Volume 2, number 2 CHEMICAL PHYSICS LETTER3 June 1968

THE DETERMINATION OF INTENSITIES OF

MODULATION-BROADENED LORENTZIAN ELECTRON RESONANCE LiNES

A. J. PARKER Chemistry Dcpartmenf , University of Manchester Institute of Science and Technology,

Manchester 1, England

Received 24 May 1968

A method is proposed for determining the intensities of Lorentzian electron resonance lines from a knowledge of the height of the first derivative curve, the modulation amplitude, and the unmodulated linewidth. The advantages of the method are: a) It does not neglect the area in the wings of the spectrum. b) It does not require the measurement of a large number of ordinates for each spectrum. c) It can he applied fcr any amplitude of field modulation so long as this amplitude is known.

Electron Spin Resonance (ESR) is now becom- ing widely used in studies of reactions involving free atoms in the gas phase. If atomic concen- trations are to be obtained from the “first deri- vative” ESR spectra, a double integration of a modulation-broadened Lorentzian line is required A common method of performing this integration is to evaluate the first moment of the observed signal, which has been shown [l] to be propor- tional to the area under the absorption curve for any amplitude of modulation. However, it has been pointed out by Ultee [2] that this method can give rise to considerable errors in the case of Lorentzian lines. These errors arise because of the slow fall-off of the Lorentzian lineshape func- tion to zero, and the consequent neglect of a por- tion of the area in evaluating the first moment.

The theoretical investigations of Wahlquist [3] have lead to analytical expressions for modula- tion-broadened Lorentzian lineshapes. These have been used by Barth, Hildebrandt and Pata- poff [4] who have proposed a potentially more useful method of obtaining the integrated area without neglecting the area in the wings of the line. The method described in ref. [4] involves setting the modulation to the vaiue required for maximum signal and measuring tine peak height of the resultant “first derivative” ESR curve. The integrated intensity may then be calculated from this single measurement if the unmodulated linewidth is known. However, it is not always convenient to set the modulation to this value, expecially if simultaneous measurements are re- quired on more than one atomic species. The work described here involves an extension or’ this

method to any moduiation setting. The accuracy needed for the setting in the method of Barth et al. is also investigated.

The equation for a Lorentzian lineshape may be written in the non-normalized form as:

k?(H) = IO $I%$

1

(+-)2 + (H-Ho)2 9

where H$ is the half-width of the unmodulated line, Ho the field at which resonance occurs, and 1, is a constant depending on the number of para- magnetic centres present. The area, A, under such a line can easily be shown to be:

A = nIo (2)

so that the problem reduces to that of finding a method to evaluate 1, from the modulated “first derivative” curve.

If a sinusoidal field-modulation of amplitude H, is applied to the resonance line described in eq. (l), the effect of this may be determined using the method of Fourier analysis and contour integration described by Wahlquist [3]. Only the final results will be given here, since the analy- sis is identical to that of Wahlquist except for certain constants which arise because a non-nor- malized lineshape function must be used. The maximum height, Smax, of the modulated “first derivative” curve is found to be:

S mz = 3&Q/H; (3)

wh,zre

Q = [%&$q]‘,

Volume 2, number 2 CHEMICAL PHYSICS LETTERS June I968

and

up = 2 + $32 + gj(p2 + ;,Q

p =gk_ W

Application of eq. (2) gives:

~~rnax% A=Tz,,= 3Q .

Eq. (4) can be used to calculate the integrated area from a measurement of Sm,, if H.$ andH, are known. H, can always be determined and, given a magnet with sufficient field-homogeneity, H+ can be measured for any ESR lines of inter- est. This measurement of HL can either be made using a very small amplitudg of modulation, or by correcting for the broadening effects of modu- lation using the equations of ref. [3].

It may be noted that the result of Barth et al. [4] can be obtained from eq. (4) by setting HL equal to Hw. In this case, p = 3 and Q = 6, &ving:

A=nSm&I+_

Table 1 shows a set of computed values of the function Q, for various values of the ratio JTw/H+, which can be used for direct substitution into eq. (4). The quantity Q is also shown graphically as a function of Hw/H~ in fig. 1. The modulated ESR signal has its ‘maximum amplitude when Hw/H$ = 1.

0 I-O 2-o 30 “%

Fig. 1. Dependence of Q (see text) on the ratio of modu- lation amplitude, Hw, to unmodulated half-width, H;.

Table 1 Values of the function Q for various ratios of Hw/H;

HW/f4 Q HW/H$. Q

0.00 0.00000 2.20 0.29412

0.10 0.08471 2.40 0.28675

0.20 0.15927 2.60 0 -27973

0.30 0.21817 2.80 0.27308

0 -40 0.26110 3.00 0 -26679

0.50 0.29062 3.50 0 -25256

0.60 0.30999 4.00 0.24OI9

0.70 0.32264 4.50 0.2293i

0.80 0.32899 5.00 0.21982

0.90 0.33239 5.50 U.Zi133

1.00 0.33333 6.00 0.20373

1.20 0.33070 6.50 0.19686

1.40 0.32482 7.00 0.19066

1.60 0.31750 8.00 0.17978

1.80 0.30967 9.00 0.17055

2.00 0.30180 10.00 0.16262

When using this method to study mixtures of atomic species, fig. 1 shows that the modulation should ideally be set to the optimum value for the narrowest line under study. This is because the Q-function falls off much less steeply fcr over-modulated lines. Fig. I also shows that it is not necessary to make a very accurate sgtting of the modulation amplitude for maximum signa when using the method of Barth et al. [4]. An er- ror of f 20% in the setting of H, = HL Leads to a maximum error in Q of less than 1_5%&This is much less than the error caused by neglecting the wings of Lorentzian lines in the first moment method of determining areas.

References

11) K. Halbach, Phys. Rev. 119 (1960) I230. [Z] C. J. Ultee. J. Appl. Phys. 37 (1966) 1746. [3] H. Wahlquist. J. Chem. Phys. 35 (1961) 1108. [4] C. A. Barth, A. F. Hildebrandt and &I. Patapoff, Dis-

cussions Faraday Sot. 33 (1962) 162.

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