was
positioned
at
the
negative signals.
the
voltage
top
of
the
MCA screen to reveal the
The signals were negative simple because
applied
to the photomultiplier was negative.
The baseline for the absorption experiment
be at
the top of the MCA screen.
was set at the bottom of
appears to also
In reality,
the screen
while the
the baseline
signal was
50
blocked.
This allows us to develop an idea of the strength
of the absorption line.
zero line,
For
instance,
if
a line
hit the
we could conclude that there was 1 0 0 % absorption
in that region of the spectrum.
To this point,
not been
the
discussed.
precision of
the measurements has
Because the raw data is read off the
Spex,
the question boils down
to
Spex
is.
a calibration experiment was
To
answer
preformed.
This
spectrum of
a hollow
this
experiment
consisted
was done, ( 2 01
of the
how
of
precise the
recording the
cathode iron-neon lamp and comparing
the results with those published by
of Physics. <19)
asking
After an
the American Institute
analysis of the calibration data
it was concluded that the wavenumber read off
Spex was shifted by -1.83 ± 0. 09 cm" 1 from the true
values throughout the spectrum.
51
CHAPTER 4
DATA ANALYSIS
Upon leaving the laboratory,
quantity of data and
it.
Therefore it
little
organization
associated with
was necessary to first sort through the
data and neatly organize it.
found for
one typically has a large
organizing the
The ■ most
efficient method
data was to plot the spectra out
and then arrange the plots in a notebook in order
spectral range.
been recorded,
Once it was clear what spectral lines had
a
table
transitions which
Then
by
of their
making
was
constructed
could have
guesses
of
all possible
produced the observed lines.
from
the
tables,
a
chart
of
experimentally determined energy levels was produced.
During
the
experiments,
spectrum was recorded by
an analog
and then stores the
digital
if we scanned
of addresses
over
from
the multichannel
instrument converts
maximum number
data
1024
scanning
analyzer.
the
This
signal to a digital signal,
count
in
an
address.
available is 1024.
wavenumbers,
would store
the total
wavenumber.
The raw data appears as a
then
signal recorded
The
Therefore
each address
after scanning one
data
the actual digital count in that address.
address,
and
To determine the
52
actual wavenumber where a spectral line
has been recorded,
it was necessary to know the initial wavenumber of the scan
as well as the
number of
wavenumbers in
each
address.
The algebraic expression was simply given by
W6 = W i - C ( A ) .
Here Wi
23
is the initial wavenumber,
per address,
c represents wavenumber
A is the address of the
represents the
spectral line
wavenumber of the line seen.
and W 6
With the data
in hand as well as a way to calculate the wavenumber of any
line seen,
it was time to plot out the data.
The absorption
and fluorescence
spectra were plotted
on the HP7470A graphics plotter which
Basic
program . run
Basic program was
student in
plot,
MCA
on
a
written
NEC-APCl11 microcomputer.
by
D.
the Physics department.
it was first necessary to
into
an
x-y
was controlled
format.
Macpherson,
by a
This
a graduate
In order to generate a
convert the
data from the
This was accomplished with a
program written in the C programming language.
The plots
can be examined in the Appendix.
With all of the data neatly plotted it was possible to
make a table containing all of
transitions.
the experimentally observed
The crux of the situation was that the energy
levels involved in each transition were yet unknown.
at this
point that
that is,
deciding on which
each transition.
the hard
two
It is
part of data analysis began;
states
where
involved in
53
In
order
transitions,
the
upper
to
know
anything
about
the fluorescence
it was necessary to know the energy
states.
This
levels Of
dictated the need to label the
states involved in the absorption transitions
first.
was
all.
because
at
very
low
temperatures,
transitions could be assumed
state,
3 H a ( 2) .
■the ground
Knowing
of
the
from the ground
Here the p=2 quantum number was assigned as
state term
that
to originate
This
the
initial
immediately implies
transition exactly
by inspection
that
of Sarup' s work. (1 3 '
value
the
of
energy
corresponds to
energy
value
is
seen
zero,
in the
the energy of the final
state.
But how is the upper state labeled?
comparing the
ion energy.
worked out
energy found
in the
The energy levels
in Wybourne's
group
which
an
transition belongs to,
to
be
assigned.
transition to the free
of the
free ion
have been
book17 ’ and an estimation of the
labeled levels is given in Figure 3.
general
This was done by
upper
After deciding on the
level
in
an
absorption
the appropriate \i quantum number had
This
was
accomplished by knowing the
polarization of the transition and then using the selection
rules presented
in Table
2.
In most cases the assignment
of the upper state could be made without
to the
fact that
spaced.
3 P 0 state
the levels
A good example of
which is
in the
this is
any ambiguity due
free ion were widely
the. assignment
somewhat separated
of the
in the spectrum at
54
20438 cm-1.
always as
Unfortunately,
clear,
particularly
the ground state.
ambiguous,
so
the assignment of states is not
if the ion did not start in
Fluorescence transitions
it is
are terminally
appropriate to discuss the assignment
of states specifically for this case.
Before the ion can
excited state.
fluoresce,
Initially
absorption transition
upper fluorescence state with
obtain a
must
start
in an
the 3Po state was pumped by the
dye laser since the absorption for
A strong
it
stronger signal.
this level
allows us
more
was strong.
to populate the
electrons
and thereby
Because the 3Po state is pumped
does not mean that all of the fluorescence
originates from
this
in Chapter 2.on
state.
Recalling
nonradiati ve transitions,
the
discussion
it is seen that transitions could
easily originate from the 1Dz state.
This state of affairs
necessitated the listing of all possible
states for
each transition.
given in Table
3.
The
An example of such a list is
list
finding the
difference between
in the free
ion
which
transition seen.
Then
would
was
to find
constructed
by first
the upper and lower states
roughly
correspond
to the
by knowing the energy level of the
upper state from absorption data,
be used
initial and final
the selection rules could
all of the possible lower states..
It was
at this point that educated guesses had to be ,made for each
transition.
55
Table 3.
Example list of possible transitions responsible
for spectral lines recorded.
Level determined
(cm"1)
Line recorded
( cm" 1 )
Possible transition
16176.6 a
=Po(O)
- = H6( 2)
=H6 (2 )
4261.7a
=Po(O)
- = H6( 2 ' )
= H 6 ( 2' )
4261.7
1 D 2 (0)
- = H4( 2' )
= H 4 (2* ) = 296. 4
1 D 2 (I)
- =H 4 (I)
=H4 M )
= 627. 4
1 D 2 (I)
- =H 4 (O)
= H 4 ( 3)
= 627. 4
- =H 4( 3')
= H4( 3' ) = 627. 4
- = H4( 2' )
= H4( 2' ) = 586. 4
. 1 D 2 (I)
1 D2( 2)
1 2304. 8 V
. 1 D2C 2) ■- = H4( 0)
=H 4 (O)
= 586. 4
=Po(O)
- 1 G 4 ( 3)
1 G 4 (.3)
= 8133. 5
= Po(O)
- 1 G 4 ( 3' )
1 G4( 3' ) = 8133. 5
-1 D 2 (I)
- = H6( 2),
=H 6 ( 2)
1 D 2 (I)
- = H6(2' )
= H6( 2' )
1 D2( 2)
- =H 6 (I)
=H 6 (I)
= 4458. 2
1 D2( 2)
- =H 6 ( V )
=H6 ( V )
= 4458. 2
1 D 2 (O)
- =H 6 (S)
= H6( 3)
= 41 6 8 . 2a
- = H 6 ( 3' )
=H 6 (3')
. 1 D 2 (O)
*■ indicates the final selection.
= 4499. 2
4499. 2
4168. 2
56
A
reasonable,
question
were the guidelines used
at
in
this point might be:
the
selections,
the expression,
educated guesses?
listed here
order
guideline
in
is
that
pumping laser.' s
of
no
could
be
that the
low as 11000 cm-1..
Therefore,
initially.
had
This
importance.
The first
state with energy greater than the
energy
transitions
legitimizing
Several guidelines are
their
condition considers
the
what
to
be
suggests
considered.
A second
spectrum was scanned only as
the electrons'
in
that
the
3Po
there
involved in
or
are
1Ds groups
four possible
initial states given by:
1 D2C 0) ,
Most tables
of possible
these initial
that
only
the
consulting
p
However,
=
0
Finally,
Sarup's
fluorescence and
1 D 2 (2),
3 Po( 0) .
transitions included
states.
transitions.
paper was
1 D2( 1) ,
in
states
the end
resulted
selections
paper
absorption
all four of
were
which
in . observable
made
listed
signals.
it was found
by
all
Reliance
again
observed
on this
kept to a minimum since the phonons in LaCla are
different from Y( OH) 3.
After the data has been taken,
compiled
listed,
and
possible,
plots made,
transitions
states involved in the transitions
final selections of the states involved in each
57
transition were made.
From these selections it was easy to
obtain the experimental energy levels.
58
CHAPTER 5
COMPUTER FITTING THE HAMILTONIAN
After
taking
and
analyzing
the data,
attention was
directed towards finding the magnitude of the parameters in
the
Hamiltonian.
In
short,
this was done by initially
guessing the magnitude for each parameter
the Hamiltonian
for its
eigenvalues in
hand,
it
eigenvalues.
was
and then solving
With the calculated
possible
to
compare these
values to those obtained experimentally.
Measuring the fit
was done by a
The fit generally
least squares
was unsatisfactory
which meant
be adjusted and the
the
full
form
parameters,
analysis.
that the parameters had to
calculation repeated.
of
the
it is clear
perturbing
that
appropriate
calculation be
Hamiltonian
calculating
even once is a time consuming task.
the
Recalling that
Hamiltonian
the eigenvalues
require
of times.
that
National
program
was
familiarity
parameters,
Laboratories. <8>
functioning
properly
with
effects
the
data from a
known
To
as
of
example
a group at
insure
well
that
as
varying
was
this
Fortunately I
was provided with a computer program written by
Argonne
18
The process of finding
may
repeated hundreds
has
the
gaining
different
given
to the
59
computer and
this done,
a fit
for the
Hamiltonian
my data was used
in
the
was found.
program
to
With
seek the
appropriate Hamiltonian for Pr 3 + : Y(OH) 3 .
The Fitting Program
The program
was written
11/780 computer.
Its
eigenvalues and
level
basic purpose
to
not calculated
the
levels,
the
program
experimental data. Based on
adjust any
since a
transition from
then repeat
to calculating
fit,
the
program would
the operator had specified, as
and
was free
to vary
desired.
Another possibility
ratio to it.
configuration.
a least squares fit to the
the
parameters which
holding
calculate the
In addition
did
variable,
while
was to
ground state would produce a spectral
line above the visible region.
the
and run on a VAX
the eigenvectors of the 4f
The 1 So level was
this
in fortran
the calculation.
The operator
as many parameters in the Hamiltonian as
one
or
was
to
vary
one parameter
more other variables in a constant
The maximum number of
iterations the program
would execute was specified by the operator.
In this work,
eight iterations was specified as the maximum.
Rarely did
the
number
program
iterations.
change in
actually
complete
the
maximum
This was because it terminated as
of
soon as the
the variable parameters became small enough such
that the fit was unaffected.
Monitoring the
performance
of
the
program
was the
60
responsibility
quantities
of
the
calculated
operator.
during
There
each
were
iteration
several
that
were
helpful in monitoring the fitting process.
Probably the
sure that
determined
the.
most Important
calculated
by.
Hund's
ground
and
number
say
was
trivalent
nothing
obtained
of
by
absorption line
The
make was to be
matched
However,
term
notation
the
Hund's
of
of
it
The p
work
on
and from examining unambiguous
cm " 1 with
seen was an
a ct polarization.
inspection of the free ion energy levels and
rules,
rules
the ground
previous
confirming transition
at 20438
the one
p quantum number.
analysis
praseodymium ( 131
transitions seen.
state
rules.
determine only the free ion
state
check to
By
the selection
is seen that the only states giving rise to this
transition would be 3 H*(2) and 3 Po(O).
Since
the line was
recorded with the crystal at liquid helium temperatures,
it
is clear that the ground state has a p quantum number of 2 .
Recalling the
discussion in
Chapter 4, it is clear that a
different ground state erodes
the
other
selections
ground state,
the run
responsible parameter
the credibility
made.
was
So
upon finding a shifted
immediately
was noted.
for most of
abandoned
and the
This situation occurred
many times.
Each time the program
it calculated
the
error
bar
an error
grew
changes a
bar for
to
a
specified parameter,
the parameter varied.
significant
fraction
If
of the
61
parameter itself,
the fit was viewed with skepticism.
quantity usually remained under
ten percent
This
of the varied
parameter and was not a significant problem during the many
runs made.
Finally,
the
followed very
least
squares
closely.
are reasonable,
performance of
the
fit
As long
fit
had
the run.
quantity
<j ,
was
as all other indications
the
last
In fact,
word
as
to the
the goal of the project
was to feed all of the data collected from the experiments,
and obtain a minimum in a.
But c will never be zero.
This
is because there are always some errors in
both the theory
and
is
the
measurement.
examining the
lines look
For
plots in
example
it
the Appendix
like delta
functions.
clear
that not
from
all of the
Broad lines can easily
produce errors in the determination of a line.
Fitting to Data from Pr3-l^ L a C l s
The first step in
to make
sure the
the computer
program was
gain familiarity with the
fitting procedure was
functioning properly and to
Hamiltonian.
This was
done by
data
for
Pr 3 + : LaCl3. from Sarup' s work, (131
using the free
ion
Hamiltonian
entering
parameters
Argonne National Laboratories (ANL)
To
begin
with,
including 1 So term)
with
the
free
ion
45
were
of
the
entered
parameters
gi ven
and
in the
report . ( 81
60 possible levels (not
into
from
the
program along
ANL.
The free ion
parameters
were
attention
was
terms.
Sarup's
parameters,
assumed
directed
so
to
first
paper
B6 &
be
and
reasonably
towards
gives
close
so
the crystal field
values
for
the
field
B6O were entered using the given
values while B2O and B4O were entered as arbitrary numbers.
The
first
run
consequently
varying B2O and B 4 o.
chosen first
These
the.
effects of
parameters were
because they have nonzero matrix elements for
B66.
This
The results
Sarup.
turns out not
of this
program quickly converged on the
reported by
on
crystal field
every term in the free ion.
case with
focused
run showed that the
values
for
Numerical results
Table 4, run number 1.
This Table
to be the
B2 o
and B4O
can be examined in
provides an
example of
the type of organization used when running the program.
The second
run concentrated
on the effect of varying
B6O with B 6 6 in constant proportion
using constant
R.L.
Cone
to it.
idea of
proportions arose from discussions with Dr.
after
crystals. <2) As
reviewing
the
results
run are
from
other
in run 1, convergence came quickly and the
values closely matched those given by
of this
This
given in
Table 4,
Sarup.
The results
run number 2.
Notice
that the values for B2o and B 4 o had been updated with those
obtained in run number 1 .
After varying the crystal field parameters,
vary the free ion
parameter.
parameters starting
Starting
with
the
I began to
with the spin-orbit
values
given in the ANL
63
Table 4: Numerical results from fitting program using data
from Pr 3 +:LaCl 3 .
RUN NUMBER : 1
VARIATIONS:
LUMPED PARAMETERS:
Ea v = 9928. 0
M2
= 0. 986 . ST ARTING
B 2 0 = 200.0
F2
= 68368. 0
M 4 . = 0. 669
B 4O = -550.0
F4
= 50008. 0
P2
= 275. 0
F6
= 32473. O P 4
= 206. 3
oc
= 22900. O b P 6'
- 137.5
ENDING
B
= -674.0
B20 =
B 2 o = 108.4
Y
= 1 520.0
B 4o =
B 4 o = -320. 8
Z,
= 750. 0
B 6 o = -677. 0
M0
= 1 . 760
B66
= 466. 0
PARAMETER
ERRORS:
B 2 0 = 16.5
B 4 o .= 39. 0
SIGMA:
<7
= 19.6
RUN NUMBER : 2
LUMPED PARAMETERS:
E a v = 9928. 0
M 2 = 0.986
F2
= 68368. 0
M 4 = 0. 669
F4
= 50008. O P 2
= 275. 0
F6
= 32473.0
P 4 = 206.3
cc
= 22900. O b P 6
= 1 37. 5
B
= -674. 0
B 2 0 = 1 08.4
Y
= 1 520. 0
B 4 o = -320. 8
C
= 750. 0
B60 =
M0
= 1.760
B66 =
PA RAMETER
VARIATIONS:
ERRORS:
STARTI NG .
B 6 o = 36.3
B 6 o =' -1 000. 0
B 6 o = -. 6 9 B 6 6
ENDING
B 6 o = -677. 0
SIGMA:
a = 17.7
RUN N UMBER : 3
PARAMETER
VARIATIONS:
LUMPED PARAMETERS:
ERRORS:
STARTING
Ea v = 9928. 0
M2
= 0. 986
F2
=13.0
,F 2
= 68368.0
F2 =
M4
= 0. 669
F
4
= 39. 0
F4
= 50008.0
F4
P2
275. 0
F
6
= 28. 0
F6
= 32473.0
F6 =
P4
206.3
<c
= 22900. O b
P6
= 137.5
SIGMA:
ENDING
B
= -674. 0
B 2 0 = 1 08.4'
F2
= 68359.0
Y
= 1 520. 0
B 4 O = “--320. 8
F4
= 50009.0 . a = 8. 2
£
=
748.0
B 6 0- = -677.-0
F6
= 32728.0
M0 = 1 . 7 6 0
B 6 6 = 466.0
* All values in units of cm " 1 .
b The program uses 1 000«:, which is expressed as cc here.
64
report,
the
iterations,
program
claimed
showing
almost
convergence
no
within
improvement
in
Numerical data from this run is not shown here,
value obtained for
The
next
£
the fit.
but the new
was used in all subsequent runs.
parameters
which
were
electron-electron interaction terms.
for these
three
parameters were
varied
The
those given
were
the
initial values
by the ANL report.
As is typical for
this program,
convergence came quickly.
Scanning
the
it
through
results,
was
found
that
the
differences between calculated and experimental eigenvalues
was not
constant in the sense of being all positive or all
negative.
produced
On
a
the
other
constant
results of this run
hand,
negative
are given
the
spin-orbit parameter
difference.
in Table
The numerical
4, run
number 3.
Notice that the improvement of the fit is considerable.
With all
of the
varied and adjusted,
smaller
correction
major parameters
attention
terms.
First
Starting with an arbitrarily
eigenvalues
showed
experimental values.
large
large
Even
was
in the Hamiltonian
directed
among
value,
positive
with, this
towards the
these
was.
oc.
the calculated
shift
from
the
large initial shift,
convergence came within three iterations.
Small correction
terms similar to
By starting with
oc
arbitrarily shifted
are given by B and Y.
values,
a
large negative shift in the
calculated eigenvalues was observed.
the
program
converged
within
Once
again however,
three iterations with very
65
little improvement in the fit.
independently was P2.
the
ANL
The last
Following the procedure suggested in
report,(8>
P4
and
P6
were
proportions to P 2 during this run.
would contribute
initial value
parameter varied
little
in
constant
Assuming this parameter
towards
was that
held
improving
the
fit,
given in the ANL report.
did show some change in the
parameter which
the.
This run
would suggest
that it should be varied when trying to fit the Pr 3 + : Y(OH) 3
Hamiltonian.
As a final test,
same time
all parameters
except for
the M's.
were set
free at the
The M's were held constant
because it was felt that these
parameters would contribute
very little to the fit.(2) The initial parameters used were
those obtained in
the
first
7
runs.
Convergence came
quickly and the fit improved little.
In
review,
it
was
observed
that the crystal field
parameters and the electron-electron interaction parameters
( F2, F4, F6) made the most significant contributions to the
fit.
Additionally,
it is
assumed that
by reviewing the
3P
and
1 I.
J. Sugar 1 1 1 1
stronger crystal fields produce bigger
shifts in energy levels,
as
work of
particularly the upper levels such
Therefore
when
experimental data to the Pr3*: Y(OH) 3 .
were made to utilize these findings.
attempting
to fit the
Hamiltonian,
efforts
66
Fitting to Data from Pr3t: Y( 0 H ) 3
Fitting the
Hamiltonian to data from Pr3+:LaCls is an
admittedly artificial example.
that the
This
is
due
to
the fact
most difficult and critical decisions had already
been correctly
constructing
spectra.
made.
a
Those decisions
table
of
were involved with
energy levels from the observed
When fitting to Pr 3 + :.Y(OH)
3
data,
it must be kept
in mind that some of the selections may not be correct.
It
only takes one mislabeled level to prevent the program from
finding an
used
to
acceptable fit.
find
the
Because
Hamiltonian
of this,
for
the process
Pr 3 + : Y(OH)
3
is more
involved.
The selection process was complicated in this work due
to the fact that
energy levels.
fit.
1 16
1 D 2 has
appeared
to
have similar
the 1 D 2 terms did not seem to
been common
been made ( 2 1 1
this work did not
refinements.
3 Pt
Additionally,
Trouble with
attempts have
and
in Pr3+:LaCls and
to eliminate this problem but
incorporate
any
of
these
very recent
These selection problems were finally solved
by excluding all experimentally determined levels from 1 D 2 ,
3 P 1 and
1 I6.
This meant
that only
the most restrictive
data was initially entered.
There is a lower limit
program
needs
to
function
on
the
number of levels
properly.
The program needs
enough data to compare eigenvalues with such that
ion
parameters
( F2,
F4,
F6-,
the
the free
C ) and the crystal field
7
67
parameters may
be roughly
ion parameters,
the program needs at least.four
gravity.
Entering
determines that
crystal
determined.
one
level
in
manifold’s center
field
acts
to
split
To adjust the free
a
free
centers of
ion manifold
of gravity.
the
Since the
degeneracies in these
manifolds,
then it is reasonable to expect that the program
must have
more than
one level
anything about the crystal
condition given 121
fitting
is
that
entered,
less
in a manifold to determine
field parameters.
The general
in order to have a reasonable chance of
the
total
number
of
observed
the number of centers of gravity determined,
must be greater than four.
This may be written as
Noi - NcofG >> 4
23
where Ni 0 represents the number of
and NcofG
levels
is the
observed levels entered
number of centers of gravity determined.
This expression gives some basis for
which must
be entered.
Because
selections,
the program was
the number
of the
initially
of levels
ambiguity in the
run
with . the bare
minimum of nine levels.
In the
final process
of obtaining a fit,
best determined levels were entered,
along
the parameters
ANL . 1 8 , 1 3 1
given by
Sarup and
nine levels are indicated in Table
One significant
that only one. or
8
on
with values for
The initial
pages 76
problem in obtaining a fit
two parameters
nine of the
were varied
and 77.
previously was
in each run.
This
resulted in taking too much time
It
was
therefore
found
find convergence.
to
convenient
to
vary
several
parameters in one run.
Varying several parameters at once was not without its
hazards.
Because not
orthogonal,
it is possible to have
the parameters.
show strong
beyond
The
interdependence between
parameters F6, cc and Y
interdependence.
physically
previous
all of the operators are completely
So
acceptable
'work, (2 2 ’the
other
while
bounds
turned out to
one
as
dependent
value grew
determined,
parameters
in
could
adjust to
compensate making it appear as if a fit had been
obtained.
This was found to
following
the
lead
of
actually
occur
others, <8 •22 >
in
y , so
y was permanently
fixed for all runs.
After much experimentation with the parameters,
found that
it was
varying all of the crystal field parameters and
the largest free ion
produced consistent
parameters,(F2,
and reasonable
obtained for the initial
nine
F4, F 6 , C > together
results.
levels,
With a fit
other
levels were
slowly added and new parameters obtained.
In order
to
be
run
to obtain
over
nonproductive and
runs.
So,
200
a reasonable fit,
times.
tedious to
this Chapter
Obviously
the program had
it
would
be
reproduce the results of ' 2 0 0
concludes
with
a
review
of the
results from the final runs.
During all
of the runs made with the final 19 levels.
69
no problems,
Therefore,
such as a shifted ground state,
were observed.
attention focused on minimizing a.
the results obtained after
parameters except
for
varying
whose
all
of
value was
Table 5 list
the
free ion
held fixed at a
value given by H. Crosswhite.(22> The initial parameters in
this run gave a least squares fit of 26 cm-1.
Within three
iterations this figure had improved considerably to 7 cm"1.
This
run
parameters,
After
provides
a
typical
example of varying several
provided that the correct levels
varying
the
free
field parameters were varied
are in place.
ion parameters,
again.
the crystal
Keeping in
mind the
large changes made in all of the parameters since the first
run with nine levels,
this point.
Finally,
very little
change was
to assure a good fit,
recorded at
I returned to
varying one or two parameters per run.
The last
run made
'
has its
numerical data documented in Table 5 run number 2.
Even though many runs had been
made between
in Table 5, very little actually changed.
the two shown
At this point,
moderately good Hamiltonian was considered found.
a
70
Table 5: Numerical results from fitting program using data
from Pr 3 + : Y( OH) 3 .
RUN NUMBER : 1
■
LUMPED PARAMETERS:
Eav
F2
F4
F6
OC
B
Y
;
M0
= 9928. 0
=
=
=
=
=
= 1 4 0 0.0
=
= 2. 300
M2 ■ =
M4
P2
=
P4
P6
B2O =
B4 o =
B 6 o .•=
B6 6 =
1 . 290
0. 874
119.3
89. 46
59.65
321.1
-1057.
-1133.
645. 5
VARIATIONS:
STARTING
F 2 = 6831 8 .
F 4 '=" 47385.
31 808.
F6
OC
= 26241. b
B = -729.
y = 748. 0
PARAMETER
ERRORS:
F 2 = 34.
F 4 = 131.
F 6 = 50.
cc = 277. b
B = 20.
= 0.9
ENDING
F2 =
F4 =
F6' =
OC
=
B =
y =
SIGMA:
68481.
47524.
321 1 5.
26566. b
-736.
.747. 9. ■
CT
= 7.1
RUN NUMBER : 2
LUMPED PARAMETERS:
M2
9928.0
Eav
M4
68481.0
F2
= 47 52 4. 0
P2
F4
P4
F6 = 321 1 5. 0
= 26566. Ob P 6
CC
=
B2O
-736. 0
B
= 1 400. 0
B4O
Y
= 747. 9
B6o
C
B66
M0
= 1.760
0. 986
0. 669
=
=
=.
=
=
321 . 9
-1 056.
-1 137.
646. 4.
VARIATIONS:
STARTING
P 2 = 119.3
P 4 = 0. 7 5P 2
P 6 = 0. 50P2
PARAMETER
ERRORS:
P 2 = 18.4
ENDING
P 2 =. 1 45. 7
SIGMA:
C t = 5.7
* All values in units of cm'1.
b The• program uses 1 OQQcc1 which is expressed as oc here.
-71
CHAPTER 6
RESULTS AND CONCLUSIONS
Having
presented
this project,
obtained.
the
I conclude
The first
ideas
by
and methods relevant to
reviewing
the
final results
results which should be reviewed are
the final experimentally determined energy levels.
the
energy
levels
presented along
Also
the
as
determined
with e,
magnitudes
by
Second,
calculation
are
the least . squares fit parameter.
of
hami ltoni an are given.
the
parameters
Finally,
a
in
the
few comments
final
are made
in review of this thesis.
Table
contains
6
experiments.
the
Examination
Appendix reveals
results
of
that the
of
the
the .spectral
absorption
plots
in the
line which did not work well in
the computer fitting procedure was not particularly strong.
The Appendix
lines
contains the
recorded.
spectral plots
However,
from all. of the
the ' magnitudes
of
the
fluorescence lines are not well represented by these plots.
The
magnitude
of
the
lines
depended
not
only
on the
population of the state and the transition probability,
also on
For
the voltage
example,
the
applied to
voltage
but
the photomultiplier tube.
applied to the photomultiplier
72
Table 6 .
Absorption lines recorded and levels determined
Linea
Polari­
zation
Transition
re s ponsible
Level
determineda
a
3 H4( 2)
- 3 P2( 2)
3 P2C 2)
= 22249. 7
ir
3 H 4 ( 2)
- 3 P 2 (I)
3 P 2 (I)
= 221 84. 7
22172. 0
a
3 H 4 (S)
- 3 P 2 (T)
3 H 4 (S)
= 14.5
221 35. 7
a
3 H 4 ( 2 ) - 3 P 2 (O)
3 P 2 (O)
= 221 35.7
21339. 2
a
3 H4( 2)
- 1 I 6 (I)
1 I 6( 2)
= 21339.2
21 086. 2
<7
3 H 4 ( 2)
- 1 I 6 CO'')
1 I 6 CO' ') = 21 086.2
21 0 2 2 . 6
CT
3 H 4 ( 2)
- 1 I 6 (OT)
1 I 6 CO' ) = 2 1 0 2 2 . 6
2 1 0 0 1 .6
CT
3 H 4 ( 2 ) - 3 Pi(O)
3 Pi(O)
= 21 0 0 1 ..6
20990.2
V
3 H 4( 2)
- 3 Pi(I)
3 Pi(I)
= 20990. 2
20436. 5
CT
3 H 4 ( 2)
- 3 Po(O)
3 PoCO)
= 20436. 5
1 6802. 2
Tr .
.3 H 4 ( 2)
- 1 D 2 (I)
1 D 2 (I)
= 16802.2
I 6761. 2
CT
3 H4C 2)
- 1 D2 ( 2 )
1 D 2 C 2 ) = 16761. 2
1 6734. 2
CT
undetermined
16471.2
CT
3 H4( 2)
22249.7
221 84.7
v
- 1 D 2 (O)
all numbers in units of cm
undetermined
1 D 2 (O)
= 16471.2
73
when recording the lines at 1 6472 and 1 6457 cm" 1 was around
900V.
In contrast*
below 16450 cm"',
the
lines
recorded
with wavenumbers
were recorded with the photomultiplier at
1 800V.
The results from, the
reviewed in
Table 7.
fluorescence experiments
Note
that some
of the transitions
recorded yielded information on the same
means
that
there
were
fewer
computer fitting procedure,
can be
eigenstate.
This
levels to work with in the
but
the levels
used were more
reliably determined.
The most strikingly contradictory line recorded was at
16457.3 cm"1.
Appendix
shows
recorded.
that
In fact,
instruments
transition
1 D 2 (O)
Examination of the plot for this line in the
fluorescence was the strongest
this line was often used to
during
believed
3 H a (S),
this
which
the
experiment.
level
had
However,
also
for
determined
3 H a (S)
determined
disappeared.
in this
this
line
from
the
was
to have an energy
seemed reasonable
it was later found that
the. fluorescence
from the
been
Originally
responsible
corresponding to 16cmTI This
adjust the
since this
absorption
when pumping
the
data.
Dz(O),
region of the spectrum virtually
This suggests that the line either originates
3Po state or from an upper level of the 1Dz term.
The logical next test would be
to pump
1 Dz.
the energy level determined
Since the
magnitude of
for this line does not
match
any
of
an upper
those
level in
expected for
Table 7:
Fluorescence lines recorded and levels determined
Li ne*
Polarization
Transition
responsible
Level
determined* .
18797.6
IT
3 Pi(I)
- 3 H3( 2 )
3 H 5 ( 2)
= 2192.2
1 6824. 0
<7
3 Pi(I)
- 3 H&( 3)
3 H 6 (S)
= 4166.4
1 6733. 0
V
3 Pi(I)
- 3 H 6 ( 2)
3 H 6 ( 2 ) = 4259.7
1 6473. 0
a
1 D 2 ( 0)
- 3 H 4 ( 2) .
'D2 (O)
1 6457. 3
TT
undetermined
16270.7
IT
1 6249. 0
<7
unde t ermined
16176.6
a
3 P 0 (O)
- 3 H 6 (2)
I 2304. 8
IT
1 D 2 (0)
3 H 6 ( 3)
12211.5
CT
'D 2 (O)
- 3 H 6 ( 2' )
3 H 6 ( 2' ) = 4261.5
1 1 394. 0
a
'D 2 (O)
- 3 F 2 (2)
3 F2( 2)
•3 P 0 ( 0)
- 3 H 4 (3')
all values have units of cm
= 16471. 2
undetermined
3 H4 O ' ) = 2 0 2 .8
undetermined
3 H 6 ( 2 ) = 4261.7
3 H6 O )
= 41 6 8 . 2
= 5079.0
75
praseodymium,
it
-may
be
that
the crystal studied had a
contaminant in it such as europium.
Both the absorption and
several
repeated
fluorescence experiments were
times.
recorded
was
This line
particular
crystal
and
of
more
than
one. occasi on.
this was a line at.11394 cm-1.
exception t O
One important
most
on
presented here were recorded
crystals.
All
an
during
was
not
experiments using a different
of
the lines
Therefore
other
using one
experiment
found in two subsequent
crystal from
conditions
the same batch
such
as
crystal
temperature,
level pumped and
identical.
The last fluorescence experiment pumped levels
1 D 2 (O),
3 Po(O)
and 3 P 4 (I)
photomultiplier voltage were
in
an
attempt
to
clarify the
select!ons.
As
mentioned
previously,
the
calculated energies to experimental
many
times
However,
and
final
artificial
parameter with
all
fitting
energies had
been run
results were moderately good.
of
value.
19 levels
the
final
experimental levels used in
energies
for
a has a
three levels were not used in this fit so
somewhat
lists
the
program
were
calculated
The final least squares fit
fitting was
7.7 cm
calculated
the
from
fitting
a
.
Table 8
energies
and
process.
hamiltonian
parameters given in Table 9 with units of cm
with
the
These
the
76
Table 8 .
Term
U
3 H*
2
3
2'
3'
1
0
3 Hs 3
2
3'
1
2'
1'
0
3 H6
3
2
3'
0
0'
1
2'
1'
0 ''
Final calculated energy levels for Pr3*::Y( OH) 3.
calculated
energy (cm"1)
experimental
energy (cm"1)
difference
( cm" 1 ) b
0 .0
5. 5
182.7
188.3
277. 5
456. 9
0. Oa
15.0'
2 0 1 .0 '
-
0 .0
9.5
. —
1 4. 5
-
2132. 6
2186. 3
2226. 1
2264. 0
2278. 4
2434. 7
2501.0
21 94. 0
—
—
7. 7
4162. 9
4252. 6
4386.9
4392. 0
4420. 9
4472.5
4504. 4
- 4709. 4
'4747. 3.
41 6 8 . 2'
4261.5'
—
—
5079. 0
—
-
5. 3
8 .9
—
—
—
—
—
6 i7
3 F2- 1
0
2
4970. 4
5037.9
5072. 3
3 F3
6302. 4
6346. 8
6377. 2
6402. 0
641 2. 7
—
~
6553. 7
661 0. 9
6653. 8
6704. 2
6713. 4
6800. 6
-
-
0
2
3
1
3'
3 F4 3
3*
2
0
1
2'
-
* one of first nine energy levels used in fit
b difference = (experimental) - (calculated)
—
77
Table 8 .
"Term
U
1 G*
3
3'
0
2
1
2’
Final calculated energy levels for Pr3*::Y( OH) 3
( continued) ■
calculated
energy (cm-1)
9229.
8
9447.9
9566. 8
9594. 3
9653. 2
9788. 6
experimental
energy ( cm"1)
-
—
difference
( cm" 1 ) b
-
2
1
1 6459. 2
1 6803. 7
1 6758. 8
16471.2
1 6804. 0
1 6763. 0
1 2 .0
0. 3
4. 2
3Po 0
20425. 6
20436. 5*
10. 9
3Pi 0
I
21 0 0 2 . 6
21001.6
20983. 0
20 9 90. 2
1 I6 0
0’
1
0 ''
1•
2
21 01 8 . 8
21 0 2 2 . 6
21 086. 2
1 D2
0
3
2'
3'
3 P2
0
I
2
21 083. 9
21 322. 4
21 324. 5
21 383. 0
21 478. 0
21559.7
21 576. 6
21 633. 0
2 2 1 18.0
22181. 9
22250. 3
21 339. 2
—
221 35.7*
' 221 84. 7*
22249. 7*
*' one of first nine energy levels used in fit
b difference = (experimental) - (calculated)
-1 . 0
7. 2
3. 8
2. 3
16.8
—
—
—
—
17. 7
2 .8 .
-0 .6
J
78
Ea v
F2
F4
F6
=
=
=
=
OC
B
Y
C
M0
=
=
M2
M4
P2
P4
P6
9928. 0
68481.0
47524. 0
321 1 6. 0
26. 566
-735. 0
1 400. 0
748. 0
2. 300
=
=
=
B20 =
B40 =
B60 =
B66 =
1 . 290
0. 87 4
I 45. 7
109.3
72. 85
321. 9
-1 056.
-1 I 37.
646. 4
Table 9: Final parameters for the Hamiltonian of
Pr 3 + : Y( OH) 3
The
However
fit
obtained
the
calculated
presented here
as a
guide
sample.
in
can be
when
energy
information on
experiment
searching
future
the
may
thereby
splitting.
thesis.
levels
for
spectral
experiments
the
line
be
to
finding
Finally,
contaminants in it,
in
was
not
for
perfect.
Pr3*: Y( OH) 3
considered close and should be used
states in the 1 D 2 manifold.
it
work
lines
in this
In an attempt to improve the determination of the
energy levels,
field,
this
same
could
pump
the upper
This should at least yield more
seen
subject
at
16 4 57
cm 1 .
Another
the crystal to a magnetic
additional
lines
due
to Zeeman
if it is felt that the sample has some
a new sample
could be
experimental procedure
obtained,
using
outlined in this
79
REFERENCES CITED
( 1)
M. S. Otteson, Ph. D. Thesis,
unpublished (1984).
(2)
Private discussion with Dr.
(3)
J. D. Jackson, Classical Electrodynamics, John Wiley
and Sons, New York (1975).
(4)
Private discussion with Dr.
(5)
R. L. Liboff, Introductory Quantum Mechanics, HoldenDay, Oakland, California ( 1 980).
(6 )
S. Hufner, Optical Spectra of Transparent
Rare Earth
Compounds, Academic Press, New York (1978).
(7)
B. G. .Wybo.urne, Spectroscopic Properties of Rare
Earths, Interscience Publishers, New York (1978).
(8 )
W. T. Carnell, H. Crosswhite, H. M. Crosswhite, " Energy
Level Structure and Transition Probabilities of
Tri valent Lanthanides in LaFs" Argonne National
Laboratory Report (1977).
( 9)
B. R . Judd, H. M. Crosswhite,
I 69 1 30 ( 1 968)
( 1 0 ) K. Ra j nak,
( 1 1 ) J. Sugar,
Montana State University
R. L. Cone.
J. L. Carlsten.
H. Crosswhite,
B. G. Wybourne,
Phys.
Rev.
IA
Phys.
Lett,
Rev.
Rev
280 (1968)
731 ( 1 965).
( 1 2 ) M. Wei ssbluth, Atoms and Molecules,
New York (1978).
M. H. Crozier,
1 32
Phys.
J. Chem.
Academic Press,
(13)
R. Sarup,
Phys.
42.
371 ( 1 968)
(14)
R. D. Chirico, E. F. Westrum, Jr., J.B. Gruber, J.
Warmkessel, J. Chem. Thermodynamics 1J_ 836 ( 1 979).
(15)
Spex 14018 .85-meter Czerny-Turner Double
Monochromator-Spectrometer, (Spex Industries Inc.,
Metuchen, New Jersy 1974).
I
80
REFERENCES ClTED-continued
(16)
J. H. Eggert, M. J. Kimlinger, R. L. Cone, "Polarization
Sensitive Spectral Response of an Optical Double
Monochromator with Holographic Gratings , unpublished.
(17)
T. W. Hansch,
(18)
Model 162 Signal Averager, (Princeton Applied
Reasearch Corporation, 1975).
(19)
American Institute of Physics Handbook, 2nd ed.
McGraw-Hill, New York (1963).
Appl.
Opt.
1_i_ 895 (1 972).
( 2 0 ) Private discussions with M. Winkley,
( 2 1 ) .Y. Y. Yeung,
( 1 987).
D. J. Newman,
J. Chem.
( 2 2 ) H. M. Crosswhi t e, H. Crosswhite,
246 (1 984).
R. Jones.
Phys.
J. Opt.
86
6 71 7
Soc.
Am.
B 1_
81
APPENDIX
„ Experimental Plots of Spectra
Th^e
plots
which
experiments
and
were
labeling of
each plot
follow
used
are those obtained from the
in
the
The horizontal
The
if any
axis of
is indexed in terms of wavenumbers and therefore
have units of inverse
indexed in
analysis.
indicate which polarization,
was used when recording the signal.
each plot
data
counts from
centimeters.
The vertical
the multichannel
analyzer.
axis is
These
counts were heavily dependent on the settings of the signal
processing
instruments
and
do
the intensity of individual lines.
not necessarily represent
I i i r
j__ i__ i
22100
22160
Figure 12.
i i i r
i I I r
I -1— L
22220
22280
22340
Absorption with sigma polarization at 22400.
22400
T I r
j— L_i
22100
22160
I
i... -I
22220
22280
22340
Figure 13. Absorption with pi polarization at 22400.
22400
170
150
130
no
21200
21240
Figure 14.
21280
21320
21360
Absorption with pi polarization at 21400
21400
2800
I
I— I T
T— I— r
2560
2320
2080
1840
I
1600
20900
20940
Figure 15.
1-1
20980
21020
21060
Absorption with sigma polarization at 21100.
21100
2700
2440
2180
1920
1660
1400
20980
20988
Figure 16.
20996
21004
21012
Absorption with pi polarization at 21020.
21020
20430
20433
Figure 17.
20436
20439
20442
Absorption with sigma polarization at 20445.
20445
3800
3540
3280
3020
2760
2500
18781
18794
Figure 18.
18808
18821
18835
Fluorescence with no polarization at 18849.
18849
3800
3740
3680
3620
3560
3500
16709
16735
Figure 19.
16761
16788
16814
Fluorescence with no polarization at 16841.
16841
16750
16780
Figure 20.
16810
16840
16870
Absorption with pi polarization at 16900.
16900
16720
16734
Figure 21.
16748
16762
16776
Absorption with sigma polarization at 16790.
16790
Figure 22.
Absorption with sigma polarization at 16490.
450
i
I
I
r
i I I r
i I I r
i i i r
390
330
270
210
150
L
16450
I l l l l l l l l l i _ I_ I_ I
16456
Figure 23.
16462
16468
16474
Fluorescence with pi polarization at 16480.
J_ I_
16480
450
390
330
270
210
150
16450
16456
Figure 24.
16462
16468
16474
16480
Fluorescence with sigma polarization at 16480.
16100
16138
Figure 25.
16176
16214
16252
Fluorescence with sigma polarization at 16290.
16290
16100
16138
Figure 26.
16176
16214
16252
Fluorescence with pi polarization at 16290.
16290
J_ I_ L
12000
12080
Figure 27.
12160
12240
I
I
12320
Fluorescence with no polarization at 12400.
12400
500
480
460
440
420
400
11200
11260
Figure 28.
11320
11380
11440
Fluorescence with no polarization at 11500.
11500
MONTANA
762 J0015948