5
I NORGANICA CHI MICA ACTA
Volume 1, Number 1
J une 1967
Department of Chemistry, Michigan State University,
East Lansing, Michigan
Metallurgical Engineering, Tomkins-Johnson Industries,
jackson, Michigan
Department of Chemistry, Harvard University,
Cambridge, Massachussetts
Paramagnetic Relaxation in Inorganic Complexes. I.
Inhomogeneous Hyperfine Broadening
L. A. Dalton, L. L. Dalton and L R. Dalton
Received March 25, 1967
The electron paramagnetic resonance (EPR) spectra of
stabilized sodium, potassium and rubidium atoms in
benzene matrices at liquid nitrogen and liquid helium
temperatures were recorded. Sodium atoms, which
exhibit two distinct 2Wa spectra of different hyperfine
and g-tensor interactions, appear to be stabilized in a
substitutional position oj the benzene lattice and in an
interstitial site. Potassium and rubidium atoms, on
the other hand, are stabilized only in the substitutional
position. The linewidths of the· paramagnetic resonance spectra of the alkali metals showed a strong
dependence upon the nuclear spin quantum number m,.
A distribution of Zeeman and hyperfine interactions
over the alkali metal environments arising from crystal
defects is capable of qualitatively accounting for this
linewidth variation.
If the high-field approximation is
made and the distribution of Larmor precession times is
assumed to arise from a distribution of hyperfine interactions, the linewidths is predicted to vary as ml, while
if both a distribution of Zeeman and hyperfine frequencies is assumed operative, the linewidth is predicted to
depend upon both m 1 and ml. When the linewidth
expression is derived from the Breit-Rabi formulism, it
is found that either a distribution of Zeeman interactions, a distribution of hyperfine interactions, or a
combination of these interactions can give rise to linewidth variations which depend upon m, at least through
the fourth power. Because of the complexity of the
expressions involved, a quantitative test of the distribution models is not possible at this time. However,
by .studying the three alkali metals as a junction of
frequency (9.2-, 17.5-, 24.3~, and 35-GHz) and
temperature, we were able to realize that the linewidth
variations are qualitatively explained by a distribution
of hyperfine frequencies. The linewidth expression
derived from the high-field approximation suitably
explains the potassium linewidths; however, the expression derived from the Breit-Rabi formulism is
required to account for the linewidths of sodium and
rubidium.
Introduction
Until recently one of the most puzzling features of
the electron paramagnetic resonance (EPR) spectra of
many inorganic materials was the dependence of Jinewidth upon hyperfine component or the nuclear spin
quantum number m 1•
This phenomenon was first
observed in solutions of copper(.l I) complexes by
McGarvey} McConnell2 confirmed McGarvey's find·
ings and proceeded to explain the effect in terms of a
strong anisotropy of the electric field at the splitting
nucleus modulated by the .c tumbling» of paramagnetic
species in solution . This « McConnell mechanism ,.
which requires the tinewidth to be dependent upon
both viscosity and frequency has successfully explained
the EPR spectra of several first- and second-row trans·
ition metal ions.3• 10
(I) B. R. McGarvey. / . Phys. Chem .• tiC. 71 ( 1956); 6/, 1232 (1957).
M. McConnell. f . Chem. Phys.• 25. 709 (1956).
E. o•Rellley. / . Chem. Phys.• 29. 1188 (1958).
N. Rogers and G . E. Pake. / . Chem. Phys., JJ. 1107 (1960).
J. Marriage. Australian / . Chem .• 18, 463 (1965).
C . Rasmu:;sen. H. A. Kuska and C . H. Brubaker Jr., /nora.
Chem .• 4. l4l (1965).
(7) D. A. McCiu!li. L . R. OaUo n and C . H. Brubaker Jr .• / nora.
Chem .. s. 1985 (1966); l . A. Dalton. l. R. Dalton. J. l . Dye and C. H.
Brubake r Jr.• /. Cltem. Phys.• 10 be published.
(8) W. B: Lewis, M. Aiel Jr. and L. 0. Mo111an, / . Chem. Phys .• 45.
(2)
(3)
(4)
(5)
(6)
H.
D.
R.
A.
P.
4003 (1966).
(9) F. A. Walker. R. l . Carlin and P. H. Rieger, /. Chem. Phys. . 45.
4181 (1961>).
(10) 0 . C. McCain and R. J. Myers. / . Phs Chem., 71. 192 (1967).
Dalton, Dalton, Dalton
I
Paramagnetic Relaxation in lnorgUitic Compl exes
6
Several refinements of the original treatment of
McConnell h ave been made. 11 • 12 Recently, Kivelson and
co-workers 13 have modified the treatment for the case
of large hyperfine interaction by including second-order
elfects and have given expressions for the dependence o[
Jinewidth upon m. through third order (see Table 1).
Coefficients of m, in the Linewidth Expression from
the Kivelson Mechanism for the Time-Dependent Modulation
o f Anisotropic Hyperfine and g-Tensor Interactions.
Table I.
spin-dependent relaxation in alkali metal-amine solutions in terms of the model developed in Ref. 16. Most
recently Catterall, Symons and Tipping17 have interpreted similar studies of metal-amine solutions in
terms of the chemical exchange model uf Dye and
Dalton. Walker, Carlin and Rieger have found a
combination of the Kivelson mechanism and chemical
exchange mechanism to be capable of accounting for
the effects of addition of bases such as pyridine to
solutions of vanadyl acetylacetonate.
Table II.
a
Coefficients of m, in the Linewidth Expression
Employing the Two-State Exchange Model fo r Time-Dependent
Modulation.
= (1/15}(~g/g) Wo 'T-(4/4 + u)
f3 =
1
1
+ (7 /90)b'l(l+ 1 )'To[3/7 + u- (a/Wo)uf]
(2/15}(~g/g)bWo'To(4/3+u)
T,-'(m,)
= p./TlA+ PafT,.+p,'p.'(-.,+-.a) X
(a.+~m 1 +ym 1 ' +1im 1'+Em,')
- ( I /45)b'l(l +
i)'T-(a/Wo)(l + u + 7uf)
- (I / 45)b''T-(a/Wo)(3/2 + u)
- (2/ 15)(~g/g} 'To(a/Wo)(4/3 + u +u()
y == ( l/90)b2"to[ 5-u + 5(a/Wo)uf)
- ( l /45)(~g/g)b"T.(a/Wo)(l 4 +5u + 12uf)
o = (l/45)b'-;.(a/Wo)(l + u +uf)
I(I+ I) F,)
[F1 + lf2 F. -I(l+ t) F,]'-2F,( F,+FJ(I+ I))
o 2F,(F,+FJ(I+I)]-2F,(F,+If2F.-I(l +l)F,]
t = 2F,(fz+ lflF,- 1(1+ I)F,] + F/
where
where
1
a = ( F,+FJ(l + l)]'
(3
y
= 2(F,+ F,I(I+I)]((F,+lf2F,=
=
= (w,•--wa•)
~g
F,
b
-..
F, = -(a.- aal
F, = -a.'/2w. +aa'/ 2wo
F,
-a,'/2w/+as'/2W32
811-Bl.
all-a.!.
the rotational correlation time for tumbling of the
molecule in solution
== nuclear spin
Wo
resonant frequency
U
(I +Wo1'To1) · '
f == w'o'To'u
=
An alternate mechanism which could also result in
a dependence of linewidth upon m1 involves the rapid
interconversion of two paramagnetic species having
For exchange slower
different contact densities}•
than the exchange-narrowed limit. this leads to an
expression for linewidths14
in which aA and aa are the time-independent hyperfine
splitting frequencies (in radians per second) of the two
species undergoing exchange, PA and pa are the respective fractions of these species, T2A and T 28 are the
time-independent relaxation times of A and B separately, and 'tA and 'to are the mean lifetimes of the two
species. Althoug Eq. 1 shows dependence only upon
ml, the high-field approximation was made, the
assumption of Lorentzian line shape was used, and it
was assumed that A and B have the same g-values.
Dye and Dalton 16 have recently generalized Eq. I for
the case of species A and B having different g-values
ahd large hyperfine splitting constants. The result is
given in Table II and shows a dependence through m,•.
Dye and Dalton have attempted to interpret nuclear
( I I) D . Kivelson, /. Chem. Phys., 27. 1087 (1957)
(1 2) D . Klve1son, / . Chem. Phys., JJ. 1094 (1960).
(11) R . Wilson and D . Kivelson. ]. Chem . Phys., 44. 154 (1966); 44,
4440 ( 1966); 44, 4«5 (1966); P. Atkins and D. Klvelson, f. Chem. Phys. ,
44. 169 (1966).
( 14) L•. R. Dallon, J. 0. Rynbrandt, F.. M. Han~n and J. L . D~e.
/ . Chmt. Phys., 44. 3969 (1966).
(15) F. Bloch, /'hys. Rev., 102, 104 ( 1956).
( 16) J. L . Dye and L . R. Dalton. / . Phys. Chem., 71. 184 (1967).
lnorganica Chimica A cta
I
1:1
I
June, 1967
=
w~· and wa• ~>re 1he th•n~ition frequencies of species A and B,
respectively, in the absence of hyperfine interaction.
a. and a. are the hyperfine splittings of species A and B,
respectively.
The formal similarity between the chemical exchange
model and the Kivelson mechanism for an axially
symmetric system should be noted. Both treatments
involve the time-dependent modulation of two magnetically nonequivalent states. Therefore, distinction
between these models requires correlation with viscosity
and frequency and examination of the pattern in frozen
solutions.
Both mechanisms involve the Lorentzian line shape
assumption (i.e., the rate of modulation is sufficiently
greater than the reciprocal of the hyperfme frequency).
If this approximation is not valid, (i.e., if the observed
line shapes are Gaussian) then one must resort to the
rather laborious computer evaluation of the rigorous
linewidth expressions.
lt should also be noted that
neither the expressions of Kivelson and co-workers13 nor
Dye and Dalton16 are valid in the limit of very large
hyperfine interactions or very small applied magnetic
fields. In such cases perturbation treatments must be
abandoned altogether and the linewidth ex.pressions
must be derived in the framework of the Breit-Rabi
formulism. 13
. Dye and Da lton'~ also pointed out that another posstble source of the dependence of linewidth upon m 1
requires a species whose hyperfine splitting is strongly
dependent upon the solvent enviroment around the
splitting nucleus so that a distribution of hyperfine
(!7) R . Catterall, ~- C. R. Sym ons and J. W. T ipping, priv ate com ·
mumcatlon to be pubhshed .
(18) G. Breil and I. I. Rabi. Phys. Rev.• 38, 2082 (1931).
7
contact densities exists. Once again, approximate
trcntaments yield dependence only upon powers of m.
In this paper we present the first
through m12•
experimental observation of the operation of the
distribution model. The distribution of contact den·
sities reported here results from the trapping of alkali
metal atoms in molecular matrices of benzene at 77"K.
The time-dependent Kivelson mechanism and exchange
mechanism cannot explain this m1 dependence while,
on the other hand, the environment-dependent distri·
bution model is an obvious choice. A theoretical
consideration of relaxation theory as applied to electron
nuclear double resonance (ENDOR) suggests a method
for mapping the distribution of contact densities and
thus of testing the distribution-dependent paramagnetic
relaxation theories of Dye and Dalton 16 and those
developed in this paper.
Experimental Section
The two basic electron paramagnetic resonance
spectrometers employed in this work were a Varian
V-4500-toA X-band spectrometer and a Varian V-4503
35-GHz spectrometer. Both crystal and superheterodyne detection were available. A range of modulation
frequencies from 20 cps to 100 kc/sec were employed.
Measurements at 24.3 GHz were obtained using a
Varian 24.3-GHz klystron and power supply. For
measurements at 17.5 GHz a homemade klystron and
power supply were used.
Measurements were made over a temperature range
from 4.2"K to too·K. EPR measurements at 4.2"K and
n•K were made using modified V-4545A and V-4546
liquid helium and liquid nitrogen Dewar accessories
respectively. For measurements between 77"K and
too· K. temperature regulation was achieved using a
modified V-4540 variable temperature control unit.
Temperatures between 60•K and n ·K were obtained by
varying tne rate of pump-off of nitrogen from the Dewar
system. Temperatures were measured to within ±0. 1•K
during the EPR measurement with a copper-constantan
thermocouple attached to the EPR cell.
The calculation of relaxation parameters and magnetic
structural parameters from EPR spectra has been
described elsewhere}4.16
Since an intensive study of relaxation processes was
made, detailed analyses of the effect of modulation
amplitudes and frequencies, power, and passage con·
ditions were performed. In general, microwave power
of about 200 m W or less incident upon the cavity was
used. Modulation amplitudes did not exceed 0.5 gauss
and slow passage conditions were employed.
The field stability of our spectrometers allowed the
use of a Varian C-1024 time ave.raging computer in
studying very weak signals.
The benzene used in this experiment was purified and
degassed by a repetitive freeze-pump-melt technique,
brought into contact with drying agents. and subsequently distilled into a vessel which allowed sweeping
with pre-purified nitrogen.
A 3600 Control Data Computer wa!> employed to
synthesize the spectra and to perform the calculations
indicated in this paper.
The samples containing stabilized alkali metal atoms
were prepared in the spectrometer cavity by the
simultaneous condensation of a beam of atoms and
benzene vapor on a surface cooled with liquid nitrogen.
The rubidium and cesium atoms were obtained from
the bromides of these metals mixed with calcium and
heated to 873•K. The lithium, sodium, and potassium
atoms were obtained by evaporating the pure metals.
During evaporation, the nitrogen in th e operating
Dewar was pumped down to freezing on the surface to
lower the temperature of the substrate, which made the
capture process more efficient, and increased the rigidity
of the matrix. Using this method, it was found possible
to stabilize sodium, potassium, and rubidium atoms in
a matrix of benzene. In the most successful experiments
the number of stabilized sodium and potassium atoms
was about 1015, and the number of rubidium atoms was
3 X 1015 • Assumil)g uniform distribution over the
volume of the sample, the concentration of stabilized
atoms was of the order of 1017 cm-3 •
Attempts to stabilize lithium and cesium atoms in
benzene were unsuccessful, which is probably due in
the case of lithium to diffusion, and in the case of
cesium possibly to a reaction with the material in the
matrix.
Because of the large hyperfine con tact
densities, the results were analyzed by using the Breit·
Rabi formulism}~
Results and Discussion
X -Band EPR. Sodium.
The paramagnetic resonance spectrum of sodium atoms in benzene shown in
Figure I demonstrates that two trapping sites are
present. The observed magnetic parameters are sum·
marized in Table Ill. The linewidths of both patterns
A and B show a dependence upon the nuclear spin
quantum number mt.
Figure I. X-band ElecJron Paramagnetic Resonance Spectrum
of Sodium Atoms Stabilized in Benzene Matrices at n•K-
Dalton, Dalton, Dalton
I
Paramagnetic Relaxation in Inorganic Complexes
8
Table Ill.
Structural and Relaxation Parameters for Gaseous Alkali Metal Atoms and for Alkali Metal Atoms Stabilized in
Benzene Matrices
Atom
% Natural Abundance
" Na
,,K
3/2
3/2
5/2
3/2
Rb
" Rb
35
100
93.1
72.8
27.2
Atom
g, trapped atom
a, trapped atom,
, Na(A)
" Na(B)
2.0029
2.0036
2 .0024
2.0046
2.0046
±
±
±
±
±
1438.8
1615.9
363.4
2450.4
4982.5
"K
35
Rb
" Rb
0.0005
0.0005
0.0007
0.0010
0.0020
m1
= +5/2
22.9 ± 0.5
m1 = +3/2
7.86
7.98
5.98
15.4
1771.75
461.75
3035.74
6834.10
% free atom splitting
Relative hfs sh.ift
81.2
9 1.2
78.7
80.7
80.7
- 18.8
- 8.8
-21.3
-19.3
- 19.3
±
5.0
5.0
± 1.0
± 10.0
± 20.0
m1 = +3/2
3.89
11.3
11.2
14.9
5.78
"K
Data valid over a temperature range (rom 60 to n•K.
impossible.
a
5.50 ± 0.5
4.59 ± 0.5
I
I: I
mr = - 1/2
b
3.55 ± 1.0
b
b
b
9.51 ± 0.5
16.2
19.4
h
/ June, 1967
b
25.1
b
b
b
9.37
m, = -3/2
7.02
7.00
5.98
8.62
± 0.5
± 0.5
± 0.5
± 0.5
ffil
= -5/.2
16.0 ± 05
m, = - 3/ 2
m, = -5/2
12.7
12.7
14.9
10.3
5.57
Overlap with the radical absorption made linewidth measurement .
The letter A designates the capture location which
occurs more often than the alternate capture location B.
In some experiments, stabilization at the location B did
not occur and the spectrum consisted of the lines of
spectrum A alone. However, spectrum B never occurred without spectrum A. T he lines corresponding to
the transition {1,- 1~2 ,0) are somewhat masked by the
wing of the spectrum of the free radicals formed during
condensation of the sample. As can be seen from
Figure 1, this spectrum is several orders of magnitude
more intense than the sodium spectrum.
Periodic recordings while raising the temperature
showed that from 60•K up to too·K the spectrum did
not exhibit any changes. This condition was found to
persist over the period of an hour at 100•K. When
the temperature was raised above too·K, the spectra
disappeared, with the lines corresponding to the capture
location B disappearing at a higher temperature than the
A lines. No detailed study of the destruction kinetics
was made, but since the spectra A and B disappeared at
different temperatures, the following conclusions can
be drawn: I) spectra A and B are due to the existence
of different capture locations of differing structural
characteristics with respect to the captured atoms in the
matrix, and 2) the stabilized sodium atoms disappear
with an increase in temperature as a result of diffusion,
rather than by reaction with the material in the matrix
since the start of diffusion depends upon the type of
capture location, while a reaction would start at the
same temperature in different capture locations.
lnorganica Chimica Acta
+1/2
Spin-Spin Relaxation T ime. T,, (sec X 10 ·•>•
m, = + 1/2
m, = - 1/ 2
m, = +5/2
,Na(A)
uNa( B)
Rb
2.00231
2.00231
2.00241
2.00241
±
m1 =
± 0.5
± 0.5
± 0.5
± 0.5
Atom
35
a, free atom, (Me)
Linewidths, 6-H(m,), (gauss)•
Atom
"Na(A)
»Na(B)
l<K
•' Rb
(Me)
g, free atom
The electron paramagnetic resonance linewidths of
sodium atoms measured between the extrema of the
derivative of the absorption line are given in Table II I.
The dependence upon m, is clearly seen.
As the
temperature was raised above too•K, the lines narrowed
appreciably. This may be due either to unfreezing the
motions of the benzene molecules, or to changes in the
structure of the matrix of the nature of phase transitions.
Potassium. Potassium has two reasonably abundant,
naturally-occurring isotopes: 39K with nuclear spin
I = 3/2 and abundance 93 % and 41K with l = 3/2
and abundance 7% . The spectra from the 41 K nuclei
were obscured by the free radical absorption, hence
only the data for the 39K nuclei are presented in Table
[[J.
The very intense radical spectrum near the free
e lectron value (gc = 2.002.3 ) made recording of the
(1,0<-+2,1) and ( 1.- 1<-+2,0) transitions very difficult.
The m,
+ 3/2 (I, I <-+2,2) and m,
- 3/ 2
(2,-2<-+2,-1) lines were however recorded very clearly
and were used to calculate the hyperfine splittings and
g-values given in Table lJ I.
The potassium atoms observed in our experiments
were strongly stabilized at only one major site. The
spectrum of the stabilized atoms d isappeared in about
10 minutes when the temperature was raised from 60•K
to n·K.
The interference of the free radical spectrum
prevented an analysis of the nuclear spin dependence
of the linewidths for this case.
=
=
9
Rubidium. A total of ten hyperfine lines were
observed for rubidium atoms stabilized in benzene. Six
strong lines arising from paramagnetic interaction with
85
Rb nuclei (abundance 73%) and four weak lines
arising from interaction with 87 Rb nuclei (abundance
27°/b) account for the stabilized atom spectra.
The
data for the two isotopes are given in Table Ill. The
weak intensity of the 87 Rb lines together with the
extreme broadening of the (1 ,1 ~2,2) and (2,-2-2,-1)
transitions made linewidth analysis of the 87Rb spectra
impossible .
In addition to the spectra from the stabilized rubidium
atoms a free radical spectrum near (g = 2.0023 was
observed.
The spectra of the rubidium atoms disappeared
slowly even with contihuous pump-off of the nitrogen
in the working Dewar (the intensity decreased by a
factor of two in 30 minutes). The spectra disappeared
several minutes after pump-off of the nitrogen was
stopped.
The linewidths given in Table Ill clearly indicate a
linewidth dependence upon m1.
It is particularly
important to note that odd as well as even powers of
m1 are important.
EPR at higher frequencies. The potassium linewidths at 17 .5, 24.3,and 35 GHz were identical to
those at 9.2 GHz within the experimental accuracy of
our instruments. For both sodium and rubidium the
EPR patterns became more symmetrical at the higher
frequencies, indicating that the coefficient of the m1 2
term has become relatively more important at these
frequencies.
Helium temperatures. The spectra of the alkali
metals in benzene at 4.2°K exhibit essentially the same
magnetic parameters and nuclear spin-dependence as at
770K.
Further studies of alkali metal atoms. The spectra
of the alkali metals in the gas phase were recorded at
temperatures of 650°K and above.
Within experimental error the linewidths of an the hyperfine components were found to be equal. Hence the linewidth
variations of the alkali metal atoms stabilized in
benzene matrices is a function of the alkali metal-matrix
interaction.
The structure of the benzene matrix.
The following
facts lend support to the assumption that the benzene
matrix is largely crystalline: 1) The deposit builds up
in the direction of the temperature gradient (i.e., along
the normal to the surface of the semicircular bottom of
the Dewar on which the sample is being condensed), and
has a granular structure upon cleavage, which may
indicate that the sample is polycrystalline, with the
grains oriented in the direction of the temperature
gradient. 2) The results of an X-ray structure analysis
of deposits of various substances obtained by freezing
out vapors at a rather high condensation Tatt:19 imlil:ate
that for the majority of substances the deposits are
crystalline, and 3) the occurrence of different capture
locations in benzene, and the narrowness of the lines
agree with the assumption that the atoms are stabilized
in a crystalline phase.
Benzene crystallizes in the space group D2h o f the
orthorhombic crystal system (face centered unit cell).19
There are four translationally inequivalent molecules
per unit cell located at (0,0,0), (1/2, Vz ,.()), (O,I/2, 1/2 ), and
( lh ,0, 1/:z ) in the crystal axis system. These molecules
are referred to as I , II, III, and IV. The molecules
II , III, and IV are generated from I by screw-axis
transformations parallel to the a, b, and c axes, respectively. At 77oK the lattice constants are
a
= 7.277 A.
b
= 9.452 A,
c = 6.728 A
and the direction cosines of the three molecular axes
of molecule I are
L=
-0.276
0.964
-0.030
'
M --
0.654
0.168 , N =
-0.754
0.709
-0.224
0.668
where L is the axis in the molecular plane which passes
through atoms, M is the in-plane axis which passes
through bonds, and N is the axis perpendicular to the
plane.
Other matrices.
The alkali metal atoms have
previously been investigated in matrices of the noble
gases at helium temperatures.20 Although no attempt
to explain the phenomenon was made, it appears that
nuclear spin-dependent linewidths were also obseTved
in these materials.
We have unsuccessfully attempted to stabilize alkali
metal atoms in a number of aliphatic and alicyclic
hydrocarbons at 77°K.
The reasons for the lack of
stabilization in these materials are not clear.
An attempt has also been made to stabilize alkali
metal atoms in a number of aliphatic alcohols.
For
these cases the reaction tO form the alkali metal alkoxide
appears to be sufficiently fast to prevent the formation
of any appreciable concentration of stabilized atoms.
In view of the existence of reasonably stable alkali
metal monomers in solutions of the amines, diamines,
and aminoethers, 14• 16 a logical extension of the present
work would be to study to· alkali metals in matrices of
these materials at n·K. Such a study is presently
underway.
Conclusion
The nature of the stabilization sites. The Van der
Waals half-thickness of the benzene molecule is 1.85 A.
The dimensions of the sodium, potassium, and rubidium
atoms are such that they are found in capture locations
where separation of the lattice has occurred.
lt is
evidently impossible to take account of this lattice
separation in theory20 • 21 so that the hydrogen and carbon
atoms forming the benzene molecule are at various
distances from the stabilized atom.
It may be assumed that the sodium atoms, for which
two types of capture locations occur, are stabilized in
a substitutional position of the benzene lattice, and in
(19) G. C. Pimentel • • Formation and Trapping_ of Free Radicals•.
edited by A. M. !loss and H. P. Broida (Academic Press, Inc .. New York.
1960). p. 73; V. L. Bronde. Soviet Phys.-Usp., 4, 584 (1961); Usp. Flz.
Nauk. 74. 577 (1961); H. C . Wolf, Solid State Phys•• 9. I (1959).
(20) C. K. len. V. A. Bowers. E. L. Cochran and S. N. Foner. Phys.
Rev .• 126, 1749 (1962); I. P. Goldsborough and T . R. Koehler, Phys. Rev ..
/JJ. 135 A (1964).
(2 1)
F. ). Adrian, /. Chem. Phys., 32. 9T2 (1960).
Dalton, Dalton, Dalton
I
Paramagnetic Relaxation in Inorganic Complexes
10
an interstitial site, in the center of a somewhat distorted
octahedron, while the potassium and rubidium atoms,
which are of larger size than the sodium atoms, are only
stabilized in the substitutional position, where there is
more room than in the octahedral position.
An important result of this work to which we will
now turn our attention is the large negative shifts in the
hyperfine structure splittings. We have previously
observed large negative shifts of the alkali metals in
solutions of amines, diamines, aminoethers, and
ethers 14• 16• 22 but in these cases strong ion-dipole and
dipole-dipole interactions are expected and indeed are
indicated by increasing spin-orbit coupling as one
considers the alkali metals from sodium to cesium. In
these cases it is doubtful that the paramagnetic electron
can be described as residing in a pure 2S., state. Indeed
a much better fit of experimental data is obtained by
mixing excited state wavefunctions into the ground
state wavefunction.22
Within the limits of experimental accuracy, the
g-factors of alkali metal atoms stabilized in benzene
matrices are reasonably close to the values for free
atoms, (or at any rate greater than the free atom values)
which shows that the spherical symmetry in the external
electron cloud, i.e., the 2S ., state of the atom, is
maintained.
In view of the g-factor results the large negative
hyperfine splitting shifts are of great interest from a
theoretical standpoint 20· 21 and indicate that Van der
Waals forces predominate in the interaction between
the captured atom and the matrix. The fact that no
monotonic character is observed in the matrix shifts
upon progressing from sodium to potassium and from
potassium to rubidium is not inconsistent theoretical
concepts,20· 21 since the increa:;e in negativt: shift resulting from an increase in the polarizabilities in the order
sodium, potassium. rubidium may be compensated by
the positive contribution to the matrix shift, which
increases with increasing size of the atom in the series
of alkali metals.
The nature of the nuclear spin dependence. Clearly
neither of the previous mechanisms 13· 16 proposed to
explain nuclear spin-dependent linewidths can be
operative in this case while a distribution of hyperfine
interactions appears to be an obvious choice for consideration. We view this distribution as arising from
crystal defects which cause the trapped alkali metal
atoms to experience a range of environments and interactions.
Theoretically such a problem can be treated by use
of either the Bloch 15 or relaxation matrix formulism.
However, for a time-independent distribution the Bloch
formulism is clearly preferable.
Stated most simply, the problem is to determine how
the line shape functions of the various hyperfine components vary with [w(m,)];; where w(m 1); is an individual Larmor frequency corresponding to a particular
hyperfine component and a particular matrix environment.
(22) f. L. Dye, L. R. Dalton, E. M. Hansen. Abstract, !49th National
Meeting of the American Chemical Soclely. Detroit . Michigan. April. 1965,
p. 455; L. R. Dalton, J. D . Rynbrandl , V. A. Nicely and J. L . Dye.
Abstract. I53rd National Meeting or the American Chemical Society.
Miami, Florida. April, 1967; L. R . Dalton. J.D. Rynbrandt, V. A. Nicely
and ) . l. Dye. f. Chern . Phys .• to be published.
Jnorganica Chimica Acta
I
1:1
·1
June, 1967
If we assume the high-field approximation and that.
the Zeeman interactions in all environments are equal
we obtain an expression for the linewidths16
(2)
whe:re j(a;) is the hyperfine distribution function and
depends upon the functional nature of the distribution.
If we assume that a distribution of Zeeman interactions as well as hyperfine interactions exists and retain
the high-field approximation we obtain
where j(w;) is the Zeeman distribution function.
For the case of large hyperfine interaction the highfield approximation is no longer valid and hence the
derivation of the linewidth expression must be based
upon the Breit-Rabi formulism. Since it is desirable to
have an explicit expression for the frequencies of the
transition involving the hyperfine component m1 we
approximate the Breit-Rabi expression by employing a
power series expansion obtaining
[ w(mi)]; =
-
0
tu;
-aimi-{a// 2w;)[ I( [ + 1)-m12 ]
(aN 2w,2) jm,[m.2 + Yz-I(I+
OJ!
(4)
Employing this equation, we arrive at the linewidth
expression given in Table IV.
Table IV.
Coefficients of m, in the Linewidth Expression
Ob1 a ined Using the Distribution Model
a = [J,+ J,T(l + I)]'
f3 = 2(J, +J,l<l+l)](h+ V2),-I(l+I)J.]
y = [h+ 112J,-l(I + !)J,]'-2J,(J,+J,l(l+ ll]
o = 2),(1,+ JJ<l+ 1)]- 2J,[J,+ V2J.-l(l+ 1)1.]
E = 2J.(J,+ 1/2 ),-f(I + l)j,) +J/
where
J, = j(w,)
J, =
)J
-j(a,)
= -j(a,'/2w,)
), = -j(a;'/2w.')
Unfortunately. to use the expression given in Table
IV for quantitative calculations one needs to know the
distribution of Zeeman frequencies, the distribution of
hyperfine frequencies, and the sign of the hyperfine
interaction. Since at this time we have no method for
determination of these parameters we make no attempt
to quantitatively evaluate the linewidths using this expression at this time.
The predominance of the coefficient of the m 12 term in
the linewidth variation in the sodium and potassium
spectra and the increasing relative importance of the
coefficient of the m12 term at the higher frequencies
indicate that the distribution of hyperfine interactions
is largely responsible for the Jinewidth variation·. Using
11
the assumption of a Gaussian distribution of hyperfine
frequencies and the expression derived using the highfield approximation, we estimate the half-width of the
frequency variation in the sodium atom environments
to be 3.0± 1.5 Me. Since at X-band a dependence
upon the odd power of m, is evident, the K-band results
were used in obtaining the above value. Moreover,
the slight contribution from odd powers of m1 evident
in the K-band results was subtracted by first performing
a computer evaluation of coefficients of the various
powers of m, by the technique described previously ! 6
The temperature insensitivity of the nuclear spindependent variation of the linewidths of the alkali metal
atoms up to 100•K is consistent with the distribution
model. The appreciable narrowing of the linewidths
in the sodium spectra above I OOOK can be understood
to result from the unfreezing motions of the benzene
molecules or from rapid modulations of the structure of
the matrices.
Evaluation of hyperfine frequency distributions by
ENDOR. We shall present a detailed analysis of the
effect of electron and nuclear magnetic relaxation
processes upon ENDOR spectra in the second paper
of the present series. For the present it is sufficient to
note that an ENDOR absorption will appear for every
hyperfine interaction. Thus the distribution of hyperfine frequencies will be mapped out in the ENDOR
spectra.
It should also be noted that ENDOR is an excellent
tool for determining long-range hyperfine interactions.
Furthermore, quadrupole splittings may possibly be
observed for the alkali metal ENDOR lines (since
I > lf2 ). Such interactions would give an indication
of electric field gradients at the alkali metal nuclei in
the various environments.
These factors suggest that ENDOR would be an
excellent method for quantitatively extending our knowledge of the alkali metals trapped in molecular matrices.
Acknowledgments.
One of us (LRD) wishes to
thank the National Institutes of Health for financial
support. We also acknowledge helpful discussions
with Professor J. L. Dye of Michigan State University,
Professor A. L. Kwiram of Harvard University, and
Professor D. Kivelson of the University of California
at Los Angeles.
Note added in proof. Since submission of this communication we have
performed detailed lineshape analyses and saturation measurements on the
samples.
Lineshapc analysis indica'tes that the lineshapes approac•l Gaussian and saturation sludics show that the lines are in~
humoj;!cneously broadened.
While these results clearly support a model
involving inhomog~neous broadening. they do not permit one to distinguish
whether this distribution of hyperflne frequencies otlses from a distribution
or interactions of the par.amagnetic electron with the protons or the
bonzene molecules or of 1he electron with the alkali metal atoms.
It
should be noted. however, that the former distribution will result in
broadening by acting upon terms containing the Zeeman g-tcnsor while the
Iauer will n:sult in broadening by acting upon terms containing the
hyperline tensor.
respectiv~
Dalton, Dalton, Dalton
I
Paramagnetic Relaxation in Inorganic Complexes