Target Polarization Measurements with a
Crossed-Coil NMR Polarimeter
Anthony Caracappa and Craig Thorn^
Brookhaven National Laboratory, Upton, NY 11973
Abstract. We have performed a complete electronic circuit analysis of the crossed coil NMR
polarimeter (CC-meter), and from this analysis we have determined the optimum conditions for
its operation. From this analysis, which is confirmed by NMR measurements on hydrogen, we
conclude that the CC-meter can be operated to give a very high signal-to-noise ratio (SNR) for
thermal equilibrium polarization while also producing a highly linear response for fully
polarized targets. In general, a well-designed and properly constructed CC-meter can provide a
larger SNR at similar non-linearity, compared to the more commonly used Q-meter.
INTRODUCTION
Accurate target polarimetry is essential to polarized target experiments. At the
LEGS facility, our goal is to achieve 1% uncertainty in the measurement of highly
polarized HD targets. To understand the conditions for achieving such high accuracy,
we have performed a complete electronic circuit analysis of the crossed coil NMR
polarimeter (CC-meter), and from this analysis we have determined the optimal
conditions for its operation. These results have been compared to the veiy well
studied and more commonly used Q-meter circuit [1-3]. The CC-meter replaces the
single coil and resonant circuit of the Q-meter with a pair of coils arranged with
orthogonal axes and a pair of associated resonant circuits, although a tuned input
circuit is not essential. In any practical realization, both the CC-meter and Q-meter
polarimeters suffer from a significantly non-linear response to susceptibility for highly
polarized samples. One important purpose of the circuit analysis is to specify
conditions under which these non-linearities become insignificant, or to allow the
computation of a correction to the polarization deduced from the NMR signal in the
presence of significant non-linearities.
CIRCUIT ANALYSIS
In order to understand and optimize the performance of the CC-polarimeter, we
have constructed an analytic circuit model based on the equivalent circuit of Figure 1.
The values of the circuit elements have been determined by direct measurement where
possible (Rp, Rs, Lp, and Ls) and by fitting measurements of the voltage gain as a
f
This work was supported by the U.S. Department of Energy under contract DE-AC02-98CH10886.
CP675, Spin 2002:15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
Sources and Polarimeters, edited by Y. L Makdisi, A. U. Luccio, and W. W. MacKay
© 2003 American Institute of Physics 0-7354-0136-5/03/$20.00
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function of frequency (see Figure 2 for an example). The values determined for two
slightly different coils are given in Table 1. The signal generator is a Rohde &
Schwarz SMY01 and the PSD is a Stanford Research SR844 RF lock-in amplifier.
Cross-Coil Transformer
Cc
Lock-In
Amplifier
Output
Cable
FIGURE 1. The equivalent circuit of the crossed-coil polarimeter.
TABLE 1. Measured Circuit Parameters
Coil
2
3
L
P
0.669
L
^
3.17
k
Cc(pF)
0.004
2.4
Rp(ohm)
Rs(ohm)
1.5
4.8
0.585
2.42
0.0001
0.8
0.8
3.4
Cables are 1.282 m of SR047FL inside the dewar and SR401 outside.
The sensitivity of the circuit to target polarization enters through the dependence of
the inductance of a coil on the susceptibility (%) of the enclosed sample:
L = LQ [1 + </>(%)} = LQ [1 + T\%\ + O(%2)
(1)
The change in inductance with susceptibility is not generally linear, but for the present
analysis we have ignored the higher order terms. The constant relating susceptibility
to change in inductance, t|, is called the filling factor.
H,dv
-1
(2)
Most importantly for the operation of the CC-polarimeter, the presence of a
polarizable sample increases the small mutual inductance coupling the two coils:
M = jL^(kQ + m) + 0(z2)
(3)
For an NMR sample, the susceptibility is a complex valued resonance function
(typically Lorentzian or Gaussian) with non-zero value only very near the Larmor
frequency.
The voltage transfer function (gain) of the circuit is a rational polynomial in
frequency with coefficients expressed in terms of the circuit elements. The response
of the polarimeter to polarization can be determined by expanding the voltage gain in
a power series in susceptibility.
G(GJ) = G,(GJ) + G1((0)% + G2((0)%2 +0(z3)
(4)
The first coefficient, G0(to), is the gain of the circuit in the absence of a polarized
target. It is an undesirable background gain on which the signal due to the resonant
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susceptibility appears. The response to the susceptibility is determined by G^to), the
transducer gain. G2(to) is the lowest order non-linearity in the susceptibility
response of the circuit, which ideally would be zero. An optimal circuit has G0 and
G2 small and Gl large.
Since the transducer gain is a complex function of the complex susceptibility, four
gains can be defined: the derivatives of the real and imaginary parts of the circuit gain
with respect to the real and imaginary parts of the susceptibility. However, the
Cauchy-Riemann conditions reduce the matrix of four values to two gains in the form
of a rotation matrix, so that a rotation in the complex plane produces a single
transducer gain. The second order non-linearity G2(to) can similarly be reduced to a
single number.
For fixed cable lengths, resonances appear in the gains G0 and Gl at frequencies
slightly higher than that for which the length is a half integer multiple of the
wavelength, K. A cable n^/2 long reproduces the terminating impedance at the
receiving end. By adding tuning capacitor to the receiving end of the cable terminated
by the NMR coil, the parallel resonance of the coil and capacitor can be made to
match the frequency at which the cable length is nA/2. The lowest frequency
resonance corresponds to 0^, that is, the parallel resonance of the coil inductance and
the cable capacitance.
1
1e-5
7
8
9
10
Frequency (MHz)
FIGURE 2. The computed background and transducer gains as a function of frequency for a typical
cross-coil circuit at the coil-plus-cable capacitance (0 X) resonance.
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PERFORMANCE
We have made NMR measurements for both the 0^ and A/2 resonances at
frequencies between 8 and 12 MHz. Typical calculated resonance curves for the
background and transducer gains are shown in Figure 2 along with the measured
background gain for coil #2 in Table 1. Also shown is the background gain curve for
a larger coil coupling capacitance, which moves the zero closer to the pole, reducing
the background gain. We have not yet exploited this mode of operation.Noise is
added to the NMR signal by four sources: amplitude noise in the RF generator,
thermal noise from the real part of the impedance, amplifier noise from the lock-in
and/or preamplifier, and flicker (l/|f|) noise from parametric fluctuations in the coil
and cable properties induced by mechanical and thermal fluctuations. The noise of the
RF generator, which is -90 dB, dominates the total error for RF levels above -40 dBm.
Below that level, the total amplifier and resistor noise, which is less than 5nV /-JJfz ,
begins to contribute. For a single scan, the circuit adds no noise to that input to the
circuit by the signal generator: the noise figure is 1.0, as shown in Figure 3. If
multiple scans are averaged the RMS noise is reduced by the square root of the
number of measurements, and the noise figure remains 1, as long as the noise is white.
However, as the number of scans averaged increases, the measurement extends to
3.0
2.5
2.0
O)
1.5
1.0
0.5
0.0
10
100
Number of scans averaged
FIGURE 3. The measured noise performance of two NMR coils at -25 dBm. Noise figure is defined as
(SNR out)/(SNR in), where the signal out is the transducer gain times the signal in. The dashed lines are
the 1/f (flicker) noise that must be added to the white noise from the RF generator to fit the data [4]. As
the bandwidth is extended to lower frequencies by averaging many scans, the 1/f noise dominates. Coil
#3 has the windings more rigidly mounted and better aligned than Coil #2.
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lower frequencies and l/|f| noise dominates, causing the total noise to stop falling, or
increase, with increasing measurement time [4]. The l/|f| noise can be reduced, as was
done for coil #3, by mounting the windings and the coil form more rigidly, and by
reducing temperature fluctuations of the cables.
The second order non-linearity is proportional to the square of the susceptibility,
and with the aid of the Cauchy-Riemann relations, can be reduced to the following
form, as a fraction of the polarization.
(5)
A summary of the measured and calculated gains, noise, and non-linearities is given
in Table 2.
TABLE 2. Measured and Calculated Gains and Noise
Resonance
Frequency
MHz
OX
9.257
Background
Gain
obs & calc
0.175
Transducer
Gain
obs (calc)
1.66(1.79)
Transducer
Gain Error (%)
obs
3.2
10.660
0.0168
0.265 (0.257)
1.3
HA,
* for 1 Hz bandwidth and protons in thermal equilibrium at 4.2 K
** for 80% proton polarization and integration over 10 line widths
SNR*
obs
(calc)
7.0(7.8)
Nonlinearity**
calc, %
9.7
12(12)
2.4
Under conditions of operation that produce similar, small, non-linearities for the
two circuits, we have found that the CC-meter has the following significant
advantages over the Q-meter. By proper design, the inductive and capacitive coupling
between the two coils can be made very small (-60dB), so that the background signal
under the NMR resonance signal is very small. This leads to small systematic errors
in the background subtraction, which result from the inevitable drift of the background
during long measurement times. Also, since most of the noise comes from the RF
signal source, the low background results in greatly improved signal-to-noise ratios for
the CC-meter. The inductances and geometries of the two coils of the CC-meter can
be separately optimized to achieve specific requirements (such as RF field uniformity
over the target) not possible with the typical embedded coils of the Q-meter. The
principal disadvantage of the CC-meter is that two coils must be carefully constructed
and rigidly mounted to avoid excess (flicker) noise. The meter is optimized by keeping
the coil couplings small, the inductance of the receiver coil at about twice that of the
transmitter coil, coil resistances low, using cables with high Q and high impedance,
and using a low transmitter source impedance and a high receiver input impedance.
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