FARADAY EFFECT IN RARE EARTH DOPED GLASSES
by
S, J. Collocott B.Sc.(Hons).
University of New South Wales.
Thesis submitted for the degree of
Doctor of Philosophy
In the Faculty of Science, University of New South Wales
March, 1978. r " 1 m 1 "" ^ UNIVERSITY OF N.S.W.
56461 17. AUG 7 8 LIBRARY This is to certify that the work embodied in this
Thesis has not been previously submitted for the award of a degree in any other Institution. iii .
ACKNOWLEDGEMENTS
I would like to express my thanks to Professor K.N.R. Taylor, my Supervisor for his guidance and continued assistance, and my fellow research workers in the Magnetic Materials Group for many useful discussions throughout this work.
I wish to thank Professor A. Runciman,
Research School of Physical Sciences at the Australian National University for making available the Cary 61 spectropolarimeter. In addition, I appreciate the assistance given by my colleagues at C.S.I.R.O's National Measurement Laboratory, namely Mr. J. Cook for helping with the low temperature absorption measurements and Mr. R. Abel for grinding and polishing the glasses. The assistance of Mr. M. Withers, School of Chemistry at the UNSW with room temperature absorption measurements is also acknowledged. I am indebted to the Australian Government for a Commonwealth Postgraduate Award. I gratefully acknowledge the co-operation of all the staff in the School of Physics, particularly its Head, Professor E.P. George.
Finally, I thank all others who have given support, in particular Mrs. Barbara Ellis for typing the manuscript. ABSTRACT
The Faraday rotation and Magnetic Circular
Dichroism (MCD) have been measured about the broad absorption bands in Erbium, Neodymium, Holmium and
Praseodymium doped soda-silicate glass. The measurements were made at temperatures between 4.2K and 100K and in magnetic field strengths of 5T. The relationship between the Faraday rotation and MCD is explored by use of the Kramers-Kronig transforms and the MCD dispersion spectra is interpreted by use of the method of moments. The method of moments enables deduction of the Faraday parameters, which give the ground state g-values for the rare earths and information pertaining to the magnetic moment of the excited states involved in the optical transitions.
The relative contributions of the A, B and C terms to the MCD spectra are examined, where it is found that C terms dominate in rare earth doped glasses. Some comments on the spread of crystal field parameters for the rare earth ion in the glass are made, with specific reference to the existence of trigonal sites for the rare earth ion in the glass matrix. It is also shown that by selective doping with the rare earths that it is possible to produce a glass that has a zero Verdet constant or a glass that has both a positive and negative Verdet constant at differing wavelengths. Some suggestions for the use of these glasses in practical devices are also made. The entry recording the discovery of an effect of magnetis on light. September 13, 1845. Par. 7504 (full si{e) CONTENTS
Page
Acknowledgements iii Abstract iv Frontespiece v
Contents vi
List of Figures
List of Tables xii
CHAPTER 1 : INTRODUCTION 1 1.1 Introductory 1 1.2 Previous Investigations 3
1.3 The Rare Earths 9 1.4 Rare Earth Spectra 12 1.5 The Nature of the Glassy State 15 1.6 The Nature of the Rare Earth Ion 19 Site in Glass
CHAPTER 2 ; THEORETICAL CONSIDERATIONS 26 2.1 Phenomenology 26 2.2 The Relationship between the 29 Faraday and Zeeman Effects
2.3 The Theory of M.C.D. 35
2.3.1 Basic Theory 35 2.3.2 Dispersion Calculations : The Rigid Shift Model 41
2.3.3 Overlapping Bands 52 2.3.4 Saturation 59
2.3.5 Moments 62 2.4 Group Theory and Symmetry Considerations 71 vii
CHAPTER 3 : NUMERICAL TECHNIQUES 79
3.1 Calculation of the Kramers-Kronig 79 Transforms 3.2 Analysis of MCD Spectra 85
3.3 Analysis of Faraday Rotation 90 Saturation Measurements
CHAPTER 4 : EXPERIMENTAL METHODS AND APPARATUS 93 4.1 Sample Preparation 93 4.2 Optical Absorption Measurements 98 4.3 Faraday Rotation Polarimeter 98 i 00 — • • 1 General Description 98
4.3.2 Lamp Power Supply 103 4.3.3 Detecting System 104 4.3.4 Step Scan System 109 4.3.5 Cryostat 112 4.4 Measurement of Magnetic Circular 114 Dichroism
CHAPTER 5 : RESULTS AND DISCUSSION (PART 1) 119 FARADAY ROTATION IN NON-ABSORBING REGIONS
5.1 Faraday Rotation of Soda-Silicate 119 Base Glass
5.2 Faraday Rotation in Non-Absorbing 112 Regions of Rare Earth Doped Soda Silicate Glass
5.3 Saturation Effects 126
CHAPTER 6 : RESULTS AND DISCUSSION (PART 2) 134 FARADAY ROTATION AND MCD ABOUT ABSORPTION BANDS 6.1 Absorption Spectra 134 6.2 Magneto-Optical Results 148 6.2.1 Holmium 148 viii
6.2.2 Erbium 163
6.2.3 Praseodymium 174 6.2.4 Neodymium 180
6.3 Summary 189
CHAPTER 7 : SOME CRYSTAL FIELD EFFECTS 193
CHAPTER 8 : SUGGESTIONS FOR FUTURE RESEARCH 207
REFERENCES 209
APPENDIX 1 : Computer Programs 222
APPENDIX 2 : A Versatile Digital Phasemeter 240
APPENDIX 3 : Publications 261 ix
LIST OF FIGURES
Page
CHAPTER 1
1.1 Observed energy levels of rare-earth 11 ions. The thickness of each level represents the total crystal field splitting in LaC£^. After Dieke [1.32] 1.2 Models of Glass Structure after (a) 16 Zachariasen[[1.49], random network theory (b) Valenkov and Porai-Koshitz [1.50], crystallite theory. 1.3 Structure of the rare earth ion site in 22 silicate glass. After Reisfeld [1.72] 1.4 Octahedral complex constructed from 24 undistorted SiO^ tetrahedra. Expansion in the direction of the arrows leads to trigonal fields at the void, After Robinson et al [1.74]
CHAPTER 2
2.1 Zeeman splitting of eigenstates and 30 their selection rules. After reference [2.9]. 2.2 The Formation of the Faraday A term 32
2.3 The Formation of the Faraday C term 35 2.4 Contributions of A and C terms to 50 MCD spectra
2.5 MCD dispersion for overlapping bands 56
CHAPTER 4
4.1 Glass making furnace layout 94
4.2 Schematic of Faraday Polarimeter 99 4.3 24V/10A DC power supply 102
4.4 Photomultiplier Dynode Chains and 105 Low Pass Filters
4.5 Step Scan electronics 108 X
4.6 Helium Cryostat 111 4.7 The Faraday Polarimeter 115
4.8 Schematic of Cary 61 116
CHAPTER 5 5.1 Faraday Rotation of Soda Silicate 118 base glass 5.2 Background Faraday Rotation at 4.2K 121 3 + for Nd doped soda-silicate glass 5.3 Background Faraday Rotation at 4.2K 123 for Ho 3+ doped Soda Silicate glass 5.4 Background Faraday Rotation at 4.2K 124 3+ for Pr doped soda silicate glass 5.5 Background Faraday Rotation at 4.2K 125 3+ for Er doped soda silicate glass 5.6 Saturation Faraday Rotation curves at 127 4.2K for 10% Nd doped soda glass
5.7 Saturation Faraday Rotation Fitted 129 to equation (3.24).
CHAPTER 6
6.1 Absorption Spectrum for 10wt% Ho^+ 133 doped soda glass at 273K and 77K 3+ 6.2 Absorption Spectrum for 10wt% Er 135 doped soda glass at 293K and 77K
6.3 Absorption Spectrum for 8.5wt% Pr^+ 136 doped soda glass at 293K and 77K
6.4 Absorption Spectrum for 10wt% Nd'*+ 137 doped soda glass at 4.2K and 293K
6.5 Absorption Spectrum for lwt% Ho^+ 138 soda glass at 4.2K and 293K.
6.6 Absorption Spectrum for lwt% Nd^+ 139 doped soda glass at 293K
6.7 Beer's Law for RE doped glasses 141 3 + 6.8 Faraday Rotation at 4.2K for Ho 147 doped soda-silicate glass xi
6.9 Faraday Rotation at 4.2K for 10wt% Ho 149 doped soda silicate glass about the 450nm band 3+ 6.10 Faraday Rotation at 4.2K in 5wt% Ho 151 doped soda silicate glass
6.11 Concentration dependence of Faraday 153 rotation for all the rare earth doped glasses 3+ 6.12 MCD of 10wt% Ho doped soda silicate 154 glass at 77K
6.13 Relative contributions of A and C terms 158 to the MCD in 10wt% Ho^+ doped soda silicate glass 6.14 Faraday rotation at 4.2K for 10wt% Er^+ 162 doped soda silicate glass 3+ 6.15 MCD of 10wt% Er doped soda silicate 165 glass at 77K 6.16 Relative contributions of A and C terms 168 to the MCD in 10wt% Er^+ doped soda silicate glass
6.17 MCD dispersion about 450nm in 10wt% Er^+ 170 doped soda silicate glass 6.18 The Character of A and C terms for the 171 two near neighbour bands at 45 0nm,.in 10wt% Er^+ doped soda silicate glass 6.19 Faraday rotation at 4.2K for 8.5wt% Pr^+ 174 doped soda silicate glass
6.20 MCD of 8.5wt% Pr^+ doped soda silicate 175 glass at 77K
6.21 Relative contributions of A and C terms 176 to the MCD in 8.5wt% Pr^+ doped soda silicate glass
6.22 Faraday Rotation at 4.2K for 10wt% Nd~*+ 179 doped soda silicate glass
6.23 Faraday rotation about 430nm, 475nm and 181 530nm in 51wt% Nd3+ doped soda silicate glass at 4.2K
6.24 Change in lineshape of 575nm Faraday 183 rotation band as a function of concentration
6.25 MCD of 10wt% Nd^+ doped soda silicate 184 glass at 77K
6.26 Relative contributions of A and C terms 186 to the MCD in 10wt% Nd3+ soda silicate glass xii
LIST OF TABLES
Page
CHAPTER 1 1.1 General properties of the rare 8 earth elements
CHAPTER 2 2.1 Character Tables for 0^ and C^v 73 symmetries 2.2 Character Tables for 0^ and C^v 76
CHAPTER 3 3.1 Comparison of calculated value and 83 theoretical value for Kramers-Kronig transform of a gaussian
CHAPTER 4
4.1 Summary of rare-earth concentrations 95 in soda-silicate glass samples 4.2 Impurity ion concentrations in base 97 soda silicate glass (Analyses performed by Analytical Chemistry, Australian Atomic Energy Commission)
CHAPTER 5 5.1 Saturation Results 130
CHAPTER 6
6.1 Transitions responsible for absorption 143 bands in RE doped Glass 6.2 The Faraday Parameters for 10wt% Ho^+ 156 doped soda silicate glass at 77K
6.3 The Faraday Parameters for 10% Er±>ium 16 7 doped soda silicate glass at 77K xiii
6.4 The Faraday Parameters for 8.5wt% Pr^+ 177 doped soda silicate glass at 77K
6.5 The Faraday Parameters for 10wt% Nd^+ 187 doped soda silicate glass at 77K.
CHAPTER 7 7.1 The energy eignevalues for the three 195 4 lowest doublets in the ln /0 neodymium y/z ground state for a variety of crystal
field parameters 7.2a) Irreducible representations of ion levels 204
under C^v symmetry, b) Theoretical reduction of the values of C/D and A/D for the transitions among each species of C3v symmetry (After reference [7.13]). 1
CHAPTER 1._____ INTRODUCTION
1.1 Introductory
On the 13th September, 1845, Michael Faraday made the following entry in his diary [1.1]:-
"Today worked with lines of magnetic force3
passing them across different bodies (transparent in
different directions) and at the same time passing a
polarized ray of light through them ......
A piece of heavy glass which was 2 inches
by 1.8 inches3 and 0.5 of an inch thick3 being a
silico borate of lead3 and polished on the two shortest
edges3 was experimented with. It gave no effects when
the same magnetic poles or the contrary poles were on
opposite sides (as respects the course of the polarized
ray) - nor when the same poles were on the same side3
either with the constant or intermitting current - BUT3 when contrary magnetic poles were on the same side3
there was an effect produced on the polarized ray 3 and
thus magnetic force and light were proved to have relation to each other. This fact will most likely prove exceedingly fertile and of great value in the
investigation of both conditions of natural force”.
Faraday in this experiment and subsequent experiments demonstrated the connection between light
and magnetism by establishing that a medium becomes optically active, rotating the plane of polarization of
a polarized light ray when it traverses the medium in the direction of the applied magnetic field. As
Faraday suggested the connection between light and magnetism has proved "exceedingly fertile" as an 2
analytical tool for probing the structure of a vast variety of materials.
With the advent and development of quantum mechanics, an increased and more detailed understanding of the Faraday effect followed making Faraday Effect
Spectroscopy [1.2] a powerful tool in spectral analysis. In recent years work has concentrated on the dispersion of the Faraday effect through broad absorption bands in
solids. Such studies [1.3] yield both qualitative and quantitative information pertaining to both the ground and excited states and their associated g-values, the symmetry of the states taking part in the optical
transition and the local site symmetry of the absorbing centres. This work concentrates on the measurement of the
Faraday rotation (the rotation of the plane of polarization of polarized light) and its companion Magnetic Circular Dichroism (MCD, the differential absorption of left circularly and right circularly polarized light) about the broad absorption bands in the visible of rare earth doped soda silicate glass. The only rare earth doped glasses that exhibit these intense absorption bands in the visible [1.4] are those doped with either Erbium, Holmium, Neodymium or Praseodymium. The magneto-optical properties of these glasses are studied in detail with information being gained on the electronic energy level structure of the rare earth ion
in its host environment as a function of concentration and temperature. Comments are also made on the possible applications of rare earth doped glasses as light modulation devices in laser systems. 3.
1.2 Previous Investigations
The study of magneto-optics in solids unquestionably had its birth with Faraday’s observation on the connection between light and magnetism. A bibliography of some 1238 references tracing the development of magneto-optics in solids is given by Palik et al in reference [1.5].
The early theories of the Faraday effect were chiefly concerned with providing a satisfactory explanation for regions far removed from absorption bands. Historically, the Faraday rotation has been measured in terms of the Verdet constant V,
(1.1) where (p is the rotation angle in minutes of arc, J£ the path length in the medium in cm, H is the applied magnetic field strength in Oersted and n the unit ~P vector in the direction of propagation giving V the units of min/(Oe.cm). Equation (1.1) is only an adequate description in transparent regions of a solid as in regions of absorption the rotation of the plane of polarization of the linearly polarized light is accompanied by the development of an ellipticity of the originally linearly polarized light.
The first attempt at the derivation of a theoretical expression for the Verdet constant was made by Becquerel [1.6], who derived the following expression for V in transparent regions from classical theory, 4 .
V = (ev/2mc^)(dn/dv) (1.2) where e and m are the charge and mass of the electron respectively, v the frequency of the incident light, c the speed of light in vacuum and dn/dv the dispersion of the media. The development of quantum mechanics through the nineteen twenties and early thirties resulted in Kramers [1.7, 1.8] producing a quantum mechanical expression for the Faraday rotation cf>,
2 4ttv N [
* =------+ l------2—!_ hen £exp(-hv /kT) g'n v2- v2 g g B ^ ng
(1.3) where
Kramer’s dispersion formula [1.10]. The relationship between the paramagnetic term and the magnetic susceptibility was deduced by Van Vleck and Hebb [1.11], giving the ratio between the paramagnetic Verdet constant 5.
and magnetic susceptibility as:
C. V (1.4) L 2 2 X 93chg n v -v
where g is the Lande splitting factor, 3 is the Bohr
magnetron, Cn denotes the transition moments and the
other symbols have the same meaning as before. Further
significant contributions to these early theories on
the Faraday effect outside absorption bands were made
by Rosenfeld [1.12] and Serber [1.13].
The experimental work during this period
reflected the development of the theory, being chiefly
directed at the measurement of the Faraday effect in
non-absorbing regions of solids. For the case of
glasses most effort was aimed at understanding magneto
optical properties of optical glasses. Cole [1.14]
studied Verdet constants in a variety of barium crown
glasses in an attempt to estimate the contribution of
the paramagnetic terns to the Becquerel formula,
equation (1.2). More recent investigations [1.15] on
optical glasses showed a close correlation between the
Faraday rotation and reciprocal dispersion.
Traditionally the glasses with large Verdet constants
have been of the heavy lead type but Borrelli [1.15]
in his investigation on rare earth doped glasses
obtained a Verdet constant of 0.203 min/(cm.Oe) from a
praseodymium doped borate glass. Observations of the
Faraday effect in rare earth doped phosphate glasses
[1.16, 1.17] demonstrated the same temperature dependence for both the magnetic susceptibility and 6 .
Verdet constant. Recognizing this connection Alers [1.18] measured the Verdet constant as a function of magnetic field strength up to 7T for cerium doped phosphate glass at liquid helium temperatures to determine the ground state g-value of the cerium ion.
Attempts have been made [1.19] to fit experimental results to a functional relation of the form, V = G/(A2 - x£) (1.5) where G is a parameter independent of the incident light's wavelength, but a function of the transition wavelength Afc, effective electric dipole matrix element C^, temperature and concentration. The expression assumes a single transition A is responsible for the rotation but in general several levels are involved necessitating the interpretation of \ as a weighted mean value of all transition wavelengths. The values obtained for \ have been of questionable value making it difficult to attach any physical significance. These early investigations outlined have been aimed solely at seeking to explain the behaviour of the
Faraday effect away from absorbing regions. There remained a paucity of work relating to the Faraday effect about absorption bands in solids. Only in the last ten years has interest swelled in the dispersion of the Faraday effect about absorption bands in solids primarily as a consequence of the successes of natural optical activity work in 7. interpreting the behaviour of ions in crystals.
(For an excellent exposition on the theory of natural optical activity see reference 1.20). Observations of the Faraday effect about absorption lines in rare earth doped fluorides were made by Shen [1.21, 1.22] and Margerie [1.23] who derived the g-factors for
Sm++ in CaF2. It was not until the development of the modern theory by Buckingham and Stephens [1.24] that satisfactory techniques became available for interpretation of experimental data.
For greater detail on the Faraday effect due to ions in crystals, the reader is directed to reference 1.25 which contains a bibliography of some 400 references. An important reason for the acceleration of interest in magneto-optical dispersion about absorption bands in solids was the availability of large magnetic fields produced by superconducting solenoids, enabling the experimentalist to gain data on these generally small effects for the first time. Additional impetus has arisen from the technological demand for glasses (or crystals) that possess large Verdet constants, for use in light modulation devices [1.26,1.27,1.28)].
This work presents the magneto-optical spectra about the absorption lines in rare earth doped soda- silicate glasses, interpreted in the context of
Stephen's theory. Ths discussion will be confined almost exclusively to magneto-optical properties in absorbing regions as generally these are more informative. 8
TABLE 1.1
General Properties of the Rare Earth Elements
Element Atomic Electronic strucutre Colour due Ionic and number (outer electrons only]) to absorp radius Symbol z tion in
Ln Ln3 + LnT 3+ Ln3+
.jl_ 2 Yttrium Y 39 4d 5s (Kr) Colourless 0.880 *-1.. 2 Lanthanum La 57 5d 6s (Xe) Colourless 0.061 112 Cerium Ce 58 4f±5dJ-6s 4f3 Colourless 1.034 3 2 Praseodymium Pr 59 4f J6s 4f 2 Green 1.013 „ ,.4 2 Neodymium Nd 60 4f 6s 4f 3 Blue/Purple 0.995 .-5- 2 Promethium Pm 61 4f 6s 4f4 Pink/Yellow 0.979
Samarium Sm 62 4fAJ> 6sr 2 4f5 Nearly 0 .964 Colourless 7 2 Europium Eu 63 4f 76s 4f6 Nearly 0 .950 Colourless 7 12 Gadolinium Gd 64 4f75d 6s 4f7 Colourless 0 .938 Terbium Tb 65 4f96s2 4f8 Nearly 0 .923 Colourless
Dysprosium Dy 66 4f106s2 4f9 Pale Yellow 0.908 Holmium Ho 67 4f116s2 4f10 Peach 0.894 12 2 Erbium Er 68 4fX^6s 4f11 Pink 0 .881
Thulium Tm 69 4f136s2 4f12 Pale Green 0.869
Ytterbium Yb 70 4f146s2 4f13 Colourless 0.858 Lutetium Lu 71 4f145d16s2 4f14 Colourless 0.848 9 .
1.3 The Rare Earths
The rare earths are so chemically alike that as a group of elements they occupy only one entry in the periodic table. The fifteen elements that compose the lanthanide group (Z = 57 to Z = 71) owe their affinity due to similar electronic structures.
The electronic configurations of the ground state of the elements are dictated by Hunds rules with the placing of electrons in hydrogen like orbitals, commencing with those of lowest energy. For a variety of elements in the periodic table the orbitals filled first are not necessarily those with the smallest radial expectation value as it is energetically more favourable to fill the outer orbitals first. Such a situation occurs in the rare earths with the sequential filling of the 4f orbital by the extra-nuclear electron rather than the 5d orbital. The rare earths are created by the stepwise filling of the fourteen vacancies in the 4f shell.
A summary of some of the basic properties of the rare earths is contained in Table 1.1. The element yttrium is included because it is similar in many physical characteristics to the rare earths.
The table details the electronic structure of both the neutral and trivalent ions and gives the principal colour associated with the trivalent lanthanide ion.
Only the outer electrons are tabulated as all the elements share the common Xenon core, which is
n 20 20 6 2 6_a(K 2. 6..10,. 2_ 6a 2 Is 2s 2p 3s 3p 3d 4s 4p 4d 5s 5p 6s
The exception is yttrium which has a krypton core. 10.
The 4f electrons which determine the electronic properties are shielded both spatially 2 6 and energetically by the closed 5s 5p levels [1.29]. It is noted from Table 1.1 that as Z increases across the series there is a resultant reduction in the ionic radius. The decrease in ionic radius is known as the "lanthanide contraction" being due to the imperfect shielding of the f electron from the nuclear charges resulting in the electrons becoming increasingly tightly bound. As a consequence of the 2 6 screening of the 4f orbitals by the 5s 5p levels the host environment plays little part in governing the properties of the rare earth ion. The properties of the ion in the host lattice are almost the same as the free ion. A feature of the 4f electrons is the strong spin orbit interaction which couples the angular momenta S and L to give the total angular momentum J. As a result J is a good quantum number with the crystal field from neighbouring ions in the host lifting the (2j + 1) degeneracy of the free ion energy levels. The crystal field splittings are generally small and may be treated by use of perturbation theory. On substitution into a host lattice bonding occurs via the interaction of the strongly electropositive rare earth ion with the surrounding crystalline electric field of the neighbouring ions.
Minimal contribution to the chemical bond [1.30] is made by the 4f orbitals as they are shielded by outer electrons in the n = 5 shell. 11
*p.
_____ — T——*/2 X •'V2"/2 y------*4 __ X M——- •p« w------V. rA
*jz% -1 c
*P., o 0_JU,0 “ -“•/» |------“V. ‘F, -*Q. 'V2 2P JH. P------v, J------s------— 2 __L 17 -1 m * ------o
. ~,l/2 „.X *0i,2 P K’ 28 * =^’l'v2 fz2£b m-- {*==• o^V, °*«. J; ___ L------6 M------^ _l.o *K, L_A'/, N— V- 26 K ——V2 __'_D2 M"— El ..y.c *G. *H, _!2s K—-% _!o»/2 .A. iZa./2Di *P 1------Va "F, I----- “% 'E __L>^ 1 ■ ” — 6 FI" «G G=^% 'i. ------X C- A 22 —*3? __X G------** F_Aj G------HIo *'* ..'A* 20 c—-w F—•/, r *H’v. E------% B——*/2 B ’s, ------3 4g - A------•/, E ‘S, E A-
ax n-. c_ _ “F, 4F D D-- B—~*j .... 4 *f2
*s. X~ A mum '2 ------3 ,3\J.
R------»/2 '/2 b- •f ,, 0------h/2 ’G. "'/2 tsm p-A. JH. o-2L *F. •I. S "V2 ?F0 dh ~ ■'V2 ------7 v-^ , tF.
J3WM5 W ' ------6 .”/2 | saw", ------"/2 ------6 — 3 Y tmmm \ . -2 ------I ■M If*** ____ Z —— ------—— Z ——- ZWM Zra» X *H. •s 7f. ‘h,% -*i, 4l *X A % %" ■v2 X Pr Nd Pm Sm Eu Gd Tb Dy Ho Ei
Fig. 1.1. Observed energy levels of rare earth ions. The thickness of each level represents the total crystal field splitting in LaCl~. After Dieke [1.32].
T m 12.
For the case of rare earth silicates, borates and phosphates the rare earth cation is surrounded by polyhedra of (MO )n , where M = Si, P or B and x and n are defined by the stoichiometry of the polyhedra. It is reasonable to assume that a similar situation pertains in glasses [1.31].
1.4 Rare Earth Spectra
The optical absorption spectra of rare earth ions in various host media arises due to the modification of the free ion spectra by interaction with the electrostatic crystal field. The rare earths are chiefly trivalent on substitution into the host media though cerium, praseodymium and terbium may be tetravelent with samarium, europium and ytterbium often being divalent. The most complete compendum of the spectra of rare ions in crystals is contained in Dieke [1.32] and the review of Gorlich et al [1.33]. Figure 1.1 shows the observed energy levels of the rare ions with the thickness of each level representing the crystal field splitting in LaCl^. Transitions from 3 -1 the ground state to excited levels between 25x10 cm 3 -1 and 14x10 cm correspond to absorption bands in the visible portion of the spectrum. The rare earths that produce intense absorption bands in the visible are then praseodymium, neodymium, holmium and erbium.
Broer et al [1.31] showed that the spectra due to rare earth ions in condensed material was predominantly due to electric dipole transitions between states of the 4fn configuration or between 13.
that configuration and a higher energy configuration
n i such as the 4f 5d configuration. However,
electric dipole transitions within the same configuration are forbidden, necessitating some
forced electric dipole mechanism to account for the
observed spectra. It was not until the introduction of the Judd-Ofelt theory [1.35, 1.36] that a full
understanding of the mechanism of forced electric
dipole transitions was obtained. Judd and Ofelt
showed that electric dipole transitions were permitted if odd order terms in the expansion of the
crystalline electric field potential admix states into the 4fn configuration with configurations of opposite parity. The theory calculated in the
intermediate coupling scheme, (J is a good quantum number) gives the following selection rule for J, in the rare earths, namely|AJ| = 6.
Forced electric dipole transitions are allowed, if the rare earth ions nucleus is not situated at a centre of inversion. Such a situation occurs in glass as the rare earth ions occupy non- centrosymmetric sites, making electric dipole transitions possible [1.37]. In general the absorption spectrum is intensified in glasses due to the effects of the covalent bond between the rare earth ion and surrounding glass tetrahedra. Application of the Judd-Ofelt theory to the calculation of optical absorption intensities between the various Stark levels requires the availability of reliable crystal field eigenstates. This condition is 14.
often difficult to fulfil when ions reside at sites of
low symmetry. An appreciation of the lifting of the
energy level degeneracy by the crystal field of a
particular symmetry may be gained from group
theoretical techniques [1.38] , but the complete crystal
field calculation, entailing the diagonalization of the
crystal field matrix is needed if quantitative
information is to be gained. Such a calculation yields
the energy separations of the various crystal field
split levels (energy eigenvalues) and a set of
eigenvectors for use in oscillator strength
calculations.
Electronic energy level calculations for rare
earth aquo ions have been performed by Carnall et al
[1.39] who found little difference between the
solution parameters and those for the free ion. The
differences were of order 5% and attributed to
covalent bonding effects in solution. However, the
set of energy level assignments and matrix elements
derived by Carnall provide a good starting point for
further analyses of optical absorption spectra of
rare earth ions in materials where covalent bonding
predominates.
In glasses a further complication arises in
the interpretation of optical absorption spectra as
the amount of splitting of a level is of the order
of the inhomogeneous broadening of the absorption
line. This situation occurs as a result of the multiplicity of sites [1.40] than may exist for the
rare earth ion in the glass host. 15.
A later chapter will discuss in detail crystal field effects and their influence on the intensities of forced electric dipole transitions in the context of the various site symmetries that the rare earth ion may occupy in the glass matrix. A complete calculation of the relative intensities of the optical transitions between the various Stark levels will be sketched out.
1.5 The Nature of the Glassy State A variety of attempts [1.41, 1.42, 1.43] have been made over the last five decades to produce a satisfactory definition [1.44] as to what is a glass. There is however no universally accepted definition of glass that encompasses the structure of glasses and their relationship to amorphous solids [1.45], The definition of a glass due to Mackenzie [1.46] is adequate in the context of this work and is used here as a working description:- "A glass is any isotropic material, organic or inorganic, in which three dimensional atomic periodicity is absent and the viscosity of which is 14 greater than 10 poise". Glass may be viewed as a frozen liquid being produced by the rapid cooling of the glass melt to below the glass transition temperature Tg. The rapid cooling freezes in the liquid disorder that distinguishes glasses from crystals. Debye [1.47] has shown that even though no regular crystalline environment exists in glass, X-ray diffraction is possible producing several diffuse halos instead of 16 .
Fig. 1.2b.
Fig. 1.2. Models of glass structure after (a) Zachariasen '[1.49], random network theory (b) Valenkov and Porai-Koshitz [1.50], crystallite theory. 17.
the sharp lines characteristic of X-ray powder
diffraction from crystals. The diffuse halo of X-ray powder lines from glass demonstrates the absence of well defined lattice parameters. Most commercial glasses are based on silicon, phosphorous and boron together with oxygen, and in cases where glasses of specific properties are required amounts of alkali metals and, or alkaline earth metals are added. A detailed review of glass types is located in reference 1.48.
The first great contribution to the theory of the structure of glass appeared in Zachariasen's classical paper published in 1932 [1.49] where he proposed the random space atomic (ionic) network. Zachariasen argued that glass was composed of "glass formers" such as the oxides B202 ,Si02 ,Ge02/p2°5 an<^
As20^ and "network modifiers" which consisted of alkali metal oxides and the alkaline earth metal oxides.
Other oxides such as A1202 could be introduced into the glass to play an intermediary role but the glass formers remained essential for the formation of vitreous compounds. The glass was envisaged as a disordered isotropic network of randomly placed atoms having overall some statistical order (Figure 1.2a) . The other major hypothesis on glass structure was contributed by Valenkov and Porai-Koshitz [1.50] who developed the "crystallite theory" shown in Figure 1.2b.
This theory proposed regions of a high degree of ordering in localized domains or crystallites which 18 were separated by zones of less order. A similar hypotheses was supported by Grebenschchikov and Favorskaya in 1931 [1.51] who argued that silicate glasses consisted of a rigid silica sponge saturated with silicates. Since the pioneering work of the 1930's there has been continuing debate on the relative merits of the "crystallite" and "random network" theories shown in Figure 2. More recent contributions to the discussion include the work of Partel in 1972 [1.52] who observed regions of ordered silicate in glasses 3 with a density of 2.32 gms/cm whilst support for the random network model comes from Bell and Dean [1.53]. A particularly innovative view of glass structure due to Goodman [1.54] is the strained mixed-cluster model based on the idea of "clusters" of non-related polymorphs which associate on cooling but cannot nucleate, resulting in inter-cluster thermal strains.
In his latest review article [1.55] Porai- Koshitz surmises that glass has a homogeneous structure except for thermal density fluctuations that correspond to the glass transition temperature Tg. These fluctuations are also fluctuations in ordering with supporters of the "crystallite" theory terming these crystallites, remembering that the supporters of the random network theory never denied the possibility of thermal fluctuation ordering. It suffices to say that neither theory fits all glasses in all details with the selection of which 19 .
theory to use in practice depending on the system
under consideration and the nature of the experimental predictions required. The problem of inferring glass structure lies in the poor understanding of the
structural properties of vitreous silica. Though a common feature of most contemporary glass theories is the acceptance of some short range order [1.56] , typically in the range of 1 nm to 10 nm. More detail on the physical structure of glass is required, and will undoubtedly become available through the increased use of advanced diffraction techniques LAXS, MAXS,SAXS and SANS [1.57, 1.58, 1.59, 1.60,1.61]
For the production of high technology glasses various ions may be introduced into the glass matrix. Typical examples are the doping of glass with Ce3+ ions to form a photochromic glass [1.62] or the addition of Ti [1.63] as a nucleating agent in the production of glass ceramics. In this work the rare earth ion is introduced into a soda-silicate glass matrix in small concentrations (less than 15% by weight) and acts as a local non-interacting absorbing centre in an isotropic solid. The optical and magneto-optical properties of the glass are then pursued.
1.6 The Nature of the Rare Earth Ion Site in Glass On introduction into the glass matrix the rare earth oxides behave as glass modifier oxides. This is evident from observations of the behaviour of rare earth 20, oxides in glasses when compared to those in crystals and rare earth doped silicates. Kan Fu-Hsi et al [1.64] showed that the trivalent rare earth ion does , 4+ not replace Si in the frame structure of silicate glasses but rather remains in the interspace between the silicon-oxygen tetrahedra. As a glass has no long range order it is reasonable to assume that the dopant ion will only affect its local surroundings [1.65], The immediate environment about the ion will be perturbed by the strong electropositivity of the rare earth ion but long range effects will be minimized by the shielding effects of the electrons of the ligands. Local ordering is then expected about the rare earth site but this is confined to within the first few nearest neighbours. It is anticipated that from site to site the surroundings of the rare earth ion will not differ greatly with all sites overall having some definite microsymmetry. Mann [1.66] inferred that due to the close correlation in behaviour between the energy levels of
Nd 3+ in silicate glasses and those in oxide crystals that the ions occupy sites of low symmetry. Since neodymium sesquioxide has trigonal symmetry (C^)/
Mann and DeSchazer [1.67] reasoned that the site symmetry for Nd 34- in glass was similar. A distorted 34- icosahedral site symmetry for Nd ions in alkali- alkali earth silicates glasses has been proposed by
Snitzer [1.68] .
Of late there has been increased use of 21. optical spectra interpreted by ligand field theory to probe the nature of the rare earth ion site in glass. With the aid of group theory and its symmetry concepts many of the properties of lanthanide glasses have been explained in terms similar to those of impure crystals.
The principle difficulty is that in the glass the amorphous disorder produces an average site which manifests itself in the optical spectra as inhomogeneous line broadening. Ideally group theory should allow the deduction of the site symmetry from emission and absorption spectra but the situation is confused by the large number of levels into which the free ion levels split, being lost in the glassy broadening.
The optical transitions characteristic of rare earth ions in solids, are predominantly intra-fn transitions being mainly electric dipole in nature. For the free ion case transitions between the same configuration are strictly parity forbidden, hence in solids the observed spectra must result from noncentro- symmetric interactions leading to a mixing of opposite parity states [1.69]. The existence of optical absorption spectra in glass then immediately implies a site symmetry lower than octahedral.
Reisfeld et al [1.70, 1.71, 1.72] doped a variety of glasses with Eu 3+ ions and observed fluorescent spectra to ascertain the nature of the rare earth ion site. They observed slight changes in frequency of the luminescence emission bands when changing from phosphate to silicate to germanate 22.
^ \ 1
(~) = nearest oxygens O = rare earth ion
O = next nearest oxygens 0 = silicon ions
Fig. 1.3. Structure of the rare earth ion site in silicate glass. After Reisfeld [1.72]. 23. glasses, concluding that the asymmetric part of the crystal field is greatest in germanate glass, decreasing in silicate glass and hence least of all in phosphate glasses.
Reisfeld [1.73] showed that in glasses the rare earth ion is surrounded by non-bridging oxygens of phosphate, borate or silicate. We may view the rare earth ion in a cube (Fig. 1.3) with eight oxygens at the corners. The four MO^(M = B, P,Ge or Si) tetrahedra are relatively undistorted with each edge of the cube common to the cube and tetrahedron. The cube is not perfectly regular, being twisted by the tetrahedra, that may be located at angles other than 90°, thus producing a low symmetry Cg site. For this proposed rare earth site model the tetrahedra in silicate glass has a Si-0 bond length of 0.148 nm and 0-0 bond length of 0.240 nm. Assuming the rare earth ion is at the centre of the cube its distance from the nearest oxygen in silicate glass is 0.229 nm. Robinson et al [1.74,1.75] has investigated 3+ optical properties of Yb doped phosphate, silicate and germanate glasses and found the observed crystal field splittings were compatible with a distorted octahedral field. Their final idealized model suggested a structure based on an undistorted tetrahedra which has trigonal (C^ ) symmetry. Figure 1.4 shows the octahedral complex constructed from the glass forming tetrahedra with arrows indicating the direction of the distortion to produce trigonal crystal fields. Fig. 1.4. Octahedral complex constructed from undistorted SiO^ tetrahedra. Expansion in the direction of the arrows leads to trigonal fields at the void. After Robinson et al [1.74]. 25.
Accordingly in glass the Ln 3+ ion occupies the centre of a distorted cube introducing a trigonal
(D^) distortion. As site differences may occur throughout the glass due to slight changes in the twist angle of the tetrahedra, small deviations from trigonal symmetry are expected but the overall symmetry will be trigonal in nature. The rare earth doped glasses were chosen for this work as they are becoming, technologically increasingly important, and as such more information is required on the structure and physical behaviour of the rare earth doped glass. The Faraday effect is particularly important as these glasses show promise of being useful in light modulation devices. 26.
CHAPTER 2 : THEORETICAL CONSIDERATIONS
2.1.______Phenomenology It is known that magneto-optical phenomena is observed when the wave vector of a propagating light
(electromagnetic) wave traverses a solid, parallel to
the magnetic field or transverse to the magnetic field [2.1], The former case is known as the Faraday scheme
and the latter the Voigt scheme. In the Faraday effect
one observes the difference between the refractive indices of the solid for left circularly and right circularly polarized light which manifests itself as the
rotation of the plane of polarization of linearly polarized light (Faraday rotation or magneto-optic rotation, MOR) or the difference between the absorption coefficients for left circularly and right circularly polarized light known as the magnetic circular dichroism. For the case of the Voigt scheme birefringence of the solid may be observed due to the difference of the refractive indices for linearly polarized light parallel
(tt) and perpendicular (a) to the magnetic field (the Voigt effect and the Cotton-Mouton effect which is quadractic in magnetic field). These magneto-optic effects involve the interaction of magnetic fields with matter in a similar manner to the well known Zeeman effect.
Natural optical activity in a medium occurs as
a consequence of the very low symmetry of either molecules, or in the case of a crystal the unit cell. Magneto optical activity arises [2.2] as the magnetic field possesses helical symmetry producing a circular dissymmetry in the 27.
medium resulting in a different response to right and
left circularly polarized light. When natural optical
activity and magneto-optical activity occur in an
isotropic media the magnetic and natural rotations are
additive [2.3],
The initial task is to relate the
observables to the differences in refractive indices
(and absorption coefficients) of the right and left
circularly polarized light.
Consider a plane polarized electromagnetic
wave of angular frequency oo which propagates in the z
direction through some isotropic media. The isotropic
media may be a simple cubic crystal with a complex
refractive index n+)n+= n - ik where n is the real
refractive index and k the absorption coefficient. Such
a wave may be represented by the sum of two circularly
polarized waves of equal amplitude
E(a)) = E+ (go) + E_ (w) (2.1)
the + and - denoting right and left polarizations
respectively. Rewriting equation (2.1) in complex
notation, n±z ioo [t c ] (x + iy)E e (2.2) E±(w) ~ ~ o
In the presence of a static magnetic field
H, the oppositely circularly polarized components propagate with different velocities (n+^n_) and are
absorbed at different rates (k+7^k_) . The Faraday
rotation
0) (n_-n+) 2c (2.3) 28, and the MCD 0 ,
(2.4) which may be combined to form the complex optical rotation $
$ = ,() - ie = ^ (n_-n+) (2.5) where all the above follow the sign convention of natural optical activity.
The real part,
Faraday rotation and the imaginery part 0 the MCD. If we regard
Faraday rotation. The character of the dispersion relations relating real and imaginery parts of a complex expression was established by Kronig [2.4] and
Kramers [2.5], being known as the Kramers-Kronig integral transforms. (For a general discussion of integral transform relations see reference [2.6] ) . The Kramers-
Kronig transforms [2.7] relating 4> (oj) and 0 (co) are,
oo
(2.6) and 00
V J 2 9~ 77 n 2 o J ft -CO
In principle then measurements of either 0 (co) or (j) (co) are sufficient as knowledge of one enables the calculation of the other via application of the appropriate Kramers-Kronig transform. Experimentally it 29 is desirable to measure the MCD as it is uneffected by background effects due to the host media, whereas contributions to the Faraday rotation arise from the anomalous Faraday rotation about the absorption band as well as the host media. Additionally the MCD (0) is more rapidly varying than the Faraday rotation making it easier to separate overlapping bands or eliminate the influence of near neighbour bands.
The quantities discussed in this section are macroscopic variables relating the refractive indices and absorption coefficients to the observables, the MCD and Faraday rotation. To explore the microscopic significance of the Faraday effect requires an appreciation of the behaviour of the atoms in the solid and the manner in which they contribute to the macroscopic variables.
2.2______The Relationship between the Faraday and Zeeman Effects Before proceeding to a detailed theoretical analysis of the Faraday effect it is prudent to consider its relationship to another well known phenomena, the Zeeman Effect. After all the Zeeman Effect [2.8] is simply the interaction of a magnetic field with matter.
By recognizing the association between the Faraday and Zeeman effects a number of qualitative features of the MCD spectra about absorption lines are easily categorized [2.9].
Consider an atom where the angular momentum of the system is described by the usual quantum numbers 30.
0 0
Fig. 2.1a. Zeeman splitting of eigenstates of total angular momentum J=0, J=1 in a magnetic field.
4* r S' s' i J s' s' 1 s' 1 N | V/0 o 1 1 1 i 1 \ i \ i \ 1 i rep | Slop i 1 i 1 i 1 i 1 i 1 nu 1 o Fig. 2.1b. Selection rules for left and right circularly polarised light (lep and rep) radiation to m states split by a magnetic field. J
Fig. 2.1. Zeeman splitting of eigenstates and their selection rules. After reference [2.9], 31.
J and irij. J represents the eigenvalue of the total
angular momentum and mT the component of the angular momentum in the z-direction. Such a state in zero field is 2J + 1 degenerate but splits on application of a magnetic field into the various m_ states of distinct J eigenvalues, mT = J, J-l, J-2, ... - J. The relative J energies of the splitting will be determined by the Ldnde splitting factor, g.
Figure 2.1a shows schematically the situation for absorption of electric dipole radiation for a transition between a J = 0 and J = 1 state. On application of the magnetic field the J = 1 state splits giving rise to three transitions governed by the selection rules AmT = 0 ± 1. The AmT = 0 transition is J J labelled tt indicating that it is plane polarized in the z-direction whilst the a transitions corresponding to Airij = ± 1 are polarized in the x-y plane. Thus in the presence of magnetic field we see three lines, one polarized parallel (tt) to the field and two perpendicular (a) to the field. A similar set of selection rules may be derived from quantum mechanics
[2.10] for left and right circularly polarized light; these are given in Figure 2.1b. From the Law of conservation of angular momentum, circularly polarized light may be envisaged as being composed of electric dipole photons having J = 1 and mT = -1 for right circularly polarized light and mT = +1 for left circularly polarized light. On absorption of the photons by the system, for angular momentum to be Fig. 2.2. The Formation of the Faraday A term. (a) Effect of magnetic field on *S->*P transition m t (b) Absorption curve (c) Absorption of left and right circularly polarized light (d) Resultant MCD showing Faraday A term line shape 33 .
conserved the total angular momentum of the system must
change by 0, ±1 and the z-component of angular momentum by —1 for right and +1 for left circularly
polarized radiation. From these selection rules a
pictorial picture of various features of the Faraday
effect may be deduced.
Turning to Figure 2.2(a) we see a transition
from a non-degenerate ground state (^S state, J = 0) to 1 the degenerate P state (J = 1). The zero field absorption (ZFA) is shown in Figure 2.2(b) whilst
Figure 2.2(c) shows the absorption of the left and right circularly polarized light on moving from one line to the other separated by the excited state Zeeman splitting. On measurement of the circular dichroism
(k - k ) a lineshape of the form given in Figure 2.2(d) — “r is obtained providing the Zeeman splitting is less than the linewidth. The dispersion of this form is known as the Faraday A term and can only arise if the transition is to a degenerate excited state, that is J > 0. The A term yields information on the Zeeman splitting as its sign (positive or negative at short wavelengths) depends on the sign of the Zeeman splitting and the selection rules for circularly polarized light.
The opposite situation of a degenerate ground state and "'"S, hence non-degenerate excited state is shown in Figure 2.3(a). However a new feature is now apparent as in the presence of a field the ^P_^ level will have a higher population than the ^P+^ level, determined by the appropriate Boltzmann factor. The 34.
Fig. 2.3. The Formation of the Faraday C term. (a) Effect of magnetic field on * S transition , (b) Absorption curve (c) Absorption of lcp and rep light (d) Resultant MCD showing Faraday C term line shape 35 .
difference in relative populations between the
sublevels means the transition from the level has
an increased intensity over the P+^ transition as
shown in Figure 2.3(c). The resultant lineshape in
Figure 2.3(d) is termed the Faraday C term. Again the
sign of the C term depends on the sign of the Zeeman
splitting and on the appropriate selection rules, but its magnitude is a function of the inverse of the absolute
temperature and magnitude of the Zeeman splitting.
The A and C terms arise as a consequence of the Zeeman splitting of the levels by the magnetic
field. A third term also exists due to the mixing of levels by the magnetic field and is known as the Faraday
B- term. The lineshape of the B term is similar to that of the C terms but generally much smaller in magnitude. In any real situation all three terms may be present at the same time though their relative magnitudes will differ, depending on the temperature and magnetic field.
This simplified treatment of the dispersion of the Faraday effect gives a valuable insight into the expected lineshapes from a purely physical approach before embarking on a more detailed theoretical explanation.
2.3______The Theory of M.C.D.
2.3.1. Basic Theory
The most systematic theoretical discussion of the MCD dispersion through absorption bands has been undertaken by Stephens in references 2.11,. 2.12 and 2.13. 36.
These three papers represent the theoretical
cornerstone of modern theories on MCD. Stephens's
original approach [2.3] to the theory of MCD was based
A on the expressing of the complex refractive index n+ in terms of polarizabilities and calculating these
quantum mechanically, the method being an extension of
the early quantum mechanical techniques. The
contemporary approach [2.11, 2.12] is based on
semiclassical theory of radiation absorption and its
extension through use of time-dependent perturbation
theory. The approach adopted here is based on reference 2.11 (using c.g.s. units in line with the
literature) with the discussion confined to impurities in solids where only electric dipole transitions are important. It is also assumed that the impurities are rigid and dilute and their absorption localized. In the MCD experiment the sample is placed in a static magnetic field and circularly polarized light propagates through the medium in the z-direction. The
light is absorbed by the medium as it induces
transitions within the impurity centres in the solid. To calculate the contribution of each impurity centre
to the circular dichroism (CD) it is necessary to
firstly solve Maxwell's equations for the circularly
polarized (CP) light propagating through the sample and secondly treat the interaction of the light with the impurity centre quantum mechanically.
A circularly polarized wave made up of electric and magnetic fields E and H propagating in the z- direction through the absorbing medium is of the form 37. iUt-n z/c) E± = E° exp [------] (x ± iy)
i£(t-n z/c) H± = n±EQ exp [---- ^---- ] (±x + iy)
where the symbols have the same meaning as defined
earlier, with the addition £ = hv(hw) is the photon
energy. Note that E, H, D and B all lie in the x-y plane and D = eE, B = yH where e and y are the complex
dielectric constant and magnetic permeability in the x-y
plane respectively, with the approximation y = 1 made in equation (2.7). The intensity I(z) of the light is
the energy passing through a unit area normal to the
z-axis in a unit time, being given by the time average
of the Poynting vector (c/4tt) (E x H) . From equation (2.7) c o2 _2?k±z Vz) ■ <4¥)n±Eo exp[—(2-8)
-2£k z = I± (o)exp[-^c - ] (2.9)
The absorption coefficient relates to the rate of intensity diminution 81 in the z-direction. From equation (2.9)
i, _ r~ "fic i 3l(z) K± L 2£l(z)J 8 z (2.10)
thence from equations (2.7) and (2.8)
-31. (z) k (2.11) + [ --- —--- 1 n±C|E±(z)|2 8z I+(z) The quantity ----- is the energy absorbed per unit 8 z time per unit volume at z and is simply related to the transition probabilities of the absorbing centres. 38.
Suppose a set of eigenstates a and j corresponding to the ground and excited states respectively are responsible for the transition a ■* j. The probability of absorption per unit time at energy £ is P ., given N is the number of centres per unit volume in state a, then
ai(z) N P . (z) £ (2.12) 9z l a a+j af j where h k (2.13) ± ^ . N P* . (z) n±|E± (z) | a,j a a->3
The problem now degenerates to the standard calculation of P .. The Hamiltonian of the centre plus radiation a+j * is
H = HQ + Hx (2.14) where Hq is the unperturbed Hamiltonian and the interaction with the radiation. Assuming only electric dipole transition are present, then becomes
= - m.e (2.15) where m = is the electric dipole operator and e is the electric field due to the light wave at the absorbing centre. It is assumed e is uniform and is proportional to the macroscopic E field, that is e = aE, then
M*(z) = -/2 a | E+(z) | Re{m+exp [^^] } (2.16) where m+ = (1//2)(m + m ). Application of time dependent perturbation theory gives [reference 2.11] 39.
Pa+j(z) = V1 |E±(z) | |
k± = I. Na|
Recalling equation (2.9) and rewriting it as follows,
I (z) -2 gkz -A -ecz exp{ (2.19) 1(0) where 4 is the absorbance (= optical density) , e the molar extinction coefficient, c the concentration in moles per litre of absorbing centres and z the path length. Recasting equation (2.19) by taking the logarithm and application of equation (2.18) yields,
A+ e + r = (r> cz
Y + { l (vf“) | | o (£■■"£) }cz af J
NA7T2a2'Log10e (2.20) Y. 250hcn. where N is Avogadro's number and (N /N) is the a a fractional population of state a.
It is evident from equation (2.20) that the absorption for the transition a •> j is proportianal to the concentration N , the transition moment squared and the pathlength z. It is independent of light intensity and has a delta function dispersion. In deriving equation (2.20) linear optics approximations 40. were assumed giving an intensity independent A, as well equation (2.20) prohibits any non-linear optical phenomena at high intensities. The assumption of negligible interaction between absorbing centres is merely a statement of Beer's Law (4aN ) whilst the a. electric dipole approximation leads to a simpler form of the transition moment, by excluding magnetic dipole and electric quadrupole transition mechanisms.
Using equation (2.20), further basic formula describing the CD are simply derived. Defining
0 = A_ - A+ (2.21) As = e_ - e+ and further approximating n+ = n n' Y+ = Y_ = Y /
e_ cz £
=yI.(nS’) [I
(2.22) X 6{($. -£)}cz J3-
Equations (2.20) and (2.22) respectively relate the observable quantities, the absorption of the circularly polarized (CP) light and CD to the eigenstates of HQ ” the absorbing centres. The interaction term HQ of the
Hamiltonian may be rewritten as
H. H° + (2.23) O O where H° is the zero-field Hamiltonian and the o o magnetic field perturbation. To calculate the zero-field 41.
absorption A° and the CD, 0° in the absence of a
magnetic field, the eigenstates of are required, whilst in the presence of a magnetic field the
eigenstates of + are needed. As is real H o o H°o its eigenfunctions may be selected to be also real, such that
|< a|m+|j >|2 = |
thence
A° = A° , 0° = 0 (2.25)
thus in the electric dipole transition the CD in the absence of a magnetic field does not exist, indicating all CD is MCD and providing the definition of the zero field absorption (ZFA) where
A° = A° = A°_ = + A°) (2.26)
2.3.2 Dispersion Calculations: The Rigid Shift Model The problem now is to calculate the entire dispersion of the MCD - the ^-dependence. As in the last section the discussion is restricted to electronic transitions involving dilute non-interacting absorbing centres, rigidly located in the media. To simplify the theory it is further assumed that the Born-
Oppenheimer [2.14] approximation is obeyed for both ground and excited states and the Franck-Condon approximation suffices for matrix elements of electronic operators. The Born-Oppenheimer (BO) approximation permits the use of perturbation theory [2.15] as it assumes the 42.
incident wave is not seriously distorted by the scattering potential, in this case the absorbing centre.
The Franck-Condon [2.16] principle states that electronic transitions occur so quickly that the ions involved have no time in which to change their position or velocity.
Should the ion adjust its position in some manner in the lattice an energy shift occurs - the Stokes shift.
Again an electronic transition from A -*■ J is considered where A and J may both be degenerate. The zero field eigenstates of A and J are written down in terms of the B-0 [2.11] approximation and the transition matrix elements simplified by the Franck-Condon (FC) approximation. It is assumed that ground state vibrational functions of A are significant in amplitude only for R in the region of the equilibrium configuration Rq if nuclear motion is small. The electronic matrix elements are evaluated at R = R and o substituted in equation (2.20), giving
y vd I I
X { I spl
o,2 f j— d£ = y l a A A 43.
as
Equation (2.27) then becomes
-f- = YDf(£)cz (2.30) where
4- I | °|2 * A a L A 1 a1 ±1 A
o,2 ST l. [|
f (?) = 1 (/) l
J f(5)d5 = 1 (2.32) o
The shape of the zero field absorption (ZFA) is given by equation (2.30), noting that the shape is temperature dependent through the populations (N /N) with a the integrated intensity temperature independent.
Contributions to the ZFA shape are due solely to the ground and excited vibrational functions with the intensity dependent on only the equilibrium electronic functions (R ). o In the presence of an applied magnetic field,
H = Hz the Hamiltonian is perturbed, the magnetic field perturbation to first order in H being: 44.
H1 = ~y ~n~— (Z. + 2s. ) H o v 2mc 1 i 1 Z 2
5 -pHz S $(L_z + 2S z ) H (2.33)
where e and m are the electron charge and mass respectively, i sums over all electrons (but not nuclei), y is the electronic moment operator, L z and Sz the total electronic orbital and spin orbital angular momentum operators in the z-direction and 3 is the Bohr magneton. Any contribution from the nuclei is ignored as generally the nuclear magneton is an order of magnitude less than the electronic magneton. A set of ground and excited state manifolds is chosen so that Hq within the Franck-Condon approximation is diagonal as long as the equilibrium electronic wavefunctions diagonalize Any mixing between states is ignored and the Zeeman splitting of each vibronic state is independent of a vibrational level, being identical to the pure electronic splitting at Rq. Intermixing between different electronic states is introduced by perturbation theory to account for the wavefunctions in the presence of It is further assumed that the energies an(^ where k denotes a vibrational state are large compared to the Zeeman energy. Using these assumptions the transition moments [2.11] perturbed by H are: 45.
(2.34) L[ ° ±' A + 'h]
jk
(2.35) K ak which are the contributions of the vibrational levels where W° is the Rq energy of state K. The Zeeman splitting of the ground state leads to population changes N* A a a exp r?'Aaa/kBT} I exp {-C R a ABT a, a cl
exp{-ga/kBT)exp{
At high temperatures the Zeeman energies are small compared to k„T, so
N' A a exp{-ga/kBT} ^ ^ +
N* Aaa where Having established the contributions of the magnetic field to transition energies, moments and ground state populations, the CP absorption and MCD can be derived from equation (2.20). From equation (2.20) N' = Y { Z S2-) l » O 1 x |[ Here f’aX(5)= l |2«(Sja-[ In the presence of the field the shape function (?) 47. is identical in shape to the ZFA shape function f(£) but shifted rigidly along the £-axis by the A Zeeman shift - [ Collecting terms in zeroth and first order in H in equation (2.40) gives h\<\\<\Jx> \ ^xK\Jx>° r = r + Y{ a,i. a A 0 2 + l Re[ , °°|2f + ^ 1_ a' z1 a a1 ± 1 A 1 } Hcz (2.43) , dA a, A A whence | = Y (A(- ||) +(B + j^)f} (BH)cz (2.44) 5 where + 3" I U x • [ - J- y Re{ [A |me|J^>° - + l [ °< J. |me | K, >°] a 1I +1I A A 1 -1 k W°K - W°A I [| It is seen from equation (40) the manner in which the CP absorption is modified by the applied field. The Zeeman shift modifies the energy of the band whilst the intensity is changed by the ground state Zeeman effect and by the intermixing of the zero field electronic wavefunctions. The movement of the ZFA in the applied field without changing in shape leads to the naming of this situation as rigid shift. Equations (2.44) and (2.45) are derived for only high temperature conditions and large bandwidth with the various Zeeman components A J, of the transition contributing additively and linearly in magnetic field to the CP absorption and MCD. The three terms A,B,C 49 . are the Faraday parameters as discussed earlier and their behaviour determines the dispersion of the MCD through an absorption band. The existence of the A, B or C terms immediately . yields information pertaining to the electronic states. The existence of A terms requires either excited or ground state degeneracy whilst the C term only exists if there is a degenerate ground state. The B term is always present regardless of excited or ground state degeneracy. The three terms are physically separable by virtue of their dependence on either energy or temperature. Should the Faraday rotation dispersion be required instead of the MCD, equation (2.44) would be transformed by performing the Kramers- Kronig transform [2.17]. If all terms exist, then in order of magnitude, A : B : C ~ 1 : ^7 : 1 (2.46) where AW is the magnitude of an electronic energy gap. As „ (f) , 3 f N t max (2.47) 3 £ max I where T is the width of f, the maximum contributions of A, B and C terms to 0 are 11 1 A : B : C ~ r : AW : kDT (2.48) JO Clearly then narrow bands, closely spaced electronic states and low temperaturss favour A, B and C terms respectively. For a typical band at room temperature the Faraday parameters would be in the ratio 50. Fig. 2.4a. <■ MCD dispersion calculated from equation 2.44 showing varying magnitudes of the A and(B+C/k T) terms.A=l, and (B+C/k^T) in 1) 0 2) |jxl0""4 3) 10~3 4)2.4xl0~3 5) 5x10 ~3 Fig. 2.4b. The variation of C terms with temperature with A=B=0, C=1 and the ratios of kBT from 1) to 5) 20:10:5:2:1 i.e. 1/T temperature dependence. Fig. 2.4 Contributions of A and C terms to MCD spectra. 51. A : B : C * 10 : 1 : 50 (2.49) The dispersion of the MCD in a variety of circumstances calculated from equation (2.44) is given in Figure 2.4. As a result of the assumptions that the Zeeman energies are small compared to the band width r, k_.T and electronic energy gaps, changes in the CP absorption in the presence of a field are small. Typically the ratio between D and the A and C terms is -3 -4 of order 10 and 10 for B terms. Reviewing the information content available from MCD we see it provides only the Faraday parameters A, B and C which depend entirely on Rq electronic function properties. The presence of an A term indicates ground and/or excited states degeneracy and the C term arises if thermally accessible levels of the ground state undergo a Zeeman splitting, and in the event of ground state degeneracy alone A = C. As the A and C terms depend on magnetic moments and on CP transition moments, the symmetries of the electronic transitions involved are of importance. Information from A and C terms then enables the assignment of ground and excited state symmetries as well as the magnetic moments and electronic wavefunctions. Stephens [2.11] considers the problem of an absorbing centre of octahedral (0^) site syjnmetry and shows the g-values of the ground and excited states in terms of the Faraday parameters are, C I- (2.50) gex 2D ' 1 D 52. The MCD then provides a method for calculation of the g-values of ground and excited states in magnitude and sign. 2.3.3_____Overlapping Bands The last section considered the ideal situation of only a single band arising from a single transition giving the ZFA and MCD. In practice the situation of overlapping bands arises when two near degenerate excited states or an excited state and ground state lie close together. The first case may occur when the separation of the two near degenerate excited states is comparable to or much less than the linewidth whilst the second case requires the excited state to be thermally accessible by the ground state, hence the energy separation must be within k T of the ground state. The problem of calculating the MCD dispersion is analogous to the method used in the last section with the additional assumption the two near excited states J 1 and J 2 have parallel potential surfaces W 9 (R) - W (R) = AW 9 , (2 .51) J J J J where J 1 and J 2 replace the single J excited state. The procedure continues as before commencing with the zero-field eigenfunctions, which now contain identical sets of vibrational functions due to the 53. identical potential surfaces. The ZFA is a simple extension of equation (2.30) , y{D1f1 + D2f2} cz (2.52) where and f^ (i = 1,2) refer to transitions A -* and A J2, are defined as in equations (2.31) and (2.32) and f^ is identical in shape to f^ but shifted AW 9 , in energy, that is J J f2(U = f1(? - AW 2 ]_) (2.53) J J The matrix of is found in the context of o 1 2 the F-C approximation and the mixing of the J and J states treated by perturbation theory as their separation W 9 n is much greater than the Zeeman J J energies. Interactions with non-near-degenerate electronic states in the presence of the field are 2 again included with the resultant influence of J on 12 1 J being as though J were distant from J and vice versa. The MCD calculation follows the same direction as before with the result 0 ■ r. 1, 3f. ^ ,2 , Bfv £ Y^a ( a^) + A ( 1 2 + [B1 + +[B2 + SHcz (2.54) B B A^, B^ and C"^ for A -* being parameters as 1 2 defined in equation (2.45). As J and J approach in energy the MCD remains a simple additive function of the individual contributions. 54. The analysis of the MCD of overlapping bands leads to the A, B, C and D parameter for each band and the energy gap AW « , . However frequently J J1 the bands are completely unresolved (AW 2 n << r, J J the bandwidth) and AW 9 is unobtainable from the J J1 ZFA, and its extraction will be dependent upon the sensitivity of the MCD to its value. The MCD in turn varies according to the sign and magnitude of the component terms, giving rise to a number of interesting examples. Considering only C terms, then equation (2.54) becomes 0 Y[C1f1 + C2f2] (|—)cz (2.55) K B and the additional constraint that AW 9 , << T. To J J1 a first approximation (2.56) fl “ AWt2t1 J J and -3f 0 1, •, ,0H \ y{ [c1 + c2]f1 + c“aw 0 n ( ) > (£-m) CZ (2.57) J2J1 3? 'V 1 2 Should C 2 C and be of the same sign, and 2 8fi f^ >> C AW 2 2. (g-^1) t^ien the MCD is independent of J J 1 2 W 9 . However, should C and C be opposite in sign J J1 2 and of similar magnitude, C AW 9 , (3f,/3£) may be com- 1 2J J parable or greater than (C +C )f^ and the MCD is then 55. 1 2 sensitive to AW 9 ,. For the extreme case C =~C J J1 A 9 ^ 1 BH f = yC AW , , (- -=±) (jr^)cz (2.58) ? J^J1 35 'V The MCD in this situation is a "pseudo-A" term with the dispersion of the derivative of the ZFA and proportional in magnitude to AW 9 ,. J J1 Another interesting effect arises from the 1 2 coupling of J and J which manifests itself in the 1 2 behaviour of the B terms. J and J are inversely proportional to AW , and for the sake of the argument assume they and A are non-degenerate. Under these conditions the MCD is = y{B1(J2)f1 + B2(J1)f2>(3H)cz (2.59) where e i ,1^ 0-2. e i , v o B1(J2) = - 2Re{[ , 0 y{B1(J2)[f1~f2]} 6Hcz (2.61) 2 1 The contributions of J and J to the respective B terms are opposite in sign causing the overall MCD to change sign. For AW 9 , << T equation J J1 (2.61) becomes e C yb1(j2)AWj2ji (!!i) teal.ez af (2.62) yA' (- g|±) PHcz 56 . Fig. 2.5a) C terms for bands of decreasing separation (the ZFA has the same shape). Fig. 2.5b) B terms for bands of decreasing separation, giving pseudo A terms. 1 2 -C = C 1, note pseudo A term. o o Fig. 2.5. MCD dispersion for overlapping bands 57. where A = 2 Re{[ - The effect of the decrease in AW 9 n is the J J1 production of a "pseudo-A term" by the oppositely- signed B terms. The "pseudo-A term" has the dispersion of the derivative of the ZFA and is independent of AW 9 r The implication of this is J u1 that the MCD is identical to that obtained when 1 2 AW 9 n = 0 and J and J are exactly degenerate. It J J1 , may be shown [2.11] that A = A (A -*■ J) . Under the circumstances of AW 9 , << r the MCD dispersion may J J1 be evaluated via equations (2.62), (2.63) or by 1 2 treating J and J as components of a degenerate state and use of equations (2.30), (2.40) and (2.45). Illustrations of pseudo-A and B-terms are given in Figure 2.5. Overlapping bands also occur when an excited state lies close to the ground state and is thermally 1 2 accessible. Suppose A is replaced by A and A representing two thermally populated states involved in a transition to a single degenerate excited state J. Again the potential surfaces are assumed parallel so W 0(R) - W ' (R) = AW 9 , (2.64) 7T A1 A A1 The ZFA absorption is obtained as before to be = y{51D1f1 + (S2D2f2> cz (2.65) 58. where <5^ and 6^ are the fractional populations of 1 2 A and A , such that 5^ + ^ = 1* aRd f2U) = + Aw 2 ±) (2.66) A AL The MCD becomes 0 1 1 rl r= (- 3T-) + (B + 9 f 2 + 5 2 [A2 ( - + (B2 + £-^)f2]} eHcz (2.67) B and is a simple additive function of the MCD of the individual transitions A 1 -* J and A 2 -*■ J. Again the information yielded by the MCD and ZFA consists of A, B, C and D parameters of the two transitions and the energy gap between the near degenerate states. For near-ground state degeneracy the dependence on AW ^ i depends on the dispersion shift between f^ and f^r B terms due to coupling of A 1 and A2 and the population factors 6^ and 6^. Because of these various contributions the sensitivity of the MCD and ZFA to AW ^ ^ may vary markedly. The behaviour of the B-terms due to the interactions of A 1 and A2 as AW , , decreases, results in the sum of A A1 pseudo-A terms and pseudo-C terms when AW 2 ^ << kgT, A A± For small values of AW 0 , where AW 0 << kDT the 1 A A 2 A A MCD behaves as if A± and Az were exactly degenerate (AW 0 ^ = 0), with the consequence that equation A A (2.67) degenerates to equation (2.44). 59. 2.3.4 SATURATION In the discussion of the MCD in the previous sections, it was assumed that at high temperatures that k T was much greater than Zeeman energies. Under ±3 these conditions the population changes in the ground state are linear in H/T. On lowering the temperature or on increase to high magnetic field strengths, a redistribution of the populations among the ground state and Zeeman levels occurs. As the Zeeman energy and k T become comparable in magnitude the population B variation decreases very slowly with H/T until a point is reached where there is no further population change, that is saturation occurs. Quantitatively, at low temperatures equation (2.38) is replaced by V = exp{z|Aa>°H/kBT} {Na (2.68) N ]>exp{°H/k T} N a At saturation if a = the component of lowest energy (- °H is most negative), ot^ ^ g ^ N A a a (2.69) N With the prevalence of saturation conditions N A a g may be magnetic field dependent, requiring the modification of equation (2.44) to e_ * 3 -f r< = YN a -U (- ||) +(B + :r=) f} (8H) cz (2.70) € rv ^ 60. Use of equation (2.68) in place of equation (2.38) leads to the CP absorption being given by = y{ y exp{ a,X l exp{ x | ° + H | 2f' , (£) }cz (2.71) CX ± A CX x A OtA Dominant changes occur in the populations of the ground state Zeeman levels whilst changes in the transition probabilities and band shape remain fractionally small. As a good approximation equation (2.71) becomes exp{ 0 Y{ y exP{ x [| At the saturation limit equations (2.72) and (2.73) become T~t, = Yi l\ |°| 12} fez (2.74) f = Y{ I[| which are both independent of magnetic field and temperature. It is instructive to examine the saturation behaviour produced by a ground state that is a Kramers doublet. Writing gives exp { exp(g3H/2kBT) 5a + exp (-g£H/2kBT) 6 exp(g3H/2kBT) + exp(g£H/2kBT) (2.76 Also, the following must be satisfied ni = - I [| (2.77) so the MCD vanishes in zero magnetic field. The MCD is then given by _0 y tanh <20) [j> | °|2 £ BA -L |°|2}f (2.78) 1 a-^1 +1 A 1 62. and depends on H and T as tanh (g(3H/2k T) . For this situation the MCD is solely dependent on the ground state g-value and independent of the nature of the electronic transition. The ground state g-value may also be calculated from the Faraday rotation [2.3] where 4. = where 2.3.5_____Moments The previous sections entailed a number of simplifying assumptions to enable the calculation of the MCD dispersion curve about an absorption band. The most significant assumption was that the BO approximation holds for ground and excited states. The BO approximation breaks down as the Jahn Teller Theorem [2.18] states that spatial-symmetry-derived electronic degeneracy is removed by at least some nuclear displacements. Under these conditions a vibronic state can be no longer associated with one electronic state necessitating a more generalized approach [2.11] to the problem. A consequence of the 63. revised approach is that the calculation of the ZFA and MCD is much more difficult, in particular, the diagonality of is not maintained as the magnetic field perturbation scrambles different vibronic levels of the same electronic state. The calculation of the Zeeman effect within an electronic state then requires detailed knowledge of the Jahn-Teller phenomenon. Unfortunately the problem cannot be handled by resort to perturbation theory as matrix 1 elements of )-( may exist between states separated by energies greater or less than Zeeman energies, making diagonalization of an infinite continuum problem. The problem may be circumvented by assuming the Jahn-Teller effect is equal to zero thus retaining the BO approximation. Such an approach may be pursued with safety for either non-degenerate states or Kramers doublets as the Jahn-Teller effect is exactly zero. No problems then occur in breakdown of the BO approximation for singlet-singlet or Kramers doublet-Kramers doublet transitions of even or odd electron systems respectively. For other systems the BO approximation only holds if special reasons exist for the Jahn-Teller effect being accidently zero. A second alternative is to include explicitly the Jahn-Teller effect in the vibronic wavefunctions of a degenerate electronic state in the ZFA and MCD calculations. Again problems arise as each vibronic 64. level is free to interact with all other vibronic levels making the diagonalization of an infinite continuum problem. In particular for cases of near-degeneracy the model becomes increasingly complex and calculation of the ZFA and MCD is eventually unsurmountable. The problems associated with Jahn-Teller effects and BO breakdown unfortunately occur for cases of ground state and excited state degeneracy which give potentially the most interesting MCD spectra. To enable the analysis of these more complex models which are unable to be treated by dispersion calculations an alternative technique has been developed. The method of moments is an approach which largely overcomes the difficulties imposed by BO breakdown when electronic transitions are strong functions of vibrational states. The method of moments was first introduced by Henry et al [2.19] as an aid to analysis of MCD spectra due to F centres in alkali halides and was later generalized by Stephens [2.20] to apply to other systems. The moment of a spectrum is an averaged property. The n'th moments of the ZFA and MCD are defined by °>^ U - Ond£ n I CS - C)ndC where £ is chosen to be an average energy of the ZFA, 65. defined in the following manner A± 1 A°az (2.81) T~ <35 Clearly, moments are only defined for finite absorption bands which go to zero on either side of the band. The zeroth moments of A+ and 9 respectively a ™ + 0 the integrated areas of and Each particular moment is a single number reflecting some averaged property of the band, for example, Explicit theoretical expressions for moments are obtained by substituting equations (2.20) and (2.22) into equation (2.80), <0> l (jp) [ I |2 (2.82) a* D Wo ' 5ia = = l [ N N N y{ l jj2- [ x [< j | Hq I j ’ > - The preceeding expressions have the fundamental property of being invariant to a unitary transformation on the excited manifold j. That is, on substituting I j> ZIj°>U.O. (2.85) -i 1 D D where Uj°j is anY unitary transformation, equation o o * (2.84) retains an identical form with j and j replacing j and j' where ever they appear in the equation. The advantage of moments is that when the dispersion of A+ and 0 was calculated earlier the explicit j functions and hence diagonalization of The calculation of f required the solution of the nuclear motion problem and explicit forms of ground state and excited state potential surfaces. From the dispersion equations (2.30), (2.32) and (2.44), (2.45) the zeroth and first moments for the rigid shift model are ^>1 0 ; l <0> = y{B + (3H) cz (2.86) O K_a T <0>^ = yA(3H)cz where E, is the average energy of A°/E, about which <4±>^ C = 0* The parameters A, B, C and D are independent of f, being dependent on A similar set of equations relating the moments to overlapping bands arising from two near- 1 2 degenerate excited states J and J may also be obtained. They are from equations (2.52) and (2.54). y{D'*' + D2} cz 1 9 <0> ytA1 + A2 + (B1 + ^ + (B2 + j=-^) (l2 ~ 5)}(6H) cz (2.87) B where fi(^-^i)d^ = 0 (2.88) and from equation 2.66 (2.89) ^2 “ ?1 + AWt2t1 J J for <4°>^=0 pl +1)2 h h (2.90) D1 + D2 The zeroth moments for the ZFA and MCD and A-terms of <0>£ are independent of the band shape function 1 70. and excited state splitting AW 9 .. whilst the 7 J J1 B and C-terms of <0>^ depend on f^(£) and AW 2 J J Equation (2.87) may be modified by making the following substitutions 1 12 1 B1 = BX(J J + BL(K) B2 = B2(J1) + B2(K) (2.91) 12 2 1 where B (J ) and B (J ) are the B-terms originating from the interaction between J 1 and J 2 . As B1(J2) + B2(J1) = 0 (2.92) and B1(J2)(C1 - 5) + B2 (J1) (C2 - 5) E A' (2.93) equation (2.87) becomes <0>o = y{B1(K) + ^-^) + (B2(K)+£-^)}(BH)cz B B The effect of recasting equation (2.87) is to produce a pseudo A-term (Ay ), made up of 12 2 1 contributions from B (J ) and B (J ) which is independent of f-,(£) and AW ? , . Equation (2.94) 1 J J1 may also be derived from the direct calculation of the moments. For the example of overlapping bands the moment analysis avoids the task of calculation 71 of the excited state vibrational functions and the diagonalization of the Zeeman perturbation within the excited state vibronic manifold which are both required in the dispersion calculation. It should be noted that for overlapping bands the moments only yield sums of parameters, not providing the A, B, C and D parameters for the individual A + and A transitions - in real situations, in solids this is often the best one can hope to achieve. The power of the method of moments lies in the fact that the values of the Faraday parameters obtained are independent of any assumption of band shape and hence fitting procedures. The Faraday parameters are simply related to the integrated areas under the curve with the added advantage of the A term contribution integrating to zero, enabling the separation of C terms in the presence of sizeable A terms. 2.4______Group Theory and Symmetry Considerations The ZFA spectrum and MCD spectrum are sensitive indicators to the site symmetry of the absorbing centre. The nature of the observed spectra will be dependent on the energy level structure of the rare earth ion due to the surrounding crystal field. The crystal field partially lifts the (2J + 1) degeneracy of the multiplet levels of the free ion. Group theory enables the calculation of the splitting of the levels and their degeneracy 72. without resort to a full numerical calculation. Such insight is valuable as it allows consideration of the ion at sites of varying symmetry in the glass for comparison to crystals where the site symmetry is precisely defined. The symmetries of interest are cubic (0^) and trigonal (C^). The procedure of calculating the splittings is outlined in [2.21], requiring the calculation of the characters for a given J and formation of the irreducible representation. The decomposition of the irreducible representation yields the information regarding the splitting of the free ion energy levels. The method is best considered in light of a simple example, say the ground state of praseodymium, 3 in 0^ crystal field. For this example J = 4 and the characters are given by the following formulae sin (J + ±»> Xj(a) (2.95) sin(d/2) thus Xj(E) = 2 J + 1 o co II 1 • • • = 0 J=1,4, ... XJ = X(f) -1 J=2,5, ... XJ(CV = x(^) = ( -1) J Xj(c4) = X(tt/2) = 1 J=0,1,4,5 .... (2.96) Calculation of the characters for J = 4 from equation 73. 0, E 8C~ h 3 3C2 6C2 6C4 1 1 1 1 1 r 1 1 1 1 -1 -1 r 2 2 -1 2 0 0 r 3 3 0 -1 -1 1 r 4 3 0 -1 1 -1 r 5 (4) Table 2.1a. Character table for 0, and irreducible h representation for J = 4. E 2C0 3C~ 3v 3 2 r l l l r2 i i-i r3 2 -i o D(4) 901 Table 2.1b. Character table for C_ and irreducible 3v representation for J = 4. (Character tables from reference 2.23). Table 2.1. Character tables for 0^ and C2v symmetries. 74. (2.96) gives the irreducible representation D^, shown in Table 2.1a. Decomposition of D by reference to the 0^ character table in Table 2.1a gives D(4) = r! + r3 + r4 + r5 (2.97) 3 Thus, in the presence of a cubic field the ground state for praseodymium splits into a singlet state, a doubly degenerate state and two triply degenerate states. The case of trigonal symmetry (C^v) is treated in an analogous manner. The characters for J = 4 are calculated from equation (2.95) and the irreducible representation for D (4) formed. Decomposition of D (4) by reference to the C^v character table results in deduction of the crystal field splittings. The example of C3v symmetry for J = 4 is given in Table 2.1b, where the decomposition of D (4) is D(4) = 2T1 + r2 + 3T3 (2.98) giving three singlet states and three doubly degenerate states. The calculation of the energy level splittings for integral J values is perfectly straightforward, however for half-integral J values the situation is more complicated, requiring the use of crystal double groups, first presented by Bethe [2.22]. Assuming that the formula for calculating a character is valid regardless of whether J is integral or half-integral, then for a rotation through an angle a: 75. sin(J + ^)a (2.99) Xj(oO sin(a/2) Considering a rotation through (a+27r) , equation (2.99) becomes sin (J + -=-) (a + 2tt) 9 Xj(a + 2 7T) .~sin(a/2+~)------^ (2.100) Thus, for a half-integral J there is a double valued character, since a rotation through 2tt should leave things unchanged, whereas Xj(a + 2tt) = -Xj(a) (2.101) however, for a rotation through 4 tt Xj(a + 4 tt ) = Xj(a) (2.102) A rotation through 4tt is now thought of as being the identity. To cope with this situation a new group element R is introduced to represent a rotation of 2tt. The new group formed, the crystal double group has twice as many elements as the original group, though not necessarily twice as many classes, since Xjdtf^) = Xj(^2) = 0, making it possible that RC2 is in the same class as C2. The derivation of hhe character table for the cubic (0^) crystal double group is detailed in Tinkham, reference [2.23]. Application of the crystal double group to determine the energy level splittings requires the formation of the double-valued irreducible representation and subsequent decomposition into the 76. Table 2.2. Character tables for 0? and C? h 3 E R 8C3 8RC3 3C2+3RC2 6C2+6RC2 6C4 6RC4 1 1 1 1 1 rl 1 1 1 1 1 1 1 1 -1 -1 -1 r2 2 2 -1 -1 2 0 0 0 r3 3 3 0 0 -1 -1 1 1 r4 3 3 0 0 -1 1 -1 -1 r5 2 -2 1 -1 0 0 r 2 -F2 r6 2 -2 1 -1 0 0 -T2 r 2 r7 4 -4 -1 0 0 0 0 0 r8 d(9/2 } 10 -10 -1 1 0 0 T2 T2 Table 2.2a. Character table for crystal double group D 9/ 0^ and irreducible representation for J= 2. 2RC. 3a 3Ra '3v 1 1 1 1 1 1 1 1 -1 -1 2 -1 -1 0 0 X -2 1 -1 0 0 Y -1 -1 1 i -i -1 -1 1 -i i u 9/z^ 10 -10 -11 0 0 Table 2.2b. Character table for crystal double group D 9 / C^v and irreducible representation for J= 2 r5 and always form a doublet which is denoted Y. Tables from reference (2.24). 77. normal single valued representation, as though the group had not been doubled. Again, using an example, the ground state of neodymium where J= /2, the method is demonstrated for symmetries of 0^ and C°3v. The characters are calculated from equation (2.100), noting X (E) = 2J + 1 = -x(R) X(C2) = X(RC2) = 0 1 J = I 1 1 J 2'2 ..... X () = X (3—) = “1 J = 2 '.... = “X (3) » j - §■■§.... (2.103) /2 J = |'|...... X(C4) = X(f) = 0 J = . = - (RC4) 5 13 -n j = ...... The irreducible representation for D^^2^ and the (9/ ) character table is given in Table 2.2a- D 2 decomposes to d(9/2) - y + 2r 1 6 8 giving one doubly degenerate level and two fourfold degenerate levels. 9 / D The case of J = / 2 for C3v symmetry is given in Table 2.2b. D (9/ 2 ) decomposes to d(9/2) = 3r^ + 2{r^ + p^) yielding five doubly degenerate states. An ambiguity arises with the reduction of C3v with Y^ and always 78 . forming a doublet [2.25]. The preceeding examples allow the observation to be made that rare earths with half integral J values and hence an odd number of 4f electrons, at most split into levels that are doubly degenerate. This is the verification of Kramers Theorem. The Jahn-Teller effect is also implied as rare-earths with integral J values and hence an even number of 4f electrons split forming a singlet state. These simple group theory calculations give an immediate appreciation of the energy level structure of the rare earth ion in the glass without resort to a full numerical calculation. The Kramers ions are instantly identified and the ground state splittings of the ion established, giving insight into the expected MCD spectra. 79. CHAPTER 3; NUMERICAL TECHNIQUES 3.1._____ Calculation of the Kramers-Kronig Transforms The Faraday rotation and the MCD (from section 2.1) form the real and imaginary parts of the complex optical rotation, - ie (3.1) and are related by the Kramer' s-Kronig integral transforms att (3.2) 0 (w) (3.3) Jrft -wl2d0 Thus the knowledge of either Kramers-Kronig transforms when measurement of the Faraday rotation about weak absorption lines is masked by the background rotation of the host media or nearby intense absorption bands. Since the shape function of the MCD spectra is non-analytic, the calculation of 4> (cj) from 0(ft) requires the numerical solution of the Kramers-Kronig transforms. Such a numerical technique has been developed by Collocott [3.1]. The main problem with numerical calculation of the Kramer's- Kronig transform is the need to consider the Cauchy principal part in some special manner. Various attempts [3.2,3.3] using numerical integration to 80. overcome problems associated with the Cauchy principal part have met with partial success. To surmount the problem of the Cauchy principal part several authors [3.4,3.5,3.6] have suggested the use of Fourier series to compute the transform. In order to use a Fourier series cf> (ca) and 0(a)) must be band limited [3.4], that is significantly non-zero in the frequency range of interest. Under these conditions equations (3.2) and (3.3) may be written as follows: 00 mTTO) Fourier coefficients, b are found from equation (3.5) m and substituted into equation (3.4). The Fourier coefficients b are calculated by m the technique of interpolation of a function of a sine series [3.7]. Equation (3.5) may be written as a sum of n trigometric terms, u(x) = b,sinx + b0sin2x + b sin3x...b ,sin(n-l)x 1 2 x n-l (3.6) which will take for given values of (n-l) for equally spaced values of the argument x, say, 2tk _ .. „/3ttx _ .. „ (n-l)Tr _ „ u(l) - ur u(^) = u2, u(^) = U 3 * * * n-l where un, u0, u0 . . . . u ,. 1 2 3 n-l 81. Lagrange (see reference 3.7) noted that the sum sin — sin ^ + sin sin ...... n n n n + sin i£^£¥sin (3.7) n tt where p and q are positive integers less than n, has the value %n for p=q and zero for p^q. Then the function — (sinx sin E_L' + sin2x sin + ...... n n n + sin(n-l)x sin (n (3.8) has the value u when x = — and zero for x = 2. where q p n n ^ is different from p. It follows directly then that the Fourier coefficients b in (3.6) are given by the equation bm = n (ulsln n~ + u2sln ~ + ...... + u , sin (3.9) n-1 n Expression (3.6) is periodic with period 2tt, is an odd function of x and has values u^, ...... un-l at the given values of the argument. However should the function u(x) be not periodic and not odd (3.6) would have a graph agreeing with the graph of the function u(x) between x=o and x=tr. Thus, an expression agreeing very closely with a given function may be obtained over a certain range of values but outside that range, the values give a poor representation of the function. The numerical calculations of the transform were performed on a Control Data Cyber 72 computer using a FORTRAN computer program, PROGRAM FOURT written 82. for the task. The listing of PROGRAM FOURT is contained in Appendix 1 with the detailed working of the program described by comment cards throughout the deck. The program reads in the data from punched cards, treating the MCD spectra line by line. Various scaling parameters are read in categorizing the MCD spectrum of the sample under consideration, such,as temperature, magnetic field strength and the wavelength range. The program's output is fed directly onto a computer plotter, producing graphs of the MCD curve, the experimental Faraday rotation curve and the Faraday rotation curve calculated from the MCD curve. All the graphs produced are in the desired units and contain corrections for the background rotation due to the host media. The transform from the MCD to Faraday rotation is accomplished by a call to SUBROUTINE KRONIG (ZD,ZR,IN) where ZD is the input array to be analysed, ZR is the synthesised array (Kramers- Kronig transform of ZD) and IN is the number of elements in the array ZD. Two additional subroutines are called, MAXVAL and MAXMIN which respectively normalize the calculated Faraday rotation curve whilst preserving its shape function and calculate the range of values for the plotting routines. To test the accuracy and effectiveness of the numerical method data was fed in to simulate an MCD gaussian curve of the form 0 ( 03) = 5exp (-(30.5-03) 2/0.16) (3.10) TEST GAUSSIAN uo CM o o O H £ Z E PS H Q 0 PI W E K 2 D P 2 E w PS 1 H I pq ps H 1 II h h m h 83 c\° I o Q P P s u H w w in !5 D § P3 PS O > E 53 pq — PI P=q Z % m m h I CM vo LD ro CM Os o ro I • • vo CM ro O o O c- o vo r" CM o i ro O — • I CO VO CO i o\ LD ro o in o — I • • I vo O' CM ro o o o ro • vo un 00 i ro ro o — I • • I CM in i i ro ro i r-' o — — — I • I I l ■H ro •H ■H p «H r MH X! i T5 ,3 MH i i I u E -P us -p -P « — — — — cn P Q) u fd 3 0 o u P I a) cn U 3 3 CD fd P 0 3 o o fd O 3 — 3 p 0 tn 3 CD > rd 3 fd 3 H I h t I I I I ip •H m -P cn P 3 3 0 0 3 3 cn cn tr> 3 3 3 • 84 As the Gaussina is an analytic function the transform may be calculated by standard complex contour integration techniques [3.8] and compared to the solution given by the numerical method. The results for the test gaussian are given in Table 3.1 where it is seen the method is in very good agreement, being within ±1% of the exact solution. The exception occurs at 30.5 where the curve is rapidly falling through zero from a value of 99953 to -10075. Ideally the value should be zero but when -95 is considered in the context of the rapid transition the error from zero is only some 0.96%. Secondly towards the end there is a series termination error increasing the errors by some 1.5%. This Fourier technique has the advantage that fewer data points are needed when compared to the Fast Fourier Transform programs [3.9,3.10], As the Fourier coefficients are determined by the simple sine series given in equation (3.9)> the mass storage requirements are minimized and the central processing time improved. In a given production run of the program approximately forty three computer plotted graphs are producedjeach graph representing one absorption line. As all the data is on punched cards all samples and all absorption lines are dealt with in the one run. The program by a simple modification produces a complete spectrum for the sample under consideration in addition to the line-by-line output. 85. 3.2______Analysis of MCD Spectra Before proceeding to a discussion of the numerical techniques used in conjunction with the method of moments (see Section 2.3.5), it is prudent to briefly consider the alternative methods available for interpretation of MCD spectra. All the methods by a variety of means seek the values of the Faraday parameters A, B, C and D. The simplest method was one proposed by Badoz et al [3.11], with the fitting procedure utilizing a graphical technique. The method commences from the special assumption that the band shape function is of Gaussian form, f , giving the A ^ ^ + 0 expressions for ZFA, ^=- and MCD, under the rigid shift approximation: -£=- = 108.9DfG = Y (3.11) | = - 33.53[Af' + (B + C/k T)f ] = AY (3.12) t, vj Bo where fG = l/(/FS)exp[-(C-C0)2/S2] (3-13) The units used in the above equations are those of the molar extinction coefficient for A+ and molar ellipticity for 0. It can be easily shown from equation (3.11) that A-/Y = -(2/.S2) C5-50) (3.14) AY/Y = 9/S = 0.3080 [ (2/52) (S-S0) (VD)-(B + C/yi/D)] (3.15) 86 . The values on the left side of equation (3.15) are obtained from experiment and are linearly related to the energy £(£=hv) and the half width, 6 at i. This simple linear relationship, allows the determination of the ratio of the Faraday parameters to D to be deduced graphically. The chief difficulty associated with this technique is the necessity of calculating an accurate value of the derivative of the absorption curve. Particularly, when dealing with sharp absorption curves the calculation of the derivative is an awkward and often dubious task. The graphical method appears suited to only very broad, slowly varying MCD spectra. ' The technique of curve fitting which is often used in spectral analyses [3.12] may also be utilized for the analysis of MCD spectra. Curve fitting requires the assumption of a specific shape function for the band and the fitting of an explicit expression for 0 to the experimental curve, involving the Faraday parameters, frequency at maximum absorption and the half-width. These parameters are fitted using a least squares method with the parameter correction being evaluated interatively from the difference between the measured and calculated MCD. Again, the limitation of curve fitting is that it requires a specific assumption detailing the band shape. Returning now to the method of moments we can see the power of the technique as it requires no special assumptions pertaining to band shape, position 87. or width. Recalling equation (2.44) and (2.70) and rewriting them in terms of the frequency v rather than the energy £ gives the MCD dispersion through an absorption band as (3.16) The symbols are as defined in Chapter 2. If Stephen's [3.13] so called new "parameters" are used, A, B and D 2 2-1 are in D (Debye units squared) , B in D v , all energies expressed in cm \ H in Tesla and the MCD and absorption in absorbance units, equation (3.16)becomes 0 = 152.5Na[A(- ||) + (B + j—)f (V) ] Hcz (3.17) B The path length is z, and c is the concentration of ions, but as the ratios of tha Faraday parameters are only of interest they cancel out. It should be remembered that N , the population factor of the ground cl state may be a variable when saturation conditions apply. The nth moment about an absorption band A+(v) is defined as (from equation 2.80) U+/v) (v-v) n dv (3.18) band and for the corresponding MCD band 0(v) (0/v) (v-v)n dv The equations for zeroth and first moments of the ZFA and MCD in terms of the above units are A V II +1 0 326.6Dcz , (3.21 < 0 > 152.5[B + ] Hcz (3.22) O <0>1 = 152.5AHcz (3.23) The Faraday parameters are obtained from the moments via numerical integration of the experimental MCD and absorption curves. The separation of C and B terns is achieved by measurement of the MCD as a function of temperature. In the case of the rare earths the B terms are much less than the A and C terms and in certain situations may be neglected. The validity of neglecting the B terms will be pursued in discussion of the experimental results. A FORTRAN computer program, PROGRAM INT was written to compute the moments of the MCD and absorption spectra and calculate the Faraday parameters. The listing of the computer program is given in Appendix 1 with its detailed working described by comment cards throughout the deck. The program was run on the Control Data Cyber 72 computer. The experimental MCD and absorption curves were digitized with the numerical information being entered onto punch cards. The data was processed absorption line by absorption line with the Faraday parameters being calculated for each particular transition. Peripherals used by the computer program included a line printer for output of the Faraday parameters and other alpha-numeric 89. information categorizing the absorption and MCD spectra and a computer controlled plotter for graphical representation of the input data and results. The graphical output consisted of separate graphs detailing the experimental absorption curve, the MCD calculated from equation (3.16), the experimental MCD curve and the relative contributions of the A and C terms to the MCD, all being plotted over a given frequency range. The numerical integrations in PROGRAM INT are calculated by a call to SUBROUTINE PARBL which uses the repeated application of Simpson's rule to solve the integral. The Faraday parameters once determined are used as a basis for the calculation of the MCD from equation (3.16). However, the calculation of the MCD requires the derivative of the shape function f(v), which is found by numerical differentiation of the absorption curve by SUBROUTINE LAGDIF. LAGDIF performs the numerical differentiation by using Lagrangian interpolation on an array of four points (about the point where the derivative is required) to create an equispaced auxiliary table. The derivative is then calculated by application of finite difference formulae to the auxiliary table. Both LAGDIF and PARBL are standard math-science library routines. SUBROUTINE INTER is a linear interpolation routine to ensure that f(v) 8 f and are evaluated always at the same frequency. The value of the population factor N , is calculated cl 90. by assuming it is a constant independent of frequency relating the calculated and experimental MCD curves in a linear fashion. The value of N a is then computed by use of normal regression analysis [3.14] from the arrays containing the calculated and experimental MCD values. To gain a measure of the goodness of fit between the calculated and experimental MCD curves the correlation coefficient is computed. The calculation of Na and the correlation coefficient are performed by SUBROUTINE CORREL. The program also contains the facility to simulate variations in temperature so the behaviour of the Faraday parameters may be explored. In a production run the program deals with all samples and all absorption lines producing approximately sixty graphs. If desired graphs of composite spectra detailing the MCD, absorption and relative contributions of the A and C terms to the MCD of a particular sample between 400nm and 700nm are plotted. 3.3______Analysis of Faraday Rotation Saturation Measurements. The ground state g-value may be determined from measurements of the saturation behaviour of either the Faraday rotation or the MCD. From equation (2.79), the Faraday rotation where the symbols have the same meaning as defined in Chapter 2 with the addition that (j) is the saturation value of the Faraday rotation. The data was fitted to equation (3.24), using (j)^ and 2^=- as the fitting parameters. The KB fitting technique used was quite insensitive to b, which is not surprising considering b is — 6 approximately 10 A computer program fits the Faraday rotation data as a function of magnetic field strength to equation (3.24) by a normal regression technique. The computer program, PROGRAM GVALUE is listed in Appendix 1. Since the Faraday rotation was measured, it was necessary to correct the raw data by subtracting the diamagnetic contribution from the host glass? to leave only the paramagnetic rotation due to the rare earth dopant. The correction is accomplished in the program by SUBROUTINE BSODA which simulates the Faraday rotation behaviour of the base soda-silicate glass as a function of wavelength. The corrected data is fed to SUBROUTINE CORREL which fits it to equation (3.24) using normal regression analysis. By finding the regression coefficients the ground state g-value is determined. The calculation of (f)^ requires the assumption of some initial value which is substituted into equation (3.4) and the g-value calculated. The experimental and calculated values are then compared by calculation of the correlation coefficient which serves as a test of goodness of fit. Successive corrections are made to d> , until given values of <(> and the g-value produce a correlation coefficient which is closest to one. The output from the program gives the ground state g-value, normalized to a 1cm thick sample, the wavelength at which the Faraday rotation saturation curve was measured and the final value of the correlation coefficient. 93. CHAPTER 4 : EXPERIMENTAL METHODS flJD APPARATUS 4.1._____ Sample Preparation The rare earth doped soda silicate glasses were prepared in a specially constructed furnace built from morganite K28 refractory insulating bricks. The furnace was built in a dexion frame on a trolley with access to furnace via a top entrance. The furnace was electrically heated by six Morganite silicon carbide "Crusilite" elements, grouped in two rows on either side of the furnace chamber, approximately 10cm apart. The rods are connected in series to give a total resistance of 18ft. Voltage is supplied to the elements by a 10A, 250V Variac with the maximum voltage across the elements being limited to 180V, enabling the furnace to attain a maximum temperature of 1900K. Temperature control is performed by a Honeywell ON/OFF controller which operates a high current contacter in series with the supply to the elements, maintaining the furnace temperature to within ±10K of the preset value. The temperature sensor is a platinum/rhodium thermocouple. The furnace layout is shown in Figure 4.1. The glasses were prepared by initially crushing commercial soda-silicate glass. The crushed glass was then ‘'ball-milled" for 24 hours to produce a white slurry. The slurry was dried in a furnace at 393K for 24 hours and the resultant white cake of glass powder broken up and 94. layout furnace making Glass 95. R.E. Dopant Concentration by weight % Colour Praseodymium 1%,5%,8.5% green Neodymium 0.5%,1%,3%,5%,7.5%,10% blue-purple 15% Holmium 1%,5%,10% pale peach Erbium 5%,10% pink Table 4.1: Summary of rare earth concentrations in soda-silicate glass samples. 96. passed through a 300 ym sieve. The finely ground glass powder and various quantities of rare earth oxide were added together to the composition required. The rare earth oxides were of better than 99.5% purity being supplied by Koch-Light Laboratories. Before melting the two powders were thoroughly mixed by continuous tumbling for 24 hours. 3 The resultant powder was fired in a 50 cm recrystallized alumina crucible (or 95% platinum- 5% rhodium crucible) at 1773K for 8 hours. The molten glass was poured into a preheated mould and allowed to cool slowly over a period of 8 hours to room temperature. The samples were prepared in 50gm batches. No attempt was made to change the atmospheric conditions of the melt, as it was considered the air environment was a suitable oxidizing atmosphere. All the samples were annealed in a muffle oven at 773K for 24 hours to remove any internal stresses. After annealing ,the discs, 1.5cm in diameter and 0.5cm thick were ground and polished to give optically flat and parallel faces. Inspection of the samples under polarized light revealed them to be strain free. Table 4.1 lists all the samples made with the various compositions being expressed in weight percent of the appropriate rare earth dopant. In 97. Trace Element Nominal percent Cu, 1 A£ <0.005 Ni <0.005 Co <0.005 Fe 0.1 Mn 0.005 Pb 0.01 Cr 0.01 Co 0.01 Table 4.2. Impurity ion concentrations in base soda silicate-glass. (Analysis performed by Analytical Chemistry, Australian Atomic Energy Commission) 98. addition to the rare earth doped glass an undoped soda-silicate glass optical flat was prepared to enable measurement of the background Faraday rotation of the base glass. Chemical analysis was performed on the undoped glass to ascertain the type and quantity of any impurities that may be present. The result of the analysis of the base glass and concentration of impurity ions is given in Table 4.2. The glassy nature of the samples was confirmed by X-ray powder diffraction. 4.2 ______Optical Absorption Measurements The optical absorption spectra between 400nm and 700nm was measured for all samples at room temperature using a Cary 17 spectrophotometer. Low temperature measurements were made at 77K using an Oxford instruments continuous flow optical cryostat placed in the sample chamber of the Cary 17. In the case of Holmium glass it became necessary to measure the spectra at helium temperatures (4.2K) which was accomplished by using the same cryostat. 4.3 ______Faraday Rotation Polarimeter 4.3.1. General Description There are a variety of techniques [4.1] which may be used to measure the Faraday rotation, ranging from the early static methods [4.2] to the more recent double beam techniques. A double beam method introduced by Mitchell [4.3] is based on a 99. 250 W Quartz iodine lamp Vacuum Liquid Reference light i i • \ Polarizer 3 c Rotating o . ■*;; ______E polarizer 2 Reference Sample O' photomultiplier c beam a> photomultiplier o § X-Y recorder Analog 0 Y X 0 -a Pen control X Step/scan H field control monitor • monochromater motor step signal Fig. 4.2. Schematic of Faraday Polarimeter 100. direct comparison of the phases of modulated sample and reference beams. A phasemeter is used to measure the phase difference between the two beams, enabling the rotation to be recorded directly as a function of wavelength or magnetic field. An alternative arrangement proposed by Pidgeon [4.4] balances out the reference and sample signals in zero magnetic field at a fixed wavelength to give a signal proportional to the rotation when the magnetic field is applied. Several other methods utilizing an existing spectrophotometer [4.5] and low magnetic fields are described in the literature [4.6,4.7]. A double beam Faraday polarimeter of the rotating analyser type [4.8] with a detection system similar in design to the one used by Mort et al [4.9] was especially constructed to measure the Faraday rotation. Before proceeding to a detailed discussion of the various components, a general outline of the method is given in relation to the schematic of the system shown in Figure 4.2. Light from a 250W quartz iodine lamp is passed through a monochromater, polarized by polarizer 1 and focussed onto the sample. On traversing the sample the transmitted light passes through the rotating polarizer, polarizer 2 and onto the detecting photomultiplier. A second light beam generated by a green LED is polarized in the same sense, passed through the same rotating polaroid disc to be detected by a second photomultiplier. The signals from the two detectors are AC signals whose frequency (5Hz) is twice the rotational frequency of 101. the disc and differ in phase by an amount corresponding to the rotation of the plane of polarization of the light on traversing the sample in the magnetic field. The two AC signals are passed respectively through low pass filters to a digital phasemeter. Detection occurs at twice the rotational frequency of the motor, so noise due to misalignment or vibration of the rotating polarizer is easily filtered out. A digital-analogue converter provides an analogue output for the XY recorder. This is a dynamical technique allowing the direct recording of the Faraday rotation as either a function of magnetic field strength or wavelength. A difficulty arises when recording phase as a function of wavelength across the broad absorption bands as the rapid variations in absorption (as much as 1.5 absorbance units per nm) attenuates the transmitted light sharply reducing the signal from the photomultiplier. The rapidly changing signal is differentiated by the input decoupling capacitor of the phasemeter's amplifier producing a transient which causes an erroneous phase reading. To overcome this problem the monochromater is stepped across the spectrum in 0.5nm steps, being controlled by appropriate electronics and servo loops. As the Faraday rotation is small, large magnetic fields strengths are required/necessitating the use of a 5T superconducting solenoid and liquid helium cryostat as indicated in Figure 4.2. The 102 if> o \\ \\ \\ o—\AAA/——0 N O supply. power DC 24V/10A 4.3. Fig. CD jO 10 3. Faraday rotation was measured at 4.2K at a fixed wavelength by programming the field from 0-5T, the magnetic field being determined by the voltage across a precision wirewound resistor in series with the superconducting solenoid. 4.3.2,____ Lamp Power Supply The light source for the Faraday polarimeter is a 250W, 24V quartz iodine lamp. The lamp is powered by an especially constructed high current DC power supply with low ripple, as any AC would modulate the filament of the lamp producing an erroneous phase reading. The circuit diagram of the 10A/24V DC power supply is shown in Figure 4.3. The power supply is a conventional series regulator with the control section utilizing a yA723 regulator. Separate transformers are used to supply the low currents required by the control circuits and high current (high voltage) required by the lamp. The control circuit drives the series pass transitors (2x2N3055), with the voltage sensing element of the control circuit being connected to the output. In the top • corner of Figure 4.3 is the simple circuit of the supply to the reference beam, comprising four green LED's. The power supply was constructed in a well ventilated metal box with care being taken to ensure proper heat sinking of the power transistors. The control circuitry was located on a printed circuit board away from the various power elements. The 104. mains input and 10A/24V output were both protected by fast blow fuses. 4.3.3.____ Detecting System As may be seen from Figure 4.2 the detecting system is situated below the cryostat. The photomultipliers and rotating polaroid disc are located in an aluminium light tight box with a matt black interior to eliminate any stray light reflections from either the reference or sample beams. The rotating polaroid disc is 43cm in diameter being formed from a sheet of HN22 polaroid, chosen for its high extinction coefficient. To prevent sagging the polariod disc is sandwiched between two perspex sheets 38cm in diameter, leaving 2.5cm protruding to intersect the light beams. The sandwiched disc is attached by eight brass screws to a central brass spindle through which the shaft passes. The shaft has bearings at three separate points, firstly a ball bearing thrust bearing at the base of the shaft, a brass bush just below the polaroid disc and a brass bush bearing at the top of the shaft. A pulley is located on the shaft with a rubber belt connecting it to a 240V AC induction motor. The reduction drive from the motor to the polaroid disc gives a rotational frequency of 2.5Hz and hence an AC frequency of 5Hz. The reference and sample light beams pass through the rotating polaroid disc, on a diameter 105 . Sample PMT Reference PMT Philips XP1017 EMI 9661B -1.6 k V -900V - _acc I20kn 0.22jjF Fig. 4.4.Photomultiplier Dynode Chains and Low Pass Filters 106. on opposite sides of the rotating disc and are detected by two photomultipliers. The sample beam photomultiplier is a Philips end window tube, type XP1017 having ten stages, a Sb-K-Na-Cs cathode, and S-20R spectral response. The tube is operated with the cathode at -1.6kV below ground and has a soft iron shield to eliminate the effect of the fringing magnetic field from the superconducting solenoid. The reference beam uses a side window tube, EMI type 9661B, which is a nine stage tube with an S5 spectral response. Its cathode is held at -900V below ground, and shielding from the fringing magnetic field was found to be unnecessary. The dynode chains for both photomultipliers are shown in Figure 4.4. As a photomultiplier behaves like a current source a large value resistor (120kft) was connected between the anode and ground with the voltage across the resistor being fed to the phasemeter. To filter out all frequencies above 5Hz the resistor is shunted by a 0.22yF capacitor. When scanning across a spectrum the sample photomultiplier's gain is adjusted by varying the negative high voltage supply to the cathode. The anode current of the sample photomultiplier tube is monitored by an AC microammeter (mounted on the phasemeter) to aid in keeping the gain constant and also to give a relative measure of the change in absorption so the peaks of absorption bands can be accurately located. An especially constructed phasemeter containing several novel features not available in 107. commercial units, receives the two 5Hz signals from the reference and sample photomultipliers. The salient features of the phasemeter, which operates on the zero-crossing principle will be discussed here, with a detailed description including circuits contained in Appendix 2. Each time the reference signal passes through zero a clock is started, being stopped when the sample signal passes through zero. The time delay T^ measured by the clock between the two zero crossings is proportional to the phase difference A two signals. For a signal of period t, the phase difference in degrees becomes T A The phasemeter measures both T^ and t and calculates their ratio by use of Decimal Rate Multiplier units. The advantage of measuring the period of the signal is that any fluctuations in motor speed are eliminated as for each measurement of phase there is a corresponding measurement of frequency. To improve accuracy the phasemeter calculates a phase value averaged over ten periods to cancel out any small random phase errors. The value of the phase is displayed on a five digit LED display to 0.01° with an analogue output available for an X-Y recorder. To align the optics and ensure the sample beam impinges directly onto the sample 108 BY 178 bridge ^r-i Hh CL 0> c ro 2: o in fO Monochrometer motor -H LT) P I ■H W -P -p — G 0 CL u 0 0 c m o u 0 o m I 10 9. photomultiplier, the quartz iodine lamp is switched on with the rotating disc stationary. A digital voltmeter is then used to measure the DC voltage across the 120k resistor between the photomultiplier's anode and earth. The DC voltage is maximized by adjustment of a small knurled boss above the top window of the cryostat (see Figure 4.6), which has the effect of sweeping the beam laterally across the face of the sample photomultiplier. 4.3.4. Step-Scan System When scanning an absorption spectrum (as outlined earlier) erroneous phase readings arise due to the rapid changes in absorbance about the edges of absorption lines. To overcome this problem a series of feedback and servo loops are used to govern the polarimeter's various operations. Suitable electronics were designed and built to control the wavelength scanning mechanism of the monochromater, acquisition of data by the X-Y recorder and to allow a settling time for the phasemeter before it makes the phase measurement. The heart of the step-scan system electronics (shown in Figure 4.5) is a 555 timer running as a non-symmetric astable multivibrator which acts as the system clock, ensuring all the operations occur in the correct sequence. The mode of operation is as follows; initially the monochromater motor is gated on and the wavelength scanned through 0.5nm whereupon relay 1 opens allowing the 470yF capacitor 110. to charge. The charging time of the capacitor allows the phasemeter to settle, before relay 2 operates the pen lift on the X-Y recorder to record the data. Thence relay 1 closes, shorting out the 470yF capacitor and lifting the pen, allowing the wavelength to scan another 0.5nm to the next sampling point. Any variations in the scanning wavelength intervals are unimportant as a separate wavelength monitor on the monochromater is used to derive a signal for the X-Y recorder. The wavelength monitor on the monochromater is a Bourns precision ten turn potentiometer driven directly by a worm and pinion gear from the shaft that rotates the grating and wavelength indicator drum. Should a scan be required at a greater rate, as is the case of a non-absorbing solid the step- scan control can be over-ridden by the monochromater controls. Using the step-scan it takes approximately two hours to scan between 400nm and 700nm, though the process can be speeded up by increasing the wavelength interval. All the Faraday rotation curves as a function of wavelength were produced using the step- scan technique with the resultant curves being drawn as a solid line, as the interval between adjacent points is to small to discern over a 300nm (i.e. 600 points) wavelength range. 111. Window Removable central tube 0. rings with sample top plate exhaust Vacuum 5 T superconducting solenoid block Indium O.ring Radiation shield Provision He dewar for window base plate Vacuum space Nylon washer Fig. 4.6. Helium cryostat 112. 4,3.5____Cryostat As the Faraday rotation about absorption bands is small and measurements of its saturation behaviour were made, it was necessary to have both low temperatures and high magnetic fields. To achieve these conditions a liquid helium cryostat and superconducting solenoid were assembled. A schematic (not to scale), showing the various features of magnet and cryostat is shown in Figure 4.6. The cryostat is an Oxford Instruments type MD4A, heavily modified to allow the passage of a light beam along its vertical axis. The alterations were performed in such a manner to avoid the problem of locating a window, and hence glass to metal seal at liquid helium temperatures. A new base for the helium dewar was machined from a single ingot of type 304 stainless steel with a small sleeve protruding into the helium can into which a thin wall stainless steel cryogenic tube was silver soldered. Within this central tube another tube was placed with an aluminium block on the end to which the glass sample was attached. The aluminium block was free to slide on the central tube but in good thermal contact with the tube walls to ensure the sample reached a temperature of 4.2K. The vacuum seal on the sample carrying tube is a double O-ring seal located at the top of the central tube which is at room temperature. A new radiation shield and base plate were also machined to allow passage of the 113. light beam along the cryostat's vertical axis. The windows were cut from soda-silicate float glass with the top window being embedded in an araldite seal whilst the bottom window is secured by the nylon washer and O-ring arrangement shown in Figure 4.6. -7 The cryostat was evacuated to 10 torr by a conventional vacuum pumping system located on a moveable trolley. The superconducting solenoid was manufactured by Oxford Instruments from NbTi superconductor, having a maximum magnetic field strength of 5.37T. The central bore of the solenoid is 2.6cm in diameter with the central tube passing through the magnet bore as shown in Figure 4.6. The magnet and its stainless steel support tube are simply removed through the top of the cryostat by removing the sample carrying tube and sliding the magnet out over the central tube. The magnet was supplied with its own 40amp DC power supply and ramp generator to allow programming of the magnetic field to a particular value in periods varying from 1 minute to 1000 minutes. The magnetic field was monitored by measuring the voltage across a precision wire wound resistor in series with the superconducting solenoid. The voltage across the resistor was fed to the X-Y recorder when recording the Faraday rotation as a function of magnetic field at a fixed wavelength. When filled with liquid helium (approximately 114 4 litres), the magnet could be run for three hours before refilling was needed. When Faraday rotation measurements were carried out at 4.2K as a function of wavelength the magnetic field was set to 4.72T. A general view of the Faraday polarimeter is given in Figure 4.7. All the various components forming the polarimeter are mounted on a rigid, welded steel frame. Note that non-magnetic materials such as brass and aluminium have been used in construction in areas adjacent to the magnet and cryostat. The cryostat is situated on a wooden bench in the centre with the aluminium box containing the rotating polaroid disc and photomultipliers, immediately below it. The lamp power supply, quartz iodine lamp and monochrometer sit on the shelf above the cryostat with the phasemeter located in the console on the right. 4.4______Measurement of Magnetic Circular Dichroism The MCD was measured using a Cary 17 spectropolarimeter combined with a 6T superconducting solenoid and cryostat at magnetic field strengths of up to 0.5T and at temperatures in the range of 30K to 100K. The Cary 61 is a commercial instrument which uses an electro-optic polarization modulator to convert linearly polarized light into a beam whose state of polarization alternates between left- and right circular polarization. On passing through the sample the right and left circular polarization t\S Polarimeter Faraday The 4.7. Fig. 116 d> c ■o oE o ot C L- 0) cr. ^ C {/> ^ o >. Q- o >» ~o > ^ u 61. Cary of Schematic 4.8. Fig. 117. components are unequally attenuated resulting in intensity modulation of the beam. The intensity modulated beam is detected by a photomultiplier whose signal is fed to appropriate electronics to extract the circular dichroism information. Output is via a chart recorder as a function of wavelength. A schematic of the Cary 61 is given in Figure 4.3. The cryostat and superconducting solenoid operated in a similar manner to those discussed in conjunction with the Faraday polarimeter. As the Cary 61 cryostat, and magnet are commercial units, not being especially constructed as part of this work, the reader is referred to the manufacturer for further information. 118 CM nm W avelength I ✓ fC A r^5 o CO <4-1 o c o *H -P rd 4-> O PC 13rd rd U rd P4 ■—i LD Cr» -H&4 E c o X E a> > o 119. CHAPTER 5: RESULTS AND DISCUSSION (PART 1) FARADAY ROTATION IN NON-ABSORBING REGIONS 5.1._____ Faraday Rotation of Soda Silicate Base Glass It is necessary to examine the Faraday rotation properties of the undoped soda-silicate base glass to enable the separation of the contributions by the base glass and rare earth dopant to the Faraday rotation of the doped glass. The behaviour of the Faraday rotation dispersion of soda-silicate glass and similar optical glasses is well known [5.1,5.2,5.3,5.4],and the measurements presented in this section should be viewed in the context of the previous work. The Faraday rotation dispersion at 4.2K of the soda silicate base glass is shown in Figure 5.1, with a maximum positive Verdet constant of -2 8.6x10 min/(Oe.cm) being attained at 420nm. The positive direction of rotation and increase of the Verdet constant with decreasing wavelength is characteristic of diamagnetic Faraday rotation, being primarily due to the silica present in the glass[5.5]. Robinson [5.5] obtained values of the Verdet constant at room temperature for a similar -2 glass of 2.9x10 min/(Oe.cm) at 500nm and 1.4x10 -2min/(Oe.cm) at 700nm whilst values of order -2 4x10 min/(Oe.cm) have been measured by Borrelli [5.6] for a Corning type 8413 optical glass. To enable comparison with Robinson's results the 120. Faraday rotation of the base glass was measured at room temperature using a 2T electromagnet which gave a value for that Verdet constant at 500nm of -2 2.0x10 min/(Oe.cm). On cooling the Verdet _2 constant increased to 7.0x10 min/(Oe.cm). The increase in the Verdet constant was surprising as if the rotation is purely diamagnetic the Verdet constant should be temperature independent or if paramagnetic contributions occur the rotation is opposite in sign, decreasing the value of the Verdet constant. It was noted that the Verdet constant increased most markedly at temperatures below 40K indicating that vibronic effects or the introduction of thermal stresses may be contributing to the rotation, though this remains a matter for conjecture. It is not unusual to observe a spread in values for the Verdet constant in optical glasses as the exact composition of the glasses are often jealously guarded commercial secrets making it difficult to compare similar glasses from different companies. Within the same batch of glass variations in the Verdet constant may occur due to the introduction of strains by differential cooling. Because of these problems the Verdet constant may vary from 12-30% for allegedly similar glasses [5.5]. 121 glass. soda-silicate ro m doped Nd for 4.2K at rotation Background Faraday Background 5.2. Fig. O c 122. 5.2._____ Faraday Rotation in Non-Absorbing Regions of Rare Earth Doped Soda Silicate Glass As the rare earth ion is added with increasing concentrations to the glass matrix, the Verdet constant decreases in value, eventually passing through zero and becoming negative. The behaviour of the background rotation as a function of wavelength and concentration for the various glasses is shown in Figures 5.2, 5.3, 5.4 and 5.5. The room temperature Faraday rotation in non-absorbing regions of trivalent rare-earth- containing glasses has been investigated by Borelli [5.6], Berger et al [5.7] and Rubenstein et al [5.8]. As the rare earth ions have degenerate ground states (except Eu 3+ , 7 F ), the rotation is primarily paramagnetic in origin, characterized by the wavelength dependence shown in Figures 5.2 through to 5.5. Borrelli's [5.6] results for a 32wt% neodymium doped silicate glass, yielded a Verdet constant of -2 3x10 min/(Oe.cm) at 650nm. Allowing for the concentration differences and increase of the paramagnetic Faraday rotation due to the low -2 temperature, a value of 2.2x10 min/(Oe.cm) (from Figure 5.2) is obtained for the Verdet constant at 650nm for the 10wt% neodymium doped glass. This result compares favourably with that of Borrelli. Data from various glass systems [5.6,5.7, 5.8] indicates that for a given rare earth dopant the Faraday rotation is greatest in a borate glass, 12 3 to CO fd rH tn Q) -P fd u •H i—I •H CO td TJ O CO Td d) A O A + 00 O as p o 4-1 « CM • - ■p fd G o 'H -p fd -p o P >i fd Td fd p fd P4 Td G G O P tn fd CQ oo uo tn •H Fn X e F i g .5 . 4 . B a c k g ro u nF d a r a d ar y o t a t i oa nt 4.2K f o rP r d o p e dsoda-silicate g l a s s . 124 125 o co CO ptnH a> 4J 0Cd •H i—I ■H CO tO1 TOO CO TO CD a. TOo + CO pau u MHo nm wavelength, TO (Ou F(0h TO o SHCn u CQto uo m tr> •H P>4 CVJi E O c o x E CL) > O 126 . decreasing on progression to a phosphate and thence silicate base glass. This variation occurs as the diamagnetic contribution to the Faraday rotation varies with glass type. From Figures 5.2 through to 5.5 it is evident that praseodymium is the strongest Faraday rotator, followed by holmium and neodymium whose rotations are nearly the same with the weakest rotator being erbium. This hierarchy remains regardless of the glass type. It is noted from Figure 5.5 that for the case of 10wt% erbium doped silicate glass the Verdet constant is nearly zero. Such a situation occurs when the diamagnetic rotation from the glass matrix is comparable in magnitude to the paramagnetic rotation of the Er^+ ion. Under these conditions it is possible to produce by selective doping a glass that possesses a Verdet constant of zero, which is independent of both magnetic field and wavelength. Compared to the heavy lead glasses such as Coming's 8363 glass [5.6] the rare earth doped glass (in particularly praseodymium)gives a Verdet constant twice as large, making it particularly suitable for Faraday rotation devices. 5.3______Saturation Effects The Faraday rotation was measured at 4.2K in magnetic fields up to 5T for all the glass samples at wavelengths away from absorption so that only the ground state properties contributed to the observed 127 . 5 0 r MAGNETIC FIELD H ( te s la ) 128. effect. Figure 5.6 shows the magnetic field variation of the rotation for 10wt% neodymium doped soda glass, showing distinctly the characteristic curve due to saturation. In Figure 5.7, tanh ) ^ 00 has been plotted as a function of H to indicate how well the data fits equation (3.24). Figures 5.6 and 5.7 are particularly descriptive examples of the saturation behaviour, whilst Table 5.1 contains all the information pertaining to the samples studied, the wavelength at which the Faraday rotation was measured, the ground state g-values calculated and their associated correlation coefficients. All the rare earths in Table 5.1 possess degenerate ground states with the (2J+1) degeneracy being partially lifted by the crystal field. From Group Theory (Chapter 2) we know that erbium and neodymium are both Kramers ions having a doublet as their lowest level, whilst holmium and praseodymium are non-Kramers ions with the lifting of the (2J+1) degeneracy giving a series of doublets and a least one singlet level. Equation (3.24) was derived for the special case of a degenerate ground state (hence C terms are non-zero) , where a Kramers doublet is formed. However saturation behaviour is observed for both the Kramers and non-Kramers ions indicating that for the non-Kramers ions the lowest lying doublet and singlet levels must be close together. A doublet may be the lowest lying level or alternatively the singlet may be just below the doublet, but on 129 R.E. wt% Wavelength /cm g round Correlation Dopant nm °° gr u Coefficient ro + o 03 03 cr* noooorvj^^c^ro uno^t^^uorooo * ooir) HcocoHOOotNorNiLnounH^mM^n^ro^h i OMnvD^*HO> ro CN n * o o — — H — p • • • i I I 03 03 co cm o * i o lo o — — i — • ••••••• H i I i 0 * o vd rH o co o 1 — H H m >m^)uo^)voyoooo^^o*tNivDV£)Ln^r^cofon \ CN l uo h in rr i oooooooooo 03 03 03 vo - h ro + • • * ro t cn * ^ i o H h 03 03 03 r- uo S — — ••••••••• • i l cn i CN ^ uo o'* o 03 03 03 oo — • ^ I i — * CN a* uo in m o i 03 03 03 i-* H — — cr* \ l • CN l i in ■ o ■^r uo CN CN ro 03 O'* 03 03 — • I * CD CN 00 cm 03 O'* o\ 03 o o m ! O ro — — • • I I cn CT* ^ 03 o ro i O'! 03 03 03 r0 — •••••• • i LncocTiOor-' o CN rH N* ^ I o — • 03 03 03 cn I ^ i in o 03 CM CN — 03 03 03 vo • I ld O'! 03 03 i vo o ^ CN VO i I — — — • ro i i + I i — O 00 in CN iH vo CO o 03 03 03 U I • • co N* 00 in O uo O O ro o 03 03 03 03 • ro ••••• CN i uo h cn o CN 03 o cTirocovocNiH-^co 03 03 o 03 SJ — h • 1 i uo uo o 1 r-- vo rH t-" ro o 03 03 03 03 . ro ••••••• + I O h o cn CN tc ro er* m 03 03 03 03 o — 0 • 1 I i I O in o CN o O e' o 03 03 03 03 — — in i I VO I O rH 03 vo 03 r- o 03 03 03 H — oo • I cn UO in O H oo r o C?i 00 03 — • • ! co cn in o* rH uo cn O CN CN o 03 CTi • • co co o JQ i E uo i •H — i C/3 -P — -P (X 4-> — fd a> ns 3 P o c fti CD 03 h 03 2 i I I i — i * — i 130 * — i * — i i — i i — i i — i co co 131. application of a magnetic field the doublet splits forming a level below the singlet level. These situations can only be explored by doing a crystal field calculation and varying the applied magnetic field to observe how the levels split. This calculation is presented in Chapter 7 in a more detailed discussion of crystal field effects. It is important to realize the values of the ground state g-values (9grouncj^ tabulated in Table 5.1 are a spatial average of g in all possible directions as the glass has no long-range order. Nevertheless the values in Table 5.1 reflect an excellent fit to the data. A number of trends are evident from the results, namely with increasing concentrations of rare earth dopant the g-value decreases in value and the fit is marginally better at high concentrations. Within each concentration range the value of g remains relatively constant, with the best example being 15wt%Nd. In similar investigations of saturation behaviour in cerous phosphate glass, Alers [5.9] found g to be independent of composition whilst Berger et al [5.10] and Wertheimer [5.11] in their investigations found g to be composition and temperature dependent. Wertheimer [5.11] attributed the variations in g to differing degrees of short range order in the glass. The variation of the g-value is interpreted as reflecting changes in the crystal field in the glass with increasing rare earth ion concentration which in turn implies changes in the site symmetry of the 132 . rare earth ion as the twist angle of the SiO^ tetrahedra varies. If this interpretation is correct the variation of the g-value determined from Faraday saturation measurements provides a powerful method of detecting changes of ordering in amorphous materials. The close fit of the Faraday rotation curves to equation (3.24) also suggests a well defined total angular momentum for the optical excited states in the allowed transitions [5.12]. Comments relating to values obtained for the ground state g- value for particular rare earth ions in relation to crystals of similar symmetry are withheld at this stage, being dealt with in detail in the following chapter as g-values for both ground and excited states are derived from moment analysis. Absorption Absorbance/cm 400 Fig . 6.1. Absorption glass 500 at Wavelength, 293K spectrum and 77K — for 77 293 600 10wt% nm K K Ho J doped 700 soda 134. CHAPTER 6. RESULTS AND DISCUSSION (Part 2) Faraday Rotation and MCD about Absorption Bands 6.1. ____ Absorption Spectra The absorption spectra at both room temperature and at liquid nitrogen (77K) for all the 10wt% rare earth doped glasses are shown in Figures 6.1, 6.2, 6.3 and 6.4. All the rare earth doped glasses exhibit broad steeply rising absorption bands, typically 40nm wide, which on cooling to 77K intensify and decrease in width. The modifications in bandshape of the spectra at 77K are indicated by the dashed curves, where it is observed that the changes in the spectra predominate on the long wavelength side of the absorption bands. The diminution of the satellite bands on the long wavelength side of the absorption bands, occurs at low temperatures as there is a redistribution of the thermal populations among the thermally populated states resulting in only the lowest state being populated. As optical absorption is due to transitions from filled states to empty states, and at low temperatures only the lowest states are populated, the energy gap increases, which manifests itself as changes on the long wavelength (low energy) side of the observed absorption band. There is some evidence of fine structure at 77K on the short wavelength side of the 3+ absorption band centred on 430nm in the Nd doped soda silicate glass (Figure 6.4), but it is difficult to assess its importance due to the inhomogeneous 293 K — 77 K 400 500 600 700 Wavelength , nm. 3+ Fig. 6.2. Absorption spectrum for 10wt% Er doped soda glass at 293K and 77K. 136. 293 K 400 500 600 700 Wavelength, nm 3+ Fig. 6.3. Absorption spectrum for 8.5wt%Pr ' doped soda glass at 293K and 77K. Absorption Absorbance/cm 400 Fig . 6.4. Absorption glass 500 at Wavelength, 4.2K spectrum and 293K. for 600 n 10wt% m Na -- + doped 77K 293 700 K soda A b s o rp tio nA b s o rb a n c/c e m Fig. 6.5. Absorption at Wavelength 4.2K and --4.2K spectrum 29 3K. 293K X. for lwt% (nm). Ho 3+ soda glass A bsorption Absorbance/c m Fig . 6.6. Absorption at 293K . . spectrum wavelength 500 for lwt% X Nd , 3 + nm doped soda glass 139 . 140 . broadening of the absorption bands by the glass host. The MCD was measured about the absorption bands of the 10wt% rare earth doped samples as all absorption bands are of sufficient intensity to give meaningful results, whilst the Faraday rotation about the absorption bands was measured for samples of differing concentrations to establish the concentration behaviour of the magneto-optical spectra. It is seen from Figures 6.1 and 6.4 that some of the absorption bands are so highly absorbing that it becomes difficult to observe features about the top of the bands as the measuring instrument (in this case a Cary 17, see Chapter 4) lacks sufficient sensitivity. Similar problems of lack of instrumental sensitivity about these highly absorbing bands were encountered with the MCD and Faraday Rotation measurements. To overcome the problem of loss of spectral information about the highly absorbing bands, the absorption spectra of holmium and neodymium at concentrations of lwt% were measured and are shown in Figures 6.5 and 6.6 respectively. The absorption spectrum of the lwt% holmium doped glass was measured at both room temperature and at 4.2K, to check if any significant spectral changes occurred between 77K and liquid helium. By comparison of Figure 6.5 with 6. it is seen that changes, excepting line narrowing and intensification are minimal, with the only small change in structure occurring with the diminution of the satellite line at 480nm on the long wavelength side of the 450nm absorption band. The absorption spectrum of the lwt% neodymium doped soda glass (Figure 6.6) 141. 0.75 □ Ho o Nd u)t % of R.E dopant Figure 6.7. Beer's Law for R.E. doped glasses 142. shows clearly the absorption band centred 575nm and an additional band in the near ultra-violet. The band in the UV in neodymium has an intense absorption edge at 350nm making it particularly difficult to perform Faraday rotation measurements on the absorption bands between 400nm and 500nm as the large rotation from the 350nm absorption edge masks the Faraday rotation from the weaker absorption bands. Under such circumstances the advantages of MCD measurements are evident (see Section 2.1), as they are less effected by intense neighbouring absorption bands. The variation of absorbance of the absorption bands as a function of concentration (normalised to 10wt%) for all the rare earth doped glasses is shown in Figure 6.7 where the linear relationship between absorbance and concentration is consistent with Beer's Law. The obeyance of Beer's Law by the glasses is of importance as it is one of the principle assumptions made when developing the Theory of MCD in Chapter 2 (equation 2.20) namely, that there is negligible interaction between absorbing centres. Beer's Law also provides a convenient method for checking that the relative weights of the raw materials used in the glass making process,correspond to the correct rare earth concentrations in the glasses after firing. It is noted from Figure 6.7 that there was little loss of rare earth oxide in the glass making procedure, with the possible exception being praseodymium where the result for the 5wt% concentration appears to correspond H Dopant Transition DcxlO D Wavelength, nm (Debye^.10wt%) + oj tO 0J o 4s> o 2 • 0J a p to t-> \ 4^ tO tO K) 4i> to Vl to VI • OJ o Q O 0J to to to to vo ■o \ \ \ * 1 ro 45» to Vl O Vl h- M ■o O • O to to Vi -o \ \ VI OJ OJ OJ 00 1 oj M h- 4^ vo O'! o • o tc O to + tO OJ + VI fd M 4^ O EG n -O • to \ 1 4^. — 4^ t VI O *1 • V)to VO to VI \ \ 4s» CT\ 4* 1 4^ 4* VO h • F O — tO on I to -O \ \ * tO to to O'! VI O to • H CO to to CO t- M \ \ 4i» VI o 9 to vo \ On + oo EG -o Q O o • VI * Vl Ln Ul U> 00 VI VI 45> 4S» 2 Vl to o *1 o • ** 143 V) 00 1 On VI On I- to 01 M U> H VI • CO 00 tv to ^F5 8.9 650 Table 6.1. Transitions responsible for absorption bands in R.E. Doped Glass. 144 . more closely to 4wt%. The assignment of the optical transitions responsible for the observed absorption bands were made by reference to the spectra of rare earth doped crystals [6.1,6.2,6.3,6.4,6.5,6.8] and rare earth ions in aquo [6.6,6.7], where the site symmetry of the rare earth ion is similar to the rare earth ion's environment in glasses. The various optical transitions between the ground state and excited state (note that where excited levels lie energetically close together, it is impossible to separate them due to the inhomogeneous broadening of the absorptions bands by the glass) and the wavelength of the corresponding absorption band are tabulated in Table 6.1. Additionally tabulated in Table 6.1 is the electric dipole strength times concentration (Dc) for each transition. For the 10wt% samples (8.5wt%Pr) to enable comparison of the relative strengths of the transitions. The value of Dc is calculated from equation (3.21) , Chapter 3 by numerically integrating the area under the curve. The evaluation of Dc from equation (3.21) has the advantage, that it obviates the need of calculating the electric- dipole transition moment integrals accurately, a notoriously difficult problem, and secondly it eliminates the need to consider solvent corrections as only the ratio of the Faraday parameter (Sections 2.32 and 3.2) to the Dipole strength is of interest, and any solvent corrections enter into A,B,C and D in the same way [6.9], The values of Dc tabulated in Table 6.1 are 145 . accurate to within ±3%, with the errors being due to the digitizing of the curves and their subsequent numerical integration. In general most of the f«-*-f transitions of the trivalent lanthanides have intensities which are little affected by the environment of the rare earth ion. However, there are a group of transitions which are highly sensitive to small changes in the environment, these are termed hypersensitive transitions [ 6 JO ]. The hypersensitive transitions are indicated in Table 6.1 by an asterick in the wavelength column. Note that these transitions correspond to particularly intense absorption bands (see Figures 6.1 through to 6.4). The 4 5 hypersensitive character of the ^9/2 G6 trans^t:‘-on in holmium and ^1^5/2 ^ ^Hll/2 transit;*-on in erbium were established by Jorgensen et al [6.11] and Carnall [6.12], who noted all hypersensitive transitions obeyed the selection rules | AJ | ^2, |AL|<£2 and S=0 . A summary of the various properties of hypersensitive transitions is given in reference [6.10], with the most important in the context of this work being: (1) The intensity of the hypersensitive transition is zero when the lanthanide ion is at a centre of symmetry. (2) Very small deviations from inversion symmetry may alter hypersensitive transitions dramatically while others remain virtually unaltered. This has been demonstrated [6.12,6.14] for small deviations from 0^ symmetry. 146 . (3) The intensity of a hypersensitive transition can be up to 200 times greater [6.15,6.16] than that of the corresponding aquo ion transition. Because of the symmetry requirements (points 1 and 2 above) hypersensitive transitions only occur for the following point groups c5,C1,C2,C3,C4,Cg>c2v'C3v'C4V and CrT7 6V. Referring to Figures 6.1 and 6.5 which show the absorption spectrum for holmium doped soda-silicate glass, we see the most intense absorption band located at 450nm is due to the hypersensitive transition from 5 5 the Ig ground state to the Gg excited state. Absorption spectra for holmium in CaF2 [6.17] which has 0^ symmetry shows the 450nm absorption band to be approximately of the same intensity as the 420nm band. The rapid variation in intensity of the 450nm absorption band on going from CaF2 to glass is an indicator of the change in site symmetry of the rare earth site model outlined in Chapter 1, where the SiO^ tetrahedra are octahedrally arrayed but expanded along the C2 axis of the tetrahedra to produce a trigonal (C^y) site symmetry. Any variation of expansion or twist of the tetrahedra modulates the hypersensitive transition which is observed as a change in intensity of the appropriate absorption band in the absorption spectrum. Faraday rotation in 10% holmium doped soda glass 1 xIO' min/(oe.cm) constant, verdet R. F. o Wavelength, nm Fig. 6.8. Faraday rotation at 4.2K for Ho^+ doped soda-silicate glass. 148 . 6.2______Magneto-Optical Results 6.2.1.____Holmium. The Faraday rotation at 4.2K in a magnetic field of 4.72T about the absorption bands of 10wt% holmium doped soda silicate glass is shown in Figure 6.8, in conjunction with the Faraday rotation calculated from the MCD via the appropriate Kramers-Kronig transform (Section 3.1). The Kramers-Kronig transform was calculated for each individual MCD band but is presented in Figure 6.8 over the complete spectrum so that the two curves may be easily compared. For the case of simple MCD bands where the MCD band shape closely approximates a Gaussian (the 535nm band being an example) the agreement between the calculated and experimental Faraday rotation curves is excellent but worsens slightly for the more complex bands. The Faraday rotation dispersion about the broad absorption band centred at 450nm exhibits some fine structure but the calculated curve reflects only the overall features of the band. Problems arise in the calculation of Kramers-Kronig transform for the more complex bands as there are insufficient terms in the Fourier expansion to obtain an accurate representation of all the features of the band shape function.(Section 3.1.) From Figure 6.8 it is observed that the bandwidth of the Faraday rotation bands are identical for both the experimental Faraday rotation and the Faraday rotation calculated from the MCD by the Kramers- Kronig transforms, yet the former was measured in a magnetic field of 4.72T and the latter in a magnetic - 149. -0.061 min Oe.cm -0.145 -0.220 X n.m. Fig. 6.9. Faraday rotation at 4.2K for 10wt% Ho^+ soda-silicate glass about the 450nm band. 150. field of 0.2T. This indicates that the line broadening mechanism arises solely from inhomogeneous broadening of the spectral lines by the glass, being larger than any line broadening due to the Zeeman effect. The problem of inhomogeneous line broadening occurs in all the glasses and hampers observation of fine spectral detail in both the Faraday rotation and MCD spectra. The experimental Faraday rotation about 450nm for the 10wt% holmium doped soda silicate glass is shown in greater detail in Figure 6.9, without correction for the background rotation of the base glass. The variation of the Verdet constant with wavelength in this region is marked by a very rapid change at 442.5nm, which occurs over a wavelength interval of lnm. The amplitude of the change in the Verdet constant corresponds to a change in the rotation angle of approximately 70° in the sample studied. The peak rotation of 70° occurs about the rapidly rising shoulder on the short wavelength side of the 450nm absorption band which is away from maximum absorption. It is also clear from Figure 6.9 that there is a much higher resolution of structure in the Faraday rotation dispersion lines than there is in the absorption bands from which they arise. An interesting situation occurs in the 5wt% holmium doped soda-silicate glass as the background diamagnetic rotation of the host glass is approximately zero (see Figure 5.3) leaving only 151 . 152. contirubtions from the Faraday rotation dispersion. Under these conditions the Verdet constant changes sign at the Faraday rotation "edge" located at 442.5nm. The change in sign of the Faraday rotation is demonstrated clearly in Figure 6.10 where the rotation at 442.4nm and 454.3nm is shown as a function of magnetic field. The unusual behaviour of the Faraday rotation about the 450nm region has potential application in Faraday effect devices. Firstly, the rapid change in the Verdet constant at 442.5nm in the 10wt% sample produced a rotation of 70°, indicating that this glass may be of practical use in Faraday light modulation devices. Secondly, by selective doping of the glass it is possible to produce a glass which has both a positive and negative Verdet constant at different wavelengths. Indeed, should linearly polarized white light impinge on the 5wt% glass, its various spectral components are rotated in opposite directions, suggesting a use for this glass in the field of ellipsometry. The concentration dependence of the Faraday rotation for holmium doped soda-silicate glass is shown in Figure 6.11 where the peak-to-peak rotation, normalized to the 10wt% sample is plotted as a function of the wt% of the rare earth dopant. As with the concentration dependence of the absorption spectra the Faraday rotation concentration dependence obeys Beer's Law, there being a simple linear relationship between the peak-to-peak rotation and the concentration of the rare earth dopant. The Peak to peak F. R Fig. 6.11. rotation Concentration cot % for of all dependence R.E. Rare earth dopant □ o a of Nd Er Ho doped Faraday glasses. 153 . 10% H olm ium d o p e d soda glass O Fig. 6.12. MCD of 10wt% Ho~ ’^ doped soda silicate glass at 77K. 155 . shape of the Faraday rotation dispersion curves about the absorption bands in holmium remain unchanged, regardless of concentration, indicating that any co operative effects are of minor importance. The MCD spectra at 77K for the 10wt% holmium doped soda-silicate glass is shown in Figure 6.12. Included in Figure 6.12 is a second curve labelled "Theory" which is the MCD curve reconstructed from the Faraday parameters which are derived using the techniques of moment analysis discussed in Section 3.2. Once the Faraday parameters are calculated they are substituted into equation (3.17) and the MCD dispersion calculated. During the calculation of the theoretical curve from equation (3.17) it is necessary to numerically differentiate the absorption curve. The numerical differentiation method is extremely sensitive to rapid changes in the gradient, resulting in the severe rippling observed in Figure 6.12 about the 450nm band. Excepting the problems associated with the 450nm band, there is excellent agreement between the experimental and calculated MCD curves. The Faraday parameters calculated from moment analysis and their ratio to the dipole strength, the ground state g-value, N the population parameter and a the correlation coefficient which measures the agreement between experiment and theory are tabulated in Table 6.2 for all transitions in the 10wt% holmium doped soda silicate glass. The population parameter Na reflects the changes in relative populations of the thermally accessible ground state levels when rH u tn o 0 u 54 fd c 5 o 3 T r a n s i t i o n W a v e le n gnm th /D /D /A C (d eb y e cn m 44 4-1 u 0 o • in in tt O H i CM o o o r 00 in i>* VO m 00 CM 00 G vo ro 1 in o o uo rH oo ro i rH o rH in in rH CM o ro o o r- — X i 1 1 m • • • i • • • 1 • F ro UO in uo 1 vo uo ro in t H t P rH C/3 o O !\* CM o in CTi CM ro VO CM rH VO o o VO 1 ro ro ro ro X m 00 • CM • • • • ■ ' UO in 1 I 00 VO in o i [■"- r'- H P o — i H 1 VO o VO rH CO o — o i i — — — — X m 00 m. uo 1 • • i 1 • i 1 • I i ro + 1 VO CM rC T5 E4 E4 — rH P eu O -P 44 -M o\° •H •H K T5 'd i fd 0 i ■P 0 fd -P fd 54 fd — fd 54 fd e 0 « 0 — 54 03 54 O £ O 0 u ft 03 fd (0 o 03 o fd 0 03 0 fd m tn I I I I 156 15 7. saturation conditions apply at low temperatures, as in these circumstances may be magnetic field dependent (see Section 2.3.4). Before discussing the Faraday parameters for the holmium glass, a number of general comments can be made that are applicable to all the rare earth doped glasses. The existance of an A term indicates that the ground or excited state in the transition is degenerate whilst C terms are only possible when thermally accessible levels of the ground state are degenerate. The ratio of the A and C terms to the electric dipole strength D, allows determination of the ground and excited state g-values in the manner outlined in Section 2.3.2 and reference [6.18]. It should be also noted that expressions given for the Faraday parameters (equation 2.45, Section 2.3.2) are for absorbing centres oriented in the z-direction and for the case of randomly oriented absorbing centres in the glass the equations have the more general form given in reference [6.19]. The calculation of Faraday parameters from experiment by moment analysis and their physical interpretation is of course unaltered. All the rare earth doped glasses exhibit broad absorption bands, where in many instances each absorption band may be due to several electronic transitions producing a series of unresolved overlapping absorption bands. Under these conditions moment analysis gives sums of parameters (i.e. A and C terms, Section 2.3.5), that are composed of the and C term s for 10% Holmium doped soda g lass 2 xIO* cm) . T ( / ance b bsor A D. C. M. o Wavelength, nm 1 1 - + co 1 1 - 3 co r+ H- H- cn 3 O tr c Hi H- M h ft O o CD rt CD CO H- tr 3 a H- 'y h - >-Q O 3 ft — o\o ffi CD CO o a o I Q H- CD H- CD a ft CO CD ( o CD CO CO a iQ 159 , individual A and C terms of the transitions that compose the absorption band. Because of this it is often impossible to gain quantitative values of the excited state magnetic moments for each particular transition, but rather an average value of A/D that reflects the general magnitude of the Zeeman splitting of the excited states involved in the transition. The problem of overlapping bands is most severe in amorphous materials but eases for crystals. Stephens [6.20] estimates that the Farday parameters calculated from moment analysis have an uncertainty of 4 30%. The relative contributions of the A and C terms to the MCD spectra for the 10wt% holmium doped glass soda silicate glass is shown in Figure 6.13. The A term was separated out by subtraction of the C term contribution in equation (3.17) from the'theory curve in Figure 6.12. It is immediately noted from Figure 6.13 that the C terms dominate in holmium glass. This is not surprising as it was shown in Section 2.3.2 that C terms are favoured by low temperatures (equation (2.4.8)) and holmium has an open-shell ground-state. The character of the A terms which go as the first derivative of the absorption curve and the C terms which have the same shape as the absorption curve are easily identified in all bands, with a particularly descriptive example being the MCD dispersion about 535nm. The separation of the MCD into 160 . the component A and C terms was based on the assumption that any contributions from B terms were negligible and could be neglected. To check this assumption the MCD was measured as a function of temperature between 30K and 100K for the 650nm line and fitted to equation (3.22). The values of B and C obtained are respectively (-5.3±1.8)xlO“ 6 Debye nm“1 -4 2 and (3.6±1.4)xl0 Debye which gives a ratio of C to B of approximately 70. The reason for the large error in the C and B terms calculated from fitting the zeroth MCD moment to the temperature data was due to difficulty in controlling the temperature of the cryostat. For the various parameters tabulated in Table 6.2, it is estimated that A/D, C/D and C/A have errors of ±8% whilst the errors in N a and C are of ±5%, with the chief source of error in all cases being due to the problem of calculating the numerical integrals which are used in moment analysis. Nevertheless, the excellent agreement between the experimental MCD and theoretical MCD (excepting the 450nm line) as demonstrated by the various correlation coefficients in Table 6.2, verifies that the B terms are negligible in holmium at low temperatures. It was shown in Section 2.3.2 that the ratio of C to B is approximately AW : k T. Since AW is the magnitude of a the energy difference between electronic states, and -1 -1 is of the order of 5000cm and k T at 100K is ^70cm , ±3 the ratio of C to B becomes 71 which agrees with C/B value derived from experiment. In general then the 161. B terms may be neglected at low temperatures for the rare earths [6.18,6.28]. Ideally the ground state g-values tabulated in Table 6.2 should be self-consistent rather than exhibit a spread of values. EPR data for non-Kramers rare earth ions in tetragonal crystals [6.21] where the lowest level is a doublet give a ground state g^y of 15.3 for (Y,Ho)AsO^ and 15.3 in HoAsO^ [6.22], Remembering the g-value for the glass is averaged over three dimensions, where g^v = 1/3 (g^y + 2gA) then a g-value of about 5 would be expected for the glass. Data from holmium in Calcium Tungstate [6.23] suggests a g-value of 4.5 whilst EPR investigations of 3+ Ho in CaF2 [6.24] indicates a g-value of 4.9. The ground state g-values for the glass from moment analysis are between 3 and 6, (excepting the g-value for the 650nm line) whilst from the saturation measurements (Chapter 5) g-values of order 4 are obtained for low concentrations of holmium, decreasing to approximately 1.5 for the 10wt% holmium doped glass. A feature of the holmium doped glass (and other rare earth doped glasses) is the small A/D parameter when compared to the dominant C/D parameter. It was shown in Section 2.3.3 that A:C^i f thus 1 . Kgl_i_ C/A = T/kgT where T is the band width. For typical bands T is of order lO^cm ^ whilst at 77K k_.T is xi approximately 54 giving a C/A value of 20 which is consistent with the values of C/A tabulated in Table 6.2 which vary from 4, for the broad band at 450nm to 225 for the narrow band located at 535nm. The presence of the A term confirms that the excited states Faraday ro ta tio nin 10% erbium doped soda glass 1 xIO' ) cm oe. /( min constant, Verdet R. F. . o 162 Wavelength, nm Fig. 6.14. Faraday rotation at 4.2K for 10wt% Er doped soda silicate glass. 163. are degenerate but little quantitative information is obtained owing to its smallness. Its smallness may be due to covalency effects of the glass, lowering of symmetry or a combination of both [6.25]. Both the ground and excited state magnetic moments contribute to the A term (Section 2.3.2), with its sign being determined by the overall symmetry of the transition. Quantitatively the excited state magnetic moment may only be separated out for the case of a non-degenerate ground state, confining its use to identifying degenerate excited states in the presence of a degenerate ground state. Summarizing, we see that the MCD dispersion about absorption bands in holmium doped glass can be adequately explained in terms of A and C terms, with the C terms dominating. Moment analysis is satisfactory method for deriving the Faraday parameters, enabling the deduction of the ground state g-values that are in moderate agreement with the values of g-values obtained from EPR measurements in crystals of similar symmetry. However, problems still remain in the interpretation of the A terms in the presence of the dominant C terms. 6,2.2,____ Erbium The Faraday rotation spectrum for the erbium doped soda-silicate glass at 4.2K and the Faraday rotation calculated from the MCD via the appropriate Kramer *s-Kronig transform is shown in Figure 6.14. The data was treated line-by-line to calculate the 164 . transform but is presented in Figure 6.14 over the complete spectrum so the experimental Faraday rotation and calculated Faraday rotation from the MCD may be easily compared for all lines. As may be seen the agreement between the two curves is excellent, the errors being less than the thickness of the curve in most instances, which demonstrates the success of the Fourier technique for calculating the Kramer's-Kronig transforms. Erbium glass has the advantage that the absorption bands are conveniently spaced and less intense when compared to other rare earth doped glasses, making it ideally suited to Faraday rotation measurements. Figure 6.14 reveals that the Faraday rotation about 520nm changes sign, giving rise to positive and negative Verdet constants in a similar manner to the 5wt% holmium doped glass. This situation occurs as the paramagnetic rotation due to the erbium ion is nearly equal to and opposite in sign to the diamagnetic rotation of the base glass, resulting in a background rotation of approximately zero on which the Faraday dispersion about the absorption bands sits. The shape of the Faraday rotation dispersion about the absorption bands remains unchanged as a function of concentration, obeying Beer's Law in an analogous fashion to the Faraday rotation dispersion for the holmium doped glass. The concentration behaviour is shown in Figure 6.11 in conjunction with the other rare earth doped glasses. 10% Erbium doped soda glass -1 xIO cm) Absort/ance/(T. D, C. M. o W a ve le n g, thn m Fig. 6.15. MCD of 10wt% Er^+ doped soda-silicate glass at 77K. 166 . The MCD dispersion at 77K for 10wt% doped soda-silicate glass over the complete spectrum is given in Figure 6.15 with the solid curve the experimental and the dashed curve (labelled Theory) the MCD calculated from moment analysis. Table 6.3 contains a summary of all the identified transitions, the Faraday parameters for each transition, the ground state g-value, the ratio of the C to A terms, the correlation coefficient calculated to test the agreement between experiment and theory and the population parameter N which reflects the populations of the thermally accessible levels involved in the transition. The Faraday parameters for erbium were calculated in exactly the same manner as for holmium, with errors in A/D, C/D and C/A of ±8% and in N a of ±5%. (Note these errors are the same for all the glasses). It is immediately noted from Table 6.3 that again the C terms dominate in erbium doped glass at low temperatures with the C/A ratios for both holmium and erbium being of similar magnitude. This is to be expected as all the rare earth glasses have absorption bands of similar width, so the C : A = Tsk^Ta ratio will be much the same for all cases. The C/B ratio also behaves in an analogous manner, as was shown for holmium. Generally the agreement between the calculated and experimental MCD is excellent with a correlation coefficient of better than 0.8 for all but the 4 4 1-^/2 F9/2 transiti°ns • Problems arise with this band as it is highly absorbing, making it difficult to observe all the structure in the MCD and absorption i 53 u — ^ < o d I (TJ m u Transition Wavelength /D 'D /A Aground, o 0) • CM S3 H + CM o CM \ r^- uo \ CM o o CM 00 i CM o vo o 'sf CM VO d p o o CTi 00 LO co — • • • > • i t \ CM I iH m CM VO CM o in o o C 00 o 00 i in 00 o cr> •'3* CM r-> — m - \ 00 CM \ in CM • . • • F I CTi (T> o CM H o o CM C r* o 00 00 O i t in — m \ \ CM CM • i in • • • — 1 i + \ rH CM I in CM 00 1 CM in o in o o CTi o VO rH — S3 00 rH o VO o o o 1 1 oO ** \ CM \ CM 1 1 • 1 • • — s — 1 1 r- 1 1 m VO in d o in o 00 o — — CM o o H C 1 1 m 1 \ \ CM 1 CM cr\ • 1 ■ in • • • — i Table 6.3. The Faraday parameters for 10% Erbium doped soda-silicate glass at 77K. 167 M. C. D, Absorbance/(T.cm) x I0'2 I I > — 169 curve as the measuring instrument lacked sensitivity. As a consequence the data gained was questionable and the Faradays parameters derived from it are less reliable. The relative contributions of the A and C terms to the MCD dispersion at 77K are given in Figure 6.16. The C terms show the characteristic shape of the absorption curve whilst the A terms are a function of the first derivative of the absorption. Figures 6.17 and 6.18 give the MCD dispersion and A and C terms at 77K in greater detail for the transition centred on 450nm. The success of the moment analysis is demonstrated as the two overlapping bands are treated without resort to their deconvolution. In particular the relative magnitude of the A-terms contributing to each overlapping band is exemplified in Figure 6.18. Ideally the ground state g-values given in Table 6.1should be self-consistant rather than exhibit a spread of values. EPR data for trigonal erbium centres in CaF2 [6.26] gives g^y = 2.18 and g^ = 8.83 whilst for tetragonal symmetry g^y = 7.78 and g^ = 6.25 Later work by Zverev et al [6.273 on crystal field parameters of trigonal erbium centres obtained a value of g_j_ = 1.475 and g^y = 10.29. Remembering that the g-value for the glass is an averaged value over three dimensions as the erbium centres are randomly oriented a g-value between 4.7 and 6.6 seems probable. The g- values for the lines at 407nm, 450nm and 490nm come into that range however agreement is poor for the lines . C. D. A b s o rb a n/( c e T. cm ) x 10" Fig. 6.17 MCD soda dispersion silicate Wavelength glass. about 450nm , in nm 10wt% Expt Theory Er doped 170 . 171. C - term — A- term Wavelength, nm Fig. 6.18. The character of A and C terms for two near.neighbour bands at 450nm in 10wt% Er^ doped soda-silicate glass 172. at 520nm and 650nm. As was noted earlier the data for the 520nm was less reliable due to its highly absorbing nature, but reasons for the poor fit of the 650nm band are less clear. It may be that B terms are significant for this particular band, but as a general rule B terms are small for all but closely spaced electronic states (Section 2.3.2) and in this case the 4 Fg/2 state is well separated from neighbouring absorption bands. The results from the saturation measurements are less encouraging giving a ground state g-value of about 2. The reason for the disappointing result from the saturation measurement is almost certainly due to the difficulty in separating the paramagnetic contribution to the Faraday rotation from the diamagnetic contribution of the base glass as they are equal and opposite in magnitude. The effect of this is to produce a very small background rotation -3 (V»10 min/(Oe.cm)) which makes measurements difficult. Again, the A/D parameter is very small and serves to confirm the degeneracy of the excited state and indicating the components of the excited state are only slightly separated. As was seen with the holmium doped glass the C terms dominate in erbium at low temperatures and the ground state g-values are in moderate agreement for those from crystals of similar symmetry. 173. F. R. verdet constant, min/(oe. cm) x 10 .02r- k> -si T" Faraday rotation in 8.5% praseodymium doped soda glass 174. 6.2.3_____Praseodymium The experimental Faraday rotation for the 8.5wt% praseodymium doped soda silicate glass in conjunction with Faraday rotation calculated from the MCD are shown in Figure 6.19. The Faraday rotation about the absorption bands in praseodymium is small and is shown in Figure 6.19 as part of the paramagnetic rotation background curve, whilst the curve calculated via the Kramers-Kronig transforms is shown minus the background curve. No Faraday rotation dispersion curve was observed about the 600nm absorption band in either the 8.5wt% or 5wt% sample. The concentration dependence for Faraday rotation of the praseodymium doped glass is included with the data from the other rare earth doped glasses in Figure 6.11. As with the other rare earth doped glasses Beer's Law is obeyed. The MCD dispersion curve for the 8.5wt% praseodymium doped soda silicate glass rat 77K and the MCD curve aalculated from moment analysis are shown in Figure 6.20. The agreement between the experimental and calculated MCD curves is poor about the 460nm band but improves for the 600nm band. The poor agreement between the two curves for the 420nm band is due to problems encountered in numerically differentiating the region about the three absorption peaks (see Figure 6.3) where rapid changes in shape occur. The instability in the differention technique results in the severe rippling that is most noticeable in Figure 6.20. Figure 6.21 shows the relative contributions of the A and C terms to the MCD dispersion for 8.5wt% praseodymium at 77K, with 8 .5 % Praseodymium doped soda glass t>-0|x ( lu o * i) / aouDqjosq\/ *a*D *IAi 175. o Wavelength, nm Fig. 6.20. MCD of 8.5wt% Pr + doped soda silicate glass at 77K. and C terms for 8.5% Prasedymium doped soda gloss s-01 * (uuo i)/souDqjosqv Q D'IAI o Wavelength, n m 3 + Fig. 6.21 Relative contributions of A and C terms to the MCD in 8.5wt% Pr doped soda silicate glass. 177 rd VO i—1 LO CM • ( -e- 1 • l—1 • •^r vo CD M-l •—I vo 77K. u m o vo 5-1 o CO 00 • o 0 • at u u o o glass T5 g p in r- 0 • • u o o tr> tn silicate soda 00 C \ • U co in 1 l doped Q in r- \ • • U o o Pr co Q i—I i—I 8.5wt% • • C o o 1 i for e 3 .p -p o o tn VO o c vo i—i Parameters 0 Pa CO The 1—1 fi Pa 0 CO •H -P CM CM 6.4. *H Pa a cn CO 1—1 g + t rd 'VF 5-1 E 1 EH co CO Table 178 . the A term prominently displaying the rippling in the 460nm band. The Faraday parameters derived from moment analysis for the praseodymium MCD spectrum at 77K are tabulated in Table 6.4. The values of the ground state g-value calculated by moment analysis are given in Table 6.4 are more consistent than for the other rare earth doped glasses. A similar investigation by Kato et al [6.28] of the MCD spectrum of PrCl^ in polyvinyl alcohol film where the site symmetry of the rare earth ion is trigonal yielded values for the ground state g- value of between 0.68 and 0.81 whilst ESR data from the rare earth ethyl sulphates and trichlorides [6.29] suggests a g-value of between C.4 and 0.5. From Table 6.4, the ground state g-value is in the range of 0.45 to 0.75 whilst the saturation measurements give a value of about 1.0 and 1.6 for the 8.5wt% and 5wt% doped 3 glass respectively. The results for g-value obtained by moment analysis for the glass are consistent with Kato's values, which is reassuring as in both cases the rare earth is substituted into an amorphous material, but the saturation values for the g-value remain high. There is also general agreement between the glass, polyvinyl alcohol film and rare earth crystals of low symmetry in as much that they all give a g-value that is less than one. The A/D parameter for the praseodymium doped glass remains small when compared to the C/D term but larger than the A/D values obtained in the holmium and -1 .0 2r F a ra d a yro ta tio nin 10% neodym ium doped soda glass xIO cm) min/(oe. constant, verdet R. F. . 179 Fig. 6.22. Faraday rotation at 4.2K for 10wt% Nd^t doped soda silicate glass. 180 . erbium. The reason for the greater prominence of the A term in praseodymium doped glass is that the 3 transitions from the ground state to the degenerate 3 3 3 ?2 and states and the non-degenerate Pq state all overlap and are of approximately equal intensity. Under such conditions of overlapping bands moment analysis yields sums of the Faraday parameters (Section 2.3.5), so the A/D parameter in Table 6.4 should be viewed as being composed of separate A terms from the 3 3 P2 and P^ transitions. Moment analysis of the MCD spectra for the praseodymium doped soda-silicate glass gives more consistent results for the value of ground state g- value when compared to the other glasses but difficulties remain in obtaining quantitative information from the A/D parameter due to the strongly overlapping bands. 6.2.4____ Neodymium The Faraday rotation behaviour of 4.2K of the 10wt% neodymium doped soda-silicate glass and the Faraday rotation calculated from the MCD are shown in Figure 6.22. The measurement of the Faraday rotation about the absorption bands in the 10wt% neodymium glass is severely hampered by the dominant effect of the intense absorption band located in the near ultra violet. The Faraday rotation about the shoulder of this band is shown in Figure 6.22 as a rapid increase in the Verdet constant between 430nm and 550nm. The dominant nature of the Faraday rotation in this region 181. 418.5 wavelength, nm 486.5 wavelength, nm V f 2.4 x I0*3 min * Oe.cm wavelength, nm Fig. 6.23. Faraday rotation about .430nm, 475nm and 530n:n 34- in 15wt% Nd doped soda silicate glass at 4.2K. 182. from the absorption band in the UV masks any Faraday rotation about the 430nm, 475nm and 530nm absorption bands. In this case the shape of the Faraday rotation dispersion curves may only be deduced by calculating them from the MCD through use of Kramers-Kronig transforms (Section 2.1) with the dashed curve in Figure 6.22 the Faraday rotation calculated from the MCD. Because of these problems it was only possible to measure the Faraday rotation about the 575nm absorption band, where the agreement between the calculated Faraday rotation and experimental Faraday rotation is less than perfect. The absorption band located at 575nm is the most absorbing of all the absorption bands, making it difficult to observe the structure of Faraday rotation, hence producing Faraday rotation results that are of a dubious nature. The Faraday rotation about the 430nm, 475nm and 530nm bands were detected in a 15wt% neodymium doped soda silicate glass (Figure 6.23) as fine structure on the rising absorption edge of the absorption band in the near UV. Again, the concentration dependence of the Faraday rotation for the neodymium doped glass obeys Beer's Law and is graphed along with the other rare earth doped glasses in Figure 6.7. However, unlike the other glasses the lineshape of the Faraday rotation about 575nm varies markedly with concentration. The changes in lineshape are detailed in Figure 6.24 where there is evidence of loss of fine structural detail of the line as the concentration of neodymium increases. 183. 2.4 x 10 Oe. cm Oe. cm 540 550 560 570 580 590 600 610 wavelength, nm ig. 6.24. Change in lineshape of 575nm Faraday rotation band as a function of concentration. 10% N eodym ium doped soda glass 1 I0" x ) cm (T. / bsorbance bsorbance A D. C. M. I 184 o - W avelength, nm Fig. 6.25. MCD of lQwt% Nd3+"doped soda silicate glass at 77K. 185. There is a complete loss of fine structure spectral information about 580nm in the 10wt% neodymium doped glass when compared to the glasses that contain lower concentrations of the rare earth dopant. The absorption band at 575nm is so intense that only meaningful Faraday rotation measurements can be obtained about the shoulders of the band as the measuring instruments are not sensitive enough to explore the central regions of the band. The MCD dispersion for the 10wt% neodymium doped soda silicate glass at 77K and MCD calculated from moment analysis are shown in Figure 6.25. The relative contributions of the A and C terms to the MCD dispersion are shown in Figure 6.26 whilst the Faraday parameters are tabulated in Table 6.5. An interesting feature of the MCD spectrum for neodymium is the MCD dispersion about the 530nm absorption band which corresponds to the 4 4 4 2 Ig/2 ** G9/2' G7/2' K13/2 transiti°n • The MCD for this transition exhibits a "pseudo-A" term dispersion. It was shown in Section 2.3.3 that a "pseudo-A" term arises for the case of a transition to near degenerate excited states whose separation is much less than the linewidth. Under these circumstances a "pseudo-A" term is formed as the C terms for each of the separate transitions to the closely lying excited states have the same magnitude but are opposite in sign. It is seen from Figure 6.25 that the MCD dispersion about 530nm is composed of two components approximately symmetrical about zero. The two components of the 530nm and C term s fo r 10% Neodym ium doped soda glass i — 01 x ( UJO i)/aouDqjosqv ‘ a D IAI o W avelength, nm 186 Fig. 6.26. Relative contributions of A and C terms to the MCD in 10wt% Nd^+ soda silicate glass. . Transition Wavelength /D /D /A Aground Table 6.5. Faraday Parameters for 10wt% Nd doped soda silicate glass at 77K. 187 188. band were separated out and the Faraday parameters calculated for each component band, being tabulated as A and B lines in Table 6.5 where A refers to short wavelength component and B the long wavelength component. The Faraday parameters were also calculated for this band using the knowledge that for a "pseudo-A" term =. , with the Faraday parameter for this case being given in parentheses in Table 6.5. The extreme 4 case of yields a g-value of 1.5 for the ^9/2 ground state and it is this g-value that most accurately represents the character of the band as it is arrived at by considering both components, not component A and component B in isolation. The values 9groun<3 and hence C/D tabulated in Table 6.5 show a degree of consistency that is not shared with the other rare-earth doped glasses. The saturation measurements for the 10wt% and 15wt% neodymium glasses give a g-value of 1.6 which is in reasonable agreement with the ground state g-value obtained by moment analysis. MCD measurements by Kato et al, [6.18] on neodymium ethylsulphate nanohydrate where the local symmetry of the rare earth ion is / obtained 4 g-values for the *9/3 9rounc^ state in the range of 1.14 to 1.5 whilst their calculations suggest g-values between 1,21 and 1.96. The ground state g-values obtained in this work for the neodymium doped soda- silicate glass where the site symmetry of the rare earth ion is predominantly C^v, is consistent with Kato's results. 189. Again the C terms dominate the A terms as is expected at low temperatures. The A/D parameter serves only to confirm the small energy separation of the components of the excited states and the low symmetry of the glass. A transition of interest is the 4 2 2 I9/2 ** Pi/2 transition as the P}./2 excited state is unsplit regardless of the symmetry of the crystal field, the degeneracy being only lifted by the magnetic field. For this ideal case it should be possible by measurements at various temperatures to obtain useful information for the crystal field split levels which correspond to the three possible transitions 4l9/2(n=±5/2) ' Cn=±3/2) and (n=±5/2 ’) ■> 2P1/2(n=±l/2) . However due to the inhomogeneous broadening of the lines by the glass at no temperature are any of the transitions between the crystal field split levels and 2 state resolved except for the possible evidence of some fine structure in the absorption spectrum (Figure 6.1) at 77K. The likelihood of the structure being real or an artifact will be discussed in the next chapter. 6.3______Summary Having discussed each of the rare earth doped glasses in turn we are now in a position to make some general comments about the behaviour of the glasses and the interpretation of the Faraday parameters. The numerical procedures used for calculating the Kramers- Kronig transforms, the Faraday parameters, the theoretical MCD curve and its separation into component 190. A and C terms, are satisfactory in describing the character of the MCD and Faraday rotation spectra. It is clear that for all the rare earths the C terms dominate and that the B terms may be safely neglected at low temperatures. As the lanthanides are so chemically alike it is not surprising to find that the A, B and C terms and their ratios are of similar magnitudes. Indeed all the rare earth glasses studied are characterized by well separated, intense absorption bands in the visible which favour the existence of large C terms and small A terms with the B terms being of almost negligible importance. Moment analysis satisfactorily explains the behaviour of the MCD spectra but difficulties remain in interpretation of the Faraday parameters, particularly for intense bands made up of two or three overlapping bands. The values of A/D, C/D and the ground state g-value must be viewed as a value averaged over three dimensions that reflects a general property of a particular electronic transition or group of transitions. The g-values obtained for the ground state by moment analysis at any one concentration should be consistent among themselves as they reflect an average property of the ion in the glass and its surrounding crystal field. The ground state g-values obtained for glasses of differing concentration of the same ion may show a variation in g-value due to the interactions of ions in the glass and the subsequent 191. modification of the site symmetry of the ion. In the discussion of the rare earth site model in glass (Chapter 1, Section 1.6) it was shown that the glass forming tetrahedra may twist to produce sites of varying symmetry. Should there be any interaction between the rare earth ions in the glass the glass forming tetrahedra distort to produce a site of various symmetries that are a function of rare earth concentration. The uniform decrease in the value of the g-value obtained from the saturation measurements (Chapter 5, Section 5.3) suggests that there are a variety of sites present in the glass that vary as a function of concentration, with some local ordering evident for the higher concentrations of the rare earth dopant. However, referring to the graphs showing Beer's Law (Figures 6.7 and 6.11), it is seen that there is no significant departure from linearity. Note that for these Figures the peak absorbance and peak to peak Faraday rotation have been plotted as the width of the absorption bands and Faraday rotation bands are the same regardless of concentration. This fact is easily seen by comparing Figures 6.1 and 6.5 which give the absorption spectrum for lwt% holmium and 10wt% holmium at room temperature and likewise Figures 6.4 and 6.6 for neodymium. Thus, any evidence in the absorption spectra for co-operative effects are masked by the inhomogeneous broadening of the bands by the glass, forcing the use of magneto-optical data which is a more sensitive method for detecting symmetry variations. 192 . The g-values obtained for the ground state for all the rare earth doped glasses are in moderate agreement with those obtained from EPR measurements of ions in crystals which have a similar rare earth ion site, namely trigonal symmetry. The agreement between the ground state g-value from saturation measurements and moment analysis improves for glasses that have a large paramagnetic Verdet constant, with the most consistent results being obtained for the 10wt% and 15wt% neodymium doped glass. The results obtained are less reliable when the paramagnetic rotation of the rare earth ion is approximately equal and opposite in sign to the diamagnetic rotation of the base glass. The largest ground state g-value is obtained in holmium and the smallest in praseodymium with neodymium and erbium in between, this pattern also occurs in rare earth doped crystals, so in the broadest sense the results obtained from the glasses are reasonable. Finally, there remains the all prevailing problem of the disordered nature of the glass host. It is not possible in the glass to resolve any of the crystal field split levels due to the inhomogeneous broadening of the absorption bands by the glass. The amorphous character of the glass serves to smear out any fine structure, making it difficult to obtain quantitative information from the Faraday parameters relating to the individual crystal field split levels. CHAPTER 7. SOME CRYSTAL FIELD EFFECTS A group theoretical approach was adopted in Chapter 2 to explore the energy level structure of the rare earth ion in the glass host. This shows how the crystal field of a particular point symmetry lifts completely or partly the (2J+1) degeneracy of each J level, but it can give no indication of the energy separation of the crystal field split levels. To obtain a measure of the crystal field splittings in the glass host, requires the crystal field Hamiltonian which describes the symmetry of the ion site, and the crystal field matrix which is diagonalized to calculate the energy eigenvalues and eigenvectors. The crystal field calculation is performed in this chapter for trigonal rare earth ion sites, as these have been shown to be the predominant site in glass (Chapter 1, Section 1.6) to investigate the line broadening mechanism in the glass and the energy level structure of holmium and praseodymium. The calculation is then extended to determine the relative oscillator strength for a typical transition. In crystal field theory the interaction on a given ion due to other ions is represented by an electrostatic potential V(r). The form of the potential is dictated by the point symmetry of the lattice and can quite generally be written as an expansion in spherical harmonics, Y™(6, V(r) = l A(r)” Y™ (6,<(>) (7.1) £ =2,4 ,6 For rare earth ions equation (7.1) can be terminated at £ = 6, because higher terms have no matrix elements with the 4f wavefunctions. The functions A(r) are difficult to calculate ab nitio and are usually employed as empirical parameters. The crystal field Hamiltonian for trigonal symmetry (C^) is given by Judd [7.2],and in terms of Stevens [7.3] operator equivalents is:- where (7.3) n with A™ ® td td td td td 2 H{ O 0 CD CO CTi 4i> cn 4^ to + ^ 1 1 1 , 1 1 on o o 00 to 4s> o to • 00 'O 00 . to • 4^ vo o o On to vo o O • to 4^ o 00 • on CD 2 ft cn ft H{ OJ • tr H-* CD H- 2 Cb 0 — , 1 1 1 1 1 — to on vo 4^ o o 1 o M cn 1 • to to to cn • — O • to CTi 1 00 I- VO 00 • (T\ • OJ t Hi H- O ft — , vt to 1 , 1 1 to 1 00 M O 4^ to 4^ l o 4S» i 4^ to cn • CTi on — o o 00 -o • O'! O 1 • 00 cn a\ to • -o • 2 Cb O cn — , to to i H-* > 2 CD n OJ H- Hi (D CD 2 CD H{ H- CO M vt + 00 OJ CD CD Hi > < 2 Cb CD O vt Hj 2 M OJ OJ O H- Hi 0 i-3 2 a vQ to to to CD Hi 2 ft cn (D 3 H- ft tr Cb 0 2 1 1 1 l to 45» to o 1 00 VO I 00 I- to to on VO VO to I- O 1 1 1 H-* o — 1 00 h- 00 00 t VO I to on to VO to VO O ll ll td > II td M 00 > 00 vo to t> td M 1 1 1 1 1 tO tO -O 00 to I- 1 M 00 CO VO tO I- VO 4^ 1 1 1 1+ 4^ 00 1 h- -J 1 to M I- 00 tO tO M VO VO II II 4i» M VO OJ M II > 00 o td 00 > td cn 00 00 > o -J 00 M [ o on 00 VO on cn 1+ tO o 00 1 M o on VO 00 On Ch 195 Table 7.1. The energy eigenvalues for the three lowest doublets in the 4i neodymium ground state for a variety of crystal field parameters. 9/2 196. calculation. A listing of the computer program is given in Appendix 1, with SUBROUTINE CRYST handling the crystal field calculations. All the absorption bands in the glasses exhibit inhomogeneous broadening which is due to the spread in crystal fields in the glass. To examine the crystal field contribution to the line broadening 4 2 mechanism, the I^^ Pl/2 transit:*-on in ‘the neodymium doped glass was investigated in detail by calculating the energy level structure for a variety of crystal field parameters. This particular 2 transition was chosen as the P^/2 exc;*-te<^ state is not split by a crystal field of -C symmetry, making it 4 only necessary to consider the I /0 ground state. y/z The crystal field was taken to be C^v and the crystal field parameters were taken from published work on crystals which exhibit a closely related absorption spectra to that of the glass and where it is known that the rare earth ions reside at trigonal sites. The three sets of crystal field parameters used in the calculation are from the double nitrates [7.5], trigonal CaF2 [7.6] and Nd2C>2S [7.7], with the results of the calculation for the three lowest doublets presented in Table 7.1. The column labelled average in Table 7.1 is an average eigenvalue calculated for each particular crystal field split level with the error reflecting the energy distribution of that level. Also tabulated is the energy difference between the crystal field split levels. It is seen from Table 7.1 that at temperatures below 120K (i.e. 84cm only the two lowest doublets 197. will be populated and the spread in energies due to the variation in the C^v crystal fields present is of order 1300cm , thus the width of the absorption band corresponding to the -*■ ^p^/2 transit:*-on will be of order 1300cm ^. The value obtained experimentally at 77K is 1214cm ^ which compares favourably with the value estimated from the crystal field calculation, indicating that the line broadening mechanism is almost solely due to the spread in crystal fields. Each of the crystal field parameters in the Hamiltonian for C^v symmetry (equation (7.2)) was varied to ascertain their relative contributions to the calculated crystal field splittings. Not surprisingly the most important terms were found to be those that corresponded to off diagonal matrix 3 3 6 elements, namely the and B^ terms. The reason for the importance of these parameters is that the trigonal crystal fields are formed by the distortion of a cube along the <111> axis (Section 1.6) resulting in the breakdown of the relationship between these crystal field parameters, namely 35/2 and (7.3) 4 that hold for the cubic Hamiltonian in the <111> direction, where H Of the three crystal fields discussed the crystal field parameters from the double nitrates seem 198 . to describe the features of the crystal field in the glass most closely. These parameters were chosen in preference to the others as the g-values for the ground state doublet calculated by Judd [7.5], are g, = -2.72 and 9 ,y~ -0.32, giving an average g of 2.03, which corresponds moderately well with the value of 1.6 ± 10% obtained for the g-value from the MCD measurements detailed in Chapter 6. However a problem remains if the crystal field parameters from the double nitrates are used to calculate the energy level structure of the 4 I9/2 ground state, as the energy difference between the two lower doublets (from Table 7.1) is 298cm ^, indicating that only the lowest level will be populated at temperatures below 430K. It is evident, however from the absorption spectrum for neodymium doped glass (Figure 6.4) that there is a small satellite line present at room temperature on the long wavelength side of the 2 Pf/2 band at 430nm which disappears on cooling the glass to 77K. This indicates that the separation of the two lower doublets must be less than 205cm 1 (or 293K). On deconvolution of the two component bands it is found they are separated by 175.3cm-1 (or 252K) . Using this information on the energy separation of the two lowest levels and the crystal field parameters from the double nitrates as a starting set of crystal field parameters, the crystal field parameters were systematically varied until a set of crystal field parameters were found that most accurately described the observed effects in the p 1/2 transition for the 199 . neodymium doped glass. The proposed set of crystal field parameters calculated on this basis are (in cm -1). ; - R° D° B2 B4 B6 CM 00 0.324 0.2208 0.004592 0 -0.252 -0.0805 which locate the lowest doublet at an energy of -975cm, separated by an energy gap of 196cm ^ (282K)from the upper doublet. Note that this fit was obtained by 3 3 varying only the B^, and B^ parameters. In addition to the calculation of tne energy eigenvalues which show the spread in crystal fields, the corresponding eignevectors were calculated and used as input for the relative oscillator strength section of the program. This enabled calculation of the relative intensities for the transitions between the individual crystal field split components of the 410 -*■ ^P, /0 transition. 9/ ^ ±/2. The theory of electric dipole transition intensities of f f spectra was formulated independently by Judd [7.8] and Ofelt [7.9). The Judd- Ofelt theory shows that electric dipole transitions are permitted if the nucleus of the rare earth ion is not situated at a centre ofinversion, that is the symmetry must be lower than cubic for transitions to observed. Judd [7.8] shows that for the rare earths the forced electric dipole transitions have the selection rule 200 Aj<6 and arise as odd harmonics in the static or dynamic crystal field admix configurations of opposite parity into the 4f level. To calculate the intensity of a spectral line corresponding to an electric dipole transition from the component i of the ground level to component f of the excited level requires matrix elements of the form (Judd [7.8]):- < i (') f > (7.5) where D ( ') is written as a sum over spherical harmonics (see Judd [7.8]) . The theory commences by writing down the free ion states of an ion having an fn electronic configuration, as being composed of a linear combination of Russell-Saunders states and where J is a good quantum number. After much discussion Judd shows (see [7.8] for detail) that the line strength of a transition may be expressed as the sum of three intensity parameters,T^ and reduced matrix elements of tensor operators . As we are only interested in the relative intensities of the transitions between the individual Stark levels, the calculation is greatly simplified. We require matrix elements of the form f> = a, |j'm'> and Ii> J ,m> fm' 1 i and f denote the initial and final states, J and m are the total angular momentum and magnetic quantum numbers respectively with the prime denoting an excited 201. state and, a^ and a^ are the eigenvectors of the initial and final states. The matrix element i i < J,m|H. |J',m'> = £ E (-l)P+q (2A+1) 1 q q A = 4 with the 3j symbols [7.10] being written as and q = 0,±1, p = 0,±3 and t = 3. The results of the relative intensity calculation for the average crystal field energies discussed earlier gave relative intensities for transitions from 4Ig//2 ^Pi/2 1 : 0.93 : 0.89 : 0.83 : 0.16 (7.8) which demonstrates that more transitions of equal intensity are likely if a wide variety of crystal fields are present. If the observed line is considered as being composed of a number of Gaussians all of similar shape and intensity but separated by a distance much less than their width at half-height and 202. summed together in the ratios given in (7.8), then to a first approximation the resultant line will have a bandshape, approximating a sharply peaked gaussian. It 2 is seen from Figure 6.4, Chapter 6 that the Pi/2 kand at 430nm has the shape of an elongated Gaussian. The calculation of the complete energy level structure for all the levels in neodymium glass is considered beyond the scope of this work and so no comments pertaining to the relative energy separation 4 2 of the centres of gravity for the ^9/2 Pl/2 transition are made. However, attention is drawn to recent calculations for the energy level structure of Nd:LaC&3 by Crosswhite et al [7.11] who used 20 adjustable parameters to fit 101 levels with an error 8.1cm ^. Similar calculations have been performed for Nd2C>2S [7.7] where the difference between 2 experiment and calculation for the ^i/2 level was 69cm -1 4 and 33cm for I0. Should a similar calculation be y/2 attempted for glass then the parameterization scheme would certainly be more complicated as it would be necessary to consider the disordered nature of the glass in some special manner. A crystal field calculation using Judds [7.5] crystal field parameters from the rare earth double nitrates was also performed to investigate the energy level structure of the non-Kramers ions, holmium and praseodymium. The calculations revealed that for praseodymium the lowest crystal field level is a doublet located at an energy of -278.4cm ^, with an energy gap 203. of 36.4cm between it and the next lowest lying level which is a singlet. The Zeeman splitting for the lowest doublet in praseodymium in a magnetic field of 5T is 8.2cm . A similar situation occurs in holmium where the two lowest lying crystal field levels are doublets of energy -1006cm”1 and -927.8cm”1. The Zeeman splitting of the doublets in a magnetic field of 5T is of order 10cm 1. It should be remembered that the spread in energy eigenvalues will be similar to those given in Table 7.1. These crystal field calculations show that the non-Kramers ions, praseodymium and holmium will have only the lowest doublet level populated at helium temperatures. This result is consistent with the saturation measurements presented in Chapter 5. Before levaing the absorption spectra to discuss the MCD and its relation to the crystal field a final general comment pertaining to the inhomogeneous broadening is made. The absorption spectrum of neodymium in crystals (see Kato [7.12]) shows clearly fine structure that can be identified with transitions between crystal field split levels, the band being typically 400cm"1 in width for the 2?1/2 level but for glass the width of the 2^-^/2 kand is 1214cm 1 wide and thus precludes any observation of fine spectral detail that is simply related to the crystal field split levels. Further insight into the nature-of the crystalline field, and hence site symmetry of the rare Ion Level C3V symmetry 3p Jr r.± O + r3 r2 3pi 1 3 r. + 2I\ 1 3 D2' P2 3tt 2r. + vn + 3r0 1 2 3 ^4 Table 7 . 2a. Irreducible representations of ion levels under C^v symmetry Transition C/D(3) A/D(3) -y - (i//2) u (i//2)y r3 r2 y -y - (i//2)u -(i//2)(u -u ) r3 r3 ^ y 0 (i//2)ye h r3 0 - (i//2)ye r2 r3 Table 7.2b. Theoretical reduction of the values of C/D, and A/D for the transitions among each species of C^v symmetry. (Both tables from reference [7.13]). 205 . earth ion in the glass may be obtained from the MCD by considering the sign (positive or negative) of the C/D parameter. Kato [7.13] has shown that for praseodymium in a crystal field of symmetry that the signs of the A and C terms may be deduced from group theoretical considerations. The free ion level splits into the sum of several irreducible representations of C^v symmetry, as is shown in Table 7.2a. Kato then proceeds by use of the Wigner-Eckart theorem to calculate the expressions for C/D and A/D for the transitions among the crystal levels, which are given in Table 7.1b where u y and ye are a reduced matrix element c | T ^ | |y| |T3> of the r3 species in the ground and excited states respectively. By assuming that the populations of ail the r3 species in the ground state are equal, the overall sign of C/D for both the -*■ ^Pq, transition and 3 H4 -*■ 1 D2 transition are found to be positive. Recalling the C/D values for praseodymium from Table 6.4 we note that they are both positive which is consistent with the estimate of their signs from group theory. The sign of the C/D parameter for praseodymium serves to confirm that the rare earth ion is at a site of C3v symmetry in the glass. This simple group theoretical approach lends further evidence to the rare earth ion occupying a site of C3v symmetry, but further theoretical calculations are needed if both the magnitude and sign of the A and C terms are to be considered in detail for a variety of crystal symmetries. Suchacalculation for glass would be 2 06. exceedingly complex as it would be necessary to consider both the amorphous nature of the glass and the variety of site symmetries in the glass in some special manner. CHAPTER 8: SUGGESTIONS FOR FUTURE RESEARCH It is innevitable that any research work poses more questions than it answers, and thereby suggests areas in which research should be undertaken in the future. The results of this thesis suggest two avenues for future research, firstly the application of rare earth doped glasses to physical devices and secondly a theoretical investigation of the MCD about absorption bands that arise from hypersensitive transitions. The rare earths are already used in a variety of applications, including colourants for glasses and ceramics, phosphors in the electronics industry, the neodymium doped glass laser and magneto-optic modulation devices. The rare earth doped glasses are well suited for use in Faraday modulators as their Verdet constant is larger than the Verdet constant of the heavy lead glasses presently used. The principle limitation of magneto-optic modulation devices is that their frequency range is limited when compared to electro-optic modulators. An area of research that could be pursued in relation to improving the performance of magneto-optic modulators is the effect of ESR on the Faraday rotation about the absorption bands. It may be possible to modulate the electronic transition by the perturbing influence of the microwaves in a microwave cavity. If the electronic transitions can be modulated by ESR, resulting in a significant change in the Faraday 208. rotation, the frequency range of operation for the Faraday effect modulators would be increased into the giga-hertz regions. A theoretical aspect of the work that could be developed in the future is an investigation of the MCD about hypersensitive transitions in crystals of differing symmetry. Such an investigation may take the form of doping a material of cubic symmetry, such as Ca?2 and then applying strains in the crystal and observing the absorption and MCD spectra. As hypersensitive transitions are only observed in crystals where the lanthanide ion is not at a centre symmetry, the applied strain would remove the centre of symmetry, with MCD being used to probe the nature of the strain induced transition. 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Appendix 1 : Computer Programs PROGRAM FOURT PROGRAM INT PROGRAM GVALUE PROGRAM OSCST ooonoooooonnn n o 300 200 100 17 1 X y labelling PRINT** CALL LABELLING CALL PRINT** BMIA=8MI-PAN CALL FORMAT SFT SCALE CALL WRITE PANGE RAN=(BMA-BMI)/7.0FINDING AR=A-XRA RANGE CALL FORMAT NORMALISATION PLOTTING XPA=( CALL CONTINUE WRITE X-AXIS ZFRD(IJ)=(ZFRD(IJ)+OFSET)*(b0.0 DO CAL^ GO 7R(IJ)=((7RUJ)^(FRMAX/FNORM) SFLF FRMAX=1 CALL CALL CONTINUE DATA M.C.D DO IF(NFD.GE.I) ZFRD(I)=7FRD(I)*FSC FORMAT READ ZD CONVERSION hfcqr H H=0 FORMAT(II) KKK=1 FORMAT RF.AO READ PROGRAM OFSET SAMTH FWA NFP SH FTEM HFLS FSC TEMP READ SF A FORMAT IFUN.EQ.O) DIMENSION PROGRAM ZFRD ZD IN = =? HFLD a IS Cl) IS d ZZZ TO 100 VS . IS IS IS IS 7 ORIGIN b900S5 IS (1 IS (1 (1 (1*1) ORIGNV(-AB#-BMIA) AXIS(A*BMI AXIS SCALE(XPAfPAN) i MAXMIN(ZD» NGRAPH(lf8,0*9.0t0.4) NORMALIZATION MAXVAl MAXVAL(ZRtIN*FNOPM> KRONIG(ZD*ZR»IN) — = (3* ( (3 IN INTERVAL WAVELENGTH IS IS IS IS FACTORS OF FLOAT OF ) 300 hfld PLOT IN M.C.D * CApy NUMBER IS IS M.C.D. * (ZD »S) IS IS 6 *2» •1• 1 IJ=T 3) (6(4 .0 (//» 2) 1 F.R NUMBER ( ( ( MAX 86) 4) AS = 2F9 5A TERMINATES FIELD TEMP F.R TEMPERATURE FOURT I4*F6.1,2F4.I,Fb.l*F4,l Y-VALUES X-VALUES M.C.D INCHES! ( UNITS 1 ABSOLUTE SAMPLE H ( INTERVAL *INfAtSF*HFLDtlEMPfHFLSfNFDtFSCfFWAVStFTEMfSAMJH X-AXIS Y-AXIS A»BMI»24HM,C.Df ZD # KKK I • 10) ZD(200)»ZR(200)»ZHD(b)»ZFRD(200) (ZHD(IA) BMAiBMI*RAN*XRA*At X MM OF * ( GRAPHS ( o )/(254.0*33.0 (ZFRDfNI .4) IN SCALE FIELD SCALE * AND 7HD IN)-1.0)*H)/6.0 * IN GO SAMPLE . EXPT ( F GO * I i AT OF DATA ( FOP 30X AND 13•6)) FROM M.C.D WIDTH )» OF IN INPUTtOUTPUTtTAPE » INtBMAfBMI OF ( TO MIN 14HWAVELFNGTH* IA TO * ZFRD THICKNESS WHICH LOWER DATA Y-AXIS NFD * FACTOR DATA KG FACTOR DATA IN PLOTTING 5A CALCULATION ) ABSORBANCE/ F.R 999 WIDTH * TO 200 *IA=1*5) CARY THICKNESS IA=1*S) IF VALUE 10) * H PEP KG AT AT ( FRMA I POINTS FIELD 1 ) F.R POINTS LIMIT IN=0 IN WHICH WHICH AT 7 CHART *HFCOR*SH) ?MM FOR DEGREES/MM FOR X ) DEGREES ABSORBANCE/ INCHES! ) RECORDED WHICH IN TO IN F.R M.C.D IN IN ZD ) CM F.R M.C.D < IN IN OF +OFSET AND T NM /(SA NM TESLA (1) . NM M.C.D 1 f CM CM EXPT F.R M.C.D = 14, *-14 VERDET AT WAS 1 CONVERTED 1 ) ) NPUT.TAP£3=0UTPUT*TAPE22fPtOT) m *SF RECORDED F IN TH*HFLS) )*(60.0/( EXPT A 6,2, * ( INCH DATA RECORDED DATA 6,* T DEGREES RECORDED . CM CONSTANT F8 0.0 DATA SPARE )* NM/INT .4, 24 * TO )*0,001 4 SAMTH*HFLS F4 * * KELVIN XRA 7. DEGREES .1 ALROUND t * 90.* ) Fb .4, BMI 223 ))*0.001 F6.2, * . . RAN OFSFT F4 ) .?) . SH c C c n o o no o o o o n C 400 999 700 101 33 32 87 31 l CALL CALL N DO AMIN DIMENSION END PRINT** GO CALL CALL PLOT CONTINUECALL Y=ZFRD DO CALL CALL CONTINUECALL CALL labelling AMIN=Z FINDS STOP PLOTTINGCALL Y=ZR(IL) DO CALL m CALL CALL AMAX=Z AMAX Z subroutine Y P S = A M A + ( A R A N / 2 . Q ) X=A>( Y=ZR(1) CALL RANGE FINDING FORMAT CALL NOW PRINT** CALL CALL X=A+<(FLOAT(IG)-lt0)#H) DO MOVE x X=A WRITE APAN= A ama CALL HEADING CALL CALL PLOTTING CONTINUE y CALL AMIA=AMI CALL YPS=BMA+( IF IF(NFD.LT. X H = A + 1 0 , 0 * H IPEN=1 IF(CMI.GE.AMI)IF IF(NFD.LT = mi = i a zd IS IS * ( (CMA.LE.AMA) TO +<(F Z 33 32 ~ = aran 31 100 cm cma PLOT ( < OF PLOT PLOT PLOT(A*ZFRD S Y M B O LPLOT ( A * Z R ( 1 ) PLOT(X*Y*IPEN) NUMBER CLOSE SYMBOL(XH*YPS S Y M B O L ( A f Z F P D ( 1 ) * 0 . 0 8 * » £ X P T PLOT O P l G N V ( - A B * - A M I A ) A X I S ( A * A M I * 1 4 H W A V E L E N G T H * SCALE(XRA,ARAN) MAXMIN(ZFRD*NFD*CMA*CMI)MAXMIN(ZP*IN*AMA*AMIN G R A P H ( 1 * 8 . AXIS PLOT PLOT(X PEN PL0 SYMBOL(XH*YPS SYMBOL I IS IS INPUT ig (FLOAT (3*87) ( MAX OF AMA-AMI ) IM=1*NED 17 IL i IG ( (1) (1) loat 1 IM) .LE.AMAX) ’ ) > (5 *3* = MIN MAX F.R — = FINISHED = RANGE M.C.D t THEORECTICAL Y-VALUES p M.C.D ARAN F.R 1 TO 1 •1) 1 ( ( ( ( ( (4X ( X Z(N) MIN X X axes A * A M I * 3 3 H F , P . X AN/2.0 1) X * * * N maxmin ARRAY AMA*AMT IN IN ♦ * * * * * ( ( ( Y OF EXPT Y Y Y VALUF VALUE Y* A*ZD STARTING IM)-1 Y il *F13*6) GO GO CURVES * •1) * * ) * OF PLOT 0) /7.0 VALUE 0) 1) )- 0) 1) ELEMENTS GO GO DATA TO TO 0*9.0*0.4) i ) (1) (1)* GO PLOTTING DATA Y-VALUES .0) . ( z g TO TO *0.08* * *0.08* 400 ARAN*ZR 700 COMPLETE )* * * ) * TO OF n *FWAVS) 0.08, 1) 0.08 h POINT 700 * 101 F.R amax ) AN 10 ZHD* ZHP* IN *« VERDET tTHEORY ) GRAPHS CURVE EXPT ARRAY * (1)* ARRAY am WITH 0.0*50) 0.0*50) ; n ZFRD ’ NM ) * C O N S T A N T * M l N / ( O E . C M ) OF PEN 0.0 •* ’ ’ *-14 *0.0»4) 0.0 (1) REAL * UP 4) * * 6) 6.* NUMbfcRS 0.0 „ * A ^ , XRA . ) * 33*7.0 224 * 90. O * A ° o o o o o o o o o 100 700 _A A00 200 300 100 10 „ END RETURN CONTINUEAMAX=ABS DO DIMENSION N AMAX=ABS(Z AMAX subroutine END SUBROUTINE RETURN 7R IF CONTINUE CONTINUE 7R(II)=PK PK DO PK=0 AIE=FLOA A DO FOURIER USE 7BM CONTINUE CONTINUE BM 7BM(IB)=(2,0/AIN)*B BM=BM+(ZD(ID)*SIN((ATB*PI*AM)/AIN) DO BM=0 DO AM=FLOAT(ID) CALCULATION AlB=FLOAT(IB) PI AIN=FL0AT(IC) 7BM DIMENSION 7P ZD SUBROUTINE END TRANSFORMS IC RETURN CONTINUE AMIN«Z(I) IN AMA IF(Z(I) IIfFLOAT(II) = IS ( x -3 A 100 PK+(ZBM(lE)<*COS( A IS 700 300 200 IS IN+ IS IS IS x 6 CONTAINS 00 7BM ,0 IS =7 ,0 * S NUMBER IS 1 COSINE FOURIER ( AIS92b54 NUMBER COSINE 1 INPUT ( Z I=1»N IE 1 FpURIER ID= IB=1*IN .GE.AMIN) I ABSOLUTE t TO ( (7(D) ANALYSIS 1 ) I = = ( Z(N) ))• (1)) IE) 1*IN 1 ZD(IN)#7P(IN)*ZBM(200) 1 maxval MAXVAL SYNTHESISE . FROM KR0NIG(7D*ZR»IN) * OF IN OF ARRAY IN LE SYNTHESISED FOURIER SYNTHESISED OF COEFFICIENT ELEMENTS • FOURIER COEFFICIENTS AMAX ELEMENTS IMAGINERV VALUE COMPLETE ( GO finds Z * m (AIE^PI^AII N ) TO • COEFFICIENTS GO AMAX COSINE OF COEFFICIENTS 100 IN TO max ARRAY ARRAY IN NUMBER OF ) PART ARRAY 100 ARRAY absolute SINE SERIES ) /AIN) TO SERIES REAL ) ) IN value SINE PART of SERIES an array 225 of . numbfrs r o n o no n nonnn n o nn non no 101 22 70 21 90 10 12 11 31 50 b 7 2 , 5 1 1*//) CONTINUE AB5N(I)=S0R(I)/F CALL WAVL(I)=F DO F=F STPA=(FMAX-FMIN)/FLOAT(NA) CALL CONTINUE M.C.D CORRECTED DO 7EROTH f STPM CONTINUE ABSORPTION GO A FORMAT READ hfld STPM=(FMAX-FMIN)/FLOAT(NO) ASN(I)=(ASN(I)/<254.0*33.0 CORRECTED SOR DO CONTINUE FORMAT 6=1.0 G=1.0+(X/(2.0#AJ) AMS=(AMS S DO hfld AP AA=0 WRITE AMS SAMPLE HFLD FR FREQUENCY PFAD#,FMlN*FMAX,TEMP,FRtAR,HFLD,FRG,STHM*STHA PRINT#*' X READ# heading NA ISKIP=1 IF SOR PRINT**' READ*, = =AJ+AMS#(AMS+1.0)-AL#(AL+1,0) J=A 1 f * (IFIX(AMS) (IFIX(AJ) MIN + 22 S2 90 TO 70 10 IS IS 31 IS ( IS ( stpa J# = I IS •0 IS IS I (1 < (1«101) PARBL(ABSN,ISKIP PARBL CONVERTED hfld IS )=( IS , I (2*7) ) IS IS (2,2) (2 ABS FULL 1=1. 1=1, IN 1=1, IS =SOP 12 1=1,10 NUMBER THEN * NUMBER ( 1=1,120 ) ION (SOP(I) ABSORPTION AJ+ MULTIPLICITY.AL 6) THICKNESS RAW RAW =THETA (11X*'THEORECTICAL (2A10.F3. ( * (7Al0t2I3) ( — STEP MAGNETIC SOR STOPS ABSORPTIONM.C.D. 11X,7A10*///) 5) INT * UNITS LEVEL # 1•0) » ' 1 NA M.C.D NO NA S RANGE RANGE ABSORPTION 1.0) o WAVL AMCO (S0R*ISKIP*STPA,NA*VABS*IR1) ( • ( Z H D ( I ) » I = I S1.S2.G RANGE ABSORPTION M.C.D IN • PLOT 1 . I ( ( ) ( EQ « ZHO KKK GO I i * ) INPUT E FOR /2 S2 =AMCD )/( OF OF * ( I Q UNITS • WHEN 1 I ( 1 2(120) 0 ) * T H E T A < 1 2 0 ) , T H C A L ( 1 2 0 ) LABEL OF 0) to = = •0 .0) TO DATA * ) ( DATA AMS 254 SWITCH 1 =THCAL IN I GRAPHS POINTS • 1 POINTS 1 M.C.D CM FIELD SWITCH DATA ) ,F4. )*1=1*7) * 1 * MOMENT,VABB GO ABSORBANCE/(CM.TfcSLA) tesla NA NO OUTPUT 99 ( GO NM , NO • DATA I .ABSN OF Al STHM 0#STHA ) ) ) IN FROM DATA TO , =ABSN IE 1 f STPA,NA TO DATA AND ,F7.4) * EQUALS IN 7),N0.NA ( AND ABSORBANCE/CM A SETTING IS IN OF OF DEGREES I GROUND J SETTING 11 IN )=0,0 . ( 1 2 0 ) * Z H D ( 7 ) , A S N ( l 2 0 ) , S O R < 1 2 0 ) M.C.D 21 MM TAPE MM *HFLD*STHM) FROM L G KG TEMPERATURE )) M.C.D ABSORPTION ( STPA IN A 8 S 0 R P T A N C E I #AR ON VALUE )=0.0 , ZERO 1 VABB MM IS = MM AND STATE FOR SAMPLESSTHA 1 FOR IS NPUT DATA VALUE FOR , STEP IR2 AJ ABSORPTION M.C.D * 419/2 DATA ) T A P E 2 = 0 U T P U T #FR THE ) IN IS FOR J K ».2A10,» ON ABSORPTION .CHANGED ABSORPTION * T A P E 2 2 1 P L O T LEVEL TO SAMPLE CM-1 226 IS',F11.6 ) n o 200 44 41 42 14 40 lb 13 15 30 20 bO 4 „ 3 9 b CALL 2• 1 , 1 0 X , * M # C . D llON 1 1CJTED* l4.b, F=F DO f F O R M A T ( l l X , • C O N S T A N T S * , F 1 4 , M.C.D T H C A A = TT H H C C A A L L ( ( I I ) ) = T H C A L ( I ) * C O N S T WRITE CALL CONTINUt GO GO F O R M A TWPITE ( / / / * 1 1 X , THETA G E X 4 = - ( A B S ( (G A E / X D 2 ) = - ( A B S ( ( A / D ) THCAL FORMAT DO F O R M AWRITE T ( 6 X , F 9 ,GEX3=AbS 4 , 2 X , 4 ( 2 X . F 9 .G 4 G ) R = A B S ( ( C / D ) T H E T A ( I ) = A S N ( I ) # A A T H C A L ( I ) = ( ( - F D E R V ) * A ) + ( WRITE FINDING CALL F=F+STPM S P A C F =FORMAT S T P A /WPITE 2 . 0 GEX F=FMIN ADIV=C/A CD=C/D AD=A/D C = ( V M C D * T K ) / 1 5 2 . 5 T K = 0 . 7 0 * T E M P FORMAT A = F M O M / 1 5 2 . 5 WRITE FIRST IF(NO*NE,NA) F=FMIN CALL A M C D ( I ) = A M C D ( I ) / F CONTINUE IF(IER.EO.O) CALL CONTINUE F=F>STPMDO AMCD ZEROTH CONTINUE F=F+STPMDO F=FMIN A M C D ( IDO ) = A S N (AA=0 I ) ^ A A FORMAT finding vbar FORMAT WRITE D=VAB8/3?6.6 WRITE IF = RAT IN fmin ( + TO TO 200 100 1 30 fcO 20 (IR1 STPM TOOK =ABS 1 ORDER = .2 12X, ( CORPEL(NO.THCAL,THETA,CC,CONST)0 LAGDIF I N T E R ( W A V L , S O R , N A , F , 5 0 R L ) PARBL(AMCD,I (2, P A R B L ( A M C D * I S K I P t S T P M * N O t V M C D * I P 3 ) (2,14) ( vabs (2*15) I (2,13) (2,4) (2,3) (2*9) (2,8) 44 200 IN IS I M.C.D (/,* 1=1, I 1 (/,1 ) * C 4 X * » A / D * * I 0 (* (11 E X P T / C A L C * , 5 X , 1=1, = )=( + =1• = 1=1, M.C.D (11 DERIVATIVE 1 8 ) C 0 N S T , C C AMCD vbar I R 2 + I R 3 + I R 4 ) . E Q . ( ( 1, « T E M P E R A T U R E = * UNITS EXPT PLACE*,/) / (A/D) (A/D) X * ARE X IX* NO NO NO (-FDERV vabb 11 • NO NO G G R , G E X 1 , G E X 2 , G E X 3 , G E X 4 , A D , C D , A , C , A A • I R 1 , I R 2 . I R 3 , I R 4 ,* VAbS ( ’ EXPT WAVL ADIV MOMENT,FMOM ,20X,*C/A RELATIVE OX ( GO GO PROBLEM MOMENT » ) *,4( I M.C.D PROBLEM ) ) ) * OF -GGR +GGR ’ MF-VBAR) * , CALCULATION TO TO * )) )) FARADAY D ABS/(T.CM) SOR ) 2X S K I P , S T P M * N O . F M 0 M t I R 4 ) • -GGR ABSORBANCE/ *A +GGR TEMP 41 42 x , , ) ,4, I VMCQ .»C/D» ♦ 2)//) WITH » OSCILLATOR ABS RAJ WITH ( ,F7. NA ( S 0 R L * C ) / T K ) ( SOR 0) IS PARAMETERS 10=• , IS ABSORBANCE/CM*,/) F INTEGRATION 2. , l l X , * A * , 1 2 X , » C * , 1 0 X , 6, DIFFERENTIATION GO VALUE », , ( OF SPACE I VALUE , »K 4X 10X 10X, ) tF ( CM *C TO » M.C.D 15,6•//) , ,//) F9 . ,» ) STRENGTH: TESLA /TK 40 ’ , CORREL CALCULATED FDERV .5, 1 ///* ) CURVE 7x ENCOUNTERED ) . , l=»,EI4.6»10X F9 IER »,11X,»G COEFF= »,//, .4,2( ^ ENCOUNTERED ) , M.C.D A 2X llX •, ’ M.C.D F , GROUND , E „ ERROR 10.6,/) ,* „ ABS/ 13.5) , WAVELENGTH ^ * »D ^ FACTOR*/) , NO ( » T PARAMETFRS VALUE=*,E1 , ,20x _ 5X • 227 CM CQMPlJTAT , F7 )* ,» . , G •4) n 5X M* EX , 228. IF (THCAA.GE.l.OE-14) GO TO *3 THCAA=1•OE-14 43 DIFF=TH£TA(I)/THCAA 100 continue PLOTTING ABSORPTION CURVE CALL NGRAPH(1*8.0*9.0,0.4) X-AXIS IS 6 INCHES * Y-AX1S IS 7 INCHES.1 INCH SPARE ALROUND CALL MAXMTN(SOR*NA*AMA*AMI) FINDING MAX AND MIN VALUE OF ABSORPTION YRA =(AMA-AMI) /7.0 XpA=(FMAx-FMlN)/6.0 RANGE OF VALUES FOR X AND Y VALUES CALL SCALE(XRA.YRA) SCALE FACTORS FOR PLOTTING YPA6=AMI-YRA xrab=fmin-xra CALL ORIGNV(-XRAB*-YRAB) SET ORIGIN CALL AXIS(FMIN.AMI*14HWAVELENGTH* NM*-14*6.0*0.O.FMIN.XRA) CALL AXIS(FMIN.AMI*25HABS0RPTI0N. ABS0PBANC£/CM.2b*7f *90,*AMI *ypA) labelling X AND Y AXES CALL PLOT(FMIN*SOR(l)*1) PEN TO STARTING POINT DO 23 1=1*NA CALL PL0T(WAVL(I)*SOR(I)*0) 23 CONTINUE CALL PLOT(wAVL(I)•SOP(I)*1) PLOTTING ABSORPTION XH=FMIN+(XRA/2,0) YPS = AMA + ( YRA/2•0) CALL SYMBOL(XH•YPS* 0.08.ZHD.0.0*70) HEADING OF ABSORPTION CURVE NOW PLOT M.C.D CURVES CALL NGRAPH(1*8.0*9.0*0.4) CALL MAXMIN(THETA,NO*CMA*CMI) CALL MAXMIN(THCAL*NO*CTMA*CTMI) IF (CTMA.LE.CMA) GO TO 24 CMA=CTMA 24 IF (CTMI.GE.CMI) GO TO 25 CMI=CTMI FINDING RANGE OF Y VALUES 2b YPA=(CMA—CM I)/7,0 CALL SCALE(XRA.YRA) YRAb=CMI-YRA CALL ORIGNV(-XRAB.-YRAB) CALL AXIS(FMIN*CMI*14HWAVELENGTH. NM.-14.6.0.0.0 *FMIN *XRA) CALL AXIS(FMIN.CMI.24HM.C.D* ABSORBANCE/(T•CM)* 24,7.*90..CM I.YrA) LABELLING X AND Y AXES PLOTTING FXPT M.C.D CURVE CALL PLOT(FMIN.THETAU) *1) F=FMIN DO 26 1=1,NO F=F + STPM CALL PLOT(F•THETA(I).0) 2b CONTINUE CALL PLOT(F,THETA(I),1) CALL SYMBOL(FMIN*THETA(1)*0.06*»EXPT*.0,0,4) PLOTTING THEORECTICAL M.C.D CURVE CALL PL0T(FMIN.THCAL(1)*1) F=FMIN DO 27 1=1,NO F=F+STPM CALL PLOT(F * THCAL(I)*0) 27 CONTINUE PLOTTING THEORECTICAL M.C.D CURVE CALL PLOT(F.THCAL(I)*1) XPOS=FMAx-XRA CALL SYMBOL(XPOS*THCAL(I)*0,06* * THEORY **0.0*6) YPS=CMA+(yRA/2.0) CALL SYMBOL(XH.YPS*0.08*7HD*0.0,70) heading GOING TO PLOT A AND C TERMS f=fmin DO 32 1=1,NO F=F+STPM THCAL TO contain A TERM THETA to contain c term IF(NO.EQ.NA) GO to 33 n n n n n o n n n n C 100 99 37 36 3b 34 32 33 GP S X Y = S X = S Y = S X S = S Y S = 0 . 0 ZX N CORREL S Y = S Y + Z Y ( I ) DO 7Y SX=SX SXY CC SUBROUTINE DIMENSION N FINDS END GO SUBROUTINE END DO PRINT***CALL CALL CALL CALL RETURN CONTINUE AMlN=Z AMIN DIMENSION STOP CALL YPS=DMA+CALL CONTINUE F=FMIN CONTINUE DO yrab CALL DMl=EMI NOW AMAX=Z AMAX Z f DO CALL CALL F=F+STPM F=FMIN CALL CALL AMIN=Z = IS f A IS IS = 100 IS IS TO IS + 36 = IS 37 100 TO S X Y + ( Z X ( I ) * Z Y = = stpm DO ( NUMBER + CLOSE NUMBER SYMBOL(XH*YPS PLOT Z S Y M B OPLOT(F#THETA(I L ( X P O S t TP H L E O T T ( A F * T H E T A ( I ) » 0 ) P L 0 T ( F M I N » T HS E Y T M A B ( O 1 LPLOT(F*THCAL(I) ) ( F M I N t T H C A L P L 0 T ( F M I N * T H C A L ( 1 ) ORIGNV A X I S ( F M l N t D M l » 2 4 H Mdmi . C . D * SCALE(XRA*YRA)DMA luj MAXMIN(THETA*NO»EMA*EMI)N G R A P H < 1 * 8 . 0 * 9 . 0 * 0 . 4 ) A X I S ( F M I N > D M I * 1 4 H W A V E L E N G T H * M A X M I N ( T H C A l * N O * D M A f D M I ) IS IS INPUT INTER(WAVL#SOR*NA*FtSORL) X Y ZX SLOPE CORRELATION MAX 50 a 32 1=1 I (1) ( (1) DOES — 1 PLOTTING I 1 • (I = VALUES VALUES GE = MIN MAX — - ) 1 ( 1 YPA/2. yra DMI 1, .NO * ( MIN ZX(N)*ZY(N) F Z(N) NO * ) • N CORREL N MAXMIN FINISHED AM ARRAY * < THCAL OF OF NORMAL OF VALUE VALUE -XRAB ) / IN ) VALUE 7 LINE DATA ELEMENTS GO 0) GO ) .0 GO GO (I) * ( < ( COEFFICIENT N Z*N.AMAX -YRAB TO TO I *0.06, REGRESSION * )* )*0) PAIRS ZX TO OF PLOTTING TO ) *1) 3b 1 34 * ) ( (1)*0,08,» ZY*CC AN 10 100 I ) *1) ,1) ZHD* IN )•0,08* ARRAY r . ARRAY AMIN) 0.0*70) * GR GRAPHS* „ A B S O R b A N C E / ANALYSIS ) * OF C-TERMt A-TERM NM*-l REAL *•0•0 4,6•0, *0•0 NUMBERS ( T.CM A * * 6) 6) Q ) *24,7.«9.0 * FMIN * XRA 0.* 229 DMI ) * YPA ) n o n n o 100 46 47 45 44 END RETURN GO S O R L = ( E * S L O P E ) + C E P T K=hA- C E P T = S O R ( I ) - ( S L O P E * W A v L S L O P E = ( S O P ( I + l ) - S O R ( I ) NA DO FMIN=WAVL DIMENSIONSORL F DOES CEPT S L O P ECONTINUE = S O P ( l ) / ( W A V L ( l ) - F M I N ) WAVL END SUBROUTINE IF RETURN GR^bTOP/BL A L P = ( S Y - ( G R ^ 5 XBR=(FLOAT ) BT0P= IF B L = ( F L O A TCONTINUE ( N ) * SSYS=SYS+(?Y X S SXS=SXS+(7 C C = B T O P / ( S O R T ( B L ) * S Q R T ( B R ) IS ( ( TO 45 IS (F.GE.FMIN).AND.(F.LE.WAVL (F.GE.wAVL =SOR X LINEAR 1 IS IS < 46 NUMBER 1 FLOAT(N)*SXY = VALUE X 1 INTERPOLATED (1)- *K (1) < N ) # S Y S ) - ( S Y # * 2 ) WAVL(NA)*SOR(NA) VALUES* x INTER ( ( -(WAVL INTERPOLATION I I (SLOPE*WAVL OF AT ) ) ( I *»2) #*2> )) ) WHICH X-Y /FLOAT(N) )-( ( WAVL .AND.(F.LE.WAVL SOR (2) )-( SX**2 PAIRS -WAVL Y SX*SY * SOR )/ INTERPOLATION IS VALUE ( (1)) I (WAVL ) Y )) * NA (1)) ) ) VALUES • F (1+1) (1))) f SORL (1+1))) -WAVL GO ) OCCURS TO ( GO I 44 )) TO 47 230 o o o n o n n n n n n o o o n o n o n 999 400 300 ioo 100 21 40 20 3 2 1 2 F I C I E N T ' / / 4 2 X » F 5 . 2 * » C M 1NGTH ZD N END gr 7X WL RSODA SUBROUTINE RETURN C C = B T O P / ( S OR R P T = ( ( B F L L ) O * A SB T O L ( R = N T ( ) ( F * B L S R OB Y ) A T S T O ) ( P - N = ( ) ( Scontinue * F Y S L * X O * S A 2 ) T ) - ( ( N S ) X * * S * X 2 Y ) ) ~ ( S X * S Y ) CC N A L P = ( S Y - ( G P * S X ) S X = S X + SXY=SXY+ Z X ( I ) DO GP 7Y CORREL DIMENSIONSUBROUTINE END GO GVAL=GR*6. S Y S = S Y SS + X ( S Z = Y S ( X I S$ + Y ( = 7 S X Y ( + I Z ) Y * ( * I 2 > ) sxy 7 STOP CONTINUE ^ CONTINUE CALL taking Z Y ( I A ) =DO 0 . 5 * A L O G ( ( 1 . 0 + ( Z D ( I A ) / A ) ) / ( 1 . 0 - ( Z D ( I A ) / A ) aa CL NN A=A+STEPDO RANGE FORMAT A=7D(N) IF S T E P = ( A M A - A ) / 5 0 . 0 CALL RFAD CONTINUE AMA=7D(N) PRINT**' READ*.WL WRITE FORMAT DO WL heading ZX(I DIMENSION PROGRAM ZHD 7D 7X IF(N.EQ.O) X F = -CC = = IS IS (NN.NE.99) = btop IS TO 400 IS IS a 100 300 IS IS IS EO IS 0 IS 100 IS IS sx / A)S(FLOAT(I •0 is (1 NUMBER NUMBER S FR h = WAVELENGTH CORRECTS CORRELATION SLOPE Y X 40 OF WAVELENGTH TO FR H 1 DOES / sy 1=1. inverse heading I IB=i 1) X '•21 (*1». ( 6 A 1 Q * F 4 , 2 ) FLD bl VALUES VALUES A FLO GVALUE(INPUTtQUTPUTfTAPElsINPUT A (7 *N»(ZD DATA = — A-VALUE + 99 = IN ♦ Z X ( N ) t Z Y ( N ) sxs 25242718 X * 1 ZX 1»23 ( Z D ( N ) / 2 . 0 ) (ZHD(IA) BSODA(N.WL.ZD.ZX.STH) N CORREL lT#ABS(Cl) GO * X * ( (ZHD(I PTS OF OF OF NORMAL N 200 NEW I DEGREES PTS •' (23) = GO )*7 ) 2 0 X . 6 A 1 0 . ' S A M P L E **2) sys I TO A-VALUE ( n PTS DATA LINE DATA tanhs ) A) IA) /FLOAT(N) Y TO IN SAMPLE IN .7Y DEGREES = IN FROM -1.0)*0.2146 999 A ) * I A = 1 . 6 ) . S T H . S T H ( IN ( COEFFICIENT N u I . *IA=lf6) TFSLA REGRESSION 20 * )) . TESLA IA=1*N)»NN (23) NM FOR PAIRS ZX o NM! AT > FRMAX * DEGREES 00 ZY CALL THICK 1CM »7HD BASE ZY . CC T0 ) fSTH H IS (8) TO * », GLASS 21 GR FID '.1 ANALYSIS FR 7y THT .ZD FRMAX+100% ) 9 X t F 7 . 2 . 9 X » F 8 . 4 , 2 3 X f F 9 . 6 ) SX . DATA CKNESS= INTEVALS ' (23) 1 • CM ♦ MOMENT THICK') * • TAPE3=0UTPUT * F5.2 IN 1 • 1% 1 5 X t ' C O R R E L A T I O N * • STEPS CM )) • /// ) l S X . ' W A V F L E 231 . CoEF n o n o 100 30 20 10 NOW GO GO END RETURN CONTINUE DO AINF=(PHI*STH)/(TANH FLO I 1 ( ( =79.705946-(0.06511 = TO WL TO 100 I)=(AINF#TANH(GVAL*C0N$T*ZX CORRECT 145.82974-(0.17941 IS OF IS » GT, 20 20 IS ,9916 ___ SAMPLE 1=1 FP FP 4.7212T 580.0) 445.0) ZD iN AS INFINITY DATA F(JNC THTCKNF5S GO_T GO AT T OF FIXED 0 VALUE 7801 WAVEL 96»WL) 30 10 19#WL) IN b9#WL WAVELENGTH AT CM AT FIXED ) ( MAX I ))) 7212)) -ZD(I) H WAVELENGTH FLO 232 C C C C C C ononoonnnnnon 600 200 300 A0 bOO 100 80 6 5 0 A l l F O R M A T t / ^ l X t 2 lx eigenvalue HEADING WRITE RJt=bJE-DO BJE=EJ+ FORMAT NORMALIZING WRITE FORMAT WRITE CONTINUE BJG=GJ+1CONTINUE DO AMAT DO CALL DO CONTINUE N=(IJG* DO T E M P ( I ) = B J G - F L 0 A T ( I ) S C R A T ( N ) = A M A T DO CONTINUE AMAT CONTINUE DO A AMJGC=GJ+ DO DO DO MATRIX X A M J E C = A M J E C - 1 X= AMJEC=EJ+ WRITE CONTINUE N=(17*(I-1DO DO COMPLEX A M A T ( I , J ) = S C P A T ( N ) = 0 , 0 CALL FORMAT WRITE J x A CALL READ READ*,GJ,GLFORMAT COMPLEXDIMENSION MAY EIGENVECTORS(COMPLEX) NOMINATED FOR IJE=IEIXIJG=IFIX HEADING*PPINT KEEPING PROGRAM AND CALLING SYMMETRY PROGRAM SUBROUTINE IF = MJGC=AMJGC = l ( G J . L E . 0 . 0 ) 0,0 A VALUES o (( 700 600 200 200 300 300 bOO A BOO A00 m 100 100 • GVEC 00 = A B S ( G J - E J ) 0 G-VALUES,RANGE FIELD BE ( ( ( M A A y A L ( S C R A T I IA,IB)=X*X (6,3) (6,6) (6,5) CRYST(GVEC)CPYST(EVEC) B (6,1) (6,2) * (1 (7 * (* (8 J)=AMAT J J= 1=1 1,0 FORMED 1=1, 1=1, 1=1, J=l,17 ID= IC= 10=1, IA=1 (1-1)) 1=1*17 A VARIED SPECIFIES H x = 1« .0 ARRAY E V E C ( 1 7 , 1 7 ) , G V E C ( 1 7 , 1 7 ) OSCST A10) SUBROUTINE CALLING TO ( 1* ) ( ( 1, , 1,1 18 1,0 1•0 GIVEN AND ( 2 , 0 * £( J 2 ) . + 0 1 * , G 0 J ) ) + 1 , 0 ) 0 l 1 7 ( F 6 . 3 * i x > 1, 1 CRYSTAL AND ))+ ZHD * 1 0 X * 8 A 1 0 * / / ) (ZHD(I) BJE, — IJE IJG IJG IJE IJE , ( , * CRYSTi CALCULATE »FJ*EL ( MAX TEMP JG ID)*CONJG(EVEC 1•0 IJG IJE IJE IJG 7HD + j •0 L ( ’ (8) ( J C A L C U L A T l O N I OF IS GO ( OPTIONS ANGLE IN INPUT I , ( VALUES IN ( OSC AMAT J) * ( I THE , J)/AMAX I *289, EIGENVECTORS TO THREE ( )*1=1*8) AMAT ,1 STATES )* IJE-ROW CALCULATES FIELD SUBROUTINE = OF EIGCH * STRENGTH 1 ( 999 OUTPUT»TAPE5=INPUT AND APPROPRIATE 1,8) = I ) RELATIVE (17,17) AMAX) * 1 FOR PARAMETERSFOR AND J * DI ) IJG) CRYSTAL HAMILTONIAN, ( INVOLVED , * MENSIONS IMSLIB) J OF GROUND IJG-COLUMN MOMENTS EIGENVALUES ( : I 'It A,IC) • TO RELATIVE SCRAT CUB, THE IJG) WHERE OSCILLATOR ONE FIELD SUBROUTINES )) AND FOR CRYSTAL AND . CRYST FOR FINDS (289), *HI ROW ) EXCITED FIELD PROGRAM THENCE NT OSCILLATOR PARAMETERS AND X * TAPE6=0UTPUT , VECTOR TEMP READS THE Y ( ALAM FIELD STRENGTHS * EIGENVECTORS AND AN^ EIGENVALUES TRACES (17) * STATES OSCST GJ Z IN CORRESPONDS MATRIX ANGLtfSTEP * DIRECTIONS EJ STRENGTH THE • * USING THt DOES ) AMJGC FOLLOWING FOR . J-VALUES APPLIED AND HOUSE * A AMjKr MATRIX 51 THE TO By 233 7E )) 1 + / o n o n o o o 400 401 300 999 100 700 20 90 91 60 81 27 „ 2b 14 49 b , 1 3 3 2, 1 12=' 1.2** HSM(KA)=VHEX(I,JA) KA=K DO KA DO FORMAT CONTINUE WRITE CONTINUE AM FORMAT DO CONTINUE DO CONTINUE WRITE DO DO FORMAT WRITE CP=C0S(P1*3. WRITE(6,401) AM(JA)*REAL(VHEX CALL ST=SIN(T CT=COS(T SP=SIN(Pl*3» FORMAT CONTINUE VHEX DO IF(H1.NE.0.0) WRITE DO FORMAT FORMAT NJ=2*A+1 WRITE(6,2b) WRITE B PEAD*,B2,B4,66,B43,b63*B66P H1=H read PHI READ**H*HMAX*THETA*TMAX*PHI*PMAXfTEMP A T1=THETA STEPPING H READ* heading IA=IR=IC=0 FORMAT WRITE FORMAT P£AD END COMPLEX COMPLEX DIMENSION DIMENSION SUBROUTINE GO V STOP CONTINUE FORMAT(2x*FA.l*lX*17(F6.4»lx) IST0P=2 ISTOP=l ISTOP=0 IF C-AxiS 3X* 1 = =PHI VALUES ( IS IS IS ( 4 4 JA)=AIMAG(VHEX 90 91 BO 0 81 1 I *E9.3,3X,«B4=»,E9.3*3x,*B6=»,E9.3,3x, TO ISTOP.EQ, A+ DEGREES *, ANGLE '666=* ( (5 JA= C3VSYM(NJ*A,G*VHEX*H1 J=1*I 1=1, I FLD (6,400) 1 J-VALUE EIGENVALUES (6,400) (6*300) (6*?7) * (8*100)A,G*TEMP (6*5) JA=1,NJ J * 1 = 1=1, I ( * A (' ( if 80 (/ (* = A J lH lH 1* 3) * (• (« ( = ANDERS.!,'DEGREES 1, I 1*3.14159/180.0) 1*3.14159/160.0) G , ) AND PRINTS PRINTS PROGRAM X VHEX(17tl7)»VEC(17.17) , 7A NJ =VHEX(J*I)=VEC(I*J)=VEC(J*I)=CMPLX(0.0* NJ V FOR ' 1 * PARAMETERS * IN it * * NJ N I ,I2,9(F13.4,1X) */) OF *5X.'ORIGINAL , ,E9,3//) EJXP( Y « •* *10X*7A10///) 10 (ZHD(IZ) NJ J »*1X*'APPLIED CRYST(VEC) * * (ZHD(IZ) (17) H1,T1*P1 bX B2,B4,B6,R43,B63*B66KELVIN'//) It(AM(JA),JA=1,NJ) It(AM(JA),JA=1,NJ) 14159/180.0) KOE 14159/160,0) STOP 10X* ip 0) * CRYSTAL APPLIED AND I GO 2 •' EIGENVALUES BOTH GO * ) 17*17) ,AM( CRYSTAL THET STOPS ( • TO I J=» G COMMAND ( I * TO •17 JA * IS *17=1*7) 17) JA) 20 A EIGENVALUES *F5.2,5X*'G-FACTOR=» ) FIELD FLO FOR = 99 ,EJYR(17,17),EJZR(17,17),ZHD(7)*VTEMP(17> ) G-VALUE ANGLE 1*7) tWK ) MATRICES'*/) FI F ,/» H WITH (51) FLU * I * *ISTOP B2 ELD TO THETA ONLY IN OF »,2 * ) B4 X-AXIS'/) PARAMETERS DEGREES Y-AXIS*TEMP ♦, tHSM( APPLIED X F , AND B6 ,8{ AND 10.2,» * 289) F B43 EIGENVECTORS 13,4,1 PHI KELVIN FIELD • IN B63 ,Fb.2,bX* »R43=* IN KOE IS X * B66 DEGREES )) WITH TEMPERATURE AT tE * SP 9.3, 0.0) •* » ^ F5 TEMPERATURE*» Z-AXIS * CP K»*//» 3X ,1,» , * CT .» DEGREES B63= * ST 234 ) »,3x.»B » ,F9,3 .F6 TO 235 CONTINUE CALL EIGCh(HSM.NJ.1.VtVEC.17.WK*IER) WRITE(6*600) 600 FORMAT(lH *5X*/*lX*’EIGENVALUES ARE • •»/) WPITE(6*101) (V(I)*I=1*NJ) 101 FORMAT(t t.2X.9(F13.4,lX)./* ».2X.8(F13,4,lX)) DO 6 K=1.NJ X=CMPLX <0,0*0,0) DO 7 JA =1 * M J X=(VEC(JA.K)#CONJG(VFC(JA.K)))+X CONTINUE Y=CSGRT(X) DO t JA=1,NJ VEC(JA*K)=VEC(JA*K)/Y continue CONTINUE NORMALISATION of eigenvectors IF(IST0P.E0.2) GO TO 10 WRITE(6*700) 700 FORMATUh *1X./.1X*’EIGENVECTORS ARE * • • /) DO 110 K = 1 * NJ DO 111 JA = 1 * NJ am(jA)=REAL(VEC(JA*K)) 111 CONTINUE WRITE(6*400)K»(AM(I)* 1 = 1tNJ) 110 CONTINUE WPITE(6 * 401) DO 120 1 * NJ DO 121 JA=1*NJ AM(JA)=AimAG(VEC(JA*K)) CONTINUE WRITE(6.400)K* ^DIME^SION l 4 0 . * S 4 - 6 0 , * S 2 VMAX=X DO $= F X=AfeS VMAx=0 COMPLEX DIMENSION SUBROUTINE END RETURN CONTINUE C=1 VHEX 0 3 6 = ( (B=Il.*Z3 P * C ) + ( P * B ) ) / 4 , 0 3 4 = ( (CONTINUE P » Z ) + (P=P*$GPT P * ( Z + 3 . 0 ) DO GO IF(X.LE.VMAX) 7 Z 1 = Z + F L O A T ( K7 ) = - F 1 L , O 0 A TP=1 ( J ) - A - 1 . 0 L=J+3 K=J GO VHEX V H E X ( I *Y J = ) H = 1 C * M S P T L * X S ( P X * * 0P=SORT Y . ) 0 6 7 1 77 * = G F * L P O / A 2 T . ( 0 J ) - A - 1 . 0 VHEX O66=P/2.0CONTINUE P k 04=35• X = B 6 3 * 0 3 6 + 6 4 3 * 0 3 4 p=p*SQRTZl DO z GO V H E X ( I f J ) = C M P L X ( X * 00 , 6 0 = ) 2 3 1 , * 7 6 - 3O 1 2 5 = , 3 . 0 # Z 2 - S S 2 = A * ( A + 1 . 0 ) S6=S4*$ Z = F L O A T ( J ) - A - 1 , 0 DO C NJ X = - H 1 * S T * C P * G * 0 . 0 6 7 1 7 * P / 2 y=B66*066 X = B 2 * 0 2 + B 4 * 0 4 + B 6 * 0 6 - H l * G * C T * 0 . 0 6 7 1 7 * Z S4=S?»52 7^-22*2272-1*2 DO FORMAT WRITE VHEX HAMILTONIAN 76=Z2*Z4 SUBROUT IF(I.Nt.L) IF(I.NE.K) IF IF 3 = =J+6 = omplex FUNCTIONS 15 = flqat ( 1 ( ( i? TO IS 79 + TO Z+FLOAT TO 74 70 1 • I • I IS f + 0 . 0 * Z * 7 * Z - 3 . * 7 * A * ( A 0 1 0 • ( • ( ( - 3 J 1=1, NE NE ItJ)=VHEX ( I IS V K •0 1 70 70 K 70 *74-30. ORDER J=1 * 1 * (( (6,1) = = J ,) 2 = I ( • • IMF < (• VEC * I j A-Z) K vhex MATRIX 1 J ) l,6 ) 1* ( ( /2 R =CMPL =VHE IS z )) * )- ) ) (A-Zl « EJYI V^1 3 * Z 3 - 3 . * Z 3 * A * ( A 3 NJ I * ( TRACES GO GO K a GO 2 GO i C3V$YM(Nj*A*G*VHEX*Hl,B2*84*R6*b43*b63*b66*5P*CP*CT,^T) (17 * ARE - *( £>X OF FOP ) l (17*I7) 7?! -1,0 X ( J * I ) = C M P L X C *S2*Z2+25. i GO TO A+Z+l TO X * TO ) TO , (17*17) )* * ( J* * 17) * S 2 * Z 4 + 7 3 5 , * Z 4 + 1 HAMILTONIANMA q ( * UNNORMALISED e X CRYSTAL C3V (A+Zl+1 (A+7 TO JXR( 70 77 I 75 72 TRIX » < “ IS , ) =CMPLX(X Y .0)) 15 , SYMMETRY-TRIGONAL ) 1+1.0)) V 17*17) ,EJZP ) * + • A )/4 V E C * F J X J X , F J Y J Y * F J 7 J Z t F J X J Y , F J X J 7 * T H J y *72-6 i.0)-59.*7 *0) FIELD IS + t # 0 l 0 X ) (17*17) J-VALUE*G ,£ »0.0) . * 0 )-59.*Z3 ,*S2+3.*S4 JXI • OF 0 05. ) (17*17) C3V *EJZI *S4*Z2-525 SYMMETRY*/) IS CRYSTAL (17,17) , E J Y R < 1 7 , 1 7 ) G-VALUE . *S2*Z2 * EJX2 FIELD * Hi (17) + *EJZ2(17) IS 294 FIELD ,EJY2(17) , #Z2-S 236 * , TRJY *<*6 + 237. continue FPROPl=VMAX*ltOE-8 ERROR2=EPPORl*10.0 DO1 1 = 1 * IS DO 1 J=ltIS F JXR(11J)=EJXI(I* J)=EJYR (It J)=E JYI(11J)=EJZR(I»J)BEJZI SUBROUTINE XRAGE(V.NJ) C checks numbers are in machine range DIMENSION V(17) lb AMAX=AMIN=v(1) DO 100 1 = 1»N J IF(V FUNCTION HINT(ALAM,GJ,EJ,AMJGtAMJE) C HINT EVALUATES MATRIX ELEMENT x = 0.0 p=—6 » 0 Q=-2•0 DO 300 IC=3,9,3 P=P+FLOAt(IC) DO 300 ID = 1*3 0=G+FLOAT(ID) YY=P+G XX=(-P)-0 N=IFIX(YY) SIGN=(—1.0)**N X =((<2.0*ALAM)+1,0)*SIGN*WIG3J(1.0,ALAM,3.0,Q,XX•P)*WIG3J(GJ,A| AM, 1FJ*AMJG»YY»AmjE))+X 300 CONTINUE HINT=X RETURN END FUNCTION WIG3J(AJ1,AJ2,AJ3,AM1,AM2,AM3) C BOTH INTEGRAL AND HALF INTEGRAL ANGULAR MOMENTA C VALIDITY CHECKS IF ( (ABS(AMl).GT.Ajl) .OR. (ABS(AM2)•GT * AJ2).OR.(AbS(AM3) .GT.AJ3) ,nR. 1(AJ3.GT,(AJ1+AJ2)).OR.(AJ3.LT.ABS(AJ1-AJ2)>.OR.(IF IX(AM1+AM2+AM3). 2NE.0)) GO TO 25 C SET CONSTANT FACTOR N1=IFIX(AJ1+AJ2-AJ3) N2=IFIX(AJ1-AJ2+AJ3) N3=IFIX(AJ2+AJ3-AJ1) N4=IFI X(Ajl+AMl) N5=IFIX(Ajl-AMl) N6=IFIX(AJ2+AM2) N7=IFIX(AJ2-AM2) N8=IFIX(AJ3+AM3) N9=IFIX(AJ3-AM3) N10=IFTX(AJ1+AJ2+AJ3+1.0) XX=(FT(N1)*FT(N2)*FT(N3)*FT(N4)*fT(N5)#FT(N6)*FT(N7)*FT(N8)*FT(m9) 1)/FT (N10) XX=SORT(XX) C SET SIGN K=IFIX(ABS(AJ1-AJ2-AM3)) SIGN=1.0 IF (K.EQ.0) GO TO 122 DO 12 1=1,K 12 SIGN=SIGN#(-1.0) 122 XX=XX*SIGN C K SUMMATION KI=IFIX(AJ1+AJ2-AJ3) K2=IFIX(Ajl-AMl) K3=IFIX(AJ2+AM2) K4=IFIX(AJ3-AJ2+AM1) K5=IFIX(AJ3-AJ1-AM2) C SELECT UPPER LIMIT OF K SUM IF(K2.GE.KMAX) GO TO 14 C n o o n 100 26 25 5 r 6 4 2 RETURN END CONTINUE DO DIMENSION N DO RETURN- A AMAX=Ab$(2 m — MIN 1 AX=ABS IS = IS KMIN A (( b TO TO 100 TO TO K TO TO v *0 - ~~L( K 1 NUMBER J=1 = IS INPUT 4• 0 (-1.0) 6 6 26 22 — 0 21 21 (740*0) 1=1 •0 LE LOWER 1 ABSOLUTE *K SUMMATION (7( FACTORIALSFT •0)• Z *N MAXVAL M A X V A L ( 7 * N t A M A X ) ARRAY GO I GO ) ( ( ( N OF J) K )) tLE.AMAX) 0 ) ) ) GO LIMIT TO TO GO R ELEMENTS GO • TO ( VALUE 4 2 K TO FINDS 5• TO OF 20 24 LE 16 60 K •0)) OF IN SUM TO MAX #FT(N3> NUMBER ARRAY GO 100 ABSOLUTE TO * F T ( N 4 ) * F T ( N 5 > IB VALUE OF *FT(N6) AN ARRAY > 239 OF NUMBFRS 240 . Appendix 2 : A VERSATILE DIGITAL PHASEMETER by P. Young, S.J. Collocott and K.N.R. Taylor ABSTRACT A digital phasemeter is described which has been developed directly for use in measurements of the Faraday Magnetooptical effect in glasses, in which the input signals to the meter comes directly from photomulitpliers operating at low signal to noise ratios. The instrument was designed for low frequency operation, however extension to a wide range of frequencies is possible by adjustment of amplifier bandwidths and clock rates. The theory of operation is discussed along with the circuitry. The phase accuracy is 0.05° ± 0.01° over an input range of 600V p.p. to 100 mV p.p. 241. Reference Signal Phase shift = x 360° Z Fig. 1. Measurement of Phase Shift Variable 242. 1_.______INTRODUCTION There are a variety of methods (Brown and Laramore, 1967, Starostin and Feofilov, 1969) available for measuring the Faraday rotation in solids. The spinning analyser technique used by Suits (1971) and Mort et.al., (1965) utilizes a rotating Polaroid disc to modulate the plane polarized light at twice the rotational frequency. This is a dynamical method, giving directly the angle of plane of polarization of the light by measurement of the phase difference of two A.C. signals. A digital phasemeter has been designed and constructed to measure the phase difference between the two A.C. signals, displaying the phase difference in degrees on a five digit display and giving an analogue output which is fed to an X-Y recorder. 2_.______THE PHASEMETER 2.1. Basis of operation The instrument described in this paper was designed to measure the phase shift between the two low frequency (5Hz) sinusoidal waveforms originating from the signal and reference photomultiplier channels in the Faraday measurement system. The phase shift between two signals (Fig. 1) may be found in principle by determining the time delay 0 between the positive going zero crossings of the two waveforms and the average period x of the wave. The 243. phase shift is then given by 0 o At . 360 T On the assumption that both 0 and t can be measured experimentally then their ratio can be found and the scaled result presented as the measured phase shift to an accuracy determined by the precision of measurement of 0 and t. In the circuitry, a two step digital technique is used to calculate the phase shift. Briefly, the time delay 0 and the period t are measured independently in Stage 1 and the information is stored. This information is then fed to the second stage in which the ratio is established, scaled and displayed as a phase shift between the two wave forms. While in essence all the available data can be determined by a single pair of measurements (0 and t) and two ideal wave forms, the presence of noise on the signal channel at low signal strengths causes some uncertainty in the instant of zero crossing in that channel and hence in the determination of 0. In order to reduce this effect is is necessary to average over a number of cycles of both waveforms. Since the technique reduces the measurement of angle to that of two time delays it is most convenient to establish the desired accuracy by employing digital counting techniques to determine these time intervals, (Martin 1972) the counting pulses n> > = o 2. <0 o 5 ^ 2 ■o Out of rqnge detect 244 1 - ** P> O H- Cd o I O H- iQ 245 . for both delays being generated internally by a clock oscillator whose frequency serves to partially determine the final accuracy for a given frequency of the input and reference signals. With reference to the block diagram, Fig. 2, the operation is as follows. The zero crossing point on the reference channel initiates the accummulation of counts in two counters 6 and t. The first zero crossing of the signal channel terminates counter 0 while the other continues to accummulate clock pulses until the second zero crossing of the reference signal. For signal averaging to be carried out, this second zero crossing of the reference signal starts the counter 0 again and the instrument cycles for a given number n of input waveform periods, where n is the number of periods to be averaged. In this operation, the counter t accummulates continuously until the (n+l)th zero crossing. The counter t at this time has N groups of clock pulses each of which corresponds to 0. The ratios of the two measured time periods are best determined by pulse rate techniques. In this system the accummulated counts in the period counters 0 and t are used to generate two corresponding clock signals, whose frequencies are proportional to the contents of 0 and t. This is achieved in the two Decimal Rate Multiplier (DRM) units whose operation is discussed later. Measurement of the frequency ratios of these two series of clock pulses originating from the two 246 . DRM's provides the required output information ,and suitable scaling is all that is necessary to enable the display to read degrees of phase shift. In order to achieve this, the pulse rate output of the t DRM is scaled and controls the display gate while the pulse rate from the 0 DRM is accummulated in the display unit. This count is then transferred into the LED display store. 2.2. Analysis of Operation. The analysis of the phasemeter operation is straight forward and is given below in terms of the characteristic parameters of the various components. In the following the symbollic representation is x the frequency or the pulse rate c the clock pulse rate a the contents variable o counter capacity M the order of the divider capacity STEP 1 In terms of the above parameters, the gating time into the counters after the first synchroniser is t = M— . k, sec x k acts as a reduction factor for the c^te time and is a function of the phase shift. 247 At any time the counter contents are aN = M— . k sec x and for the two counters this gives Mlclk ae = x n 0 a - ^ T x N T In this is the order of the number of periods averaged. STEP 2 The DRM devices (Texas Instruments Application Report CA-160) have a transfer function x. .a .N x = in out N where aN is a BCD number and N is the order of the counter. The frequencies generated by the multipliers are then: a c 2 x a c T 2 The step 2 gate period is 2/x^, where M2 is the scale factor required for the display, and the display counter contents are consequently ‘e ■0 aDND T 248 . Substituting for from stage 1 and assuming N. = N we have 0 T aDND kM2 Hence from this, the display counter contents are proportional to the phase shift variable k(= /t) and is independent of the clock rates, signal frequency and the number of periods averaged. The scale factor M2 and the display counter capacity NQ can be chosen according to the number of significant digits required and the measurement unit (in this case degrees of angle). Display Resolution The display resolution is a function of NQ and M2 only but in practice the phase accuracy should be comparable to this resolution. The technique of representing the waveform period information as a frequency has certain limitations which determine the minimum acceptable counter capacities. (Their maximum capacity is limited by their maximum speed and measurement time as discussed later). The DRM's in the output stages of the phasemeter generate a pulse train which is a fraction o < a < 1 of an input pulse rate, the fraction being determined by an input BCD number. The reduced output p.r.f. being achieved by the removal of clock pulses at specified times. The output p.r.f. however has a period uncertainty which may be effectively reduced by cascading a number of DRM's. 249 . A computer simulation of a DRM system shows that for x ^ = O.lx. , the output p.r.f. of N decades out in' c is x out, ± 25. This error reduces to±lasx out. -> xin. and in consequence the resolution of the period counters and DRM1s must be at least an order of magnitude greater than the required display resolution, i.e. Nq = N > N + 1 0 t D Clock Rates and Sampling Times In stage 1, the clock rate and the magnitude of the averaging counter is determined by the input frequency range, the sampling rate and the uncertainty of input periods. Because of the DRM limitations, the period counters require to be at least 10% full and consequently only a 10:1 range of input frequencies can be accommodated giving 0.1 < a < 1. The measurement time is the total of stage 1 period measurement and stage 2 conversion times. However as stage 2 is only limited by the speed of the counters, stage 1 will effectively determine the measurement time particularly at low frequencies. The step 1 gate period is given by M—1 where X x is the input frequency and from equation £1J_ we have M-, a N Period measuring time = — = —sec. x c i Mo and with the stage 2 gate period of — it follows that X the Conversion time a c~ x 2 250 detector. crossing Zero and .Amplifier Input 3. Fig. 251. and the a N M0 sec Total sampling time JE-+ a 2 c~ t 2 In practice with the CMOS devices used in this application the clock rates c^ and c2 are limited to approximately 5 MHz. 3. CIRCUIT DESCRIPTION The two input signals are fed into identical channels (Fig. 3, one channel only) which are composed of a comparator amplifier which drives a zero crossing detector to give a pulse whose leading edge is within a few millivolts of the actual zero crossing point. This input amplifier must satisfy a wide range of requirements, namely: high input impedance, high sensitivity, low drift and a wide dynamic range, to allow measurement of 0.01° phase shift. A BB3521L I.C. op amp operating with negative feedback bounding satisfies these requirements aid ensures zero potential at the summing junction. This latter point is essential as differential mode signals degrade the drift specification and generate an output offset resulting in a zero crossing (and hence phase) error. Bounding with zener diodes (Tobey et al 1971) avoids saturation of the amplifier output and the consequential delays and asymmetries. The high zener leakage currents are sourced and a pair of diodes and LU CM h* CL Ve en CL So. LU «/) Q _ 3 O O- o 3 xl.2k 15 V — ICO o o» c u A o o 253. a resistor avoids offsets (Graeme 1973) at the summing junction. To ensure a fast rate of change of voltage driving the zero crossing switches, positive feedback is provided which permits low input voltages (50mV peak) to be accommodated. The diode pair prevents feedback during the knee of the zener characteristic and hence maintains timing accuracy for the zero crossing point. The control circuit for the phasemeter is given in Fig.4. (Note, the I.C*s. are numbered from 1 in each separate Figure). Synchronisation of the zero crossing pulses is achieved by using a dual D type flip-flop (IC 1A,B), to clock the zero crossing pulses on rising clock transitions. The synchronized pulse is then fed to a J-K flip-flop (IC2A,B). In conjunction with IC12, IC2 forms a synchronous edge detector whose output pulses set and reset the phase counter control flip-flop IC11A,B. Coincident pulses on IC12 generate a system reset via STOP ensuring a stable display near zero phase. The first zero crossing after a system reset sets STEP 1 flip-flop IC4B and phase control ICll with their outputs SIG and STEP 1 controlling the gates of the phase and period counters. The number of periods to be averaged is determined by the down counter IC3. Reference pulses decrement IC3 to zero at which times the "borrow" output goes low, setting IClA and inhibiting further counting or period gating until the completion of STEP 2. The "borrow" output goes STEP! SIG ©CTR "GCTR A.DET o 254 H- Clock and Divider Circuits 255, low, setting IC1A and inhibiting further counting or period gating until the completion of STEP 2. The "borrow" output going low also serves to terminate STEP 1 and initiate STEP 2 by gating the J.K. flip flop IC4, gating the clock into the DRM counters and opening the display gate IC9A. After 36000 counts (M2) from the t DRM, a pulse ES2 from the clock board terminates STEP 2. As the clock, gating and counter systems utilize CMOS logic, buffering (IC10) is required to convert the TTL levels to CMOS levels. The dual flip-flop, IC6 and gate IC13 serves to generate a system RESET on power up or manual reset, zero input phase generating a STOP command and after a LOAD display command which is initiated by end of STEP 2. Note that on power up or manual reset, RESET is followed by a LOAD pulse to ensure zero display. The clock, scaling and alarm circuits are shown in Fig. 5, in which all IC devices are CMOS with the exception IC14 and IC15. A single crystal controlled oscillator (IC15) of 3 MHz provides all the clock frequencies necessary for the phasemeter with the various clock rates 36000 ripple counter (IC9,10,11,2B) scales the DRM output, tqut which determines the end of STEP 2. The divider gate IC7-D detects the DRM pulse which opens the display gate via the ratio control flip-flop on the control circuit. Overflow and underflow detections is performed by IC12 and IC7C which gates a 2Hz signal, 256. o Q CA G) S U i • O shown) not is and identical is Counter (Period DRM and counter Phase 6. Fig. 257 causing the display to blink. The period and DRM counters (Fig.6) consist of six cascaded decades of counters, one each for PHASE and PERIOD, with the LSD of the period counter becoming the MSD of the DRM counter. The use of CMOS dual decade counters SCL4518A coupled with the Motorola version of a DRM MC14527 (CMOS) permits cascading of decades without further combinational logic and reduction of the overall package count. Each DRM unit is a four bit synchronously clocked counter where the number of output pulses from the DRM for every ten input pulses is determined by the BCD number on the input derived from the four bit ripple counter. As the reduced output rate from the DRM is aperiodic, only an average p.r.f. need be considered. This necessitates the cascading of a number of units to produce a more periodic pulse train for the purpose of frequency measurement by the phasemeter. The circuits for the display system are not presented, being perfectly straightforward in design. It suffices to say that a five stage ripple counter accummulates a count which is a function of the phase angle measured. The BCD output is available to a DAC digit select system whilst decoder driver latches drive a five digit LED display module. As five decade (20 bit) accuracy is not required to drive an X-Y recorder, the three digits of most significance may be selected by a 12 pole three position switch whose outputs are selected by the digit switch. The outputs are latched into a buffer store to maintain a fixed analogue output until the next display update. The Digital to Analogue kQ •^1 m Phase shift, 0° O H- (D 259 . converter is a Burr Brown module, type DAC80CCD with a 12 bit BCD input. All power supply requirements are satisfied by the use of three and four terminal regulators, with 15V supplies being used for the CMOS to allow good high speed operation. 4. THE PHASEMETER IN OPERATION The phasemeter is currently being used in the investigation of the Faraday Effect in various glasses. The experiment utilises a liquid helium cryostat and superconducting solenoid to generate magnetic fields of 5T. The advantage of using the phasemeter with its analogue output is that the experiment becomes a dynamical one with the phase The Verdet constant, V, knowing the thickness of the glass may be quickly and simply calculated from the graph, where Vd being the slope of the line. 260 . REFERENCES Brown, F.C. and Laramore, G. (1967). Appl.Opt.6_, 669-673 Graeme, J.G. "Application of Operational Amplifiers - Third Generation Techniques", McGraw-Hill Book Company Inc, New York, (1973). Martin, J.D. (1972). The Radio and Electronic Engineer, 42, 285-299. Mont, J., Luty F. and Brown, F.C. (1965). Phys.Rev.137, A566-A573. Starostin, N.V. and Feofilov, P.P. (1969). Sov. Phys. Usp. 12, 252-270. Suits, J.C. (1971). Rev. Sci. Instrum. 4_2, 19-22. 'Texas Instruments Application Report CA-160' SN7497 binary rate multiplier. Tobey, G.G., Graeme, J.C. and Huelsman, L.P. "Operational Amplifiers - Design and Application", McGraw-Hill Book Company, Inc., New York (1971). 261. Appendix 3 : Publications and Conferences Attended PUBLICATIONS 1. Numerical Solution of Kramers-Kronig Transforms by a Fourier Method. S. J. Collocott. Compt. Phys. Commun., 13,203 (1977). 3+ 2. Verdet Constants near 450nm in Ho doped soda glass. S. J. Collocott and K.N.R. Taylor. Laser Interaction and Related Phenomena" ed. H. J. Schwarz and H. Hora, Vol.4A (Plenum Press, NY., 1977) . 3. Magneto-Optical Properties in Rare-Earth Doped Glasses. S. J. Collocott and K.N.R. Taylor. Proc. 5th New Zealand Science of Materials Conf. p53 (1977) . 4. Magneto-Optical Properties of Erbium Doped Soda Glass. S. J. Collocott and K.N.R. Taylor. J. Phys. C : Solid State (in press). 5. Magneto-Optic Saturation in Neodymium Doped Soda Glass. S. J. Collocott and K.N.R. Taylor. Chem.Phys.Lett. (submitted). PAPERS PRESENTED AT CONFERENCES 3+ 3+ 1. Faraday Rotation about Absoprtion Bands in Nd and Ho doped Soda Glass. S. J. Collocott and K.N.R. Taylor. A.I.P. Solid State Physics Meeting, Wagga (NSW)(1977). 2. Faraday Rotation about broad absorption bands in Rare Earth Doped Soda Glass. S. J. Collocott and K.N.R. Taylor. Ann. Solid State Physics Meeting, Manchester, U.K., January, 1977. 3. Magneto-Optical Properties of Erbium Doped Soda Glass. S. J. Collocott and K.N.R. Taylor. A.I.P. Solid State Physics Meeting, Wagga (NSW) (1978) . Computer Physics Communication 13 (1977) 203-206 © North-Holland Publishing Company NUMERICAL SOLUTION OF KRAMERS-KRONIG TRANSFORMS BY A FOURIER METHOD S.J. COLLOCOTT Magnetic Materials Research Group, School of Physics, The University of New South Wales, P.O. Box 1, Kensington 2033, Australia Received 11 March 1977 PROGRAM SUMMARY Title of program: KRONIG Method of solution The solution utilizes the result [2] that if 0(gj) and d(to) are Catalogue number. ACMN band limited (i.e. significantly non zero only within the fre quency range -oj i < to < u> j) then 0(u>) and 6 (u>) can be Program obtainable from: CPC Program Library, Queen’s written in terms of a fourier cosine series and Fourier sine University of Belfast, N. Ireland (see application form in this series: issue). Computer: CDC CYBER 72-26;Installation: Computing Ser 0(u>) = 2 Xy bm cos(mncj/u)\) , (1) vices Unit, University of New South Wales. m=l oo Operating system : KRONOS. 2.1.1 LEVEL 393 0(cj) = 2 Xy bm sin(mTru>/u>i) . (2) m-1 Programming language used: FORTRAN By finding values of bm that satisfy 2 and substituting these into 1 the values 6(u>) are computed from 0(u>). Each bm is High speed storage required: 31kg calculated by a series method obviating the need for large stor age and improving the speed of the calculation. No. of bits in a word: 60 Restrictions on the complexity of the problem Overlay structure: none The program computes 0(u>) from 0(cj) at equally spaced in tervals where each and every M.C.D. data point is used to cal No. of magnetic tapes required: none culate the Fourier coefficient bm. Currently the program is limited to 200 data points but this may be increased by modi Other peripherals used : line printer (unit 3) for testing. fying the program depending on available storage. No. of cards in combined program and test deck: 101 Typical running time Typical running time on a CDC CYBER 72-26 for a transform Card punching code: 1BM029 with 200 data points is 14 sec. Keywords: Solid state physics, Kramers-Kronig, integral Unusual features of the program transform, magneto-optical activity, Fourier series. As the program uses a Fourier technique to compute the in tegral all problems concerning the Cauchy principal part en Nature of physical problem countered in numerical integration techniques are avoided. The two magneto-optical phenomena, the Faraday rotation The program tests the accuracy of the method by using a gaus- and magnetic circular dichroism compose, respectively the sian function and comparing the calculated transform with real and imaginary parts of the complex optical rotation (0 that from theory, the gaussian having an exact analytic solu = 0 - id). The Faraday rotation may then be calculated from tion. the M.C.D. by use of the Kramers-Kronig integral relation [1] References [1 ] C.A. Emeis et al., Proc. Roy. Soc. (London) 297A (1967) n 0(u>) 0(U>) dn . 54. n2 - to2 [2] D.W. Johnson, J. Phys. A. 8 (1975) 490. 203 204 S.J. Collocott / Solution of Kramers-Kronig transforms by Fourier method LONG WRITE-UP technique of interpolation of a function of a sine se ries [6]. Eq. (4) may be written as a sum of n trigome- 1. Introduction tric terms, u(x) = by sinx + b2 sin 2x + b2 sin 3x The Faraday rotation and magnetic circular dich- roism form the real and imaginary parts of the com ... + bn_ i sin (n — 1) x (5) plex optical rotation [ 1 ], 0 = 0 — W and are related by the Kramers—Kronig [2] integral transforms which will take for given values of (n — 1) for equally spaced values of the argument x, say £2 0(n) d£2 , (1) u(-n/n) = ul , u(2n/n) = u2 , u(3it/n) = w3 , a2- w2 oo u((n — 1) tt/n) = un-1 6(0) f (2) where ux,u2,u3, ...,un_x. Lagrange (see ref. [6]) noted that the sum Normally these integral transforms are calculated pit qit 2ptr 2qtt by use of numerical integration techniques that con sin — sin — + sin —— sin---- sider the Cauchy principal part in some special man ner. However, the Cauchy principal part remains a . (n - \)ptt . (n - \)qit problem in all numerical integration techniques that + ... + sin------sin------is never completely resolved. To overcome the problem of the Cauchy principal part several authors [3—5] where p and q are positive integers less than n, has the have suggested the use of Fourier series to compute value \n for p = q and zero for p i=- q. Then the func the transform. It is this method that is presented in tion this paper utilizing the special properties of half Fou pit 2ptt rier sine and cosine series to perform fast numerical sin x sin — + sin 2x sin — calculations of the Kramers—Kronig transforms. n n (n - 1) mn" + ... + sin (n — l)x sin 2. Method of solution has the value up when x = p/n and zero for x = q/n In order to use a Fourier series 0(co) and 0(co) must where q is different from p. It follows directly then be band limited [3], that is significantly non-zero in that the Fourier coefficients b in (5) are given by the the frequency range of interest. Under these condi equation tions (1) and (2) may be written as follows mix 2mit sin — + u2 sin----- oo n n (p(co) = 2 zC bm cos^Trcu/cOi) , (3) m = 1 (in — \)mit + ... + un_ i sin______(6) OO 0(co) = 2 Zy bm sin(mTToolooi) (4) Eq. (5) is periodic with period 2n; it is an odd func m= 1 tion of x and has values ux,..., un_x at the given val for the frequency range —cu! < co < cox. These two ues of the argument. However, should the function equations form a half Fourier cosine series and sine u(x) neither be periodic nor odd, (5) would have a series respectively that may be applied directly to ex graph agreeing with the graph of the function u(x) perimental data. To calculate 0(co) from d(oo) the between x = 0 and x = 7r. Thus, an expression agreeing Fourier coefficients, bm are found from (4) and sub very closely with a given function may be obtained stituted into (3). over a certain range of values but outside that range, The Fourier coefficients bm are calculated by the the values give a poor representation of the function. S.J. Collocott / Solution of Kramers-Kronig transforms by Fourier method 205 3. Program description selected values with the calculated percent differences are given in the reproduced output. The detailed working of the program is described by comment cards throughout the deck. The trans form from the imaginary to the real part is accomplish 5. Discussion ed by a call to SUBROUTINE KRONIG (ZD, ZR, IN) where ZD is the input array to be analyzed, ZR is the As can be seen from the output the accuracy of synthesised array (Kramers—Kronig transform of ZD) this method is very good, agreeing within ±1% with and IN is the number of elements in the array ZD. the exact solution. The exception occurs at 30.5 where Currently IN is restricted to 200 but this may be the curve is rapidly falling through zero from a value increased by adjustment of the DIMENSION state of 9953 to —10075. Ideally the value should be zero ment in KRONIG. As can be seen from equation 6 the but when —95 is compared with 10,000 the error calculation of each bm uses every data point. Should from zero is only some 0.95%. Secondly towards the one wish to transform from the real part to the imagi end there is a series termination error increasing the nary part [i.e. from 0(co) to 0(co)] then SIN and COS errors by some 1.5%. are interchanged in the subroutine. It should be emphasised that very often the con stants in eqs. (3) and (4) require conversion factors owing to the units used for UJ Z ro O X CT' CO h- m Lu vO •4- CVJ x •—« Ct co lO *-• uj rvj >c > x rvj X X> • • x I CVJ x> I o I at o o o o o o o o o o O 3 o o o o o o o o o o o o 0 o o o o o o o o o o o o o o o o o • • • • • • • X X ro ^ if) o o >4 X) X h- X rvi if) J' 1 O X CO X) c> o LO co I £ nc => 2 if) O X *-• X C\J X *4 2 —• X o LD o o X X in r- X r- rvj z O' ^ X 4 X X —• X 2 o © o O o o o o o o CO r-~ • ix cr n n • H O' CT' o o o D CVJ oj co m r> n m cu H D O Z e D 2 UJ H go U_ W < H 3