Optical and electron spin resonance spectroscopy on matrix-isolated silicon and manganese species


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Optical and electron spin resonance spectroscopy on matrix-isolated silicon and manganese species
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xiii, 237 leaves : ill. ; 28 cm.
Ferrante, Robert Francis, 1951-
Publication Date:


Subjects / Keywords:
Electron paramagnetic resonance spectroscopy   ( lcsh )
Silicon   ( lcsh )
Manganese   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis--University of Florida.
Includes bibliographical references (leaves 232-236).
Statement of Responsibility:
by Robert Francis Ferrante.
General Note:
General Note:

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University of Florida
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Full Text







To my parents

Digitized by the Internet Archive
in 2010 with funding from
University of Florida, George A. Smathers Libraries with support from Lyrasis and the Sloan Foundation



The author extends his deep appreciation to Professor

William Weltner, Jr. whose encouragement, professional

guidance, patience, and support made this work possible.

Thanks are also due to all the members of Professor

Weltner's research group, particularly Dr. W. R. M.

Graham and Dr. R. R. Lembke, for their collaboration,

assistance, and advice during the research.

The author greatly appreciates the expert craftman-

ship displayed in fabrication of experimental apparatus

by A. P. Grant, C. D. Eastman, and D. J. Burch of the

Machine Shop and R. Strasburger and R. Strohschein of the

Glass Shop, as well as the maintenance of electronic

equipment by R. J. Dugan, J. W. Miller, and W. Y. Axson.

The author would also like to acknowledge the support

of the Air Force Office of Scientific Research (AFOSR) and

the National Science Foundation (NSF) during this work.









The Matrix-Isolation Technique 1
References Chapter I 9


Introduction 11
Experimental 11
Apparatus 11
General Technique 26
References Chapter II 29


Introduction 30
Atoms and the Resonance Condition 30
The Hyperfine Splitting Effect 33
2E Molecules 43
The Spin Hamiltonian 43
The g Tensor 47
The A Tensor 52
Randomly Oriented Molecules 53
Molecular Parameters and the
Observed Spectrum 61
3E Molecules 7
The Spin Hamiltonian 76
4Z Molecules 94
The Spin Hamiltonian 95

E Molecules 105
The Spin Hamiltonian 106
References Chapter III 113


Introduction 117
Experimental 118
ESR Spectra 119
SiN2 119
SiCO 125
Si2 133
Optical Spectra 133
Si and Si2 133
SiN2 137
SiCO 144
Si(CO)2 150
Discussion 152
References Chapter IV 171


Introduction 175
Experimental 177
ESR Spectra 179
Mn Atoms 179
Mn+ 179
MnO 184
MnO2 190
MnO3 196
Mn04 205
Discussion 207
Mn Atoms and Mn 207
MnO 212
MnO2 215
MnO3 217
MnO4 220
References Chapter V 232






I ESR data of SiN2 and SiCO in their 3
ground states in-various matrices at 40K 131

II Si2 absorption bands in argon matrices at 40K 135
III Ultraviolet absorption spectrum of Si N2
in an argon matrix at 40K 140

IV Vibrational frequencies and calculated
force constants (mdyn/A) for SiNN and
SiCO molecules in their ground 3E states 143
V Absorption spectrum of Si CO in an argon
matrix at 4K 147

VI Comparisgn of stretching force constants
(mdyn/A) for relevant molecules XYZ 154

VII Total density matrix elements for SiCO,
SiN2, and the free ligands CO and N2 157

VIII Spin densities in SiCO and SiNN 164

IX Comparison of vibrational frequencies and
electronic transitions of CXY and SiXY
molecules 167

X Field positions (in gauss) of observed fine
and hyperfine structure lines of Mn+:Ar
at 40K. A = 275 G; v = 9390 MHz 185

XI Magnetic parameters, observed and calculated
line positions for the I+1/2)-+-I-/2)
perpendicular transition of MnO ( E) in Ar 191

XII Magnetic parameters, observed and calculated
line positions for the I+1/2) >I-1/2)
perpendicular transition of MnO2 (4E) in Ar 195

XIII Spin Hamiltonian matrix for the states M, m)
for Mn03 (2A) including interaction with
the 55Mn (I = 5/2) nucleus 199

XIV Magnetic parameters of MnO (2A ) in Ne;
observed transitions in Ne an Ar 203

XV Magnetic parameters, observed and calculated
line positions for MnO4 (2T1) in Ne 208

XVI Summary of magnetic parameters and derived
quantities for manganese and some
manganese oxides 210




1 Basic design features of the liquid
helium dewar used for ESR studies 13

2 Variable-temperature modification of
liquid helium dewar used for ESR studies 14

3 Basic design features of variable-temper-
ature liquid helium dewar used for
optical studies 17

4 Basic design features of cryotip assembly
used for optical studies 20

5 Zeeman energy levels of an electron
interacting with a spin 1/2 nucleus 36
6 Zeeman energy levels of a SS ion with
I = 5/2 37

7 Absorption and first derivative lineshapes
of randomly oriented molecules with
axial symmetry 58

8 Energies of the triplet state in a magnetic
field for a molecule with axial symmetry;
field parallel to molecular axis 82

9 Energies of the triplet state in a magnetic
field for a molecule with axial symmetry;
field perpendicular to molecular axis 84
10 Resonant fields of a E molecule as a func-
tion of the zero feild splitting 87

11 Theoretical absorption and first derivative
curves for a randomly oriented triplet
state molecule with axial symmetry 90

12 Theoretical absorption and first derivative
curves for a randomly oriented triplet
state molecule with orthorombic symmetry 92
13 Energy levels for a Z molecule in a magne-
tic field; field parallel to molecular
axis 102


14 Energy levels for a 4 molecule in a mag-
netic field; field perpendicular
to molecular axis 103

15 Resonant fields of a 4E molecule as a
function of the zero field splitting 104

16 Energy levels for a 6E molecule in a
magnetic field for 0 = 00, 300, 600,
and 90 109

17 Resonant fields of a 6 molecule as a
function of the zero field splitting 111

18 ESR spectra of SiN2 molecules in argon
matrices at 40K 120

19 ESR spectra of SiN2 molecules in various
matrices at 40K 124

20 ESR spectra of SiCO molecules in argon
matrices at 40K 126

21 Effect of temperature upon the ESR spectrum
of Sil3CO in an argon matrix 128

22 ESR spectra of SiCO molecules in Ar and
CO matrices 130

23 Ultraviolet absorption spectrum of SiN2
molecules in an agron matrix at 40K 138

24 Infrared bands of SiN in argon and
nitrogen matrices a? 4K 142

25 Absorption spectrum of SiCO molecules in
an argon matrix at 40K 145

26 Infrared spectra at 40K of an argon matrix
containing vaporized silicon atoms and
13CO/Ar = 12CO/Ar = 1/375 148

27 Infrared spectra at 40K of an argon matrix
containing vaporized silicon atoms and
Cl60/Ar = C180/Ar = 1/375 149

28 ESR spectrum of Mn in argon at 40K 180

29 Zeeman levels and observed transitions for
Mn+ in argon 185




30 ESR spectrum of MnO in argon at 40K 187

31 ESR spectrum of MnO2 in argon at 4K 193

32 ESR spectrum of MnO3 in neon at 40K 197

33 ESR spectrum of MnO3 in argon at 40K 202

34 M = 3/2 component of MnO in argon at
various temperatures, and in neon 204

35 ESR spectrum of MnO4 in neon at 4K 206

36 Molecular orbital correlation diagram
for MnO3 218

37 Molecular orbital correlation diagram
for MnO4 223

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of
the Requirements for the Degree of Doctor of Philosophy



Robert Francis Ferrante

December, 1977

Chairman: William Weltner, Jr.
Major Department: Chemistry

The 3 molecules carbonyl silene, SiCO, and diazasilene,

SiNN have been prepared by vaporization and reaction of

silicon atoms with N2 or CO and trapped in various matrices

at 40K. The electron spin resonance (ESR) spectra indicate

that some or all sites in some matrices induced slight

bending in the molecules, and that the species undergo

torsional motion in the solids. Isotopic substitution of

13C, 15N, 180, and 29Si was employed to obtain hyperfine

coupling data in the ESR and shifts in the optical spectra.

In solid neon, assuming gl =gi=ge, D = 2.28 and 2.33 cm1

for SiN2 and SiCO, respectively. Hyperfine splitting in
14 15 29
argon yield A = 17 ( N), 21 (15N) and 95 (29Si) MHz for

SiN2 and 84 (29Si) and <14 (13C) MHz for SiCO. These

confirm calculated results, in the complete neglect of

differential overlap (CNDO) approximation, that the electron

spins in both molecules are largely in the pn orbitals of Si.

Optical transitions (with vibrational progressions) were
0 -1
observed beginning at 3680A (Si-N stretch, 470 cm ) and

0 -1 o
3108A (N-N stretch, 1670 cm ) for SiN and 4156A (Si-C

stretch, 750 cm-1; C-0 stretch, 1857 cm-1) for SiCO in Ar.

Infrared (IR) spectra in Ar indicate vNN = 1733 and
-1 n -1
V = 485 cm for SiN and v = 1899 cm for SiCO.
Si-N 2 C-0
Calculated stretching force constants are k = 2.0,
k-N = 11.8, kC-O = 15.6, and ksi-C = 5.3 mdyn/A, the
latter assuming vSi-C = 800 cm The CNDO calculations

suggest r bonding of Si to the ligand, which is stronger

in SiCO than SiN2, and some ligand + dSi back-donation,

also stronger in SiCO. An attempt was made to correlate

these vibrational and electronic data with those for CCO

and CNN. Annealing an argon matrix containing SiCO to

350K led to the observation in the IR of 1 Si(CO)2, a

n! -1
silicon counterpart of carbon suboxide, with v = 1899.6 cm1

A corresponding treatment of a SiN2 matrix did not produce

N2SiN2, nor was N2SiCO observed when both ligands were present.

The molecules MnO, MnO2, MnO3, and MnO4 have also been

prepared, by the vaporization and reaction of manganese atoms

with 02, N20, or 03, and isolated in various inert gas matrices

at 40K. ESR has been used to determine magnetic parameters

which are interpreted in terms of molecular geometry and

electronic structure. MnO is confirmed to have a 2 22 6 +

ground state with gl = 1.990(7), assuming gl = ge, and a

zero field splitting in accord with the gas phase value

IDI = 1.32 cm Hyperfine splitting due to the Mn (I = 5/2)

nucleus are JAI I = 176(8) and IA1! = 440(11) MHz. MnO2 is
S 4 2
a linear E molecule with probable configuration U6 ,


D = 1.13 cm-1 (assuming g11 = g = 2.0023), IAI I = 353(11),

IAi = 731(11) MHz. Mn03 exhibits very large hyperfine
splitting IAII = 1772(3) and JAI| = 1532(3) MHz indicative
of an sdz2 hybrid 2A ground state of D3h symmetry. The

spectrum of MnO4 is consistent with a C 3 molecule
distorted from a T1 electronic state in tetrahedral symmetry

by a static Jahn-Teller effect. The g and A tensors are

slightly anisotropic: gll = 2.0108(8), gl = 2.0097(8);

JAIII = 252(3), JAII = 196(3) MHz. The electron hole is
almost entirely in an oxygen r-bonded orbital with one

oxygen atom displaced along its Mn-O bond axis. Warming

to 350K did not induce thermal reorientation.



The Matrix-Isolation Technique

Molecular spectroscopy, the primary tool for the investi-

gation of intimate details of molecular geometry and electronic

structure, has been routinely applied to a large assortment

of stable chemical species in the solid, liquid, and gaseous

phases. The advent of high-speed electronic instrumentation

has extended the range of spectroscopic techniques to allow

the study of unstable or short-lived molecules and fragments.

However, some molecules are still very difficult or even

impossible to observe because of their short lifetimes,

reactivity and/or method of preparation. Among these are

molecules that exist only under high temperature conditions,

such as stellar atmospheres or in arcs, and fragments whose

reactivity precludes production of sufficient quantities for

normal analyses. Even when such species are observed, anal-

ysis of their spectra is often greatly complicated by their

production in a multitude of electronic, vibrational, and

rotational states, as can occur in laboratory methods of

generation of such radicals via arcs, flash photolysis, etc.

Utilization of the matrix-isolation technique can overcome

many of these difficulties.

The technique of matrix-isolation spectroscopy was pro-

posed independently by Norman and Porter (1) and by Whittle,

Dows,, and Pimentel (2) in 1954. Basically, the high

temperature species, reactive molecules, or radical frag-

ments are prepared and trapped as isolated entities in inert,

transparent solids, or matrices, at cryogenic temperatures.

They do not undergo translational motion, but are immobilized,

thus preventing further reaction and preserving the specimen

for conventional spectroscopic analysis. The common technique

of treating solid samples by diluting with KBr and forming a

compressed disk of finely dispersed solid in a KBr matrix

can be considered a crude form of matrix-isolation. It has

been demonstrated that the information gained by this tech-

nique is gas-like within a few percent.

The matrix material can be any gas which will not react

with the trapped species and which can be readily and rigidly

solidified. Many different substances have been used for this

purpose, including CH4, CO, N2, CS2, SF6, 02, as well as

aliphatic and aromatic organic, However, the solid rare

gases Ne, Ar, Kr, and Xe are usually employed because they

are relatively inert chemically, transparent to radiation

over a wide wavelength region, and offer a wide range of

melting points and atomic sizes. The choice of matrix gas

is determined to a large extent by the effect the solid matrix

will have upon the trapped molecule. Neon, because it is the

least polarizable, is expected to perturb the molecule least

and generally makes the best matrix. Unfortunately, its

trapping efficiency is not as great as the other rare gas

solids, and it is difficult or impossible to achieve isolation

of some species in solid Ne. Argon is better in this

respect, and is generally used as the matrix medium. The

heavier rare gases are found to perturb the trapped molecules

to a greater extent than Ne or Ar, and are therefore, less

desirable as matrix materials.

Temperatures sufficiently low to condense the matrix

gas can be attained with either physical refrigerants (nitro-

gen, hydrogen, or helium in the liquid state) or mechanical

closed-cycle refrigerators utilizing Joule-Thompson expansion

of high pressure hydrogen or helium gas. Readily available

and inexpensive, liquid N2 (boiling point 77.40K) is useful

for some matrix materials, but is limited to relatively

stable guest species. Liquid H2 (boiling point 20.4K)

is often used, but it entails a fire hazard in addition to

the normal dangers of handling cryogenic fluids. Thus liquid

He is the most suitable of the physical refrigerants and the

only one useful for condensation of solid Ne, which melts at

240K and permits solid state diffusion at about half that

temperature. Temperatures of liquids H2 and He can be

reduced to 150K and 1.20K, respectively, by pumping on the

liquid. Closed-cycle refrigerators which can attain temper-

atures near the boiling point of He are commercially avail-

able. Their advantages include convenience, elimination of

the need to replenish cryogenic fluids while experiments

are in progress, and low cost of operation after the initial


The substrate on which the matrix is condensed is

usually chosen to be transparent in the spectroscopic region

of interest. Some suitable materials are CsI, KBr, NaCI, and

Au (for reflectance systems) in the infrared region, quartz,

sapphire, and CaF2 for the visible and ultraviolet regions,

LiF for the vacuum ultraviolet and sapphire or other non-

conducting material for microwave spectroscopy, including

electron spin resonance. The polished crystal plates are

mounted on a cold block which makes good thermal contact

with the liquid refrigerant reservoir or the expansion cham-

ber of the Joule-Thompson refrigerator. In variable-tempera-

ture dewars, the cold block is isolated from the reservoir,

but cooled by introducing a controlled leak of refrigerant

into the mounting block.

Several methods can be employed for production of the

guest species which are trapped on such substrates. One

common procedure is vaporization of a non-volatile material

from a high-temperature Knudsen cell in a vacuum furnace.

These cells can be constructed of carbon, or the refractory

metals, Mo, Ta, or W. To prevent degradation of the cell

material from contact with the hot vapor, the crucible can

be lined with C, A1203, or BN. The cells are heated by

resistance or induction methods, and temperature up to 29000K

can be achieved in this manner. The vapor effusing through

a small orifice is collimated into a crude molecular beam

and deposited simultaneously with the matrix gas in such

proportions that M/R (the ratio of the number of moles of

matrix material to the number of moles of the trapped species)

is 500:1 or greater. In many cases the compositions of

vapors so obtained have been characterized mass spectrometri-

cally, greatly facilitating the analysis of the resulting

spectra. The matrix gas can also be doped with a reactant

gas by standard manometric techniques to produce other

unstable reaction products. Another high temperature source

is thermolysis of a gaseous compound by passing it through

a hot W or Ir tube, the resulting products being co-condensed

with the matrix gas as above. An additional common method

for generating unstable species is to subject a volatile

parent compound to high-energy radiation, as produced by

microwave or electric discharges, ultraviolet lamps, laser

sources, or gamma rays, or by electron or ion bombardment.

This exposure can be performed during or after sample deposi-

tion, and the resulting changes observed spectroscopically.

Combination of the high-temperature and photolytic procedures

can produce other unstable species for study.

A large array of spectroscopic techniques can be applied

to matrix samples, including infrared (IR) and Raman,

visible (VIS) and ultraviolet (UV) absorption and emission,

electron spin resonance (ESR) and Mossbauer spectroscopy.

Matrix isolation spectroscopy has several advantages over

gas phase work. First of all, there is the ability to

observe normally unstable or highly reactive species at

leisure, using conventional or only slightly modified spectrom-

eters. Another prime advantage, particularly important for

high-temperature species, is that the molecules are always

trapped in their ground electronic and vibrational states.

With most work done below 250K, there are no "hot" bands, as
the thermal energy is only 17 cm Because states other

than the ground state are thermally inaccessible, sensitivity

is increased over that of high-temperature gas-phase work,

and analysis is aided since the originating level for spectro-

scopic transitions is always the ground state. With long

deposition times (up to 48 hrs with automatic flow control),

sufficient concentrations of molecules can be accumulated

in order to observe species of low abundance or spectral

features with low absorption coefficients. In addition,

controlled diffusion experiments can be conducted in order

to follow the formation of new species, polymers, or clusters.

Finally, it is possible to observe preferential orienta-

tion of some species in matrices; the equivalent effect is

observed in single crystals, but for most species considered

in matrix work, these would be impossible to prepare.

Of course, there are some disadvantages to the technique,

primarily the frequency shift from gas-phase values, caused

by perturbing effects of the matrix. Frequencies in neon

matrices are generally higher than those in argon, and the

data in these solids often bracket the gas-phase value. Neon

does give the closest agreement, with vibrational frequencies
shifted 10 cm or less. Electronic transitions show a

similar trend, the Ne values differing from gas-phase by

up to 200 cm-. For trapped molecules, all transitions
up to 200 cm .For trapped molecules, all transitions

exhibit shifts of the same order of magnitude, and usually

the same direction; trapped atoms show no such regularity,

and often include absorptions that have no apparent corre-

spondence to gas-phase transitions. Magnetic parameters are

also influenced by matrix effects, usually exhibiting, trends

related to the atomic number of the matrix gas. Theoretically,

host-guest interactions causing these perturbations are not

well characterized, although some effort has been extended

in this direction. Samples of such attempts can be found in


Another common observation in matrix work is that the

shapes and widths of bands vary widely, depending upon the

extent of interaction between the absorbing species and

the matrix. Usually Ne shows the narrowest lines, up to
10 cm full width at half maximum (FWHM) in the IR, up to
20 A FWHM for electronic absorptions. The broadening effect

usually increases with the atomic number of the matrix gas,

but lines in Ne are occasionally rather broad, also. Line-

shapes are also somewhat matrix-dependent. However, for a

given matrix, the perturbations are useful in identification

of progressions of vibrational lines in a particular excited

electronic state, since such bands always show the same

shape. The departure of the lineshapes from the usual

Lorentzian form are generally attributed to occupation of

multiple sites of similar energy in the rare gas lattice,

and/or the simultaneous excitation of lattice modes (phonons).

Site splitting of a few wavenumbers, Angstroms, or gauss

are common, but can often be partially or completely

eliminated by the irreversible process of annealing the

matrix. This is done by allowing the matrix to warm up,

and then rapidly quenching it to the original low temperature.

A final disadvantage is the loss of information about

rotational levels of trapped species. Some molecules, such

as H20, HC1, NH2, and NH3 do rotate in matrices, but rota-

tional sturcture is usually lost in broad vibrational


This introduction is not designed to discuss in depth

the various aspects of the matrix isolation technique, but

to illustrate the method and possibilities of its applica-

tion, as well as enumerate some advantages and disadvantages.

More extensive details can be found in several recent reviews

(6, 7, 11, 13-22) and references contained therein.

References Chapter I

1. I. Norman and G. Porter, Nature, 174, 508 (1954).

2. E. Whittle, D.A. Dows, and G.C. Pimentel, J. Chem.
Phys., 22, 1943 (1954).

3. M. McCarty, Jr. and G.W. Robinson, Mol. Phys., 2,
415 (1959).

4. M.J. Linevsky, J. Chem. Phys., 34, 587 (1961).

5. G.C. Pimentel and S.W. Charles, Pure Appl. Chem.,
7, 111 (1963).

6. B. Meyer, "Low Temperature Spectroscopy," Elsevier,
New York, 1971.

7. A.J. Barnes, "Vibrational Spectroscopy of Trapped
Species" (H.E. Hallam, ed.), Wiley, New York, 1973,
p. 133.

8. R.E. Miller and J.C. Decius, J. Chem. Phys., 59,
4871 (1973).

9. A. Nitzan, S. Mukamel, and J. Jortner, J. Chem. Phys.,
60, 3929 (1974).

10. G.R. Smith and W. Weltner, Jr., J. Chem. Phys.,
62, 4592 (1975).

11. S. Cradock and A.J. Hinchcliffe, "Matrix Isolation,
A Technique for the Study of Reactive Inorganic
Species," Cambridge University Press, Cambridge,

12. B. Dellinger and M. Kasha, Chem. Phys. Lett., 38,
9 (1976).

13. A.M. Bass and H.P. Broida, "Formation and Trapping
of Free Radicals," Academic, New York, 1960.

14. W. Weltner, Jr., Science, 155, 155 (1967).

15. J.W. Hastie, R.H. Hauge, and J.L. Margrave, "Spectro-
scopy in Inorganic Chemistry," Vol. 1 (C.N.R. Rao
and J.R. Ferraro, eds.), Academic, New York, 1970,
p. 57.

16. W. Weltner, Jr., "Advances in High Temperature
Chemistry," Vol. 2 (L. Eyring, ed.), Academic,
New York, 1970, p. 85.

17. D. Milligan and M.E. Jacox, "MTP International Review
of Science, Physical Chemistry, Series I," Vol. 3
(D.A. Ramsay, ed.), Butterworth, London, 1972, p. 1.

18. A.J. Barnes, Rev. Anal. Chem., 1, 193 (1972).

19. A.J. Downes and S.C. Peake, Mol. Spectrosc., 1,
523 (1973).

20. L. Andrews, Vib. Spectra Struct., 4, 1 (1975).

21. B.M. Chadwick, Mol. Spectrosc., 3, 281 (1975).

22. G.C. Pimentel, New Synth. Methods, 3, 21 (1975).



The general experimental procedure including cryogenic,

high-temperature, spectroscopic, and photolytic apparatus is

discussed in this section. Specific details peculiar to

individual molecules investigated will be presented with

the discussion of those species.



In this research, three separate cryogenic systems

were employed, two for optical and IR studies and a third

for ESR experiments. An ESR and an optical dewar utilizing

liquid He as physical refrigerant were adapted from the design

of Jen, Foner, Cochran, and Bowers (1), and modified for

variable-temperature operation as described by Weltner and

McLeod (2). Both systems are comprised of an outer liquid

N2 dewar, which serves as a heat shield, surrounding the inner

liquid He dewar. The sample substrate is attached to a

copper block suspended from the bottom of the inner dewar

by a small tube which permits passage of a controlled leak

of refrigerant into the block. The inner dewar is positioned

such that the trapping surface is directly in the path of

the sample inlets.

Pertinent details of the ESR dewar, described by Easley

and Weltner (3), and Graham, et al.(4), are indicated in

Figures 1 and 2. The stainless steel inner dewar, capacity

approximately 2.1 liters, is jacketed at the lower end by

a copper shroud which is part of the outer (liquid N2) reser-

voir surrounding the upper portion. This jacket extends down

to encase the microwave cavity and maintain it near 770K.

A slot approximately 3.5 cm long and 0.6 cm wide allows the

matrix gas and furnace vapor to reach the sample substrate.

At 900 to either side of this slot, two rectangular openings

of approximately 2.4 cm x 0.6 cm are provided. These points

correspond to the location of interchangeable windows, sealed

by Viton "O" rings to the outer vacuum chamber, which permit

visual examination of the matrix and serve as ports for

photolysis or spectroscopic observations.

The trapping surface is single crystal sapphire, obtained

from Insaco, Inc. It is a flat rod, 3.3 cm long, 3.1 mm

wide, and 1.0 mm thick, securely enbedded by Wood's metal

solder into the copper block, as indicated by Figure 2.

A Chromel vs Au-0.02 at. % Fe thermocouple is also attached

to the copper block so that the temperature can be monitored.

Although the temperature of the sample substrate itself is

never determined, single crystal sapphire has high thermal

conductivity at 40K so that it rapidly equilibrates to the

temperature of the mounting.

Also shown in Figure 2 is the construction of the

variable-temperature modification. The copper lower can is

< LL J

- a.

< > 0
0 C/) < 0





o_ ( c- V .... -- -- r

C 0
Oa) 0


.-" \
)) ,

I 0 --

\ LI 0

-' -J -
U) Ua


connected to the main liquid He reservoir by a thin stainless

steel tube; a vent for the lower can is also provided to

exhaust the gaseous He as it evaporates. Outside the vacuum

vessel, the He outlet is equipped with a valve to control the

flowrate of liquid He into the lower can. The main liquid He

reservoir is pressurized to about 2.5 psi to supply an

uninterrupted flow through the mounting block. To vary the

temperature, the outlet valve is closed, and as the He

evaporates, it forces the liquid refrigerant out of the

lower chamber, allowing it to warm. The change in tempera-

ture is monitored with either a Leeds and Northrup model

8687 potentiometer or a Newport model 2600 digital thermometer.

After sample deposition is completed, the entire

inner dewar assembly is lowered approximately 3.8 cm with

respect to the fixed outer can and vacuum chamber, utilizing

a vacuum-tight bellows arrangement mounted at the top and

not indicated in the Figures. In this manner, the rod is

positioned in the center of the copper X-band (=9.3GHz)

microwave cavity; this location corresponds to the maximum

intensity of the circulating magnetic field of the microwave

radiation injected into the cavity. The front end of the

cavity is slotted and aligned with another interchangeable

window in the outer vacuum chamber located just below the

sample inlets. In this way, the sample can be photolyzed

and ESR spectra recorded simultaneously. The back end of

the cavity is fitted with a standard copper waveguide coupling

mounted just outside a mica window, which serves to maintain

high vacuum conditions in the cavity.

With the sapphire rod in position, the dewar is separated

from the furnace assembly by means of a gate valve and

disconnected from it. The entire dewar assembly is then

rolled on fixed tracks to the proper position between the

poles of the ESR magnet. When so aligned, the alternating

magnetic field of the microwave radiation is oriented

perpendicular to the static field of the external magnet.

However, the inner (liquid He) dewar can be rotated 3600

on bearings to permit detection of resonance signals with

the flat surface of the rod oriented at any angle with respect

to the external field.

Following the same basic design, the dewar used for

optical studies is diagrammed in Figure 3. The stainless

steeliriner (liquid He) dewar, of capacity 8.6 1, is surrounded by

a'liquid N2 container and copper sheath. Four circular open-

ings in the sheath, each of 3.5 cm diameter, are located at

the level of the sample substrate mounted on the lower cham-

ber of the inner dewar. At one of these openings, the fur-

nace and matrix gas inlets are attached on the outer vacuum

chamber. This opening can be sealed with a gate valve when

deposition of vapor from the furnace is not desired. At

90 to either side of the sample inlet, the openings form

part of the optical path of the spectrometric instruments.

Interchangeable windows (approximately 4 cm diameter) are

mounted with Viton "O" rings to the outer vacuum chamber at

these points. The window materials are chosen to match the

spectral region of interest; CaF2 is used for the visible


Figure 3. Basic design features of variable temperature
liquid helium dewar used for optical studies.

and UV regions (2000-7000 A), and CsI is used for visible
and IR studies (3500 A-50p). All optical crystals were ob-

tained from Harshaw Chemical Co. The fourth port is located

at 1500 to the sample inlets, and interchangeable windows

can also be mounted on the outer chamber at this point.

These are usually either quartz or LiF, and serve to admit

photolyzing radiation to the sample in the ultraviolet or

vacuum ultraviolet regions, respectively. Such sample

photolysis cannot be accomplished simultaneously with deposi-

tion. As in the case of the ESR dewar, the entire inner

assembly can be rotated on bearings through 3600, to align

the sample window with any of the above.

The sample substrate is a polished optical crystal,

2.2 x 1.1 cm, chosen to match those on the outer vacuum

casing and the spectral region of interest. It is mounted

in the lower chamber of the variable-temperature inner dewar

with all four sides in contact with the cold copper block.

At all points where the window is in contact with the copper

heat sink, a thin gasket of indium metal is inserted. This

material has good thermal conductivity and is sufficiently

plastic that it conforms to all contours of both surfaces

when the window mounting frame is firmly screwed into the

copper block from above the substrate. A chromel vs Au-0.02

at % Fe thermocouple is mounted to the copper immediately

adjacent to the window. Temperatures at this point are

measured with either the potentiometer or a Cryogenic Tech-

nology Inc. digital thermometer /controller. Variable-

temperature operation is achieved in the same manner as

described above for the ESR dewar.

The third piece of cryogenic apparatus employed in this

research utilizes an Air Products model DE-202 Displex

cryotip. This is a two-stage, closed-cycle He refrigerator

which makes use of the Joule-Thompson effect as compressed

gas at 300 psi is expanded, with a pressure drop of over

200 psi. The vacuum housing for the cryotip is very similar

in design to the liquid He optical dewar, except that the

fourth window discussed above is located at 1800 to the

sample inlets; thus photolysis can be conducted simultaneously

with sample deposition.

Internally, there are a few modifications. The cryotip

unit (shown in Figure 4) is constructed of stainless steel,

with the exception of the final expansion chamber. There

is no liquid N2 outer dewar, but its function as a heat

shield is taken over by a nickel-plated copper shroud attached

to the first expansion stage, maintained at 40-600K. This

extends down to and surrounds the sample substrate holder,

with two openings cut at 1800 apart. The entire unit is

rotatable through 1800, to align the sample window with any

two opposite ports in the external vacuum housing.

The second expansion stage is terminated with a copper

cold tip. The copper sample window holder is firmly screwed

into this tip, and good contact is assured with an indium

gasket. The circular sample windows (2.6 cm diameter) are

of the same materials as employed in the other optical dewar,









Figure 4.



Basic design of cryotip assembly used for
optical studies.

and are secured with indium gaskets and a copper retaining

ring. The chromel vs Au 0.02 at.% Fe thermocouple is mounted

on the sample holder. The temperature is varied between 100K

and ambient by an electrical heating wire wrapped around the

second stage expander cold tip. The temperature is measured

and automatically maintained at any preset value with the

CTI thermometer/controller.

The vacuum chambers of all three of the above systems

are pumped by 2 inch oil diffusion pumps, with liquid N2 cold

traps, backed by mechanical forepumps. Dewars employing

liquid He refrigerant attain an ultimate vacuum of approx-
imately 5 x 10 torr. All pressures are measured with

Bayert-Alpert type ion gauge tubes and Veeco RG-31X control


Vaporization of non-volatile materials was accomplished

in vacuum furnaces of identical design attached to each of

the cryogenic systems. These are water-cooled brass cylinders

20.3 cm long and 15.2 cm in diameter. Furnaces associated

with the liquid He dewars are pumped by an oil diffusion

pump of minimum two inch diameter intake, backed with a

mechanical forepump and equipped with liquid N2 cold traps,
attaining pressures below 1 x 10- torr. The furnace mounted

on the Displex cryotip utilized the same pumping system as

the cryotip assembly.

A schematic of the furnace apparatus is illustrated in

Figure 1. Interchangeable, demountable flanges of various

design are inserted into the furnace body. For resistance

heating of samples, these flanges are equipped with water-

cooled copper electrodes. Tantalum cell holders are securely

bolted with Ta screws to the ends of the electrodes, and

inserted in the holders are Ta cylinders 2.5 cm long, 6.4 mm

O.D., and of varying wall thicknesses, filled with the solid

to be vaporized. The open ends of the cell were sealed

with tight-fitting Ta plugs.. The effusion orifice of 1.6 mm

diameter is directed towards the target window. These cells

could be aligned at any angle to the horizontal; the vertical

position is necessary for samples which are melted to produce

sufficient vapor pressure for deposition. Not shown in the

figure is a water-cooled copper heat shield placed 1 2 cm

in front of the cell and fitted with a central 2 cm hole to

allow passage of some of the sample vapor to the target.

The cell was heated by passing up to 500 amps at up to 6 volts

through the electrodes. Cell temperatures were measured with

a Leeds and Northrup vanishing filament optical pyrometer

through a flange-mounted Pyrex window which was shuttered

to prevent deposition of a film on the window.

Induction heating of samples was performed in the same

furnace body, but equipped with a flange holding 10 turns

of 6.5 mm hollow copper coil, the axis of the helix aligned

towards the target window. A Ta Knudsen cell, 2.2 cm long,

1.4 cm O.D., 3 mm wall, is supported, coaxially with the coil,

on three W rods (1.5 mm diameter and 8 cm long) attached to

a screw mechanism mounted on the furnace flange in place of

the electrodes. In this way the cell position could be

adjusted in the coils to provide maximum coupling. Tempera-

ture measurements were achieved as above, except that the

cell was provided with a black-body hole, which eliminates

the necessity of emissivity corrections. The water-cooled

RF coils were attached to a Lepel 5 kW high frequency induc-

tion heater. With both methods of heating, the distance from

cell orifice to the trapping surface was approximately 12 cm.

Matrix gases or gas mixtures were usually admitted to

the dewars through the copper inlet shown in the figures.

This was connected to a copper manifold equipped with

fittings to connect to Pyrex sample bulbs. The manifolds

were pumped by a 2 inch oil diffusion pump, equipped with

liquid N2 trap, and backed by a mechanical forepump, which
gave pressures less than 1 x 10- torr. Flow rates were adjusted

with an Edwards needle valve, and pressure changes monitored

with a Heise Bourdon tube manometer. Gas mixtures were

produced in a similar vacuum system by standard manometric

techniques. The rare gases, neon and argon, were Airco

ultrapure grade, and used without further purification except

for passage through a liquid N2 cold trap immediately prior

to deposition.

Electron spin resonance measurements were made with an

X-band Varian V-4500 spectrometer system employing super-

heterodyne detection. A12 inch electromagnet useful from

0 13 kG provided the static magnetic field, which was

modulated at low (200 Hz) frequency. The output of the

instrument was recorded on a Moseley model 2D-2 X-Y recorder.

When signals were weak, a Nicolet model 1072 signal average,

equipped with SW-71A sweep and SD-72A analog-to-digital

converter plug-in units was used to improve the signal-to-

noise ratio. The magnetic field was measured with either

an Alpha Scientific model AL-67 or a Walker Magnemetrics

model G-502 NMR gaussmeter in conjunction with a Beckman

6121 counter. The microwave cavity frequency was determined

with a Hewlett-Packard high-Q wavemeter.
Absorption spectra were recorded from 7000 2000 A

using a Jarrell Ash 0.5 meter Ebert mount scanning mono-

chromator. Gratings ruled with 1200 lines/mm and blazed at
5000 and 3000 A gave a reciprocal linear dispersion of
16 A/mm in first order. Detectors used were the RCA 7200,
for the range 3700 2000 A, and either the RCA 1P21 or
931A, for the range 7000 3500 A, each operated at 1000 VDC.

The photomultiplier output was processed by a Jarrell Ash

82110 electronic recording system and displayed on a Bristol

model 570 strip chart recorder. Continuum light sources

were a General Electric tungsten ribbon-filament lamp for

the visible and a Sylvania DE 350 deuterium lamp for the UV.

Radiation from these sources was passed through the matrix

and focused onto the spectrometer slit with quartz optics.

The spectra were calibrated with emission lines from a Pen

Ray low pressure Hg arc lamp. A Perkin-Elmer 621 spectro-

photometer (purged with dry N2 gas) was used in the IR region

from 4000 300 cm with an accuracy of 0.5 cm-1

Photolyzing radiation in two spectral regions was avail-

able from either a high pressure Hg arc lamp or a flowing

H2-He electrodeless discharge lamp. The Hg lamp consists of

a water-cooled General Electric type AH-6 Hg capillary

lamp operated at 1000 W, the output of which was focused

onto the sample with quartz optics. The radiation from this

lamp consists of the characteristic Hg lines and a strong

base continuum. When this lamp was in use, the dewars were

equipped with quartz optical windows for transmitting the

radiation to the sample.

The flowing H2-He electrodeless discharge lamp is

constructed after the design of David and Braun (5). It

consists of a quartz tube, 15 cm long, fused 4 cm from the

end with a larger diameter quartz tube to form an annular

space 6 cm in length. The annulus has provision for inlet

of the 10% H2 in He gas mixture (Air Products), and the

central tube is connected to a mechanical pump, which can
evacuate the entire system to about 3 x 102 torr. This

effectively seals the quartz body against an LiF optical

window by means of a brass fitting equipped with "0" rings.

The LiF window is mounted in the dewar photolysis ports.

The gas flow is adjusted to give a pressure of about 1 torr

with a gas regulator. An 85W Raytheon PGM 10 microwave genera-

tor, operating at 2450 MHz, was used with a tunable cavity

to excite a discharge in the flowing gas. The emission was
characterized by the intense Lyman a line at 1216 A. Color

centers which developed in the LiF due to the high energy

radiation could be removed by annealing the windows at

AIr Or1 4:M- Ci.i^ I,^,,-,

General Technique

Preparation of the matrix samples was achieved in the

following manner. The dewars, furnace assemblies, and gas

manifolds were readied at least one day before an experiment

was run, and allowed to pump out overnight. If a good vacuum

was maintained, the liquid N2 cold traps associated with the

diffusion pumps were filled; this brought the furnace and

gas manifold assemblies near their ultimate vacuums. The

sample cells were then slowly heated and the samples allowed

to outgas at low temperatures (about 2000C below deposition

temperatures) while the dewars were prepared.

While preparation of the Displex cryotip involved only

checking the He and cooling water pressures, and switching

on the device, preparing the liquid He dewars was somewhat

more involved. First the outer, and then the inner dewar

was filled with liquid N2. The inner dewar was constantly

purged with dry N2 gas when not in use to prevent formation

of ice in the narrow channels of the variable-temperature

chamber. Filling this container with liquid N2 served to

precool it, and minimize the quantity of liquid He wasted

for this purpose. After the lower chamber, on which the

sample substrate was mounted, had reached liquid N2 tempera-

tures, that refrigerant was pumped out by pressurizing the

chamber with N2 and He gas. When the liquid N2 had been re-

moved and recovered, the dewar was flushed with He gas and

allowed to warm up 10-200K. This assured complete removal

of the N2, which could solidify as liquid He was added. It

was found to be very important that a positive pressure of

dry gas was applied to both dewar openings, especially when

it was cold. When the dewar had warmed slightly, liquid

He was transferred by a vacuum-insulated tube into the dewar,

which was sealed with a pressure cap when transfer was

completed. These preliminary activities took approximately

one hour to perform. A charge of liquid He lasted approxi-

mately 4 8 hours, depending on the rate of flow through

the lower chamber, which was set with a needle valve. The

Displex cryotip also took about one hour to reach operating

temperature, but it could maintain that temperature indefi-


With the deposition surface at a sufficiently low

temperature, the gas manifold was sealed from its pumps

and filled with the matrix gas or gas mixture. To prevent

formation of a solid residue from vaporization on the

surface, the gaseous sample alone was deposited on each side

of the substrate for approximately five minutes. The rate

of gas deposition through the entire run was controlled with

an Edwards needle valve to obtain a flow of about 0.3 mmole/

min. During this time the non-volatile sample was heated

to its deposition temperature; formation of the metal film

on heat shield and furnace viewing port indicated that

sufficient material was being vaporized. With the initial

deposit on the sample substrate, the gate valve separating

furnace and dewar was opened and furnace vapors were co-

condensed with the matrix gas. During sample deposition,

typical pressures observed in the furnace and dewar were
-4 -5
1 x 10 and 2 x 10 torr, respectively. Deposition times

varied from 1/2 to 2 hrs, depending on the species being

formed; the sample substrate was rotated 1800 periodically,

to form an even coating on the surface. When deposition

was completed, the dewar and furnace were isolated, the

gas flow stopped and high-temperature cell allowed to

cool. The matrix samples thus prepared were then observed



References Chapter II

1. C.K. Jen, S.N. Foner, E.L. Cochran, and V.A. Bowers,
Phys. Rev., 112, 1169 (1958).

2. W. Weltner, Jr. and D. McLeod, Jr., J. Chem. Phys.,
45, 3096 (1966).

3. W.C. Easley and W. Weltner, Jr., J. Chem. Phys., 52,
197 (1970).

4. W.R.M. Graham, K.I. Dismuke, and W. Weltner, Jr.,
Astrophys. J., 204, 301 (1976).

5. D. David and W. Braun, Appl. Opt., 7, 2071 (1968).



The interactions of paramagnetic atoms and molecules

with magnetic fields, which gives rise to the electron spin

resonance phenomenon, is discussed in this chapter. Details

of the theory applicable to atoms (or ions) and doublet,

triplet, quartet, and sextet state molecules encountered

in this research will be presented in separate sections.

More extensive treatments of the basic theory presented

here can be found in a number of excellent references (1-13),

Atoms and the Resonance Condition

The paramagnetic substances with which we are concerned

are those which possess permanent magnetic moments of atomic

or nuclear magnitude. In the absence of an external field

such dipoles are randomly oriented, but application of a

field results in a redistribution over the various orienta-

tions in such a way that the substance acquires a net magnet-

ic moment. Such permanent magnetic dipoles occur only when

the atom or nucleus possesses a resultant angular momentum,

and the two are related by

p = yG [i]

where p is the magnetic dipole moment vector, G is the angular

momentum (an integral or half-integral multiple of h/2n = "i

where h is Planck's constant), and y is the magnetogyric

ratio. The motion of these vectors in a magnetic field H'

consists of uniform precession about H at the Larmor precession


w =-yH. [2]

The component of G or p along H remains fixed in magnitude,

so that the energy of the dipole in the field (the Zeeman


W = -VIH [3]

is a constant of the motion.

The magnetogyric ratio which relates the magnetic moment

to the angular momentum according to Eq. [1] is given by

Y = -g(e/2mc) [4]

where e and m are the electronic charge and mass, respectively,

and c is the speed of light. The factor g = gL is unity for

orbital angular momentum and g = gS = 2.0023 for spin angu-

lar momentum. Including this factor with Eq. [1] and defining

the Bohr magneton as 8 = ei/2mc, we have (along the field


UL= -gLmL [5a]


S = -gsmS

Because the angle of the vector p with respect to the applied

field H is space quantized, only 2G + 1 orientations are

allowed. These allowed projections along the magnetic field

are given by mG where mG is the magnetic quantum number

taking the values

mG = G, G-l, ..., -G. [6]

This accounts for the appearance in Eqs. [5] of the factors

mL for orbital angular momentum and mS for spin angular

If only spin angular momentum arises, as in a S1/2 atom,

the 2S + 1 energy levels separate in a magnetic field, each

of energy

E = mSH (7)
EmS geBmSH, [7]

with equal spacing ge H. However, the angular momentum does

not generally enter as pure spin, so that the g factor is

an experimental quantity and mS an "effective" spin quantum

number, since some orbital angular momentum is usually mixed

into the wavefunction. In orbitally degenerate states

described by the strong (Russell-Saunders) coupling scheme,

J = L + S, L + S 1, ...,IL Sland

E = gi m H


S+ S(S + 1) + J(J + 1) L(L + 1)
J 2J J + 1)

is the Lande splitting factor. This reduces to the free

electron value for L = 0.

Taking the simplest case of a free spin, mj = ms = 1/2,

and there are only two levels. Transitions between these

levels can be induced by application of magnetic dipole

radiation obtained from a second magnetic field, at right

angles to the fixed field, having the correct frequency to

cause the spin to flip. Thus the resonance condition is

hv = geSHo [10]

where H is the static external field and v is the fre-
quency of the oscillating magnetic field associated with

the microwave radiation; this frequency is about 9.3 GHz
2 2
for the X-band. Thus for a S or P atom, the ESR spectrum

will consist of one line corresponding to the particular g

value of the atom.

The Hyperfine Splitting Effect

If only one line were observed in the general case, the

ESR technique could offer only a limited amount of informa-

tion, the g value. However, there are other interactions

to consider which increase the number of spectral lines and

the information that can be obtained. One of the most impor-

tant-isthe nuclear hyperfine interaction.

Usually at least one isotope of an element contains a

nucleus having a non-zero magnetic moment. The magnetic

moment of the odd electron can interact with this nuclear

moment and split the single ESR line into hyperfine struc-

ture. This effect can be pictured as follows. The magnetic

field "felt" by the electron is the sum of the applied

external field and any local fields. One such local field

will be that caused by the moment of the magnetic nucleus;

this is, in turn, governed by the nuclear spin state. It

is then clear that, in the case of nuclear spin I = 1/2,

for example, the local field in which the electron finds

itself will be one of two contributed by the nucleus, since

there are 21 + 1 nuclear levels. Hence, there will be two

values of the external field which satisfy the resonance

condition, that is,

r (H' ) = (H' AM) [11]

where A/2 is the value of the local magnetic field, A

being the hyperfine coupling constant, and H' is the

resonant field for A = 0. One example of this phenomenon is
1 2
the H atom. This is a pure spin system, S1/2, with I = 1/2;

its ESR spectrum consists of two lines separated by A = 508G,

centered around g = ge = 2.0023, as shown in Figure 5.

A more detailed look at the paramagnetic species with

non-zero nuclear spin in a mangetic field indicates that

there are several interactions at work. One is the inter-

action of the external field with the electrons, which has

already been considered. An analogous term results from

the precession of the nuclear magnetic moment in the exter-

nal field. The nuclear magnetic moment pI is related to the

nuclear g factor gI by the relation

g = [12]

where 8N = eli/2M is the nuclear magneton and M is the proton

mass. The third term describes the interaction between the

electrons and nuclei. Thus the Hamiltonian can be written

H = gH-J + hAI.J g B H-I [13]

where the underscore indicates that the quantities are

operators. Except in very strong fields, the interaction

of the nuclear moment with the external magnetic field (the

nuclear Zeeman term), which is represented by the last term

in Eq. [13], is small, and will be neglected. Also omitted

from this Hamiltonian are even smaller effects, such as the

nuclear electric quadrupole interaction.

Reference to Figures 5 and 6 will indicate the behavior

of the levels as a function of external field strength.

The two limiting cases of very weak and very strong field

are of particular interest.

The Zeeman effect in weak fields is characterized by

an external field splitting which is small compared to the

natural hyperfine splitting; that is, hAI.J > gBH-J in Eq. [13].

In this case, the orbital electrons and the nuclear magnet


o ro


I II -4

0 ,

0 0
z Z

_n) I i

N2 "-

z a
Wj- -,. 4J


0 u
w z

U LJ \0
Z -l< E



0 7-

L C\OJ C0J c\j J C\J












remain strongly coupled. A total angular momentum F = I + J

exists, which orients itself in the external field. F takes

the values I + J, I + J 1, ..., II JI. The component

of F along the field direction, mF, has 2F + 1 allowed

values, the integers between -F and +F. In the case presented

in Figure 6, with both pN and A positive, these components

are arranged, in order of decreasing energy, mF = F, F 1,

..., -F. Each individual hyperfine level splits up into

2F + 1 equidistant levels in the weak field, giving (2J + 1)*

(21 + 1) Zeeman levels altogether. Note that in both Figures

5 and 6, the levels are not all degenerate even at zero

field. This effect, produced by the hAI-J term, is called

the zero field splitting.

In the Paschen-Back or strong field region, the splitting

by the external field is large compared to the natural hyper-

fine splitting. The strong interaction with the external

field decouples I and J, which now process independently

around H. F is no longer a good quantum number, but there

exist m and mi, the components of J and I along the field

direction. In this case, each Zeeman level of the multiple

characterized by a fixed m is split into as many Zeeman

hyperfine lines as there are possible values of mi, that

is, (21 + 1). Since there are still (2J + 1) levels for

a given J, there are, exactly as in the weak field, (2J + 1)*

(21 + 1) total energy states. In contrast to the weak field

situation, the levels here form a completely symmetric pattern

around the energy center of gravity of the hyperfine multiple.

This pattern manifests itself in Figure 6. Also recorded in

that Figure are the m values of the different hyperfine

groups. Values of m are, in order of decreasing energy,

-5/2, -3/2, -1/2, 1/2, 3/2, 5/2 for mj = 0, -1, -2, -3;

this order is inverted for the remaining mj groups.

The situation in intermediate fields is somewhat more

complicated. The transition between the two limiting cases

takes place in such a way that the magnetic quantum number

m is preserved. In weak field, m = mF; in strong field,

m = mI + m In this region, the Zeeman splitting is of

the order of the zero field hyperfine splitting.

For a 2S1/2 state, as in Figure 5, the general solution

for the energy levels over all fields is given by the

Breit-Rabi equation (14). In terms of the quantum numbers

F and m = mF, it is

W(F, AW I Hm AW 4m X2 21/2 14
W(Fm) = 2(2 + 1) I + ( + 21 + 1[14a

where AW = -(21 = 1) [14b]

andX (- J/J + II /I) H0
and X = [14c]

The plus sign in Eq. [14a] applies for F = I + 1/2 and

m = +(I + 1/2), ..., -(I 1/2) and the minus sign for

F = I 1/2 when m = (I 1/2), ..., -(I 1/2). The zero

field hyperfine splitting is AW. The limiting cases of weak
2 2
and strong fields correspond to X <<1 and X >>1, respectively.

For the general case of intermediate fields, the energy

values of the Zeeman levels can be derived from the following

key equation given by Goudsmit and Bacher (15):

Xm + 1 1 (I + m + 1)(J mj + 1)]
m + 1, m 1 I


Xm 1 m (I m + 1)(J + m + 1)] = 0,
M 1, m + 1 2 I J

where A is the hyperfine coupling constant, gj and g'I = I

are the electronic and reduced nuclear g-factors, respectively,

and the XXA are coefficients in the expansion of the wave-

function; the other symbols have their usual meaning. Here,

AWH is the energy of the level with respect to the center of

gravity of the hyperfine multiple. This relation yields

one system of homogeneous equations in XX for each value

of m = mI + m The resulting secular equations are solved

for the energies of the Zeeman levels at any field. Such

a calculation was performed to produce Figure 6.

With a multitude of levels available, it is necessary

to explain the observed ESR spectra in terms of the selection

rules. Since transition between Zeeman levels involve changes

in magnetic moments, we must consider magnetic dipole transi-

tions and the selection rules pertaining to them. In the

pure spin system with I = 0, the single line observed

corresponds to the m = 1/2 -- m = -1/2 transition. In
s s
general, the criterion is that Amj = 1, corresponding to

a change in spin angular momentum of T. Since a photon has

an intrinsic angular momentum equal to I, only one spin

(nuclear or electronic) can flip on absorption of the photon,

in order to conserve angular momentum. With the fields

and frequencies ordinarily used in ESR work, the transition

usually observed is limited to the selection rules Amj = 1,

Am = 0; the opposite is true in NMR work. It is, however,

possible to observe the Amj = 0, Am = 1 NMR transition

with ESR apparatus, if the zero field splitting (propor-

tional to the hyperfine coupling constant A) is large

enough relative to the microwave frequency. If this does

not occur, only the ESR lines will be observed, resulting

in a multiple of 21 + 1 hyperfine lines for each fine

structure (Amj = 1) transition. Thus the H atom spectrum

(Figure 5) will consist of two lines, while that of Mn

( S3, I = 5/2) will contain 36 individual lines, if all

are resolved.

These interactions of the electron with a nucleus are

related to fundamental atomic parameters, which can be

deduced from the observed spectrum. They are most simply

categorized as isotropic and anisotropic interactions.

The anisotropic interaction has its roots in the classi-

cal dipolar coupling between two magnetic moments. This

interaction is given by


e'N *3( e'r) ( N' r)
E 35[16]
3 5
r r

where r is the raduis vector from the moment e to -N and
e N
r is the distance between them. The quantum mechanical

version is obtained by substitution of the operators, -gBS

and gNN I, for the moments e and N', respectively, yielding

I-(L-S) 3( r-) (Sr)
Hdip = -rg N 3 5 [17]
r r

For a hydrogenic atom with non-zero orbital angular momen-

tum (that is, p, d, ... electrons), this yields

L(L + 1) 1
aJ e N J(J + I) ; [18'

a more exact relativistic treatment also adds a multiplica-

tive factor [F(F + 1) I(I + 1) J( J + 1)]. For s elec-

trons, a similar dipolar term yields

a = gg (3cos 8 1) [19
a = geBgiSN 3 91

where 0 is the angle between the magnetic field direction

and a line joining the two dipoles. However, the electron

is not localized and the angular term must be averaged over

the electron probability distribution function. For an

s orbital, all angles are equally probable due to the

spherical symmetry, and the average of cos2 over all 0

causes the function to vanish. Thus the classical dipolar

term cannot be responsible for the hyperfine structure of

the 2S1/2 hydrogen atom.

The actual interaction in the s-electron case is

described by the Fermi contact term (16). This isotropic

interaction represents the energy of the nuclear moment in

the magnetic field produced at the nucleus by electric cur-

rents associated with the spinning electron. Since only

s orbitals have finite electron density at the nucleus,

this interaction only occurs with s electrons. This yields

the isotropic hyperfine coupling constant

a = 7T 9g Io l2 [20)
as 3 e~ (0) [20]

where the last term represents the electron density at the

nucleus. This term has no classical analog. The a value
is proportional to the magnetic field at the nucleus, which

can be on the order of 105 G. Thus unpaired s electrons

can give very large hyperfine splitting.
E Molecules

The Spin Hamiltonian

The discussion to this point has involved only those

terms in the Hamiltonian of the free atom or ion which are


directly affected by the magnetic field. However, it will

be useful to begin the discussion of paramagnetic molecules

by consideration of the full Hamiltonian, which can be

written, in general,

H = + H LS+ S + SH + IH [21]
-E -LS -SI -SH -IH

Each of the terms can be described as follows. The term

H expresses the total kinetic energy of the electrons,
the coulombic attraction between the electrons and nuclei,

and the repulsions between the electrons:

2 2
P ZAe E e
H = -- + [22]
-E i 2m A,i r. i>j r [22
1 i]

where pi is the momentum of the ith electron, r. is the
i 1
distance from electron i to nucleus A of atomic number ZA,

and r.i is the distance between electrons i and j. The

Born-Oppenheimer approximation (17) has already been invoked

to separate out the nuclear motions and nuclear-nuclear

repulsions. This term yields the unperturbed electronic

levels before spin and orbital angular momentum are con-

sidered. Eigenvalues of this term are on the order of 105 cm-1

The energy due to the spin-orbit coupling interaction

is usually expressed in the form

H = XL-S [23]

where L and S are the orbital and spin angular momentum

operators and A is the molecular spin-orbit coupling

constant. The magnitude of this interaction is of the order
2 3 -1
of 10 10 cm .

The hyperfine interaction arising from the electronic

angular momentum and magnetic moment interacting with any

nuclear magnetic moment present in the molecule may be

expressed as

-SI I= gI Nge + 3
3 5
r r

S I 8T6 (r)S*-
+ [24]
r 3

This term corresponds to the sum of the isotropic and

anisotropic hyperfine interactions discussed for atoms

above. The Dirac 6-function indicates the isotropic

part which has a non-zero value only at the nucleus. The
-2 -1
hyperfine interaction has a magnitude of about 10 cm

The electronic Zeeman term, SH' is primarily responsible

for paramagnetism. It accounts for the interaction of the

spin and orbital angular moment of the electrons with

the external magnetic field, according to

H = B(L + g S)*H [25]
These energies are on the order of 1 cm
These energies are on the order of cm-1
These energies are on the order of 1 cm

The interaction of the nuclear moments with the

external field, that is, the nuclear Zeeman effect, is

given by

H =? Y Y H), [26]

-3 -1
and is usually too small (10 cm ) to be significant.

It is evident that this total Hamiltonian (which still

excludes higher-order terms and field effects which could

be observed in crystals) is very difficult to use in

calculations. However, experimental spin resonance data

obtained from the study of the lowest-lying levels can be

described by use of a spin Hamiltonian in a fairly simple

way which does not require detailed knowledge of all

the interactions. These levels are generally separated
by a few cm- (by the magnetic field), and all other

electronic states lie considerably higher. The behavior

of this group of levels in the spin system can be described

by such a spin Hamiltonian, and the splitting, which may

usually be calculated by first and second order perturba-

tion theory, are precisely the same as if one ignored the

orbital angular momentum and replaced its effect by an

anisotropic coupling between the spin and the external

magnetic field. This was first shown by Abragam and Pryce (18).

Just as the g factor becomes anisotropic and not necessarily

equal to g = 2.0023, the S cannot represent a true spin

but is actually an "effective" spin. This is defined, by

convention, to be a value such that the observed number

of levels equals (2S + 1), just as in a real spin multiple.

Thus we can relate all the magnetic properties of a system

to this effective spin by the spin Hamiltonian, since it

combines all of the terms in the general Hamiltonian which

are sensitive to spin. Nuclear spins can be treated in

the same manner, and the spin Hamiltonian corresponding to

Eq. [21] can be written (neglecting the nuclear Zeeman


H pin= 8H-S + I-A-S [27]
-spin 0

where the double underscore indicates a tensor quantity.

The g Tensor

As alluded to previously, the anisotropy of the g-tensor

arises from the orbital angular momentum of the electron

through spin-orbit coupling. Even in the case of E states,

which have zero orbital angular momentum, the interaction

of a presumably pure spin ground state with certain

excited states can admix a small amount of orbital angular

momentum into the ground state, and change the values of

the components of q. This interaction is usually inversely

proportional to the energy separation between the states.

The spin-orbit interaction can be described as (5)

H = AL*S = A[L S + L S + LS ]
LS -- -x-x -y-y -z-z


This term is added to the Zeeman term in the Hamiltonian,


H = H. (L + g S) + AL*S .[29]

For an orbitally non-degenerate ground state represented

by IG, MS>, the first order energy is given by the diagonal

matrix element

(1) = <,MSIg HzSz G,Ms> + ISIHz + ASz Ms , [30]

where the first term is the spin-only electronic Zeeman

effect. Because the ground state is orbitally non-degenerate,

<|ILzI > = 0, and the second term vanishes. The second-
order correction to each element in the Hamiltonian matrix

is given by

H Z|I2
!M M 1 [31]
s S n (0) (0)
n G

where the prime designates summationover all states except

the ground state. Since = 0, the matrix elements of

g98H.S will vanish. The operator matrix can then be
expanded to

HM M []
S1 S (0) (0)
n W W [32]
n G

and the quantity




(0) (0)
n G
n G




factored out where the ijth element of the

tensor is given by

= ij W"" H'0
n (0) (0)
w -
n G

= A [33]

second rank


where i and j are any of the cartesian coordinates. This

simplification yields

HMs = M (MS2i H-.A.H + 2AH*-A.S- + A 2 S-A-Ms' > .


The first operator represents a constant contribution to

the paramagnetism and need not be considered further.

The other terms represent operators which act only on

spin variables. When combined with the Zeeman term of

Eq. [30], the result is the spin Hamiltonian

H = B. (g 1 + 2AA)*S + X2S*A*S
-spin e

= BHg'S + S*D.S ,



S= g 1 + 2AA [37]


D = X2A [38]

with 1 being the unit tensor.

The S-D*S term is operative only in systems with S 1 1

and will be considered later. The other term in Eq. [36]

is the spin Hamiltonian in the absence of hyperfine inter-

action. It is evident that the anistropy of the g tensor

arises from the spin-orbit interaction due to the orbital

angular momentum of the electron. This may be expanded

to show the g-tensor as

BSgq-H = B[S S S I g g g g H [39]
-x-y-z xxxyxz x [39

gyx yy yz y

zxgzygzz z

where S S and S are the components of the effective spin

along the axes. Strictly, g is a 3x3 matrix and is referred

to as a symmetrical tensor of the second order (the symmetry

implies that the unpaired electrons are in a field of

central symmetry). The double subscripts on the g-tensor

elements may be interpreted as follows: gxy is the contri-

bution to g along the x-axis when the magnetic field is

applied along the y-axis. These axes are not necessarily

the principle directions of the g-tensor, but a suitable

rotation of the axes will diagonalize it; then the diagonal

components are the principle directions of the g-tensor

with respect to the molecule. It is noteworthy that, if

the molecule has axes of symmetry, they must coincide with

the principle axes of g; if it has symmetry planes, they

are perpendicular to the principle g axes.

Three cases of interest, with regard to molecular

symmetry can be outlined. If the system is truely a spin-

only system, g will be isotropic and the diagonal elements

equal to g If it is isotropic but contaminated with

orbital momentum, the principle components will be equal

but unequal to g e In the former case

-spin -x-y-z ex

H0 g H [40

= ge[Hx S + H S + H S ].
S x-x y-y z-z

For a system containing an n-fold axis of symmetry (n > 3),

two axes are equivalent. The unique axis is usually designated

z and the value of g for H II z is called g ". For H I z,

the value is gi. Thus

Hspin = (gHS + g HYS + g HS ).
-spin Ix-x I y-y z-z


Finally, for systems where there are no equivalent axes

(orthorhombic symmetry), gxx yy g 9zz and

H spin (g H S + g H S + g H S ). [42]
-spin xx x-x yy y-y zz z-z

The A Tensor

It has already been noted that the hyperfine inter-

action is composed of isotropic and anisotropic parts.

While the isotropic coupling is that which is observed

in liquids, the anisotropy due to dipole-dipole interac-

tions can be observed in fixed systems, such as molecules

in a rigid matrix. If Eq. [24] is expanded (and the L*I

term is dropped since this is a Z state), the interaction

can be seen to assume the form of a tensor

2 2
H r / rAr2 -3x \ /3xy\ -3xz \
-dip x-z \Sf Kr5 -[x
dip = (- NSxS Sz r x r 5 r

-3 r2 -3y2 /3yz [43]
r r r [43]

2/3xy\ 3y r23z2
3" \ (--r z z
r r r

= hS*T*I

When the isotropic part is added, the spin Hamiltonian

with hyperfine becomes

H spin= BSgH + hS.A*I



A = A 1 + T [45]

with A0 being the isotropic hyperfine coupling constant

and 1 the unit tensor. Thus the element

A.. = A. + T.. [461
13 iso 13

and, as with the g-tensor, certain cases can be selected due

to the molecular symmetry. An isotropic system has A =
A = A = A. Systems with axial symmetry are
yy zz iso
characterized by A = A = A = A. + T and Az
xx yy iso xx zz
All = Ais + Tz
iso zz
Randomly Oriented Molecules

Having discussed some basic theory of ESR in mole-

cUles, one must consider how these effects manifest them-

selves in the spectrum. Anisotropy will appear in the

spectrum of a rigidly-held molecule, but there is a

difference between the effects observed in crystals and

in matrices. In a single crystal, with a paramagnetic

ion or defect site, for example, the sample can be aligned

to the external field and spectra recorded at various angles

of the molecular axis to the field. In the matrix, the

samples ordinarily have a random orientation with respect

to the field, and the observed absorption will have contri-

butions from molecules at all angles. This was first

considered by Bleaney (19, 20), and later by others (21-26).

Solving the spin Hamiltonian,Eq. [42],in the ortho-

rombic case and assuming the g tensor to be diagonal, the

energy of the levels will be given by

2 2 2 2 .2 2 2 21/2
E = HH(gl sin Ocos2 + g2 sin Osinc + g cos2)/2

= BgHSHH [47]

where SH is the component of the spin vector S along H,

gH is the g value in the direction of H, 6 is the angle

between the molecular z axis and the field direction, and

{ is the angle from the x axis to the projection of the

field vector in the xy plane. For axial symmetry g
2 2 2 2 1/2
(g1 sin + g11 cos ) and the energy of the levels

is given by

2 2 2 2
E = BSHH(gl sin 6 + g 1 cos 6). [48]

Thus the splitting between energy levels, and therefore

transitions between them, are angularly dependent.

Consider first the case of axial symmetry. As a

measure of orientation, it is convenient to use the solid

angle subtended by a bounded area A on the surface of a

sphere of radius r. The solid angle is the ratio of the

surface area A to the total area of the sphere, that is,
S= A/4wr If all orientations of the molecular axis

are equally probable, the number of axes in a unit solid

angle is equal for all regions of the sphere. If the

sphere is in a magnetic field, the orientation of the axes

will be measured by their angle 6 relative to the field.

Taking a circular element of area for which the field

axis is the z direction, the area of the element is

27(r sin6)rd6, and the solid angle dQ it subtends is

2nr sind6
dQ --2 -- = 1/2 sin6d6, [49]

and if there are N0 molecules, the fraction in an angular

increment d8 is

dN = -- sinEd6. [50]

Assuming the transition probability is independent of

orientation, which is approximately the case, the absorption

intensity as a function of angle is proportional to the

number of molecules lying between 6 and 8 + dO.

Since g is a function of 6 for a fixed frequency v,

the resonant magnetic field is

hv 2 2 2. 2-1/2
H = (g2 cos 6 + g sin 2 [51]

and from this

s2 (g0H0/H) g
sin 6 = 0 2 [52]
g 2
gi 11

where go = (gl + 2gi)/3 and HO = hv/g0B.


2 2
-g0 H0O
sinede = H

2 ( 2 0 [o 2 2H -1/2
S) H. [53]

The intensity of absorption in a range of magnetic field

dH is proportional to

dN IdN Ie


where the two factors on the right are obtained from Eqs. [50]

and [53], respectively. From the above equations

H = hv/gll = g0Ho/gl at 6 = 00

H = hv/gB = g0H0/g9 at 0 = 900.



At these two extremes, the absorption intensity varies from

IdN N 091.1
2g H0(g11 2-g12

dN = m

at 0 = 0


at 6 = 90. [57]

Tf plotted against magnetic field, this absorption (after

considering natural linewidth) takes the appearance of

Figure 7a, for gll>gj. Since the first derivative is usually

observed in ESR work, the spectrum would appear as in

Figure 7b. If the g tensor is not too anisotropic, gll and

gi can be readily determined as indicated. In general,

the perpendicular component can easily be distinguished

in such a "powder pattern;" the weaker parallel peak is

often more difficult to detect.

If there is also a hyperfine interaction in the randomly

oriented molecules, the pattern of Figure 7b will be

split into (21 + 1) patterns, if one nucleus of spin I is

involved. Such a spectrum for I = 1/2 is shown in Figure 7c,

where, because gl1 % g, and Ai < Al the lines for m =+1/2

and -1/2 point in opposite directions. If instead, gl were

shifted up-field, relative to gll and A1 % All the spec-

trum would appear as two patterns similar to Figure 7b,

and separated by the hyperfine splitting.

The spin Hamiltonian for an axially symmetric molecule

is similar to that for an atom, but now incorporating

parallel and perpendicular components of g and A:

H sp = B[g H S + g (HS + H S )]
-spin II z-z x-x y-y
+ Ai S I + A (SxI + S I )
II -z-z -x-x -y-y

0 .0

*H -H rQ

0 0
>1> >

O-r -

S*-H --

(0 -M
O -- -
0 4J

t i rnv
M -.SH -

0 0
4) 4- U U

--I Q

>U 0a)
C4 tP

0 U C

> W r-1
0) >i-

O O (*
4-' >i U>
() r-4 (d

rl (t (L) -H

0 4 -4 Q)

0 O0)


t r


- 0-7

This omits the small nuclear Zeeman term and assumes the

symmetry axis is the z axis. Equation [51] can be rewritten

to include the nuclear hyperfine effect (8) to first

order as

hv K
H -8 g mI [59]


2 2 2 2 2
g = g cos 6 + g| sin 2 [60]


2 2 2 2 2 2 2 2
Kg = All gl cos + Ai g1 sin [61]

The intensity of absorption IdN/dHI can again be derived

and is found to be

22 2
dN 0 2cos6 (g0 _g )goH
'dH' 2 2
9 2g [62]

+ m g11 Al 2 g2 A12- K(g -g ) -1
+I -- -n-----9--*I -U^L
+ ml g
2K g

Here sinOd6 cannot be solved explicitly so that dN/dH cannot

be written as a function of only magnetic parameters.

Equations [59] and [62] must be solved for a series of

values 8 to obtain resonant fields and intensities as a

function of orientation. However,

H = (g0H0/gl ) (mIAll /Bgll ) at 0 = 00 [63a]

H = (g0H0/g ) = (miA /Bg) at 6 = 900 [63b]

and again IdN/dHI -- Thus a superposition of the

typical powder pattern results, with the relative phase

of the lines determined by the magnitudes of the magnetic

parameters, as discussed above. In Figure 7c, the lines

do not overlap and analysis is simple, but this is not

always the case. In general, the best approach is to

solve the given equations by computer for a trial set of

g and A values, and match the calculated spectrum to the


A similar treatment can be applied to molecules of

orthorombic symmetry and instead of the two turning points

at g11 and qg, there will be three corresponding to gl'

92, and g3. Such a spectrum is considered, including
hyperfine interaction with a spin 1/2 nucleus, by Atkins

and Symons (11) and Wertz and Bolton (5).

As mentioned in Chapter I, molecules trapped in

rare gas matrices do not always assume random orientations.

During condensation of a beam of reactive molecules in

solid neon or argon matrices, some preferential orientations

of the molecules relative to the flat sapphire rod may occur,

and in some cases the alignment can be extreme. This non-

random orientation can easily be detected by turning the

matrix in the magnetic field; a change in the ESR spectrum

indicates some degree of preferential orientation. The

degree of orientation appears to depend upon the size and

shape of the molecule, the properties of the matrix, and

other factors which are not completely understood (27).

Two examples are the molecules Cu(N03)2 (28) and BO (29).

The latter case shows very strong orientation such that

with the magnetic field perpendicular to the rod surface,

the parallel lines were strong and the perpendicular,

weak. With the rod parallel to the field, the perpendicu-

lar lines became very strong and the parallel components

disappeared entirely. This indicated that the BO molecules

were trapped with their molecular axes normal to the plane

of the condensing surface. This orientational behavior is

analagous to that usually observed in single-crystal work.

Molecular Parameters and the Observed Spectrum

Having discussed the nature of the interactions appearing

in the spin Hamiltonian and the form of the observed spectrum,

it is time to consider the relationships between the spectral

features and the paramagnetic species themselves. This will

begin with the exact solution to the spin Hamiltonian in

axial symmetry, and presentation of the second-order solutions

which are usually adequate, and conclude with the molecular

information revealed through g and A components.

A thorough discussion of the spin Hamiltonian

Hspin g 11H S + gi (H S + HIS )

+ A I S + A (I S + IS ) [64]
jII -z-z -x-x -y-y

has been given by several authors (8, 30, 31). Considering

the Zeeman term first, a transformation of axes is per-

formed to generate a new coordinate system x', y', and z',

with z' parallel to H. If the direction of H is taken

as the polar axis and 6 is the angle between z and H, then

y can be arbitrarily chosen to be perpendicular to H and

hence y = y'. Thus only x and z need to be transformed.

With H = Hsin9, H = HcosO, and H = 0
x z Y

H = 8[gl cos8Sz + glsinS x]H. [65]

Choosing the direction cosines z = g cos9/g and x =

gisin6/g, with g defined by Eq. [60], then

H = gg[ S + S ]H [66]

and the Zeeman term becomes

and the Zeeman term becomes

H = gS''H



S = S + S
-z z-z x-x

S = -. S + 2 S
-X X-Z Z-X

S = S
-y -y

For the hyperfine terms

hf = A S I + A (SII + S I ),
-hf | -z-z -x-x -y-y

and I is rotated by

I = nI nI
-Z z-z x-x

I = nI + .nI
-X X-Z Z-X

I = I '
-y -y

where the n. are the direction cosines for the nuclear
coordinate system relative to the electronic coordinates.

By inverting Eq. [68] to obtain the S. analogs of Eq. [70],

and substituting into the Hamiltonian of Eq. [69], the

Hamiltonian of Eq.[64] is transformed into








S= ghS + KI 'S + KI I 'S '
z -z -z K -x x

2 2
+ -A A sinecoseI S + A I'S [71]
K -z --x i-y -Y

with the definitions = A1 g1 cose/Kg, = A g sin9/Kg,
2 2 2 2 2 2 2 2
and K g = All g1 cos + A g1 sin 8. Dropping the primes

and using the ladder operators S = S + iS and S = S -iS
-x -y x -y,
this can be rewritten in the final form

2 2 + -
A -A2 g S+S S
Hspin signSz + KSz +kO -
S = gHSz + KS Iz [ + -I 2 cosesin -2 z

+ I AI + ) (s+I+ + S-I )
4K 4

+ ( A+ A (S+I + S-I) [72]
4K 4

This Hamiltonian matrix operates on the spin kets IMs,MI),

and can be solved for the energies at any angle. An

example of the use of this exact solution will be presented


The exact solution is difficult to solve at all angles

except e = 0, but elimination of some of the off-diagonal

elements (those not immediately adjacent to the diagonal)

results in some simplification and is usually adequate. The

solution is then correct to second order, and can be used

when gH>>A11 and A as is often the case. The general

second-order solution is given by Rollmann and Chan (32)

and by Bleaney (20) as

2 2 2
A All + K 2
AE(M,m) = g H + Km + [ 1 [I(I + 1) m ]
8G K

A 2
+ 4(- A)(2M l)m [73]

where K is A and A1 at 6 = 0 and 900, respectively, and

G = gH/2. Also, M is the electron spin quantum number of

the lower level in the transition, and m is the nuclear spin

quantum number. Note that the first two terms on the right

result from the diagonal matrix elements and yield equi-

distant hyperfine lines; this is the first order solution

given in Eq. [59]. The last two terms cause increasing

spacing of the hyperfine lines at higher field which is

referred to as a "second-order effect." This solution can

routinely be applied because the hyperfine energy is

usually not comparable to the Zeeman energy.

The g tensor has appeared in the derivation of the

spin Hamiltonian, and it is seen, from the above equations,

how the values of the principle axis components can be

determined from the ESR spectrum. Now we shall consider

its relationship to a molecular wavefunction in the linear

combination of atomic orbitals (LCAO-MO) approximation.

The usual form applied, Eq. [37], is a result of the

second-order perturbation treatment, yielding the first-

order corrections to the g components

<0 ILn) )
g g 6. 2E [74]
ij e 13 E
n n

where the primed summation is over all excited states n

which can couple to the ground state 0 and E is the energy

of that state above the ground state.. The Kronecker delta is 6...
Because the correction is caused by the spin-orbit inter-

action, only certain states can couple with the ground state.

Specifically, these are the states such that (33)

Fn e L 00 + A lg [75]

that is, the direct product of the irreducible representations

of the ground and excited state with the representation of

the angular momentum operator (which transforms as rotations)

must include the totally symmetric representation of the

symmetry group. A specific example, which will be encountered

later, is that a Z state can only mix with a H state. Usually

there will only be one such state of energy low enough

to make the term significant. The spin-orbit coupling

constant C can be assumed positive or negative, depending

on whether the excited state involves excitation of an

electron or a "hole," respectively. Both this constant

and the orbital angular momentum operator L can be written

as sums of atomic values:


ZL. = k [76a]
k1 k k

i = k [76b]

where k indicates a particular atom in the molecule. Actually,

,k decreases rapidly for large rk (ar-3) so that k is
essentially zero except near atom k, where it may be assumed

to have a fixed atomic value Xk. Thus Eq. [74], for the

perpendicular component of a E state diatomic molecule,


g9 = Z nikk Ix ( lkI) > [77]
n k,k'

Here, E and H are the LCAO wavefunctions

@ = Eaix(i) [78a]

j = Zb X(j) [78b]

where x(i) and X(j) are A.O.'s in the ground and excited

states, respectively. Then the second matrix element in

Eq. [77] will reduce, for a diatomic, to sums of terms

involving integrals of the type

(X(i)kl kxIx(J)k) (atom k only) [79a]

x(i)kIk'. Ix(j)k ) (both atoms) [79b]

(X(i)k l,' X()k> (both atoms) [79c]

(X(i)k' X(J)k,) (both atoms) [79d]

The first matrix element in Eq.[77] can be simplified if

Ak is assumed constant near k and zero elsewhere. Then

it becomes similar to the integrals of Eq. [79a].

The integrals in Eqs. [79c, d] require that the origin

of the operator kx be moved from atom k to atom k'. This

introduces a linear momentum term according to

x = k +-i RP [80]
k ky

where R is the interatomic distance. Fortunately, the

elements involving P are usually zero or small, so that

the term can be neglected and the integrals in Eqs. [79]

are then all of similar form.

Eq.[79a, b] involve the application of the angular

momentum operator to the atomic functions involved. The

non-zero elements have been tabulated (34) and are given


(PxLy lP> ) x y zy z z x-y

(d IL Id 2 2)

(dxz Lxdyz) = (dz2 LxIdyz) =- i [81c

(dxyl dyz) = (dyz Lz dzx) = i [81d]

(d xL xd y) = (dx2 2 L Id = (dx 2 21L d z)= i


(ilL qi') = (i' Lq i> .[81f]

Overlap integrals which appear because the A.O's are
centered on different atoms have also been tabulated (35, 36).
Detailed discussions of this approach can be found in
Stone (37) and Atkins and Jamieson (38).
Second-order corrections to the g-factor have been
determined by Tippins (39), utilizing third-order perturba-
tion theory. This degree of the theory must be used to
calculate corrections to gI and the result, analogous
to Eq. [74], is

( <(nl |L 10 2
gil = g 1/2 -- E-E [82]
n 0 -

Thus it can be seen that gll is always very close to or

less than g since the correction factor is small and

squared. On the other hand, gl can be greater or less

than g and the difference can be quite significant.

Thus, if wavefunctions are available, or can be

constructed, the values of Ag = g ge can be calculated

and compared to those determined experimentally. Examples

of this approach can be found in references (3, 5, 11, 40),

and in the discussion of the MnO molecule to follow.

Alternately, one can approximate the energy separation of

the lowest interacting level from the ground state.

The hyperfine coupling constant has been shown to

consist of isotropic and anisotropic parts, and its tensor

nature has been discussed. The spin Hamiltonian, with

hyperfine interaction, can be written as

H s = HII.g.S + aL-I + bI*S + cI S [83]
-spin -z-z


a = gegNn 3~ [84a]

Sc 2 s -1 84b

SN g N \ 2r
c = 3gg KN N 3 [84c]

where the angular brackets are, as usual, quantum mechanical

averages. This definition has been given by Frosch and

Foley (41). Neglecting the small L.I term, we can compare

Eq.[83] with Eq.[58] and identify the observed splitting


A = b + c [85a]

Ai =b [85b]

The isotropic part can be written in terms of the parallel

and perpendicular components as

A + 2A
A. II c 8rr 2
Aiso 3 =b + 3 8 NBN [86]

The anisotropic or dipolar component is given by

A -A 2
Adp = c 3cos 0-1 [87
dip 3 3 gePN N\ 2r3

Thus the observed spectrum is related to the fundamental

quantities |I(0) 2 and (3cos28-1/2r3) for interaction with

that nucleus. If the L*I term is included, the values become

All = b + c Agl a [88a]

AI = b + Ag a


where the Agi = gi ge have already been discussed. These

small corrections can be approximated from the observed

Agi value and the value (1/r3) for the particular nucleus.

The dipolar coupling constant can be considered

further. The dipolar part of the Hamiltonian (Eq.[171) can

be written as

g N (3 cos -1) SI [89a]

and the energies of the levels IMs,MI) are given by

3cos 8-1
E = geN MI r3 1). [89b]

For an electron in an orbital centered on the nucleus in

question, the anisotropic hyperfine coupling follows

Eq. [89], but has an additional term to represent the average

direction of the electron spin vector in the orbital.

The hyperfine splitting is the separation between adjacent

levels IMSMI) and IMs,MI1) and equals

A 3cos20- (3cos2-l) [901
dip =e 9NcoN 3 3s-) [90]

where a is the angle between r and the principal axis of

the orbital, and 0 is the angle between the latter and the

direction of the nuclear magnetic moment vector. The

value of (3cos2a-1) can be evaluated for the atomic orbital

functions; the values are 4/5 for any p orbital and 4/7, 2/7,

and -4/7 for the dz2, dxz,yz, and dx2y2 xy orbitals,

respectively. These quantities are used to give the principle

value of Adip for an orbital, that is A Atomic values of

the quantities A. and Ap have been evaluated for many
iso dip
atoms, and a useful table can be found in Ayscough (1) or

Goodman and Raynor (42).

Equation [90] assumes that the nuclear moment vector

PI is aligned with the external field, that is, the applied
field is much stronger than the field at the nucleus due

to the electron. This strong-field approximation is actually

valid only when (a) the applied field is large, (b) the

anisotropic coupling is small (and hence the field at the

nucleus small), and (c) the isotropic coupling is large

(since the field caused by the electron reinforces the

applied field). It is found that the field at the nucleus

due to the electron is much larger than the field at the

electron due to the nucleus. If the strong field approxi-

mation is not valid, the dipolar interaction will vary as
2 1/2
(3cos6+1)/2. However, the numerical values of the aniso-

tropic coupling at the turning points will be the same, and

only the signs will differ. In a crystal, the difference

will be discernable, but for the powder patterns obtained in

matrices, the strong-field analysis is sufficient. A more

thorough discussion can be found in references (5, 42).

The calculated atomic A. and A dip values are often
iso dip
used with experimentally determined molecular values to

derive coefficients or spin densities on a particular atom

in a molecule. Using an LCAO-MO wavefunction described as

= ZaiXi, the values of As. and Adip at a particular

nucleus x can be written

dip e= g NN (iI (3cos2 -1/2rx)1> [91a]

x 87r 9 Nlg2 [91b]
iso 3g e N Nl (0)x2 [91b

Since these integrals are expected to be small except near

atom x

Ax = ai(P d )2A (atom) [92a]
dip i x x dip

A = a.(s )2 A (atom) [92b]
1so I x iso

where the ai are the coefficients. Although this implies

that atomic properties remain unchanged in the molecule, which

is unlikely, it is quite useful in comparing trends to model

wavefunctions. Examples of its utility will be given later.

A more quantitative approach to calculating A tensor components,

based on the intermediate neglect of differential overlap

(INDO) molecular orbital approximation, has been developed


3 Molecules

A theorem due to Kramers states that, for all systems

with an odd number of electrons, at least a two-fold

degeneracy will exist which can only be removed by applica-

tion of a magnetic field. This would apply to cases with

S = 1/2, 3/2, 5/2, ...; the first has been considered in


In the triplet case, S = 1, and two non-interacting

electrons can be described by four configurations:

a(l)a(2), c(1)8(2), 8(1)a(2), and 8(1)B(2). In a molecule

of finite size, interactions will occur and the configurations

can be combined into states which are symmetric or anti-

symmetric to electron interchange. These states are

a(1)a(2) [93a]

(1/T ) [a(l)B(2) + 8(l)a(2)] (1/1) [a(l)8(2) B(1)a(2)]


8(1) P(2) [93c]

The multiplicity of the symmetric states (on the left) is

(2S + 1) = 3; this is a triplet state. Because of the Pauli

principle, this state may exist only if the two electrons

occupy different spatial orbitals.

For systems of two or more unpaired electrons, the

degeneracy of these spin states may be lifted even in the

absence of a magnetic field; this is termed the zero-field

splitting (ZFS). If the number of unpaired electrons is

even (S 1 i, 2, ...), the degeneracy may be completely

lifted in zero field. Additional terms in the Zeeman

Hamiltonian H = BH.g.S are required to account for this.

The Spin Hamiltonian

It was shown in the derivation of the spin Hamiltonian

(Eq.[36]) that the anisotropic part of the spin-orbit

coupling produces a term S*D*S which is operative only in

systems with S > 1. However, at small distances, two

unpaired electrons will experience a strong dipole-dipole

interaction, such as has been considered in the anisotropic

hyperfine interaction:

Sg2 S *S 3(S *r)(S *-r)
HSS 3 2 5 [* 94]
r r

In this equation, F is the vector connecting the electrons,

and the Si are the spin operators of the individual electrons i.

If the scalar products are expanded, and the Hamiltonian

expressed in terms of a total spin operator S = S + S2,
1 -2
2 2 2 2
and considering that r =x + y + z the matrix form of


the Hamiltonian can be written

2 2 2 -
S=/2) [ s s s r 3x -3xy -3xy
H =(1/2)g 8 S [ S ] --55 S
-SS -x-y--z 5 5 5 -x
r r r

2 2
-3xy r 3y -3yz S
5 5 5 -y
r r r

2 2
-3xz -3yz r 3z2
5 5 5 -z
r r r .

=S DS [95]

This term, representing the dipolar interaction of the

electron spins, should be compared to the last term of

Eq. [36]; the latter evolved from Eq. [31] by treating the

spin-orbit coupling interaction as a perturbation on the

Zeeman energy and assuming that the space and spin parts of

the electronic wavefunctions were separable. It can be

seen that the two terms are identical in form, except for

a numerical constant. These spin-orbit and spin-spin

contributions to D cannot be distinguished experimentally.

Whatever the origin of the interaction, the D tensor can

be diagonalized and the fine structure term becomes

2 2 2
H=D S + D S D S [96]
xx-x yy-y z z-z

where D + D + D = 0, that is, D is a traceless tensor.
xx yy zz
This can be written in terms of the total spin as

2 2 2
SD.S = D[S 2 -(1/3)S(S + 1)] + E(S S 2
z _x -y

+ C/3)(D + D + D ) S(S + 1), [97]
xx yy zz

where D = D (D + D )/2 and E = (D D )/2.
zz xx yy xx yy
The last term is a constant, proportional to the trace of D

which is zero for pure spin-spin interaction, and does not

appear in the spin Hamiltonian. The terms involving D

and E account for the removal of the degeneracy of the three

triplet wavefunctions in the absence of an external magnetic


While the effects of spin-spin and spin-orbit inter-

actions are not separable, some qualitative statements can

be made about them. It is found, in general, that organic

triplets show little influence from spin-orbit coupling.

The g values of such molecules are usually very close to

ge, and the zero field splitting can be ascribed almost
completely to spin-spin dipolar interactions. Then one

would expect the value of D to be approximately inversely

proportional to the molecular volume; this trend can be

seen in the methylene derivatives C6 H C C6H5 and
-1 -1
NC C CN, where D is 0.4 cm (47) and 1.0 cm (48),

respectively. Introduction of a heavy atom increases the

spin-orbit coupling (since XaZ4 ), thus D for CN2 is 1.16 cm-
(49) but for SiN2, discussed below, it is 2.28 cm1. If the

molecule involves transition metals, which usually have

large spin-orbit coupling constants, the zero field splitting

is due mainly to the spin-orbit interaction.

The solution to the general Hamiltonian for triplet

states of randomly oriented molecules is given by Wasserman,

Snyder, and Yager (50). Here, the special case of linear

molecules will be briefly discussed.

Neglecting hyperfine and other spin-orbit interactions,

the spin Hamiltonian for a linear triplet will be

spin = g9) Hz + g(HS + HS ) + D(Sz2 2/3) [98]

where z is the molecular axis. If y is chosen arbitrarily

to be perpendicular to the magnetic field, then H = 0 and

Hspin = g BH zS + g0(HS ) + D(S 2 2/3). [99]

Choosing as a basis the orthonormal spin wavefunctions

S+ 1) =I'aal) [100a]

1 0 (') =(l/-/l2B2 +1"2) [100b]

I 1) = 1 ,2) [100c]

and considering the effect of the spin operators on the func-

tions as

S a =(1/2) S x =(1/2),a S a =(1/2)iB S 8 =(-1/2)ia
-x x -y -y

2 2
S a =(1/2)a S z =(-1/2) S a =(1/4)a S 2B =(1/4)
-z -z -Z -Z



then the Hamiltonian matrix will be

I + 1)

D/3 + G

Gx /2

S0 )



-2/3 D

Gx/ 2

I 1)


D/3 Gz

where G

= g 8Hz and Gx = gjBHx.

The eigenvalues for

Hi z (Hz = H, H = 0) are

W+1 = D/3 + gl BH

W0 =(-2/3)D

W 1 = D/3 g 11 H.

1+ 1)

I 0o

I- 1)




At zero field, the + 1) and 1) states are degenerate,

and the appropriate wavefunctions are

T =(1/2 )(I + 1> I -1)) [104a]

T =(1/V2)(I + 1) + -1>) [104b]

Tz = 0) [104c]

In these wavefunctions the spins are quantized along the

X y, and z axes, respectively: S IT ) = S T y = S IT ) = 0.
S' -x x -yy -z z
The eigenvalues for Hilz are plotted against H in Figure 8,

and all yield straight lines.

For H z, H = H and H = 0, and the roots of the secular
Sx z
determinant are

W1 = D/3 [105a]

W2 = (1/2) D/3 + (D2 + 4g 2H2 ) /2] [105b]

W3 = (1/2)-D/3 (D2 +4g 2H2) 1/2 [105c]

with the eigenvectors

x = 1/i) + 1) )- [106a]



O o
0 -



4O "0


0 0

-- I4 -0

C~~j ,-I
4 ( --
-roD Q )4

4-) G)

-- ----4" AJI I II -- o 4
-- '-N

X~ N
I- co


py = cosa jt + 1) + ) + sinal0) [106b]

1 F 1
z = -sinea I + 1) + I- ) +cosaI0), [106c]

where tan 2a = 2gl3H/D. As H approaches zero, where a = 0,

these functions reduce to the functions T, Ty, and T
x z
given in Eq.[104]. The eigenvalues for H z are plotted versus

H in Figure 9, and only at high fields, where a-r/4,

do the lines become straight. In the intermediate region,

y and iz are mixed and lead to a curvature of the energy

with H.

It is evident that the energy levels, and hence the

fields at which transitions between them occur, are very

dependent upon the orientation of the axis of the molecule

with respect to the fixed magnetic field. This will,

in general, cause the spectra of randomly oriented molecules

to be broad and difficult to observe.

The relative transition probabilities are given by

2 2 2 2

hij k = g cos i yl( IS j)I [107]

where i, j, and k are any of the molecular axes (z is axial)

and y is the angle between the oscillating magnetic field

of the microwave radiation (perpendicular to the fixed

field) and k. Then transitions are allowed between levels

characterized by the following wavefunctions:


I Ll


N 0

1 I

3 WU





S r( N o N
+ f + + I I I








*H H


34 04
)rl *

r X


H 4
a) ^
cl nj
i0 r-

(1) T T z

(2) Ty T z
y z 2

(3) y < X xy2

(4) z -- X xYI'

where the symbols on the right are the usual designations

given to the observed ESR lines. These transitions are

indicated in Figures 8 and 9, and all correspond to AM = 1

transitions. Also indicated in Figure 9 is a dashed line

representing the forbidden AM = 2 transition. This transi-

tion (0y -c- 'z) is allowed for Hosc H, but is has a finite

transition probability (9) when H is not parallel to any

of the x, y, or z axes, even if Hos IH, as in the apparatus

employed here. The AM = 2 notation is acutally a misnomer,

since the spin functions I + 1) and 1), corresponding

to the infinite field Ms values, mix significantly at

finite fields. Thus M is not a good quantum number and the

transition could actually be described as AM = 0, since

the eigenfunctions each contain contributions from spin

functions of the same M. Thus if D is not too large, the

transition will be observable.

i. Employing the exact solution to the spin Hamiltonian

matrix for triplet molecules, which may be bent (E ; 0), the

resonant fields of the transitions are (50)


H = T [(hV-D) E ] 11
zl g 8

1 2 2 1/2
H [(hv+D) E21/
z2 g [I

H -
xl gjy

Hy2 gi

y2 = 9

H = -

[(hv-D+E) (hv+2E) 1/2

[(hv+D-E) (hv-2E)] 1/2

[(hv-D-E) (hv-2E) I1/2

[(hv+D+E)(hv+2E)] 1/2

2 2 2
1 (hv) D +3E2 1/2
gB 4 3

In Figure 10, the resonant fields of these transitions

for linear molecules (E = 0) are plotted as a function of

D, for a fixed microwave frequency of 9.1 GHz, where the

energy equals 0.3 cm1. The z and xy lines are so marked.

As can be seen from the Eqs.[108], the effect of a non-zero

E term in the spin Hamiltonian is to split each xy line into














Figure 10.

I 2 3 4 5 6 7 8
Hr (Kilogauss)
Resonant fields of a 3 molecule as a function
of the zero field splitting.