THE JOURNAL OF CHEMICAL PHYSICS VOLUME 39, NUMBER 2 15 JULY 1963

Calculati~n of th; Electron Spin Resonance Line Shape of Randomly Oriented Molecules m a Tnplet State. I. The Am =2 Transition with a Constant Linewidth

PHILEMON KOTTIS AND ROLAND LEFEBVRE

Centre de Mecanique Ondulatoire Appliquee, 23 rue du Maroc, Paris 1ge, France

(Received 27 December 1962)

The electron spin resonance line shape for the ~m=2 transition of randomly oriented aromatic molecules in a triplet state is analyzed. This is an extension of the treatment of van der Waals and de Groot to mole­cules havin? less thll;n ternary symme.try. In the present paper the spectrum of a molecule is approximated by a GaussIan functl?n of constant WIdth. and the electron g tensor is assumed to be isotropic. The various stages of the calc~latl~n have ?een fitted mto a computer program, using as input data the principal values of t~e electron spm-spm couplin~ tensor. The calculations show that a characteristic structure may be pres­ent m the spectru~. The expenmental ~pectr~ of naphthalene and its deuterated homologue present in effect enough of this structure to make It pOSSIble to determine the absolute values of the two constants characterizing the electron spin--spin interaction.

1. INTRODUCTION

ELECTRON spin resonance (ESR) experiments on aromatic molecules in a triplet state were first

performed by Hutchison and Mangum,! who found that each phosphorescent naphthalene molecule, diluted and oriented in a single crystal of durene, gives rise to two anisotropic absorption lines. Soon afterwards, van der Waals and de Groot2 have pointed out that the third possible transition for a triplet, which is forbidden in the limit of a strong magnetic field, and is currently referred to as the fi.m= 2 transition, can also be ob­served. From the spin Hamiltonian, they determined for this transition the line shape for an assembly of randomly oriented molecules having at least ternary symmetry. They were able on this basis to deduce from the spectra observed in glassy materials the value of the constant characterizing the electron spin-spin interaction in a number of such molecules.

We report in this paper some calculations of the line shape for aromatic molecules of lower symmetry. Even in these cases it is sometimes possible to arrive at the values of the two constants describing the electron spin-spin coupling. In order to perform such calcula­tions the ESR spectrum of a molecule in a triplet state is first studied as a function of the angles specifying its orientation with respect to the laboratory axes. The spin Hamiltonian is restricted to the electron Zeeman term and the electron spin-spin interaction. The reso­nance fields and the transition probabilities under the influence of an oscillating magnetic field making a constant, but arbitrary, angle with the static field are calculated as functions of the three angles giving the orientation of the molecular axes. This analysis is similar to that encountered in the calculation of the ESR spectra of some metal ions.3

1 C. A. Hutchison and B. W. Mangum, J. Chem. Phys. 29, 952 (1958); 34, 908 (1961).

2 J. H. van der Waals and M. S. de Groot, Mol. Phys. 2, 333 (1959); 3, 190 (1960).

3 e.g., E. O. Schulz-du-Bois, Bell System Tech. J. 38, 271 (1959).

We then proceed to write an expression for the line shape and its derivative for the fi.m= 2 transition of a glassy sample. The denomination fi.m= 2 is in fact a rather loose one. We shall mean more precisely that the electron Zeeman energy and the spin-spin inter­action are of such relative magnitude as to lead for every orientation to three ESR transitions. The line shape is then calculated by taking into account only the lowest of the three resonance fields. The calculation is made under the assumption that the absorption line has a Gaussian shape, with a width which is independent of the molecular orientation. Some applications to the analysis of observed signals are discussed.

II. RESONANCE FIELD

In an electron spin resonance experiment carried out at a fixed frequency, in a variable static magnetic field, the quantity of interest is the resonance field, instead of the resonance frequency as in other types of spec­troscopic measurements. The resonance field is the magnetic field which is of such an intensity as to pro­duce two electron spin levels separated by an energy equal to the quantum of microwave energy. To calcu­late the resonance field of an aromatic molecule in a triplet state we follow closely the treatment given by van der Waals and de Groot.2 The spin Hamiltonian which determines the electron spin levels is4

(1)

In this expression Sx, Sy, and Sz are the components of the total spin S associated with two electrons in the molecular system of axes which diagonalizes the electron spin-spin coupling tensor. D and E are related to the principal values of this tensor, which will be denoted X, Y, and Z. Instead of (1), however, we shall make

4 A. Abragam and M. H. L. Pryce, Proc. Roy. Soc. (London) A205, 135 (1951).

393

394 P. KOTTIS AND R. LEFEBVRE

z

FIG. 1. Reference systems, Oxyz in the laboratory, OXYZ in the molecule. ON is the intersection between the xoy and XOY planes, OM the projection of oz in ~he ~OY plane. u: is ~ unitary vector in the xoz plane, along the directIOn of the oscIllatmg field. H is the static magnetic field, a the angle between the two fields.

use of the following spin operator,2

H=g/3S·H-{XSx2+Y Sy2+ZSZ2), (2)

in which all three axes play an equivalent role. The relation between the two sets of constants is as follows:

X=!D+E, Y=!D-E, Z=-jD. (3)

These quantities are precisely those we would wish to be able to deduce from observed ESR spectra. They have been defined by van der Waals and de Groot2 in terms of integrals over the wavefunction describing the two "odd" electrons, and more recently McWeeny5

has shown how to relate them to a wavefunction not based on this approximation. In writing the electron Zeeman term we assume that the electron spectroscopic splitting tensor is isotropic. This is a good approxima­tion for aromatic hydrocarbons, as shown both by the ESR measurements performed on single crystals,' and the weakness of the spin-orbit interaction in all these systems, as inferred from phosphorescent life­time measurements and theoretical considerations.6

The three Euler angles giving the orientation of the molecular axes (OXYZ) with respect to fixed axes (oxyz) are shown in Fig. 1. The electron spin levels of the system in the presence of a static magnetic field assumed to be in the z direction depend only on the two angles (j and cJ> giving the direction of H in the axes. We use as basis functions the eigenfunctions in a zero magnetic field2 :

Tx=2-1{/3(l)/3(2) -a(1)a(2)},

Ty=i2-i {/3(1)/3(2) +a(Oa(2) },

Tz= 2-l{a(0/3(2) +/3(1)a(2) }. (4)

Here a and /3 correspond to the quantization of the spin of an electron with respect to the molecular Z axis. One may then write the energy matrix using the operator (2) and the basis functions (4), and obtain

5 R. McWeeny, J. Chern. Phys. 34, 399 (1961). 6 D. S. McClure, J. Chern. Phys. 20,682 (1952).

from it the secular equation giving the three energy levels. This is

E3_E[(g/3H)L(XY+XZ+YZ)J

+ (g/3H){X sin2(j cos2cJ>+ Y sin2(j sin2cJ>+Z cos2(jJ

-XYZ=O. (5)

After some algebraic manipulation we find that the condition for two roots of this equation to be distant by the quantum o=hv of microwave energy can be written as

X sin2(j cos2cJ>+ Y sin2(j sin2cJ>+Z cos20=XYZ(g/3H)-2

=r3-J[(g/3H)-2(02+XY+XZ+YZ) -lJ X[(4g/3H)2_ 0L 4(XY+XZ+YZ)]i. (6)

For a given molecule this can be written as

l(O, cJ» =F(H, 0). (7)

The solution of this equation which determines H for a given 15 and a given orientation can be discussed graphically. If we construct the function F(H, 15) with H as the variable, and if we draw a parallel to the H axis at an ordinate equal to the values of l(O, cJ», the abscissas of the intersections (if any) give the resonance fields.

We shall examine in detail one case, that of naph­thalene, using the values of X, Y, and Z found by Hutchison and Mangum. ' ,7 This molecule is probably typical of many aromatic systems. Fig. 2 presents the function F(H, 0) for this molecule, with 0/hc=O.3095 cm-I (this corresponds to a microwave frequency of 9279 Mc/sec). It is clear at once from the graph that there are for any orientation three resonance fields. It is also clear that two of them (corresponding to the so-called" !J.m= 1" transitions observed by Hutchison and Mangum l ) are very anisotropic, and the third one (corresponding to the "!J.m= 2" transition observed by van der Waals and de Groot2) is very much less so. The largest resonance field in the quasi-isotropic region is obtained here when l(O, cJ» takes the value Z. This gives Hz= 1648 G. The lowest possible resonance field is that where F(H, 0) ceases to be a real function. This is

Hmin = (2g/3)-1[02+4(XY + XZ+ YZ) J1,

= (2g/3)-I{ i5L 4[(D2/3) + E2JI1, (8)

which has here the value 1527 G. Thus there is a spread of 121 G in the distribution of resonance fields. The abscissas of the two other intersections for j(O, cJ» =Z give the lower and upper limits for the resonance field

7 These values (for naphthalene diluted in a single crystal of light durene) are in em-I: X=0.0197; Y=0.0471, Z=-0.0669. Here X correspon'ds to the short axis, and Y to the long axis, of the molecule. We take for g the average of the three g values (2.0030) .

ELECTRON SPIN RESONANCE LINE SHAPES 395

FIG. 2. The function F(H, 15) for naphthalene at 9279 Me/sec.

in the "Am= 1" region. This gives a spread of the order of 2150 G for these transitions.

Some more results can be obtained from considera­tion of this graph. In their papers van der Waals and de Groot pointed out that in the case where two con­stants describe the spin-spin interaction, there should be a maximum in the absorption near the field given by Expression (8). This result also follows from our graphical representation, and can be obtained from differentiating Eq. (7). There is

dj=F'(H,o)dH, or dH=dj/F'(H, 0). (9)

Stationarity in the resonance field occurs when F' (H, 0) goes to infinity, which is the case for H=Hmin but also when dj=O while F'(H, 0) is different from zero. The latter condition is satisfied when the static field is along one of the three principal directions of the spin­spin coupling tensor. That F'(H, 0) is different from zero in the present case for these orientations can be seen directly on the graph. For every such canonical orientation there are three stationary fields, one in the Am= 2 region and two in the Am= 1 region (for this region the stationarity can also be seen on the curves given by Hutchison and Mangum' ). The slope F' (H, 0) being large in the first region, the accumulation of resonance fields might be effective in producing some

TABLE I. Expressions for the canonical resonance fields for the X direction.·

HxI = (2g(:J)-1[02- (Y -Z)2Ji

HxII = (2g(:J)-1[(2t5-31 X I )2- (Y -Z)2Ji

HxIII = (2g(:J)-1[(2t5+31 X 1)2- (Y -Z)2Ji

a The canonical resonance fields for the Y and Z directions are obtained through cyclic permutation of X, Y, and Z in these expressions.

structure in the absorption spectrum of a glassy sample, even if the transition probability is rather small. Ab­sorption in the Am= 1 region for a glassy sample has also been recently observed,8.9 and the features of these spectra are also related to the stationary behavior of the resonance field.

Table I gives the formulas from which one may calculate the fields for the canonical orientations. In the present example one obtains as points of "accumu~ lation" the following (in gauss) :

in the Am = 2 region, and

in the Am = 1 region. As a confirmation of this discussion, we have calcu­

lated an approximation to the function giving the distribution of resonance fields for naphthalene in the quasi-isotropic region. If the resonance field is written as H (fJ, e/», the function giving the density of molecules having their resonance fields between Hand H +dH is the following double integral over the sphere:

f2" f" ')'(H) = (47r)-' de/> dfJo[H-H(fJ, e/»] sinfJ. o 0

(10)

In view of the difficulty of evaluating this integral, a normalized Gaussian was used to simulate the Dirac

8 J. H. van der Waals and M. S. de Groot (private communi­cation) .

9 W. A. Yager, E. Wasserman, and R. M. R. Cramer, J. Chern. Phys. 37,1148 (1962).

396 P. KOTTIS AND R. LEFEBVRE

1t T

e

1500.

o-tJ-i-

1550

H1 y

leoo

FIG. 3. Distribution of resonance fields in the ~m=2 region for naphthalene at 9.279 kMc/sec.

function. The following integral has been evaluated

'Y'(H) = (47rA)-1(~Yf"dcjJ ~"d8

X exp{ -2[H-:(8, cjJ)J2} sin8 (11)

using the Gaussian quadrature method with 8X96X96 points. In this calculation A should be rather small, but too small a value would make the numerical quad­rature inaccurate. A value of 2 for A appeared to be reasonable. The resonance field for every orientation is obtained as explained in the appendix.

The function 'Y'(H) resulting from this calculation is displayed in the upper part of Fig. 3. The two first maxima correspond to Hmin and HXl. There is a slight maximum in the region H yl, while near H Zl there is only a fall to zero of the distribution function, due to the fact that no molecules have a resonance field for HZl<H <HZ2. The present calculation was not accurate enough to produce a maximum for H = H Zl. The fact that the maxima get smaller for an increasing field is to be related to the decrease in the slope of the function F(H, 0). The lower part of Fig. 3 helps in understand­ing how the resonance fields are distributed when varying the orientation. The curves giving H as a

function of 8 for different values of cjJ are all tangent to the vertical at H=Hmin. The partial derivatives iJH/iJ8 and iJH/iJcjJ at Hmin and for every canonical orientation are zero. This diagram shows also that for H>Hyl there is a sudden decrease in the distribution function.

The distribution of resonance fields which has just been discussed is not a property of the molecule by itself. It is dependent upon the choice of microwave frequency. As an example of this, Fig. 4 shows four curves representing F(H, 0) for naphthalene at other microwave frequencies (other values of 0).

(a) 0=0.75 em-I. The quasi-isotropic region is dis­placed to higher fields (Hmin=4000 G) and the spread is reduced to 45 G. The anisotropic region is also some­what translated, but the spread is still about the same.

(b) 0=0.115 cm-l (0<41 XY+XZ+YZ I). The threshold H=Hmin no longer exists, but there are still three transitions for every orientation, and the sta­tionarity inf(e, cjJ) should produce an accumulation of resonance fields for the canonical orientations.

(c) 0=0.100 cm- l (0< 1 Y-Z I). There are only two transitions for every orientation.

(d) 0=0.025 cm-1 (0<1 V-X i). Only at particular orientations are there two transitions.

II. TRANSITION PROBABILITY

The next step in the calculation is to derive an expres­sion for the transition probability under the influence of an oscillating magnetic field. We assume that the angle, say a, between the oscillating field and the static field can take any value between 0 and 71'/2 (in most experiments this angle is either 0 or 71'/2). Without loss of generality we may assume the oscillating field to be in the xoz plane of the laboratory system of axes (d. Fig. 1). The third Euler angle if; giving the orientation of the molecule will come into play here, as two different molecules may be in the same situation with respect to the static field, but in different situa­tions with respect to the oscillating field. The relative transition probability is given by the expression:

B(8, cjJ, if;, a) = 1 {<PI 1 u-s 1 <P2) 12, (12)

where U is a unit vector in the direction of the oscillating field. <PI and <P2 are two electron spin eigenfunctions, which are linear combinations of the form:

(j=X, Y,Z; k=1,2), (13)

where the T/s are those defined in the relations (4). One thus get the formula (d. Ref. 2)

B(e, cjJ, if;, a) = 2:ViPi(Cl*/\['2);(C2*/\Cl)i i.i

(i,j=X, Y, Z). (14)

The v/s in this expression are the direction cosines of U in the molecular system of axes. Formulas for the C/s are given in the Appendix. The above expressions

ELECTRON SPIN RESONANCE LINE SHAPES 397

.--r--.,.------~-.. -------------.. ---.. -----------------., , ..... : I I , I

+0.4 I I

I

, . I

-ott , , I I I I I I I I

-or-L------------------------------------------------ ------ - --- - - - ----(a)

+D.1r --- -------.... _------------------------------------... ----i

em-I! I

i of(l~ _________________________________________ _

I .: I I I ' I , , ,

X +------~---- -----------------.... ---- .. -__ .. -- .. ~-- -______________ _ : ! I I I : I H' !H!Ht :H1 IHa H' °r""----J.-'£--t::t1:-- 10rO-----1.-Jt.---15'U,------j-!.--~-f---------~ I I I I: I :: : !: : I' , :: :

-~ I , 'I I I I I " I zt------------ ----------------------------!-------------I

! .. 0Jrl ... --------- ------------_________________________________________ _

(c)

(b)

+UT- ----------------------------------... -------~-.. ------i . cm-t:

I I I I

Yt--- -------"------------i !----~ i!. :

x .... -~--.. - .. ---------+_-f I I 1,_. tf!~'" I

o+---l'.!:c:------~------1.,,--t:r·------1H!-~-~----------~-+----________ + • 100, 1000 HOO aooo lIDO I ' .r

i t i , I ' I : i iii

I I , I , I

: : ~ z· .. -----------... -------: : : I

_0.11____________ ---------------------------------------------------

(d)

FIG. 4. The function F(H, 0) for naphthalene at four different microwave frequencies (a) at 22.7 kMc/sec; (b) at 3533 Me/sec; (c) at 3000 Mc/sec; (d) at 750 Mc/sec.

are valid for any pair of levels that an appropriate choice of H has brought to a distance 0 from one another.

At this point in the calculation we take the average of B over the angle 1/;, because all molecules differing only by the value of 1/; have the same resonance field. It is this averaged probability which is used in the calculation of the line shape for randomly oriented molecules. This quantity is denoted B. We have

B(O, cp, a) = L(V;JJi)( Cl* 1\ C2L( C2* 1\ Cl) i i,i

(i,j=X, Y, Z) (17)

where (ViVi) is the average of ViVi over the angle 1/;. Formulas for the v/s and the (ViVi )'s are given in the Appendix.

There are some properties of B which will be useful in the foregoing discussion. When the magnetic field is parallel to a canonical axis, all the quantities coming

into the expression for B are stationary and B is also stationary. Its value is largest for a= 0 (oscillating fields parallel to the static fields) and drops to zero for a=7r/2 (oscillating field perpendicular to the static field) if the resonance field is given by an expression such as HxI of Table I. The reverse holds for the fields given by expressions such as HxII and HXIIl. The stationary behavior is shown in Figs. 5(a) and 5(b) which give B for the 11m = 2 transition naphthalene, for three values of a (0°, 45°, and 90°).

III. LINE SHAPE FORMULA

In the case of a poly crystalline or glassy sample, the spectrum which is measured is the superposition of the spectra of randomly oriented molecules. The calcula­tion of the line shape is similar to that of the function 'Y(H) of the previous paragraph. If we were to assume that the absorption line for a given molecule has an infinitely narrow linewidth, we would have for the line

398 P. KOTTIS AND R. LEFEBVRE

J 8z

9 .n

it T

e

2

0.5

1!. _ if! 2

0.1

~_CI> _ 0

0.05

0.01

- o

\ Bx

(a)

B (01,=0)

0.1 (b)

0.1

FIG. 5. The transition probability of the !J.m=2 transition of naphthalene at 9282 Me/sec (a) for «=0°; (b) for «=45°; for «=90°.

shape the formula

g(H, a) 0:: ["dtlt ["de/> j"dOB(O, e/>, tit, a)o 000

X[H-H(O, e/»] sinO,

j 2 .. j" 0:: de/> dOB(O,e/>,a)o[H-H(O,e/»] sinO.

o 0

(18)

However, the spectrum of a molecule is broadened by the electron-nuclei hyperfine interactions. In planar aromatic molecules, the intramolecular hyperfine inter­actions are very anisotropicl (the ratio for different orientations can be as large as 3 to 1). On the other hand, the hyperfine structure is rarely resolved, as shown by the experiments of Hutchison and Manguml on single crystals, and the spectrum is very nearly Gaussian, with an over-all width which does not show a considerable change upon varying the orientation. This is due essentially to the fact that the various hyperfine coupling tensors have their principal axes arranged in such a way as to make a direction of large coupling for some tensors a direction of small coupling for others, and vice versa. Therefore, at least for aro­matic molecules, it is reasonable to try first the assump­tion of a constant linewidth. This means using for the line-shape function the expression

g(H, a) 0:: ["de/> ['dOB(O, e/>, a) o 0

X exp{ -2[H-A~(0, e/»]2} sinO (19)

X here measures the distance between the points of maximum and minimum slope. This function of H for a given a (in practice a is either zero or 7r/2) can be calculated by the same method as that used for the function giving the distribution of resonance fields. A double Gaussian quadrature is performed, using the values of B(O, e/>, a) and H(O, e/» calculated for a num­ber of orientations. Once g(H, a) has been calculated over the range of static magnetic field intensity of interest, it can be normalized to give its usual meaning to the line-shape expression. These various steps have been incorporated into a program which uses as input quantities the microwave frequency, the values for X, Y, and Z, the linewidth X and the spectroscopic splitting factor. The output data are the values of g(H, a) at specified values of H, and also the values of dg(H, a) IdH which is given by the expression

dg(H, a) j2.. j" _ --"---0::- de/> dOB(O,e/>,a)[H-H(O,e/»]

dH 0 0

{ -2[H -H(O, e/» ]2} .

X exp smO. X2

(20)

It is, in fact, this derivative which is of direct interest, since most ESR measurements produce the first deriva­tive of the absorption spectrum.

IV. APPLICATION TO THE ANALYSIS OF SPECTRA

The previous discussion suggests that even for a glassy sample, where molecules are randomly oriented the line shape and its derivative may show some struc­ture. Many aromatic molecules, with delocalized elec­tron magnetization, should behave like naphthalene at

ELECTRON SPIN RESONANCE LINE SHAPES 399

X band (",9000 Mc/sec), that is with four stationary resonance fields in the D.m= 2 region. Other systems with smaller or larger spin-spin coupling energy could be made to behave in the same way if an appropriate choice of microwave frequency is made. We limit the discussion to this case, our main reason being that we had at our disposal some good spectra falling into that category.

The problem is essentially that of assigning some values to the stationary fields from an examination of an observed spectrum. For a given value of g any pair of these fields make it possible, in principle, to compute X, Y, and Z through the formulas of Table I. It is well to realize the limitations which are inherent in such a procedure. The first limitation concerns the sign of the principal values. The spectrum of a molecule being invariant under a simultaneous change of sign of all three quantities, this makes it possible to deter­mine only the relative signs (1) and this is also true for a glassy sample. lO Furthermore, even if we manage to deduce the three principal values of the coupling tensor (except for this uncertainty about the absolute signs), this does not tell us anything about the way the principal axes of the coupling tensor are arranged with respect to the geometry of the molecule. An exception is the case where the symmetry is sufficient to deter­mine the principal directions (e.g., naphthalene) but even in this case there is no way of establishing a cor­respondence between the principal values and the principal axes. Finally it is not possible from a spec­trum to obtain detailed information about the electron g tensor. However if consistency among the results can be obtained with the use of a certain g value, this should both confirm the validity of the assumption of a nearly isotropic tensor, and indicate a trend for the average g value.

Just as a matter of convenience, the stationary reso­nance fields will now be labeled Hx, Hy, Hz in order of increasing value. We have, therefore, the inequalities

or less of the stationary fields (for instance the maxima due to Hx and Hmin may merge into a single maximum). One may wonder, here, whether it is of importance to decide if a given field is the first, second, or third of the three stationary fields. The answer is no, because the formulas for these fields (d. Table I) are permuted into one another if X, Y, and Z are permuted. There­fore any assignment will do, and once X, Y, and Z have been calculated, they can be relabeled if neces­sary in order for the fields to obey the inequalities of (21). This shows that if two molecules have Hmin and one canonical field in common, they have also the other two canonical fields in common.

V. TWO EXAMPLES

The procedure which has been outlined above will now be applied to p-deutero-naphthalene and naph­thalene. The spin-spin coupling tensors are known for the molecules diluted in a single crystal of durene.1

However, we wish to show that the D.m= 2 spectra for glassy samples can also lead to this information.

The derivative of the ESR spectrum of d-naphthalene dilu ted in a 3: 1 decalin cyclohexane glass, in the region 1520-1650 G is shown in Fig. 6 (solid line).ll The microwave frequency is 9283 Mc/sec, the oscillating field being parallel to the static field. This is clearly the sort of spectrum we would expect on the basis of the distribution function calculated from the spin-spin coupling tensor of naphthalene in a single crystal of durene. In a preliminary analysis we may assign the first zero in the derivative (at 1528 G) to Hmin , the second zero (at 1532) to Hx and the minima at 1591 and 1646 G to Hy and Hz. X, Y, and Z can be calcu­lated from Hmin and anyone of the three other fields. We use for g the free-electron value 2.0023. This gives (with the convention that Z should be negative):

From the pair (Hmin, Hx), X = +0.0114 cm-I,

Y = +0.0530 cm-I, Z= -0.0644 cm-I ;

Hmin~HX~Hy~Hz and 1 Z 1 ~ 1 Y 1 ~ 1 X I· (21) . ( H ) From the paIr H min , y, X = +0.0188 cm-I,

As will appear below, H min is always easily recog­nizable because it corresponds to a threshold value for the absorption. Some characteristic details in the spec­trum to the other stationary fields must still be as­signed. The shape of a curve such as "/' (H) of Fig. 3 suggests that such details might consist either in a more or less pronounced maximum (d. the maximum corresponding to Hx) or to a drop in the intensity of absorption (d. the drop at H y). However in view of the complicated form of the line-shape function (19), this correspondence is not a very strict one. The computer program will be useful to investigate this point.

In some cases one may be able to identify only two

10 See, however, A. W. Hornig and J. S. Hyde, Mol. Phys. 6, 33 (1963), for the determination of the sign from very low tem­perature measurements.

Y = +0.0480 cm-I, Z= -0.0667 cm-I ;

from the pair (Hmin, Hz), X = +0.0168 cm-I ,

Y = +0.0500 cm-I, Z= -0.0668 em-I.

The results are reasonably consistent. However, the discrepancies should not be interpreted as indicating the approximate character of the analysis (due for instance to the use of an isotropic g tensor instead of an anisotropic one). As we now show a more careful study of the relationship between the stationary fields and the characteristic details of the spectrum enables us to reduce these discrepancies. For this purpose we

11 This spectrum and that for ordinary naphthalene reproduced in Fig. 9 have been kindly communicated to us by van der Waals and de Groot.

400 P. KOTTIS AND R. LEFEBVRE

o.

8 =O.S0S65cm~ 9":2,0038 A.:5,89IUSS

0, 1524.5 15 U.S

0', 1528 1528

1. 1532.51533

2. 1591.5 1592

FIG. 6. The tim=2 derivative spectra of d-naphthalene in a glass. Solid line: experimental; dotted line: calculated. The scale of the ordinates is chosen as to have the first maxima in coincidence. The scale for the detail at 1645 G is 50 times that of the rest.

have calculated the line-shape derivative for some trial values of the parameters X, Y, Z, and g, with a series of values for X (the linewidth parameter). The con­clusions are:

(a) The stationary fields fall always in rather narrow intervals. If we call Ho the first maximum of the derivative, HI the second maximum, H2 and Ha the minima which were considered in the preliminary analysis, we have the following rules:

Hmin=Ho+0.3X±0.1A,

Hx=HI+O.3X±O.1A,

Hy=H2+O.1A±O.1A,

Hz=Ha+O.1X±0.1X. (22)

Furthermore, if we call f:.Bo the distance from the first maximum to the first zero there is the approximate relation

X= f:,Ho(1.5±O.2). (23)

(b) These rules which were obtained for the naph­thalene case were found to hold also in several other examples. Although it would be better to check their validity in every case, they appear to furnish a good starting point for the analysis. But some caution must be exercised, as shown by the ternary case (X = Y).

Hz disappears and Hx and Hy can be obtained for the last minimum of the derivative (say at HI) from the relation

Hx=Hy=HI-O.3X±0.1X. (24)

(c) The relations of (22) do not depend on the value chosen for g. Therefore, they may serve to "read" the canonical stationary fields even in the case of an ani­sotropic g tensor if the principal axes of this tensor coincide with those of the spin-spin coupling tensor. This may be seen from the following argument: the calculation could be done with a g value equal to the principal value of g for one of the canonical axes. The stationary field for that axis is calculated correctly, and the line shape in the neighborhood of this field should be well reproduced.

For d-naphthalene the best value for X appears to be of the order of 6 G. From the values given on Fig. 6 for the fields Ho, HI, H2, H3 and the relations (22) we ob­tain for the stationary fields (in gauss)

Hmin' = 1527 ±2.5, H y' = 1591±2.5,

H x'= 1533±2.5, Hz' = 1646±2.5. (25)

The dashes indicate that these fields are not to be con­fused with their theoretical counterparts. We have not indicated the uncertainties such as they would result from an application of the rules in (22) (they would

ELECTRON SPIN RESONANCE LINE SHAPES 401

amount to 0.6 G at most). The 2.5 G given in (25) take into account other sources of error which are:

(a) Possible errors in the calibration of the spectrum. This may be here of the order of one gauss.

(b) The use of a constant linewidth in the derivation of (22). As the maximum deviation in these relation is 0.3~, this error is of the order of 0.3~~ where ~~ is a measure of the anisotropy of the linewidth. As this quantity amounts to about 1G,1 the corresponding error is of a tenth of gauss.

(c) The use of an isotropic ~ tensor in the analysis. We may consider that the stationary fields obtained from the observed spectrum approximate those of a molecule with the same spin-spin coupling tensor and an isotropic ~ tensor. The single-crystal studyl shows that the anisotropy of ~ around the average value does not exceed 0.0010. At the X band the corre­sponding displacement of the fields is of 0.75 G.

We may remark that the effect of the anisotropy of ~ is smaller than the other effects. This justifies the use of an isotropic ~ tensor to interpret the experi­mental spectrum. The three independent parameters which occur in the expressions for the stationary fields have been calculated from the minimization of the quantity

(Hmin-Hmin')2+(Hx-Hx')2+(Hy-Hy')2+(Hz-Hz')2

(26)

under the condition X+Y+Z=O. This has given

Y = +0.049±0.OO2, g= 2.004±0.001,

X = +0.017 ±0.003, Z= -0.067±0.OO1. (27)

The errors on X, Y, and Z are calculated from the un­certainties for the fields given in (25).

As a check of the consistency of the analysis, we have recalculated the three principal values of the spin­spin coupling tensor from the three pairs of fields which were considered before. We obtain:

From (Hmin', Hx ') X = +0.0165 cm-I,

Y = +0.0498 cm-l Z = -0.0664 cm-I ,

From (Hmin', Hy')

Y = +0.0489 cm-l Z= -0.0667 cm-I,

From (Hmin', Hz') X = +0.0180 cm-I,

Y = +0.0488 cm-l Z = -0.0667 cm-l.

The calculated line shape derivative is shown as the dotted line in Fig. 6. The numbers indicate the fields at which some characteristic points occur in the experi­mental and calculated spectra. The agreement is satis­factory except in the region of Hx. This is probably due to the assumption of an orientation-independent line­width. The spectrum in that region varies considerably, even for a variation of 1 G of the linewidth parameter.

FIG. 7. Effect of the linewidth parameter on the calculated spectra of d-naphthalene.

This is shown in Fig. 7, which gives two spectra calcu­lated for ~=4.7 and ~=6.7 G. The anisotropy of the linewidth for d-naphthalene is of that order. Therefore one might hope to get some improvement if the de­pendence of the linewidth upon the orientation is introduced. This point is being presently investigated, and will be discussed in a later paper.

Figure 8 shows the effect on the calculated spectra of d-naphthalene of varying the angle between the oscillating field and the static field. For a= 60° the spectrum is already somewhat smoothed out, and for a=90° there remains only the maximum corresponding to Ho. This is due to the vanishing of the transition probabilities for those molecules which have the static field along one of their canonical axes.

Our next example is naphthalene. The solid line of Fig. 9 gives the parallel derivative spectra in the region 1515-1600 G at 9279 Mc/see. The spectra bear some resemblance to that of d-naphthalene, but due to a much larger linewidth the structure corresponding to HxandHz has disappeared. We may still calculate the principal values of the spin-spin coupling tensor from Hmin and H y , if we use for the average g value that obtained for d-naphthalene. This gives

X = +0.020 cm-I±O.OOl em-I,

Y = +0.048 cm-I ±0.002 cm-I,

Z= -0.069 cm-I±O.OOl em-I.

The spectrum calculated on this basis is given as the dotted line in Fig. 9.

Another feature of the experimental spectrum of naphthalene is worth mentioning. In the region of Hy(·,-, 1580) the spectrum has the aspect of a quintuplet with a 7-8 G interval between successive lines. As we know from the single-crystal work, this stationary field corresponds to the static field along the long axis of the naphthalene molecule. It is for this orientation that Hutchison and Manguml were able to resolve the

402 P. KOTTIS AND R. LEFEBVRE

1. 1. 1. 1. 1524

2. 1528 1. 1523 1.. 1523

2. 1528.5 2. 1528 3. 1533

*. 1538.5

5. 1589

3. 1534-

-t. 1588.5

3. 1529

5'. 4.

2. 2. 3.

FIG. 8. Effect of (x, the angle between the static and oscillating magnetic fields on the structure of the spectra of d-naphthalene.

hyperfine structure at 23.2 kMc/sec. They found a quintuplet that they assigned to the four equivalent a protons. For this orientation, the hyperfine couplings with these protons should be much larger than with the {3 protons. The quintuplet occurring at Hy in the spec­trum of Fig. 9 suggests that this hyperfine struCture can be observed even for a glassy sample. Both the resonance field and the hyperfine couplings with the

t.

2.

s.

f. ISH.9 1515

2. 1523 1523

S. 1529.+ 1532

+. 1511.8 1581

FIG. 9. The ll.m=2 derivative spectra of naphthalene in a glass. Solid line: experimental. Dotted line: calculated. The scale of the ordinates is chosen as to make the first maxima to coincide.

a protons are stationary when the static field is along the long axis of the molecule, since the electron­electron spin-spin tensor and the electron-a-proton spin-spin tensors have this direction as principal axes. Figure 3 indicates that other molecules in the sample have their resonance fields in the region of H y • How­ever, consideration of the corresponding transition probabilities show them to be small. The spectra of these molecules are not intense enough as to blur out the hyperfine structure. The occurrence of this hyperfine structure, and the fact that the best linewidths to use in the calculation of the line shapes are very nearly equal to the average linewidths for the molecules in a single crystal suggest that the variation of the spin­spin tensor from a site to another in the glass does not exceed 1%.

Coming back to the problem of assigning the prin­cipal values to particular axes and of making a choice of the sign, it is to be noted that wave-mechanical calculations should be of great help here. For instance McWeeny5 has obtained for naphthalene:

long axis +0.0491 cm-I, short axis +0.0085 cm-I ,

third axis -0.0577 cm-I •

VI. DETERMINATION OF THE g TENSOR

When the g tensor has the same principal axes as the spin-spin coupling tensor (e.g., naphthalene), the ex­pressions for the canonical stationary fields are simple. For H y for instance there is the formula

The previous discussion has shown that one may be

ELECTRON SPIN RESONANCE LINE SHAPES 403

able to determine the canonical stationary fields from the observed spectrum even in this case. It should then be possible to determine the principal values of g from a comparison of the fields obtained from two spectra taken at different microwave frequencies. There is the relation

where Oa, and Ob are the two quanta of microwave energy and H ya and H yb the two corresponding canoni­cal fields for the Oy direction.

The error on gyy resulting from uncertainties in the reading of Hya and Hyb that we may take equal to a common quantity t:.H is given by

(30)

For naphthalene if t:.H amounts to 3 G, and if the spectra are taken at 9 kMc/sec (H yh..>1500 G) and 25 kMc/sec (Hy""-'4500 G), t:.gyyamounts to 0.0020. This shows that the procedure will be useful either when t:.H can be reduced, or when the anisotropy is larger.

VII. CONCLUSIONS

The assumption of a constant linewidth and of an isotropic g tensor appears to be reasonably successful for aromatic molecules. However there should be other classes of compounds where this might not be so. For instance, in certain carbenes and nitrenes which have a ground triplet state and for which ESR spectra have been recently obtained12 ,13 the linewidth may be much more anisotropic. In addition it would be desirable to make calculations of the entire spectrum, as the analyses of the other regions should also lead to some useful information. The program used here to calculate the resonance field and the transition probability for the t:.m = 2 transition need only minor changes to apply to the other transitions.

ACKNOWLEDGMENTS

We are most grateful to Dr. J. H. van der Waals and Dr. M. S. de Groot for communicating their spectra before publication. Our work owes much to the stimu­lation we have received from discussions on this prob­lem with them. Part of the computing time used in the present work was made available through a NATO grant.

,. R. W. Murray, A. M. Trozzolo, E. Wasserman, and 'V. A. Yager, J. Am. Chern. Soc. 16, 3213 (1962); G. Smolinsky, E. Wasserman, and W. A. Yager, ibid., p. 3220.

13 R. W. Brandon, G. L. Closs, and C. A. Hutchison, Jr., J. Chern, Phys, 37, 1878 (1962).

APPENDIX: CALCULATION OF THE RESONANCE FIELD AND THE TRANSITION PROBABILITY

(1) The resonance field. The condition (7),

f(l), cf» =F(H, 0),

can be transformed into a third-order polynomial in the variable y=H2. The system has three resonance fields when this polynomial in y has three real positive roots. According to the convention used here the "t:.m = 2" resonance field is the lowest of the three fields.

(2) The transition probability. The two energy levels involved in the transition can be given the form

E1= 0/2± (12)-![(2g{3H)L (o2+4XY +4XZ+4YZ)]t,

E2=EI -0. (31)

The indetermination in sign corresponds to that oc­curring in the condition (6). The square of the modulus of one of the coefficient e\ say exi can then be obtained from the secular equation. This gives

I exi 12= {hy2( Y - Ei)2+ (g{3hxhz)2+hx2(X - Ei)2

+ (g{3hzhy )2+ (g{3)-2[(X - Ei) (Y -Ei) - (g{3!zz)2]1-1

X [h y2 ( Y - Ei) 2+ (g{3hxh) Z2]' (32)

We adopt the convention that exi is a real positive number. The other coefficients are:

eyi= exi[ig{3hxhz- (Y - Ei)h y ]-I[(X - Ei)hx+ig{3hyhz],

ezi=exi[ig{31 (Y - Ei)hy-ig{3hxhz}]-1

X [(g{3hZ)L (X -Ei)( Y -Ei)]. (33)

In (32) and (33) hx, hy, and hz are the components of the static magnetic field in the molecular system of axes.

The direction cosines of the oscillating field in the molecular system of axes are (d. Fig. 1):

Vx = sina cosO coscf> cos1/t- sina sincp sin1/t

+ cosa sinO coscp,

Vy = sina cosl) sincf> cos1/t+ sina coscf> sin1/t

+ cosa sinO sincp,

v.= - sina sinO cos1/t+ Cosa cosI)o (34)

The averages over 1/t of their squares and cross products are given by:

(vx2 )1/t= cos2a sin21) cos2cf>+! sin2a(1- sin20 cos2cf»,

(Vy2 )1/t= cos2a sin20 sin2cf>+! sin2a (1- sin20 sin2cf» ,

(v} )1/t= cos2a cos20+! sin2a sin20,

(VXvy )1/t=! sin20 coscp sin¢(3 cos2a-l),

(vxv.)1/t=! cosl) sinO coscf> (3 cos2a-1),

(vyv.)1/t=! cosO sinO sincp(3 cos2a-l). (35)