Effect of Cavity Loading on Analytical Electron Spin Resonance Spectrometry I r a B. Goldberg” and Harry R. Crowe Science Center, Rockwell International, Thousand Oaks, California 9 1360
The effect of cavity loading during magnetic resonance absorption on the accuray of analysls by electron spln resonance is calculated. Large samples which degrade the cavity O-factor because of absorption of a large fraction of the incident radiation cause a nonlinearity of the absorptlon as a function of the quantity of material. This Is reflected by a broadenlng of the first derivative spectrum and reductions of the peak-to-peak amplitude and double integal of the ESR signal from that predicted on the bask of a dilute sample. The consequence of this effect is to underestlmate the quantlty of sample. Calculations of thls effect were verified with samples of MnS04*H20. Nonlinearitles in the order of 1YO may be obtained with samples of 0.4 to 1.0 mg of MnS04*H20or 0.1 to 0.25 mg of diphenylpicrylhydrazyl which are commonly used standards for ESR.
Electron spin resonance (ESR) derives much of its value from its high sensitivity and resolution. Hence, it is often used in the identification and quantification of trace impurities (1). However, there are many situations in which ESR can be used to quantitatively measure materials which exhibit intense paramagnetic absorption. Examples of this case include the calibration of ESR spectrometers,the analysis of fine grained iron in Lunar return samples (2),the determination of initial atom concentrations in studies of gas kinetics (3),the analysis of concentrated solutions of radical ions or transition metal ions, and the separation and quantification of solid materials in mixtures by virtue of their unique temperature dependencies such as antiferromagnetic-paramagnetic transitions ( 4 ) . While much attention has been devoted to the high sensitivity feature of ESR (5), the effect of the variation of the cavity Q-factor across the magnetic resonance has been studied only in terms of its effects on increasing the apparent linewidth of spectra (6). We report here the effect of intense paramagnetic resonance absorption on the various parameters used in analysis. These include the peak-to-peak amplitude of the derivative of the absorption, the double integral of the first derivative, and the peak-to-peak amplitude multiplied by the square of the peak-to-peak linewidth. Both Lorentzian and Gaussian lineshapes are treated, and experimental data are presented.
EXPERIMENTAL The ESR-computer system has been described in detail (7). It consists of a modified V-4502 (Varian) ESR spectrometer with a Magnion 38-cm magnet and a PDP 8/m computer with 32K words of core storage. Data were collected from the ESR spectrometer using a BASIC program with assembly language functions for control of the multiplex, analog to digital converter (ADC), clock, and other peripheral equipment. Two output registers control the upfield or downfield sweep direction, and the starting or stopping of the magnetic field sweep. A power meter was used to monitor the incident power to the cavity. MnS04.H20(Mallinckrodt reagent grade) was recrystallized from water at 80 “C and dried under N2 for 48 h. Heating two samples at 325 “C for 16 h showed a 10.62% weight loss as
compared t o 10.66% theoretical. The ESR linewidth of this sample is 200 G, peak-to-peak. Each spectrum contained 1536 points, with 44 points between derivative extrema. Samples were weighed to k0.05 mg. Integration of the spectra was carried out subsequent to the data acquisition. The intensity of each spectrum was between 70-100% of the full scale output of the ADC. Since the ADC is 12 bits, it is necessary t o minimize numerical round-off errors. Theoretical calculations of the effect of loading of the ESR cavity were also carried out using the PDP 8/m computer. Integrals of the absorption were computed to within h0.000270 by Simpson’s rule. Derivatives were calculated from the computed absorption by finite differences. Due to the finite word size, accuracy was limited to hO.001 in the relative peak-to-peakwidth and k0.0002 in the relative peak-to-peak amplitudes.
THEORY Cavity Q Variation during Resonance. The Q-factor of the microwave cavity during resonance, Q, is related to its Q-factor off resonance, Q,, and to the Q-factor of the sample Q,, according to Equation 1.
Q, is the Q-factor of the microwave cavity containing the paramagnetic sample and sample holder, but in which the magnetic field is such that there is no significant interaction of the magnetization of the sample with the incident radiation. The microwave loss caused by the sample magnetization in the ESR cavity is due entirely to its susceptibility, x”, and to the filling factor, 7, of the sample within the cavity (8). In cgs units, 1
The filling factor is defined as the integral of the square of the microwave magnetic field, H: over the sample to that in the entire cavity
(3) Filling factors of rectangular and cylindrical cavities containing a variety of sample geometries have been calculated by Poole (8). These and several other cases are presented in Table I. The filling factor can be treated as being proportional to a constant, K , times the ratio of the sample volume V , to the cavity volume V,,
(4) The microwave susceptibility for the case in which the field is held constant and the frequency is swept is given by Eqution 5 (91, 1
XI’ = p
I
rn
b x o 1+ ( w -
1 2 W,)2T;
where w and w, are the incident and resonant frequencies, T2 is the relaxation time and xo is the static magnetic suscepANALYTICAL CHEMISTRY, VOL. 49, NO. 9, AUGUST 1977
1353
Table I. Typical Filling Factorsa TE,,, (Cylindrical) t)
t)
Small sample at cavity center
v
12.33
$
0.82~7‘ 1 +( \ d / Tube of radius r , R along cavity axis
6.16
VS
(Y)’ ’
1+
vc
TE,,, (rectangular) VS --. 4 -
Small sample of cavity center
2 1+
Flat cell of thickness d’ and width b’
();, 2
1
+
VS
10.56 -
vc
rz
5.27 1 R
vc
VS 2.00 VC
. _VS
1.00-
lt($)
Tube of radius r along sample axis
if 2a = d
vc
irrl
bd
(if r < a)
. vs _
(G)’
vc
b’d’
1.00 bd
Special case Varian 0.331 TE,,, with 2.17-cm i.d. quartz tube a V, = cavity volume; V, = sample volume. Cylindrical cavity: a = radius; d = length. Rectangular cavity: a = thickness; b = width (parallel to static field); d = length.
Table 11. Relative Peak-to-Peak Linewidths, Amplitudes, and Double Integrals for Various Cavity Loading Parameters to That in the Absence of Loading for a Lorentzian Line Parameter
Peak-to-peak Amplitude ( ~ W X ’ ’ ~ ~ ~ Q ,width ) (A/A,) (AIA,) 0.000 1.000 1.0000 0.001 1.000 0.9988 0.002 1.000 0.9972 0.005 1.001 0.9928 0.010 1.003 0.9854 0.020 1.009 0.9709 0.050 1.026 0.9296 0.080 1.032 0.8912 0.100 1.046 0.8669 0.150 1.064 0.8111 0.200 1.087 0.7609 0.350 1.153 0.6377 0.500 1.210 0.5444 0.800 1.347 0.4142 1.000 1.416 0.3537 1.500 1.565 0.2530 2.000 1.722 0.1925 3.500 2.094 0.1048 5.000 2.430 0.0681 8.000 2.981 0.0371 10.000 3.277 0.0275
Double integral limit 1.0000
0.9995 0,9990 0.9975 0.9950 0,9901 0.9759 0.9622 0.9534 0.9325 0.9129 0.8606 0.8165 0.7454 0.7071 0.6324 0.5774 0.4713 0.4082 0.3333 0.3015
absorption, r, is given by h / ( T & ) , and the magnetic field H i s given by h w / ( g p ) . The peak-to-peak width, A,, is equal to 2/d/3r. The maximum value of x” occurs when H = H,. Thus the term ~Q,x”,~,where
tibility. We assume that the line is not saturated by the level of incident radiation. For a paramagnetic material,
where N is the number of paramagnetic centers, S is the electron spin, g is the spectroscopic splitting factor, p is the Bohr magneton, h is the Boltzmann constant, T i s the absolute temperature, and 8 is the paramagnetic Curie temperature. The term vx” is therefore proportional to the number of paramagnetic sites in the sample. Equations 1 and 2 can now be rearranged to give
(7) Since the ratio of the absorbed power to the incident power is dependent on AQ/Q,, nonlinearities in the response to the microwave susceptibility can result if 4q”vQ, is not negligible compared to unity. There are two consequences of this condition. The first is simply a saturation of AQ/Q, due to an increase in the denominator of Equation 7 . This is of primary interest here. The second effect is caused by the nonlinear response of the microwave detector to large changes in microwave power. Both of these effects can lead to distortions of the lineshape. RESULTS AND DISCUSSION Lorentzian Lineshape. Since Equation 6 is not useful for typical analytical purposes, it can be converted to the conventional case of a fixed frequency of incident microwave radiation and a variable magnetic field, Equation 8, 1
1
n 2
where the halfwidth a t one half of the maximum height of the 1354
ANALYTICAL CHEMISTRY, VOL. 49, NO. 9, AUGUST 1977
is dependent on the nature and number of spins in the sample and on the parameters of the ESR cavity, and can be used to characterize the performance of the loaded cavity. The parameter ~ Q o ~ ” m a is x given by Equation 10.
In order to compute the relative analytical parameters, AQ/Q, was calculated from Equations 7 and 8 as a function of 4~77Q,x”,~.The relative analytical parameter is the value of the computed parameter, e.g., peak-to-peak amplitude ( A ) or double integral of the derivative spectrum, divided by the value of the parameter if 47r7~”,~Q~
