SIGNAL-TO-NOISE IN MAGNETIC RESONANCE

MATCHING NETWORK

7. SIGNAL-TO-NOISE IN MAGNETIC RESONANCE

3. FREQUENCY DEPENDENCEOFEPRSENSITIVITY

If the two terms in

JR

are combined, and the expression for Q (= roL ) 2R

is used, the SIN can be written as proportional to

~ .

^{Since Q is}

kBTB

proportional to ^001/2, the product ooQ yields the ⁰⁰³¹²dependence at constant power in agreement with case 1 of Table 1.

The noise voltage and the signal voltage transform the same way in the impedance matching network, so if we take the ratio of signal to noise on the detector side of the impedance matching network, both VI and V_N are multiplied by

J ~

^,yielding

x"

ro L11 r.:;;;-- r.:;;;-- " PZo

SIN

=

^VS

=

---=2;=R====-VNs ~kBTBZo ⁽²⁷⁾

V_Ns

=

V_N

J ~

is the noise voltage on the detector side of the impedance matching network. Equation (27) is a general expression for SIN in EPR.

Cancellation ofZo in the numerator and denominator of Eq. (27) gives Eq.

(26) . The terms in Zo are retained in Eq. (27) to facilitate subsequent substitutions. Equation (1) can be converted to Eq. (27) by dividing by

~k

^BTBZo and using ^{Q= roL .} Expressions (26) and (27) for SIN are

2R

equivalent. After transformation the resistance term occurs only in the numerator where terms can be gathered to replace roL with Q. Since

2R

conversion of Eq.(1) to (27) requires only division by _{~kBTBZo ,}which is independent of frequency, Eq. (1) can be used to describe the frequency dependence of EPR SIN (Rinard et al., 1999a).

A Lorentzian line shape function can be assumed without loss of generality, which changes (26) into (28) , as in the conversion of(1) to (2).

ro² L

H;

XO - 1 ] -

-SIN= 6ro 2 R

~kBTBR ⁽²⁸⁾

We will use Eq. (28) ^In the following discussion of the frequency dependence of SIN.

3. FREQUENCY DEPENDENCEOFEPRSENSITIVITY 131

The usual derivation ofSIN for NMR (see section 3) also yields an ⁽⁰² dependence, where one factor of⁽⁰ comes from X" (as in Eqs. (I) and (2»

and the second factor of⁽⁰ comes from Lenz's law (Hoult and Richards, 1976).

7.1 Absolute

SIN

Numerical Examples

In this section we provide numerical examples ofSIN for CW and pulsed EPR for specific cases. A key message from these examples will be that sensitivity (SIN) differences between CW and pulsed EPR are a strong function of detector bandwidth and modulation amplitude.

For the pulsed EPR example we take results from a very detailed measurement and calculation at S-band (Rinard et al., 1999b). For an irradiated fused quartz sample, the echo signal at the output of the resonator was calculated to be 190 J..lV. This value is prior to amplification in the detection path of the spectrometer. The noise power available at the same point is -174 + 10Iog(bandwidth) dBm. For the ESE measurement the detector bandwidth was 25.7 MHz. Hence, the thermal noise voltage is 2.2 J..lV in a 50

n

load. Another way of saying this is that if all of the active devices had NF

=

0 dB, the equivalent input noise voltage would be 2.2 J..lV, so a 190 J..lV signal would have SIN

=

86, and a 2.2 J..lV signal would be detectable with SIN

=

I. The actual experimental SIN was somewhat less, due to the noise added by loss and gain stages in the detection path. The original paper gives details. The sample contained ca. 9.4x10¹⁵ spins. The extrapolated ultimate S-band sensitivity then is ca. I.lxlO^J4spins with SIN

=

1 if the only noise is thermal noise.

The number of spins detectable with SIN

=

I decreases dramatically if the bandwidth is narrower, since the noise is proportional to square root of bandwidth. One way to narrow the effective bandwidth is to signal average (Wilmshurst, 1990), in which case the effective noise bandwidth decreases with the square root of the number of scans averaged. Thus, it is not totally artificial to consider a pulse experiment with a 1Hzbandwidth due to signal averaging, and we consider the hypothetical case in which the ESE detection system has 1Hz bandwidth in order to make a rough comparison with CW EPR sensitivity specifications. The thermal noise voltage in a 50

n

load at 290 K detected with 1 Hz bandwidth would be 4.5xI0-^IOV, and one could observe 2.2x10¹⁰spins with SIN

=

1 at S-band.

It is well-known that state-of-the-art X-band EPR spectrometers are stated to have a CW sensitivity(SIN

=

I) equivalent to 0.8x10^1ospins/G at 200 mW for non-saturable, non-lossy sample, extending through a TEI0 2

cavity, assuming an S

=

'lS system with a single Lorentzian line, with 1 s time constant and optimum magnetic field modulation. Note that the

standard commercial definition of noise for sensitivity tests is peak-to-peak divided by 2.5, whereas the standard deviation noise we use is more nearly equal to peak-to-peak divided by 5. Note also that in some conventions various numbers of noise spikes are ignored.

We use Eq. (1) for Vs with best estimates of" (ca. 1%) and Q (ca. 3600).

Assuming the stated X-band sensitivity of 0.8xl0^1o spins/G, and 200 mW maximum available power, we calculate a signal voltage of ca. 6xl0-^1OV prior to amplification. This compares with the noise voltage of 4.5xl 0-¹⁰V in 1 Hz bandwidth. The standard SIN tests on commercial spectrometers use a

"1 second" filter time constant. To compare our calculation with these standard conditions, we have to relate filter time constant to equivalent noise bandwidth (ENBW). On modern spectrometers the filter in the signal detection circuit has a roll off of 12 dB/octave, and hence ENBW

=

1/8 times the reciprocal of the filter time constant. Consequently, for a nominal 1 second time constant the bandwidth used in the noise calculation should be 0.125 Hz, which decreases the predicted noise by 2.8 to 1.6xl0-^IO V.

Comparing this with the predicted signal voltage , state-of-the-art spectrometers are about a factor of two-to-four from the thermal limit. One cannot be more precise than this because traditions of using peak-to-peak instead of standard deviation noise and of ignoring some noise spikes obscure the mathematical relation between standard deviation noise and the experimentalSIN measurements.

Current X-band CW spectrometers probably have noise contributions from microphonics (including that due to use of high modulation amplitude), source noise (especially at high microwave power), and detector preamplifier noise (most spectrometers do not use a low-noise microwave preamplifier). The standard weak pitch SIN measurement is performed with magnetic field modulation larger than line width to maximize the signal amplitude (although distorting the line shape), and thus approximates the assumptions used in our treatment of the CW signal. Thus, both the CW and echo experiments measure approximately the total signal voltage. Note that in a field-modulated CW measurement in which the line shape is to be preserved, the modulation amplitude should be less than about 1/10 of the line width, so the signal voltage is substantially reduced from the maximum possible.

8. FREQUENCY AND SIZE DEPENDENCE OF SIN