Frequency Dependence of EPR Sensitivity
2. FREQUENCY DEPENDENCE OF EPR SIGNAL INTENSITY
A general expression for EPR signal intensity is Eq. (I) (Rinard et al., I999a),
(I)
where Vs is the signal voltage at the end of the transmission line connected to the resonator, " (dimensionless) is the resonator filling factor (see Poole, 1967, p: 291), Q is the loaded quality factor of the resonator,
Zo
is the characteristic impedance of the transmission line, and P is the microwave power to the resonator produced by the external microwave source. Themagnetic susceptibility of the sample, X", is the imaginary component of the effective RF susceptibility.
For a Lorentzian line with linewidth ~O) at resonance frequency 0) ,
substituting for the susceptibility in (1) yields (2) :
(2)
where N, is the number of spins per unit volume, kB is Boltzmann's constant, 13 is the Bohr magneton, /J<l
=
41txlO-7T2r
1m3 is the permeability of vacuum, Ts is the temperature of the sample, and we have introduced numerical coefficients for the case of S=
1/2.To see relationships between some of the terms in this equation, consider the effect of increasing the size of the resonator and sample while keeping both the sample concentration, Ns, and the filling factor, T) , constant. The number of spins increases proportional to the increase in volume of sample.
The increase in EPR signal is not directly proportional to this increase in number of spins, because, as will be shown below, the increase in size also affects Q through the change in resistance R and inductance L of the resonator. However, if the resonator size and frequency are kept constant, and the sample size is increased, thus changing the filling factorT),the EPR signal would increase in proportion to the increase in volume of the sample.
These statements assume that the sample is non-lossy and does not have a dielectric constant large enough to distort the RF B] distribution.
Prediction of the frequency dependence of EPR signal-to-noise involves examination of the frequency dependence of each terrn in Eq.(l). Equation (2) explicitly shows the frequency dependence of the magnetic susceptibility. We now seek the frequency dependence of the filling factor,
T),and of Q. Since an experiment may be done at constant incident power or constant B), we also derive the frequency dependence of these terms, Without loss of generality we can assume a sample and resonator geometry such that we do not have to include the physical description of end effects of coils or samples. In practical cases these effects and/or special geometries are encompassed in the filling factor or coil efficiency factor.
The results for three cases are summarized in Table 1: case 1 - the size (linear dimensions) of the sample and resonator are constant, case 2 - the size of the sample and resonator are scaled with 1/000, and case 3 - the size of the sample is constant and the size of the resonator is scaled with 1/00.
3. FREQUENCY DEPENDENCE OFEPRSENSITIVITY
2.1 Frequency Dependence of Q for LGR
119
To provide specificity and coherence to the presentation, we assume the resonator has the topology of a loop-gap-resonator (LGR). Any other resonator type whose frequency can be changed without changing the overall size of the resonator could be used to obtain the results for case I (constant sample size and constant resonator size). Resonators whose size scales with frequency, such as cavity resonators, could yield cases 2 and 3. To consider other resonator, the only change in the calculations would be to change the specific formulae for the inductance, resistance and capacitance of the resonator. As long as the scaling of size and frequency meets the criteria of the entries in the Tables, the trends displayed there should apply, regardless of the type of resonator used.
The loaded Q, QL, for a loop gap resonator (LGR), is given by
Q
L=
roL/2R .The inductance, L, is related to the length z and the cross-sectional area of the resonator loop, AR=
1td2/4, (d is the diameter of the loop) by Eq. (3) .(3)
The resistance, including skin effect, (skin depth 0
=
[cou ocr /2]-1/2) ofthe resonator loop is
R=ltdz
Jro ~o
2cr (4)where o is the conductivity of the material of the resonator. Substitution of (3) and (4) into the definition of Q, gives the frequency dependence of Q, (5) :
(5)
Since c can be assumed to be independent of frequency, the frequency dependence ofQis determined by co1/2d, which is(J)1/2if the resonator size is kept constant, and is co-1/2if the size is scaled proportional to eo-I .
This expression for Qinherently includes the resistance of the resonator.
If there were no other sources of noise in the spectrometer system, this
resistance would establish the noise floor of the spectrometer. Other noise sources will be added in subsequent development below.
2.2 Frequency Dependence of Filling Factor
The filling factor,TJ, must be determined for each sample and resonator combination, and requires a detailed analysis of the electromagnetic fields in the resonator. However, we often can make an educated guess forTJ. We will assume, as a first approximation, that the filling factor for LGRs as a function of frequency will scale as if the RF magnetic field were uniform and confined to the loop. For resonators with a constant loop size and sample size, this approximation will overestimate TJ at higher frequency because the gap spacing must increase with frequency for a constant loop size, and as the gap increases, the magnetic field will no longer be completely confined to the loop.
2.2.1 Filling Factor for Lossy Samples
When a dielectric sample is placed in a resonator the resonant frequency decreases (Harrington, 1961 , p.323). A familiar example is the decrease in frequency when a quartz Dewar in placed in a cavity resonator. In the oppos ite case of a highly conducting sample (e.g., a metal), the resonant frequency increases if the material is in a region of large magnetic field, and decreases if the material is in a region of large electric field (Harrington, 1961, p. 320). Most biological samples involve both dielectric and conductive interactions with the microwaves.
If a sample is "lossy," microwave energy is converted to heat within the sample as a result of interaction of the microwave magnetic and electric vectors with polar and charged species in the sample. In the case of living organisms, water and ions are the primary causes of loss. To a first approximation, the microwave B) decreases exponentially into the organism due to the loss. The real case is somewhat more complicated due to the heterogeneity of the organism (Halpern 1989; Jiang, 1995; Sueki, 1996).
Halpern and Bowman (1989) summarized available information as showing about 5 em RF penetration in animals at 200-250 MHz. The filling factor of a lossy organism in a resonant cavity is not simply the ratio of the volume of the sample to that of the resonator, as can be approximated for a lossless dielectric (Poole, 1967, p. 291). This is due to the fact that in the calculation of the filling factor B1varies over the organism due to the effects of the loss.
In the extreme case, a portion of the interior of the organism is beyond the depth to which microwaves penetrate to any significant degree.
3.FREQUENCY DEPENDENCE OFEPRSENSITIVITY
2.3 Frequency Dependence of RF Field
HI121
The frequency dependence of B) is given by Eq. (6) (Rinard et al., 1999a) for the special case in which only the resistance of the materials of construction of the resonator contribute to the power loss:
(6)
where: Kb =kl
~21l~cr/1t2
and k, is a constant of proportionality between square root of power and B), the rms magnitude of the circularly polarized component of the microwave magnetic field that is perpendicular to the external magnetic field and in phase with the Larmor precession. Typically, one would seek to maintain the same B) at the sample, unless the spin relaxation time changed with frequency. To keepB) constant it is necessary to scale the incident power as ro)l2zd. If the resonator size is scaled with frequency (i.e., d, zex: 1/ro),then the power required scales as ro·312•Table J. Predicted Frequency Dependence of EPR Sensitivity When Resonator Resistance Dominates'
Case 1 Case 2 Case 3
const. sample size sample sizeex: I/co const. sample size const. LGR size LGR sizeex: I/co LGR sizeex: l/co
I L l (3) co·1 (3) co·1 (3)
2 R (resonator co112 (4) COI12 (4) COI12 (4)
resistance)
3 Q co1/2 (5) co·l12 (5, 6) co·l12 (5)
4 11 1 I co3 (44)
5 EPRSIN at C0312 (41) co112 (42) C0712 (45)
constant P 6
7 8
P to maintain constant B, EPRSIN at constant BI
(6) (6) (6,40,41)
(43) (43) (40,42,43)
(43)
co11/4 (40, 43, 45)b 'The equation numbers that are the bases for the various statements are given in parentheses.
bNote that in Rinard et al, (1999a) table I, Case 3, there is a typographical error : the EPR signal is proportional toCl)1I/4, notCl)S/4as stated in that reference .