Ligand hyperfine structure in the ESR spectra of the ions MoOF2-5 and CrOF2-5

Ligand hyperfine structure in the ESR spectra of the ions

MoOF2-5 and CrOF2-5

Citation for published version (APA):

Verbeek, J. L. (1968). Ligand hyperfine structure in the ESR spectra of the ions MoOF2-5 and CrOF2-5. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR88291

DOI:

10.6100/IR88291

Document status and date: Published: 01/01/1968

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

ESR SPECTRA OF THE IONS

MoOF~­

AND Cr0F

2·

5

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE EINDHOVEN, OP

GEZAG VAN DE RECTOR MAGNIFICUS, DR. K. POSTHUMUS, HOOGLERAAR IN DE AFDELING DER SCHEIKUNDIGE TECHNOLOGIE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP

DINSDAG 5 MAART 1968, DES NAMIDDAGS OM 4 UUR

DOOR

JOHANNESLEGNARDUS VERBEEK

GEBOREN TE 'S-GRA VENHAGE

General Introduetion Chapter l Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Summary Samenvatting Heferences Epiloog Introduetion

The Theory of Electron Spin Resonance Ligand Hyperfine Structure in MOX~--ions Analysis of ESR Spectra of Polycrystalline Samples

The ESR Spectra of CrOF ~- and MoOF ~­ Discussion of Available Spectroscopic-and ESR Data

Final Remarks and Conclusion

Curriculum Vitae Appendix 5 6 8 13

25

37 53 66

69

70

71

74

75

77

GENERAL INTRODUCTION

The progress of coötdination chemistry got a new impetus when in the post-World War period, high! y sensitive electronic devices be-came avoilobie on a large scale.

Most important in this respect is the development of NMR and ESR, the latter being the main tooi for the investigations to be describ-ed in the following Chapters.

The great value of ESR in complex chemistry is its ability to recognize symmetry patterns, with unpaired electrens as an inter-mediary. That includes also information on the electronic structure in the ground state of a paramagnetic molecule, and on the hyperfine interactions,

i.e.

interachons of electrens with each ether or with the nuclei of constituent atoms in the molecule.

For several reasons, the complexes of the type MOX ;-, with M a transition metal ion and X a halogen, might draw some attention. One reasen is that their structure deviates only slightly from an octahedral configuration, a structure that has been extensively investigated during the last two decades.One may then consider those molecules as a first step in the generalization of the problem of chemica! bonding. The validity of methods to describe really octahedral molecules may be checked, and conclusions can be drawn in that respect. ESR is one of the major experimentcri aids in such kinds of research.

The well-resolved ligand-hyperfine structure, as observed in the Fluorine complexes with a structure as mentioned above, might be expected to bear subtie information on those hyperfine interactions. lt will be shown, however, that a classica! model describes the details of these spectra with a surprisingly high degree of accuracy.

A secend reasen for the choice of the present subject is found in catalysis. Intermediate compounds, formed in the course of a cata-lytic reaction, aften exhibit symmetries lower than cubic. Information on structure and stability of such intermediates, is of crucial importance for the understanding of the fundamentals of the catalytic process. A number of catalysts containing transition metal ions strongly bonded to oxygen atoms, with a remarkably short metal-oxygen distance, indic-ate the fundamental importance of model-compounds of the type we are going to describe.

CHAPTER I

Introduetion

In this chapter the shape of the molecules that are the subject of the work to be described in this thesis, will be considered first. The chemistry of the elements V,Cr,Mo,W,Nb,Ta and, probably, Re, has provided us with a number of ions and molecules in which the central ion consists of a pair MO with a remarkably short M-0-distance; well known examples of this kind of pairs are found in, for instance, V205 , Cr02Cl2 and VOS04.5H20ll.

Although only few data exist regarding the exact geometry, there is a fair amount of evidence for the structure of what we will occasion-ally call metalyl-compounds.

The structure of the blue VOS04.5H20 is known2l to consist of

a VO-group, with V-0-distance of 1.67

Ä,

perpendicular to a square of four water-c,xygens, now such that the V-0-distance is 2.3

Ä,

thus forming a square pyramid with V in the centre of the base; this pyramid is completed to form a distorled octahedron by placing a sixth 0, from a so;--ion, in the remaining axial position. This structure is shown in fig.l.l and obeys the transformation properties of the point-group C_{4 v.} The ions CrOX~- and MoOX~-. where X is F or Cl, are supposed to exhibit a similar structure; the nature of the axial ligand however, is rather uncertain. As will be pointed out in Chapter 2, we are not interested in its exact nature, so we will not discuss this point to any depth.

x

l

x

k55:7

X

I I I I I I I I

*

Ballhausen and Gray3_{l discussed the bonding scheme on the}

basis of an approximative Molecular Orbital calculation. They arrived at an energy diagram that is shown in fig.I.2. We shall use this picture as a guide for the electronic structure of the ions we are studying.

fig. I. 2: Molecular Orbital scheme according to Ballhausen & Gray.

A detailed account of the assumptions we use1 throughout Chopters l-41 concerning the chemica! bonding in the compounds just mentioned will be gi ven in Chapter 5.

Gray and Hare4_{l applied the scheme on MoOcl;- and were led}

to acceptable results1 which we will discuss in Chapter 5.

According to this level scheme1 the unpaired electron accupies the b₂ level. Static susceptibility-1 ESR- and spectroscopie data can

be explained by this scheme. It is also in accordance with the pre-dictions of the Crystal Field Theory.

Theoretica! aspects of the interpretation of ESR-spectra are found in m any text-book S1 such as those of Slichter5 l 1 P ake 6 l 1 Carrington and McLachlan 7_{l 1 while special mention should be made of the book}

by Abragam8l1 which covers the whole field of magnetic resonance in great detail.

We shall not attempt to reproduce the theory in its full length but shall confine oursel ves to a short survey of these aspects that are of direct importance for the subject we are dealing with.

The theory of electron spin resonance. a) Qualitatieve aspects.

Electron Spin Resonance (ESR), or Electron Paramagnetic Reson-ance (EPR), is the spectroscopy of the transitions between Zeeman-levels, using centimeter wavelengths (X-band, 3 cm.) or millimeter wavelengths (Q-band, 0.8 cm.).

A dynamica} description in full detail of the resonance-phenom-enon may be found in reierences

5-8.

We will present here the more phenomenological description.

A spin-degenerate doublet will be split by a magnetic field; this is the well-known Zeeman-splitting. Consiclering the spin as a mag-netic dipole with dipole moment p. , _s its. energy in a magnetic field will be

( l.l) We may write p.8

=

yS,

where S is the spin-angular menturn in units

'Îl (

=

h/27T), and y

=

wL _armo

/H

_e ff _ec ti _ve ; alternatively we write

lls

=

g,BS

so that

y1i.

= gf3,

,8

being the Bohr-magneton, and g the gyromagnetic ratio, the ratio between magnetic- and mechanica! moment.

From l.l fellows that a parallel position of magnetic moment and field is the most favourable one; the anomaly of the electron' s magnetic moment requires then the spin to be antiparallel to the field. For a free electron (S

=

Y2) we have the situation of fig.I.3, where the direction of the externcri field is taken as the z-direction. An electron passes from the lower to the upper level by "flipping" its spin. These two spin states, with m _s

= -Y2

and +'12 respectively, are not connected by the interaction H _{z z} S , since this operator has diagonal matrix elements only between the two states.

On the contrary, the operators S and S have matrix elements

x y

connecting the two spin states, and no diagorral elements. A micro-wave field in the xy-plane, that is perpendicular to the externcri field, produces a magnetic field with components H cos c.;t, and so

~·---~·----~•~

H

0

fiq.l.3: Splitting of a spin doublet in a magnetic field.

actions H cos c.;t are introduced; this time-dependent perturbation causes thexelectron to invert its spin.

The orbital motion of the electron has so far been neglected. It

is important however, since the presence of orbital angular momenturn causes the g-values to depart from the free-electron value 2.0023. When an ion is placed in the surroundings of ligands, the motion of its electrens is modified. In sectien b) it will be indicated how the action of a crystal field tends to cancel the orbital motion, or, to "quench" it. Experimentally this effect is revealed by the magnetic susceptibility of many first-row transition metal compounds, which turns out to be almost exactly the "spin-only"-val1Je9

>.

An interaction counteracting this quenching, is the coupling between the electran's spin- and orbital moment. The classica! Hamil-tonian for this interaction is

~

H

so

where E = electric field, p = momenturn and S = spin angular momentum. Assuming E to be spherically symmetrie, we write for it

whence

E(r) =_!_E(r)

r

Ex p =_lli!l r x p =!_E(r) L

and finally ~

H _so À(r) L.S. ( 1.2)

E(r) following Coulomb's law, causes À(r) to obey an inverse-cube law with respect to r. The effect of the LS-coupling can be shown to be a mixing of higher orbitals into the ground state orbital, lxy> in our case, by the x- and y-components of L.

The expectation value <L > of L will then no longer be zero, and the value of g departs from 2.0023.

T he

I

xy > ground state of our tetragenall y deformed octahedron is modified te be

The matrix elements of the Zeeman-operator ,BH. (L + 2S) are then

<xy+

I

+ xy > ,BH ( 1----=4-'--'-À -2t E _x 2 _{- y} 2 ,BH

0-_1_)

+i,BH

(1-~)

x E Y E x:a- ya ,BH

(1-~)-i,BH

( l - l ) x E Y E X2t y:<> _,BH (1- 4À :a- E 2 2 x - y

where the energies are relative to E _xy .

Solving the secular determinant (see Chapter 5) for three parti-cular direcUons of H, the following vcrlues of g are found:

H = (O,O,Hz) g = ,BH(2- 8À/E 2 2);

z x -y

H

=

(Hx,O,O) gx

=

,BH(2- 2À/Exz); (1.4)

H

=

(O,Hy,O) gy

=

,BH(2- 2À/Eyz).

Since E _XZ

=

E _'tZ we may write g₁₁ if =:g _Z and _g1

=

g _X

=

g • _y

The result now obtained is usually presented via the introduetion of a Spin-Hamiltonian

Pryce 10 _{l derived the general form of a Spin Hamiltonian}

H=E +2,L3S.H-I IA .. (\.S.+j3HJ){\.S.+/3H1) (1.5)

o nJ!oij 1J J 1

where A₁ .J = <0 IL.In><n ILJ lü>/(E -E ) _{1 n o}

acting wi thin the manifold of non-modified crystal field functions, i.e. functions that have not been acted upon by the LS operator.

Interaction between the electron spins in a particular ion, and interactions between electron spins and nuclear spins, also contribute to the height of the electronic energy levels. Abragam and Pryce11_l derived in an analogous way these fine- and hyperfine terms. Rejecting quadrupale terms and terms representing the interaction between nu-clear spins and the magnetic field, the following terms must be added

to the Spin Hamiltonian 1.5: 2:: {(-\.2_{A .. - pl}

1J) S.S1. +2/3(8 .. - À.A1J) H.S.

-. . 1) t 1) 1 J

1,)

( 1.6)

where u11. = _.!_E_{2 1} .kl _n I [ _,to <0 IL ₁ .In> <n ILJLk + LkL _J .10>/(E -E _{n o} ) ]

and P, À., p, Ç, k arE> constants.

In our case the S.S.-terms,representing the interactions between

1 J

the various spins in the ion, have no meaning, since only one electron is present in the valenee orbitals.

Usuall y the Spin Hamiltonian is written in a parametrie represent-ation that lends itself well to comparison with actual spectra.

For axial symmetry, and one electron present, the Spin Hamil-tonian is then

H=j3[g//HS +gi(H S +HS )]+A,,SI +AI(SI +SI)

1/ Z Z ~X X y y 1 / Z Z ~ X X y y

( l. 7)

The parameters gff' g.L, A and Al.. are those defined by equations

When an LCAO-description is used, the coefficients of the vari-ous molecular orbitals, as well as the spin orbit interaction with the ligand nuclei, enter into the formulae, complicating them a good deal, without essentially al tering them 12- 15 l.

Direct calculation of g- and A-tensors within the scheme outlined above, is risky. There is a delicate balance between the various inter-achons within the molecule, and the atomie wave functions we have at our disposal, are inadequate to deal with such subtie effects.

A notorious example is found in the predietien of vcrnishing iso-tropie hyperfine interactions in the ESR-spectrum of S-state ions, like Mn2 +, which turned out to be wrong. The Manganeus ion exhibits a well-developped hyperfine structure 6 _{l. T he electrens occupying d-type} orbitals, with noclal planes through the central-ion nucleus, have zero density there. The dipolar coupling between electronic- and nuclear spins averages to zero due to the spherical symmetry of the S-state wave function.

To circumvent this fundamentally wrong result, a mechanism was proposed in which the electrens were partially promoted into a higher s-type orbital, in this way obtaining a non-vanishing density at the nucleus 16 l. A Ie ss artificial treatment includes electron correlation. Watsen and Freeman 17_{l stressed this particular point of view. Their} escape out of the difficulty is to pass from a restricted- to an un-restricted Hartree-Feek treatment, which they call EPHF, Exchange Polarized Hartree Fock. The restrietion to be abandonned is the re-quirement that the orbital parts of two conesponding electrens should be the same for spin-up- and spin-down-state. Then, at the nucleus, it is no longer true that

f

(0) ~ =

f

(0)

f;

the net-density at the nucleus explains the presence of an isotropie hyperfine interaction. The ex-change interaction, causing like spins to avoid-, and unlike spins to attract each ether, gives rise to a eertcrin amount of polarization of the core of electrons, and so breaks down the simple picture that lead to the wrong predie ti on.

The computational effort and experience required by such a cal-culation are very large and far beyend the scope of this present work.

Chapter 2

LIG AND HYPERFINE STRUCTURE IN

MOX~--

I ONS

The purpose of this chopter is to account for the complicated structure, observed in the ESR spectra of molecules with the general formula MOX~-, where M = Cr,Mo and X = F in the cases to be dis-cussed presently. The arguments we use will be mainly qualitative.

A general treatment of ligand hyperfine structure ( 1-hfs) was given by Tinkham18 a,b); Watanabe also pays some attention to this.

phenomenon in his hook "Operator Methods in Ligand Field Theory"19 _l. Their way of handling the problem consists in the introduetion of an additional term into the Spin Hamiltonian

HI-hfs =

~IN.AN.s

where N

= number

of nuclei with I f 0; IN= magnetic moment of nucleus N; S

=

spin magnetic moment.

Instead of following this line we shall attempt here to describe the phenomenon encountered with the aid of a more pictorial model. In order to do this, we make some assumptions that may be somewhat crude, but seem to be justified, at least to a first ·approximation, by the good agreement between theoretica! expectations and experimentcri evidence.

The main assumptions are

l) The 1-hfs is entirely due to the dipole-dipole-interaction between ligand-nuclei and electron spin;

2) In the ground state of the ions the unpaired electron accupies the d orbital of the central ion; this assumption is based on the

re-s~fts

of Ballhausen and Gray3_{l for the Vanadyl ion, and of Gray}

and Hare4_{l for} CrOCl~- and MoOCl~-. In _{Chopter 5 we will pay}

closer attention to this aspect.

Consider the dipole-dipole interaction energy in its usual form 8 _l

with

I,S

=

the interacting dipole moments;

riJ

=

unit vector along the line connecting the mid-points of I and S;

R

=

the distance between the mid-points of

I

and

S.

As Abragam8_{l shows, this is the partial result of the solution of}

the Schrödinger equation using the Hamiltonian

l may be put equal to zero because of the quenching of the orbital

moment by the crystal field (in first approximation). The last term in

the above Hamiltonian represents the Fermi contact term which, due to

the presence of Dirac's Delta function, equals zero when the electron

is not in the immediate vicinity of the magnetic nucleus.

In spite of the assumed cl-type ground orbital, we will retain this interaction in the discussions, having in .mind the arguments given at

the end of the previous Chapter. We are not going to try to make an

estimate of the magnitude of the contact interaction separately.

The R-3 _{dependenee of the dipole term in combination with the}

d _xy - ground orbital, suggests a further assumption:

the influence of the axial halogen ligand is neglected in the

con-struction of the ESR spectrum, its distance from the interacting

electron being much larger than that of the four ligands in the xy-plane.

Befere going into more detail, we will consider the electron to be a m agnetic point dipole in the middle of the square of four halogen ligands.

We are now in a situation to predict the splittings in the energy level scheme of the electrens by an externally applied magnetic field, with special reference to some particular orientations of this field.

It should be noted first that equation 2.1 may alternatively be written

as

l) When Hd isalongtheaxisof the dipole, S, it equals +2S/R3 _(fig.2.l).

2) When Hd is on a line perpendicular to S, it equals -S/R3 _(fig.2.2).

3) When Hd ïs perpendicular to S, the angle a between S and rij equals 54°44'; taking Sin the z-direction we obtain this result by requiring the z-component of Hd to be zero:

from H

=

R-3 _{[-S +3S cos}2_{a) = 0} d,z z fellows whence I

Hel

!4---

A---~

' ,. +

.. , __ ,. ..

+~~

f3h

~---A----~

Hel

fig.2.1

s ..

s _

R3

fig.2.2

As will be pointed out in Chapter 3, the significant information

is obtained from the molecules with orientation of their z-axis either

parallel with, or perpendicular to Hext•

In order to derive the directions of the hyperfine fields at the ligand nuclei, it is convenient to start with a point dipale at the metal ion site and subsequently to correct the result for the spatial distri-bution of the electron in the actual wave function.

External field parallel with the z-axis.

The external field Hext is applied in the direction that coincides

with the C ₄v principal symmetry axis of the molecule. For this reason

we expect the four ligands 1,2,3 and 4 to be equivalent, corresponding

to case 2 of the foregoing section. This leads to the dipale fields

z

l.sh

y

fig.2.3: Local fields at ligand sites for Hext parallel with "'-axis.

Figure 2.3 has Hd>Hext' a magnitude that may be expected, as will be pointed out in the discussion in Chapter 5, and that is found exper-imentally from the results to be presented in Chapter 4. He is drawn toa, probably, exaggerated scale.

The electran's spin and magnetic moment being of opposite sign, the spin state with m8

=

-Y2

is the one with lewest energy in a

mag-netic field, i.e. 11 _e is parallel with H _ex t (cf. Chapter la). Adding or subtracting the energy of interaction with the ligand nuclei, a maximum lowering of the energy of the electron is obtained when all four ligand nuclei are parallel with the resultant magnetic field. This situation

I

4

is chqracterized by M₁

=

.:z:

m₁

=

2. Hence there will be five energy

1 : 1

levels with the level having m s =

-

Y2

as the centre. An analogous

rea~oning applies to the 11 _{spin-flipped" level with m}

=

₊

Y2

.

s

For each of the resulting situations the energy of the complex is given by

4 4

E

=

-p. _e .H _ex t - _{i : l} L: lli·. H t - L: /J.Ii'(H +Hd)

1 ex i : l c

The above conditions lead to the following level scheme (fig.2.4): Using the selection rules t.M1 = 0; t.ms = l, this scheme imposes the

five transitions as indicated in the figure. The relative intensities of the lines w ill be 1 : 4 : 6: 4 : 1, the ratio of the statistica! weights of the levels.

+2 +1

0

-1

-2

fig.2.4: Energy levels and allowed transitions for H ext parallel with a- axis.

Hext <Hd; .6E

=

Yr-11 ( IHd

I

+ !He

IJ.

Extemal field parallel with the x-axis.

Contrary to the former case there are some complications ansmg when we consider the events on bringing H t from the x-axis to the

ex

y-axis, staying in the xy-plane. Again for reasons to be explained in the next Chapter, we confine ourselves to two special cases, narnel y

H _ex 1// x-axis (or y-axis) and H _ex 1 making an angle of 45° with the

x-axis.

a) Hext // x-axis.

The x-axis lying in one of the symmetry planes of C ₄v, we

ex-peet the four ligands no longer to be equivalent. Actually, ligands l

and 3 are equivalent, and likewise ligands 2 and 4. This is illustrated

in fig.2.5.

1

y

fig.2.5: Local fields at inequiva!ent sites I and 2 for Hext parallel with the x-axis.

It has been indicated previously that Hd₂

=

-I-'2Hdl' Then we expect a larger splitting of the spin-energy levels due to ligands l and 3 and a smaller splitting, superimposed on the former due to ligands 2 and 4. For I = 1--2 this means a threefold splitting plus a second

threefold splitting, in the way as given by fig. 2.6. Maintaining the

L

_U

0 -0 0 0 ~ - 0

~'

_li.J.

~~· ~

t l t l 4 2 I

~ 1-,;._~

·~~===

-

.

- 0 0 + 0 0 0

•

•

•

0

.

-fig. 2.6: Energy levels and allowed transitlans with their relative intensities. Hext parallel with x- or y-axis.

requirem ent of non-changing nuclear spins when "flipping" the elec-tron spin, we have t.M1 = 0; t.M2 = 0; t.ms = l, leading to the nine

transitions of fig.2.6.

The relative intensities of the transitions are again in the ratio of the statistica! weights of the conesponding levels, so as 1: 2: 1:

2:4:2:1:2:1 (fig.2.7). 1/ 16 1/8 l/16 1/8 1/4 1/8 1/16 1/8 1/ 16

fig. 2.7: Relative weight of the levels arising from the ninefold splitting when Hext is parallel with x- or y-axis.

In the spectrum there are now to be expected three equidistant lines, each line hoving two symmetrically located satellites. Interchanging

M1 and M2 gives the situation for Hexl 'y-axis.

b) H ext under 45° with the x-axis.

A different situation arises when Hext' in the xy-plane, makes an angle of 45° with the x-axis. This situation is shown in fig.2.8.

x y

fig. 2.8: Local dipole;fields for Hext in the xy-plane, making an angle of 45° with x- and y-axis.

Short solid arrows: Hext; Long solid arrows: dipalefield Broken arrows: resultant field.

The dipole fields at the four ligand sites are equal in magnitude though. having different directions, that are, however, determined by the sym-metry of the molecule. Like in the case with Hext#'z-axis, there wil! be a fivefold splitting of the spin-energy levels, again leading to five absorption lines with relative intensities l: 4: 6:4: l.

The electron in a spatial distribution.

a)

So far we considered the electron to be a point dipole at the origin of the coördinate system. This is evidently a somewhat unreal-istic representation. The electron moves in a wave function of the d _xy -type. For the ease of the argument we will first completely neglect the in-plane pi-bonding, neither will we take into account the small effects on the wave function due to the LS-coupling.

The general farm of the d _{. xy} wave function exhibits four lobes poin ting between the x- and y-axes, and hoving their centre of gravity just in the xy-plane. If we know the exact wave tunetion descrihing the motion of the electron, we are able to calculate the average dis-tanee of the electron from the nucleus, or, what is more useful, the average value of r-3 , <r-3_{> :}

Provided the geometry of the molecule has been established in detail, we should be able to calculate exactly the energy of interaction be-tween electren and ligand nucleus; our present problem would then be completely solved.

Many complications arise, however, when we tried to treat the problem along these lines, the most obvious one being that we have no exact wave functions available. The best SCF functions are prim-arily correlated with the best energy, and it is a well-known fact that the wave functions, according to perturbotien theory, are one order less accurate than the energy associated with them. Furthermore, an electron in a molecule is no longer a free electron (free in the sense of belonging to one particular atom), as it is subject to an impressive number of interactions.

Any calculation of molecular wave fundions may use atomie

( =

ionic) wave fundions as a storting point (see Chopter 5) but, as was pointed out in Chopter l, only the most sophisticated calculations provide us with more or less reasonable functions *.

An attempt to evaluate the magnetic fields at a ligand site, using semi-empirica! wave functions, stands little chance of succeeding. The best one can hope is that an estimate of the wave functions from, for instance, the observed g-values of the molecule, will provide not too bad an estimate of the hyperfine interactions. But such estimates are very difficult to make and seldom unequivocal.

More complications arise from the unpleasant fact that only very few strudural data exist on the geometry of the molecules we are dealing with2 oa,b,cl, So even intramolecular distonces are to be estim-ated, thus rendering a calculation speculative.

*

Reviewing these considerations it will now be clear why the

The most advanced calculations of this type were carried out by Shulman and Sugano 21 ) and by Watsen and Freeman 22 l. Neither of these calcul-ations provides completely satisfactory numerical values of the hyperfine interactions, although qualitatively they seem to be correct.

emphasis in this thesis lies on the experimentcri data on hyperfine-interactions. The model, proposed to understand the meaning of the ESR spectra, should not be assigned any rigorous quantitative value. The estimates derived from the arguments in section b) hereafter will at most be order-of-magnitude estimates. It will also be found that deviations may occur when we try to fit the parameters to the spectrum.

b)

In order to arrive at some qualitative insight in the ratio of the hyperfine fields _{Hd,Hdl'Hd 2} and _{Hd 45 ,} we will have to consider the electron as a charge distributed over the region, covered by t)le wave-tunetion descrihing this electron. Passing from the picture of a point dipole to that of a smeared-out dipole, we reconsider the formula

from which follows, as we saw already, that H l. p. for a = 54 °44'. In fig.2.9 one can see how the interaction energy between dipoles p.₁ and

p.2 changes sign on the passage of the surface of the cone when dipole ll 2 travels from position 1 via 2 to position 3.

~t

__ _

til

fig. 2.9: Interaction between {ll and {l2 is zero in point (2); the interactlens have eppesite signs w hen fl2 is in point (I) or point (3).

In other words, the projection along the p.₁-directionof the mag-netic field at llt, due to ll 2 , changes sign. So, when the density of the distributed electron lies partly within such a cone, erected at a ligand site, the contribution to the magnetic field at that site, arising from the density within the cone, has its sign opposite to that of the density outside the cone. Thence, if a considerable part of the wave function

is found within the cone, a large deviation from point-dipale behaviour will occur. In fact the situation is not quite that serieus since the region of highest density is, for bonding wave functions, somewhere between the bonded ions, and in extreme cases only, very near the ligand. So we expect no large deviations, as has been qualitatively indicated in the figures.

i) Electron polarized along the z-axis.

In this case no significant part of the d _xy -wave function is like-ly to be covered by the cone (fig.2.10).

fig. 2.10: Covering of dxv -wave function by a cone with an opening of I 09° 28' and axis perpendicular to xy-plane.

ii) Electrem polarized along the x-axis.

In this situation we encountered two inequivalent sets of sites: (1,3) and (2,4). The xy-plane and its intersectien with the cone is shown in projection on the xy-plane, in fig.2.lla,b.

It is immediately clear that the deviation from point-dipale behaviour will be larger for sites 2 and 4 than for sites 1 and 3.

iii) Electron polarized in the 45°-direction.

T he electron' s magnetic moment may be decomposed into its x-and y-components 2-Y.y _e

1JS.

At site 1 the x-component produces a

a

fig. 2.11.

a. Coverlog of dxy -wave funclion by a cone with opening of 109°28' and axis along x-axis, through ligand 1.

b. idem, through ligand 2.

fiefd 2-Y.Hd₁ in the positive x-direction, the y-component a field 2-Y.Hd2 in the negative y-direction (fig.2.12).

(The ad dition pd to the subscripts in fig. 2.12 is to indicate the direct-ions of the fields when the electron behaves as a point-dipole.)

It was shown in ii) that, most probably, Hd₂ will be reduced

Hrosult.

fig. 2.12: Loca1 fields at ligand site 1 for Hext in xy-p1ane making an angle of 4 5 ° with the x-axis.

Hd d

=

dipo1e field from electronic point dipole

p. in origin;

Hd 45

=

dipo1e field from an electrooie dipo1e distributed over a d -wave function.

more than Hd 1 when the electron is spread-out over an actual wave junction of dxy-type. This means that the true

Hd

₄₅ lies closer to the x-axis than _{Hd 45} does. Vectorial addition of the external field Hext brings the resuftant field still closer to the x-axis, Generally speaking, the angle between the two will be of the order of 15°-25°, a value small enough to cause no serious difficulties in the analysis of a pow-der-ESR-spectrum, as will be considered in the next chapter. To con-clude this section, a rough and provisional estimate of the magnetic fields at the ligand sites, due to the electron, is proposed:

IHdll

~

1.7 Hd 1Hd21

~

0.7 Hd

1Hd45

I~

1.3 Hd

In the course of the analyses of the recorded spectra, departure from these values wil! be of not too great concern to us, regarding the naieve and qualitative nature of the arguments used. N evertheless, they will be very useful in serving as a guide in trying to fit the para-meters to the actual spectrum.

Chapter 3

ANALYSfS OF ESR SPECTRA OF POLYCHRYSTALLINE SAMPLES

The form of the Spin-Hamiltanion 1. 7 shows the convenience of the use of single-crystals to obtain the g- and A-values for the special directions separately. It is, however, difficult to obtain magnetically diluted crystals of the compounds we are interested in; this circum-stance necessitates the use of randomly oriented samples, in this case quickly frozen dilute solutions. We have then to consider the shape of the spectra due to such a random orientation of the paramagnetic een tres.

In the following review of the theory we wil! omit the hyperfine interachons for reasans of simplicity and brevity. lts inclusion in the problem is straightforward and fellows analogous lines as what is going to be clone for the g-values.

When we are dealing with samples of cubic (site) symmetry, there is no problem at all since the orientation of the symmetry axes with respect to the external magnetic field is irrelevant because of the relation gx

=

g

=

gz.

Crystals with axial- or lower symmetry require us to know the precise orientation of the crystal axes in the field as the principal g-values cease to be equal. This anisotropy makes that for any value of H _ex t such that (hv/,{:lg _max )<H _ex t<(hv/,{:lg. ),a resonance line is _m1n observed. The equivalence of the x- and y-axis in cases of axial symmetry, makes the polar angle <Pirrelevant. So, crienting a crystal with its cluster-C4 V -axis along H , we find g , in axial symmetrical Z Z cases mos tly called g ₁₁; whenever the axis is orien ted perpendicular to H , we cbserve g

=

g

=

g.L. When the crystal is rotated about,

ext x y_

say, the y-axis, the observei:i g·value varies between g// and g.L, obey-ing an angular dependenee to be explained presently.

Because in a polycrystalline sample all orientations are assumed to he randomly distributed, the ESR spectrum will show all these poss-ibie g-values simultaneously. Now the question arises as to the shape of the spectrum. Sands23 l and Kneubühl24 l solved this problem for axial and rhombic symmetry respectively.

For axial symmetry that is a rather simple problem, and for the sake of completeness, an outline of the salution will be given now. Our first task is to write g as a tunetion of the angular coordinates [) and

P._._

In Chopter l the Spin-Hamiltanion was presented to be

H

=

/3 [

g H

s

+ g4 H

s

+ g H

s ]

X X X YY z z z (3.la)

This may be rewritten in a dyadic notatien as

H

= /3H. g.

S (3.lb)

with

g

= i.g _XX .i+]' .g ·J· +k.g _{yy ZZ} .k _(3.lc)

Camparing this to the formula for the "spin flip" of a free electron

A

H = 2/3H. S

we may write the previous formula as

(3.2a) from which fellows

Heff = H. g/2 (3.2b)

Transferming now the coordinate system in the sense that the new z-axis coincides with the direction of Heff' we find

where Heft now is the length of the vector Heff. For S

=

~. the solutions are ± /3Heff' and thus

Inserting (3. 2b) into (3.4) we find

(3.3)

(3.4)

(3.5) Expressing H ,H and H in the polar angles _{x y z}

8 and

c/J,

this becomes

If we write g for the square root in (3.6), the familiar result hv

= g,BH, like tor a free spin, is found again; this g,

however, in-cludes Zeeman- and LS-effects and so no longer necessarily equals 2.0023. In the axial case we were considering, formula (3.6) reduces to (3.7a) or

(3. 7b)

Th is formula is of cru ei al importance for the anal ysis of powder spec-tra we are aiming at.

Apart from a possible 8-dependence of the transition probability, the intensity of the absorption by the polycrystalline sample is pro-porticnol to the number of crystals hoving their axes making an angle between

e

and

e

+ d8 with H t'

ex

Plotting the coordinates

e

and

cj;

of each crystal in the sample on a unit-sphere, we obtain a uniform dis tribution of points because of the assumed random orientations. The number of crystals hoving their

c

4v-axis with

e

as above, will now be proportienol to 27Tsin8d8, as is illustrated by fig.3.l.

fig. 3.1: Area on unit-sphere representing the number of molecules with z-axis making an angle between

Since the tot al number of crystals is proportion al to 4rr 1 the area of

the unit-sphere1 the fraction dN will be

dN

=

(N/2)sin8d8

=

(N/2)d( 1- cos8) (3.8) where N

=

total number of crystals in the sample.

(The choice of ( 1-cos8) instead of simp1y -cos8 is for later con-venience).

F ollow ing Sands 23 _{l we denote the fixed microwave frequency in} the experiment by v and so H = hv /2. Some straightforward algebra

0 0 0

produces the relation between dN and dH1 the field interval within

which resonance occurs at the frequency v0 :

P lotting dN/ dH vs. H 1 we find the absarptien spectrum for a pol

y-crystalline sample with zero linewidth (fig.3.2).

11

..1.

fig. 3.2: dN/dH vs. H for an axial molecule; zero-line width assumed.

Two features are still lacking in the problem. One1 already touched

upon1 is the 8-dependence of the transition probability. The ether is

the finite linewidth of the absarptien lines due to relaxation phenomena and to the mutual interactions between the paramagnetic centra in the sample.

The farmer point is dealt with by Bleaney's formula25_a1b1_cl

(3.10) which must be reexpressed in H for combination with (3~9).

The latter point is extremely important, since the overall shape of a spectrum depends very strongly on the linewidth, especially when a number of partially overlapping lines are present.

In the absence of strong interachons between a paramagnetic ion and its surroundings, the lineshape will be Lorentzian:

(3.11) Here f(H) is a measure of the intensity of the absorption as a function of the magnetic fields strength1 H is the centre of the symmetrie

0

absorption line and b is the half-width at half-power. This line width has not yet been included in fig.3. 2; the absorption line for each value of H is represented there by a "stick" of zero-width. To incorporate (3.11) in (3.9) has the effect of providing each stick with a finite width, such that each stick is broadened into a Lorentzian line. Care should be taken that the area under each Lorentz curve represents the same transition probability as the stick it is deduced from. 1bers and Swalen 26 l carried out the complete analytica! treatment of the problem and arrived at the complicated formulae

(3.12a)

I(H) ,.,_, [(A,8₁ + _{Bb,82}/2bP)]L 2}+ _[(-A,82 + Bb,81}/bP)] T 2 (8/Hlln[ H#/(H.!_-(Hl-H3/I.)] -D[(Hl-H2}/(H1H_11)]

#

where

p

=

[(H2- Hl- b2)2 + 4H2b2]

~;

[3

1

=

[Y2P + Y2-(H2 - H l - b2 ) ]Y.;

t-H-H L₁

=

ln [(t2 ₊ b2}/(8~ + <j:~}] - 1_

t :H-H#

t-H-H ,T ₁ = [tan-1_{(b/t) + tan-}1_{(<P/81)] - l.} t=H-H#

e

1 = y1 t + &1 - (t 2 - 2Ht + H2 - HD~~, 82 = y2t + 82 + (-t 2 + 2Ht +Hl- H2)Y. 'Yl (bj32 - Hj31)/P)

81

[

-Hbj32 + (H 2 - HD/31]/P

/3

2 =- [ YaP - Ya(H 2 - Hl- b2)]Y. L2 = [ln[(t2 +

b2)/(8~ +<P~

)] ]

t=H-Hl. t:H-H# T ₂ = [tan-1(b/t) - tan-1(<P:/ 8_2)] t:H-Hl. t=H- H# <Pl

=

y2t +

0

2

ct2

= 'Yl t +

81

'Y2

=

(b/3 1 + H,82)/P 82

=

(-Hb/3_{1 - (H 2 - Hi_)/32)/P}

The ESR spectrometer delivering first derivative curves of the absorption, we are mainly interested in the calculations of that line. This may be clone in two ways:

1) I(H) is calculated for a large number of H-values and the result is differentiated numerically;

2) the formulae are differentiated and subsequently the calculation is carried out.

We choose the second way to save computer time. Mr. W. Konijnendijk differentiated the formulae and wrote an ALGOL program for the EL-X8 computer of the Rekencentrum of the Technological University Eind-hoven. The punched-tape output was fed off-line into a CALCOMP plotter and plotted pointwise. The program is given in the Appendix. To show the shape of the spectra, and the influence of linewidth, some were calculated and are presented in fig.3.3a-f.

en lil Ü x

A_)

)

~vv

-4

3200

3250

3300

veldslerkle

a

veldslerkle

b y V

3350

3-tOO

C\1 I 51 x C\1 151

0 1---~-+----t-+---i---1

x

veldslerkle

c

veldslerkle

d

2~---.---.---,---~

3250

veldslerkle

8

l'il

0

x -8

3200

~

~

3250

veldslerkle

3300

3350

3400 e

~

~

-3300

3350

3400

2-fig. 3.3a-f: Simuloled ESR-spectra of Mo0Cl₅ for linewidth varying between I and I 00 Oe.

The relative ease of handling analytically the problem of powder spectra for axial molecules, is lost when we move on to the rhombic case.

That problem was treated by Kneubühl24 _{l without accounting for}

Lorentzian lineshape. His results are, with g₃>g₂>g_1:

I(H} "-' [ (H ₁ H

2

H

3

H-2}/[(HÎ-H2)~ (H~- H~)~]J K(k) for H ~ H2 (3.13a)

I(H) "-' [ (H 1 H2H3H-2 )/ [ (HÎ-H~)~ (H2 - H~)~ ]] K(l/k) H . ~ H2

(3.13b)

K(k) and K( 1/k} are complete elliptic integrals of the first- and secend kind respectively.

A rhombic "stick" diagram, analogous to the one in fig.3.2, is shown in fig.3.4a.

lntroducing the linewidth in the manner indicated before, the spectrum is modified to the one in fig.3.4b, the first-derivative of which is shown in fig.3.4c.

V

fig. 3.4: ESA-spectrum of a randomly oriented sample of rhombic molecules.

a: upper figure, breken line: absorption spectrum. Stick diagram with zero-linewidth.

b: upper figure, solid line: absorption spectrum. Line width included.

c: lower figure: first derivative of b. dS(H) as in fig. 3.6.

In all the cases we mentioned so far, hyperfine structure will complicate the pictures in s uch a way that 21 + 1 curves are going to be

superimposed (I is nuclear spin quanturn number). An example is shown in fig.3.5.

fig. 3. 5: Stick diagram of the ESR spectrum of a randoml y oriented sample of axial molecules with hyperfine structure.

g// >gj_; A// >AJ_; I

=

3/2. (From ref.27).

As long as axial molecules are considered, the only variabie of orientation is the angle

e

so that a plot of B vs. H t provides all the

ex

desired inform ation.

As soon as </> also influences the value of g, which happens in rhombic molecules, we are forced to make a 3-dimensional plot, viz.

e

and </> vs. Hext' The general shape of the curved plones that represent

the relation between

e

and </>, and the value of H t where resonance

ex

occurs, is given in fig.3.6.

dH 'f H c a1.•1-- · 1

fig. 3.6: Plot of the resonance field H vs. a(= 1-cos

e

l

and </>; M 1 and M2 are defined in Chapter 2.

dS(H) is the projection of the intersectien of the "field-sandwich" .dH with the curved surface, and a meesure of t.,he intensity of absorption. 6g = 0.

In order to deri ve the shape of the absorption line from this kind of picture, a "field sandwich" of thickness dH is moved upward along the vertic al a x is, storting at H

=

0. As soon as it intersects the curved plane, absorption of microwave energy starts. The intensity of the absorption is proportional to the number of 11 sticks" that corresponds

to the intersected area. Since the number of absorbing molecules in a rhombic case will be proportional to sin eded<j;

=

d( 1- cose)d<t(cf. 3.8),

it is advantageous to plot ( 1- cose) instead of

e

itself, as then the intersected area is a measure of the intensity of the absorption. The chöice of ( l -cose) is arbitrary; this tunetion is 0 at the origin, the same situation as w hen

e

was used as the coordinate. Th is facilitates

comparisons.

An analytica! treatment of rhombic ESR spectra including line-width and both central-ion- and ligand-hyperfine-structure, is extremely complicated. Simulation of this kind of spectra will most conveniently be performed by a method that calculates g and the intensity for a

large number of values of

e

and

<P.

foliowed by numerical differentiation of the obtained curve28_l.

However, the very many details in some spectra, require a high accuracy of the counting- and differentiating procedures; as the field range to be covered is, in our cases, also rather large (0.5 - lkOe) one

is forced to calculate some 104 _{different g-values. This causes the}

program to consume a fair amount of computer-time; experiences with simple rhombic cases, without ligand hyperfine structure, suggest an estimate of %-1 hour per spectrum. A best fit of the parameters im-poses a number of attempts, and so it is to be expected that a satis-factory computational check of the interpretation of our spectra will take at least 3 hours per spectrum, provided a lucky initia! guess was made. Not accounted for in this estimate is the linewidth, which enters as an extra adjustable parameter. This again may lead to an increase in the time.

In view of these arguments, we refrained from simulation of the 1-hfs spectra.

Returning to fig.3.6, it will be clear that each hyperfine line can be represented by a separate plane. When analyzi ng the 1-hfs in

CrOF;-and MoOF;-, in the next Chapter, We shall use this three-g-value de-scription in order to interpret the 1-hfs signals as "pseudorhombic"

Chapter 4

THE ESR SPECTRA OF

CrOF~-

AND

MoOF~-Experimental

The ions MOX~-, the geometry of which has already been de-scribed in Chapter l, have been known for quite a long time. Weinland

and Fiederer29_{l report the preparatien of (C5H5NH)Cr0Cl4 , which}

corr:-pound is the starting material for the fluorine complex.

Angell, J ames and Wardlaw30_{l prepared} MoOCl~-

5

_{with various}

cations. A summary of the prescriptions is given ir{ the following sections.

A salution of Cr0₃ in a mixture of concentrated HCl and glacial Acetic Acid is saturated with HCl-gas at 0°C. An equivalent amount of Pyridine, pre-treated with HCl-gas, is added to this salution, and the brownish-red precipitate is filtered off and dried over concentrated sulphuric acid; this is to maintain a HCl atmosphere in the desiccator,

as long as the compound is moist. Not until it is absolutely dry, the HCl is removed with NaOH. Then the powder is stored in a sealed

bottie and kept in darkness. The compound is under these circum-stances, stabie for at least a couple of weeks.

(C ₅H ₅NH)MoOC1_{4 :}

Ammonium Molybdate, (NH_{4 ) 6}Mo₇0₂₄.4H₂0, in concentraled HCl, is reduced by Zn or Hg. An equivalent amount of Pyridine, pre-treated with HCl-gas, is added to the deep-blue solution. After cooling to 0°C, a stream of HCl-gas is passed through until saturation. The

pre-cipitate, consisting of bright-green needles, is filtered off and washed with a small volume of concentrated HCl. The product may be

re-crystallized from warm concentrated HCl. Well-shaped crystals appear when the salution is allowed to cool down slowly. After tiltration the

crystals are treated in the same way as the Cr-complex. The coumpound is stabie for at least a year, probably even much Jonger.

Af ter sol ving the Chlorine complexes in 38% HF -solution, an instantaneous exchange of Cl by F takes place. The Cr-solution turns

faintly yellow, and the Mo-solution light blue. The spectra in the visible and UV region will be discussed in Chopter 5.

No attempts were made to isolate the CrOF~--containing com-pounds; the Molybdenum complex, however, was isolated from a salution of [ (MoO(OH) _{3 ] 2} in concentrated H ydrofluoric acid. Aft er ad dition of

an equivalent amount of NH₄HF ₂ the liquid was slowly evaporated, and a light blue crystalline solid separated out.

The compounds containing Cr53 _{or Mo}98 _{had to be prepared from}

less suitable storting materials. Cr53 _{was available as the oxide} Cr~3₀

₃

_{, containing more than 95% Cr53 ; Mo}98 _{was available as the}

metal, containing more than 98% Mo98 • In order to convert those mater-ials into the desired complex compounds, they were treated in the following manner.

The Molybdenum metal was heated in a stream of Oxygen at a tempercriure of 600°C. The Mo0₃ formed by this action, was dissolved in a small volume of concentrated hydrochloric acid. In .this .solution the Mo was reduced as described in the beginning of this Chapter.

The Cr(III)-Oxide was fused with N a₂ C0₃ under ample exposure to air. The Chromate then formed, was dissolved in glacial Acetic acid

and treated in the previously described way.

ESR measurements on HF-containing samples require extreme care as to avoiding severe damage to the microwave cavity. The usual sample-tubes of quartz glass being inadequate in this case, a suitable material had to be looked for. Teflon turned out to be most conven:ient to this purpose. For the X-band measurements it was sufheient to use tubes like the one shown in fig.4.l.

II:::A

-~~

j--sl---~·1

fig. 4.1: Teflon sample tube for ESR measurements.

Rod B is screwed into the sample campartment A, after A has been

completely filled with the sample solution. The liquid is then quickly

frezen by dipping it into liquid Nitrogen. After some time it is trans-ferred to the caoled cavity.

Spectra of the solutions at room temperature can be obtained by

inner diameter. The double-folded piece of tubing is then placed in cernpartment A. This manner of werking enables one to avoid the use of a special liquid-sample-cell.

For Q-band measurements the tube is essentially the same. The

dimensions however, differ appreciably. Particularly the thickness

of the walls must be as small as possible, in order to avoid an ex-cessive filling factor of the cavity, which would result in the imposs-ibility of tuning it.

In the experiments to be described, a wal! of O.lmm was used. The sample must be in "spaghetti tubing" also at low temperatures.

It is clear that the higher Q-band sensitivity is lost by the smaller

sample volume; at both frequenceis the lower limit of the number of

detectable spins is a bout 1011 •

ESR measurements at X-band were carried out with a Varion

V-4500 Spectrometer. The frequency was determined by a Hewlett-Packard Electronic Counter Model 5245L, coupled with the microwave bridge thr.ough Hewlett-Packard Frequency Converters, Models 25908

and 52538. Q-band measurements were carried out using the Varian

V-4561 Microwave Bridge, and special pole tips on the Varian 9 inch

magnet. The X-band cavity was cooled by leading the vapour of boiling Nitrogen through the Varian Variabie Temperature Accessory; the tem-perature was regulated by the Varian V-4561 EPR Temtem-perature Con-troller.

Cooling of the Q-band cavity by conduction, as prescribed by the

manufacturer, turned out to be unsatisfactory. The lowest temperature

we could obtain was -ll0°C. Considerably lower temperatures were

reached by blowing the vapour of boiling N itrogen on top of the cavity, which is located in the narrow neck of a Dewar vessel. Heat exchange with the surrounding air was reduced by placing a piece of Styropore in the neck of the Dewar.

Since no device for the meesurement of magnetic fields was

available to us, we had to preeeed in the following way, in order to

determine the field strength corresponding to various points in the

spectrum. The sweep range was determined by the use of a solution

of MnC12 in water; A was taken as 96 Oe. The sweep was assumed

linear over the whole range.

Using Diphenyl-Picryl-Hydrazyl (DPPH), one point of the recorder

chart was marked, and start- and end-values of the sweep range were calculated with the aid of this value.

In the calculations the following values were assigned to the

relevant constants: l Oe= 0.25.77-1_.103 _Am-1•

Planck's constant h

=

6.6256.10-34 _Js;

Spectra of dilute solutions.

A useful aid in unraveling the complex spectra of glassy samples is given by the spectrum of the unfrozen solution at room temperature. (sometimes a more elevated temperature may be required to reduce the viscosity of the solvent sufficiently).

In the spectra to be presented in this Chapter, it is usually poss-ibie to locate g by means of the c-hfs, that, fortunately, is rather large and thus spread over a w ide field range. Si nee a re lation exists be-t ween g _{11 ,} g.l and the isotropie <g> as measured in a liquid solution, we are able to calculate the location of g.l from the values of <g> and g //" We w ill now consider the required re1ation between the various g-values.

In Chapter I the Spin-Hamiltonian for an axial molecule was given. Lifting out of this formula ( l. 7) the Zeeman part, we retain

Writing out the inner product H.S, we find by substitution

Hz

_eeman =(3 [(gl/-g.l)H S +g.LH.S]

11 z z

Suppose the z-axis of a molecule to be oriented as in fig.4. 2. :r

fig. 4.2: Definition of axes and coördinates.

Then, for H2S2 we may take HqSq;Hq = Hcos6l and

H .S

= HS2 • Inserting this into (4.2) gives us (4.3):

(4.2)

Hz

= (3H [(6g cos2_{6l + g.l)S +} ~ _{6g sin6l cos6l (S+e-}1_{cP+S e+}1_4>)]

eeman z

When the molecule is tumbling rapidly, and does so rapidly enough for the correlation time to be small compared to the microwave frequency, the molecule exhibits an average orientation with respect to the magnetic field H. This means that we may take the average of the Hamiltonian. Averaging of (4.3) results in

<H> z _eeman .

= ,8H(6g/3

+ g, _...1... )S _z

= ,BH(g

₁₁ 'IJ/3 + 2g, _..L.. /3)S _z (4.4) Hence the isotropie g-value is given by

(4.5) Reasoning along the same lines, an analogous formula is found for the hyperfine parameters A₁₁, A1.and <A>:

The ESR Spectra.

Ligand hyperfine structure in ESR spectra has not aften been considered, in contrast to the Nitrogen hyperfine structure, which is found in many biochemically important compounds.

Tinkham lBa,b) i_{nterpreled the ESR spectra of impurities in ZnF 2} and observed Fluorine hyperfine structure. Kon and Sharpless 14 l inter-preled minor details in the spectrum of CrOCl~- as due to the elec-tran's interaction with the nuclear moment of Cl. The rather low mag-netic moment of the Cl-nucleus as compared with the other halogens (see Table 4.1) suggests a better developped 1-hfs character with the ligands F and Br (or I, but very little is known about the Iodine complexes; e.g. see ref.37).

T ABLE 4.1 (from ref. 32)

isotape abundance nu cl. magne tic I moment • F 19 _{100.00% 2.6273 l/2} Cl3S _75.40% 0.82091 3/2 Cl37 _24.60% 0.68330 3/2 Br 79. 50.57% 2.0991 3/2 BrBl _49.43% 2.2626 3/2 I 127 _{100.00% 2.7937 5/2}

•

_BOHR magnetons.

Kon and Sharpless 3 ll and van Kemenade, Verbeek and Cornaz33•34 •35l

i.nvestigated the ESR spectra of, respectively, the Bromine- and Fluo-rine-complexes of the axial type we are studying here. The lower I makes the Fluorine spectra less complicated. The nature of the Bra-mine spectra is ambiguous, due to a certain amount of doubt as to the Bramine complexes being of the same monomeric character as the Chlorine- and F luorine complexes 36 l. We will return to this important question in the next Chapter.

We will focus our attention on the ions Cror;- and Moor;-. For the interpretation of the c-hfs, data in Table 4. 2 are important.

T ABLE 4.2

Isotape Nat.Abund. Nucl.Spin Nucl.Magn. * moment Cr53 _{9.54% 3/2 -0.47354} Mogs _{15.78% 5/2 -0.9099} Mo97 _{9.60% 5/2} -0.9290 Mo 98 - -

-* _{nuclear magnetons}

The ESR spectra of MoOF~- in concentrated HF at room temperature, are shown in fig.4.3a,b for X- and Q-band respectively, and for natura}

Mo only, since Mo 98 possesses no nuclear spin (Table 4.2).

Assuming the two outermost signals to be the lowest- and highest hyperfine transitions, we find <A>x

=

68.5 Oe, and <A_{> 0}

= 68.3

Oe, and we adopt the average value <A>

=

68.4 Oe. The X-band spectrum

shows <g>x = 1.907, while the Q-band spectrum shows <_{g> 0} = 1.905. We adopt <g> = 1.906.

The assumption made here concerning the outermost signals, is a reasonable one w hen we take into account their apparent! y equal widths. McConnel1 38 l and Rogers and Pake 39 l pointed out how each hyperfine line may acquire its own particular line width, due to relax-ation effects dependent on m1 (

=

projection of I along the axis of

refer-ence). From their theory follows, that there is a particular hyperfine line with the minimum width, for two sufficiently differing magnitudes of the extern al magnetic field. If, by coincidence, our outermost si g-nals at X-band exhibit equal linewidths, then equal linewidths are improbable for the Q-band spectrum, provided this mechanism acts here. Since at both X- and Q-band equal widths are observed, we

ex-peet the effect not to be present and conclude that there are no c-hfs lines outside the region shown in fig.4.3.

fig. 4.3: ESR-spectrum of a salution of MoOF;- in 38% HF; room temperature.

a. X-band; g

=

1.907; A

=

68.5 Oe.

b. Q-band; g

=

1.906; A

=

68.3 Oe.

N ext we pay attention to the spectra of CrOF ~- under the same

conditions as above, which are given in fig.4.4a,b,c.

The complex containing notural Cr will show only a very weak c-hfs (fig.4.4a). Therefore Cr53 _{(>95% Cr53 ) was used (fig.4.4b,c). The} Q-band spectrum provides us with the values of <g>₀ = 1.964 and

<A>₀ = 23.1 Oe.

The X-band spectrum presents a peculiar picture. However,

assuming the indicated points to be the relevant ones, a good agree-ment is foundwiththepreviousvalues: <g>x=l.964and<A>x=23.3 Oe. We see no obvious reasen for the abnormal X-band spectrum. The non-linearity of the base line of both X- and

0-

band spectrum, suggests

the presence of some ether species, having nearl y the sa me g-val ue, but net showing a resolved hyperfine structure. Dilute solutions of

Mn Cl2 in water, that almest exclusively contain Mn(H

2

0)~ +, same-times show the same kind of base line deviations.

For the interpretation of the glass-spectra we now dispose of the data in Table 4.3:

TABLE 4.3

MoOF;- CrOF;

-<g> 1.906 1.964

a

b

c

2-fiq. 4.4: a. ESA-spectrum of CrOF ₅ in 38% HF; Q-band. q

=

1.964.

53

2-b. ESA-spectrum of Cr _{OF 5} in 38% HF; X-band. q

=

1.964; A

=

23.3 Oe.

53

2-c. ESA-spectrum of Cr OF ₅ in 38% HF; Q-band.

q

=

1.964; A

=

23.1 Oe.