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Broadband nuclear magnetic resonance using DC SQUID amplifiers
PII:
Applied Superconductivity Vol. 6, Nos 10±12, pp. 591±601, 1998 # 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0964-1807(99)00016-2 0964-1807/99 $ - see front matter
BROADBAND NUCLEAR MAGNETIC RESONANCE USING DC SQUID AMPLIFIERS C.P. LUSHER*{, JUNYUN LI$, M.E. DIGBY$, R.P. REED%}, B. COWAN$, J. SAUNDERS$, D. DRUNG% and T. SCHURIG% $Department of Physics, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK %Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, D-10587 Berlin, Germany AbstractÐWe have constructed two pulsed NMR spectrometers in which the signal is coupled to the input coil of a low Tc DC SQUID using a superconducting ¯ux transformer, yielding broadband response, with bandwidth determined by the SQUID electronics. A 50 kHz bandwidth commercial system has been used to observe free induction decay signals from platinum powder, bulk platinum, 3He gas and surface monolayers of 3He in the temperature range from 1.4 to 4.2 K and at frequencies from 5 to 40 kHz. The observed signal-to-noise ratio is as calculated with the noise dominated by ¯ux noise in the SQUID in all samples but the bulk metal. A second system, which operates in ¯ux-locked loop mode with bandwidth of 3.4 MHz using a SQUID with additional positive feedback, has been used to observe NMR signals from platinum powder at frequencies from 38 to 513 kHz and at a temperature of 4.2 K. The advantage of this technique in the study of systems with short T2 at frequencies below 1 MHz is discussed. In addition we discuss the bene®ts of both broadband and tuned input circuits for NMR detection and we describe the performance of a spectrometer with a tuned input circuit which has been used to obtain signals at 1 MHz from platinum powder at 4.2 K and from 02 layers of 3He absorbed on a surface area of 0.11 m2 at 1.7 K. The ampli®er noise temperature is predicted to be 60 mK in the 3He experiment. This demonstrates the potential of the tuned set-up for measurements at low millikelvin temperatures on systems with low spin density and with T2 greater than several hundred microseconds. # 1999 Elsevier Science Ltd. All rights reserved
NOMENCLATURE T1 T2 T*2 fp f0 fsq fN hf2Ni1/2 Df B1 M M0 mr m0 vs B0 g o0 w0 Li Lp Lsq Mi B1c k Q Vs is rp
NMR Longitudinal Relaxation Time NMR Transverse Relaxation Time Free Induction Decay Time Constant Total magnetic ¯ux threading NMR pick-up coil Flux Quantum ¯ux coupled to SQUID ¯ux noise in the SQUID rms ¯ux noise in SQUID per unit bandwidth measurement bandwidth Magnetic ®eld produced by unit current in the pick-up coil Sample magnetization Magnitude of static magnetization Relative permeability permeability of vacuum sample volume Static NMR ®eld Gyromagnetic ratio Angular Larmor frequency = 2pv0 Static magnetic susceptibility SQUID input coil inductance NMR receiver coil inductance (also called pick-up coil) SQUID inductance Mutual inductance between input coil and SQUID ®eld at centre of receiver coil due to unit current inhomogeneity factor allowing for ®eld variation across the coil extra subscripts t and u denote tuned and untuned systems quality factor of tuned circuit initial signal voltage initial signal current total series resistance in input circuit
*Corresponding author. }Present address: Centre for Quantum Metrology, National Physical Laboratory, Teddington TW11 0LW, UK 591
592 LT Ls r Tcoil TN kB Nc SNR ec a ae Lesq TR t i1 oc
C. P. LUSHER et al. total inductance in input circuit stray lead inductance fractional inductance contributed to total input inductance by Li Temperature of resistive part of NMR pick-up coil noise temperature of SQUID ampli®er connected to NMR pick-up coil Boltzmann's constant number of turns on NMR pick-up coil rms signal-to-noise ratio (inital rms amplitude divided by rms noise) Coupled energy sensitivity of SQUID coupling constant between input coil and SQUID eective coupling constant when pick-up coil is connected eective SQUID inductance with pick-up coil connected spectrometer recovery time natural time constant of tuned circuit maximum value of current in input circuit (limited by Q-spoiler) crossover frequency at which tuned circuit more favourable than untuned for NMR
INTRODUCTION
Nuclear magnetic resonance (NMR) is a widely used technique for the study of magnetic systems. Measurements of the relaxation times T1 and T2 give information on atomic and molecular motions in physical systems. NMR response can also be used to give information on the state of materials and structural information can be obtained from frequency shifts. In many situations the magnetic response is small and with conventional spectrometers much averaging is necessary in order to obtain usable information. By utilizing the high magnetic ®eld sensitivity of the DC SQUID one can study these systems with NMR within a reasonable period of time. We have built two broadband pulsed NMR spectrometers which use a DC SQUID as the front end ampli®er and have demonstrated their performance on several physical systems. The spectrometers measure the precessing magnetization directly, rather than the more conventional method [1] of using a SQUID to monitor the longitudinal magnetization. The superconducting NMR pick-up coil forms a ¯ux transformer with the SQUID input coil. The pick-up coil is orthogonal to the applied static magnetic ®eld and the SQUID detects the precessing magnetization. There are a number of previous reports of DC SQUIDs being used to detect NMR in this way [2±5]. These systems have a distinct advantage over conventional NMR spectrometers in that they are broadband. This makes measurements as a function of frequency relatively simple and also results in short recovery times. The thin ®lm DC SQUIDs are operated in a ¯ux-locked loop (FLL) mode. The bandwidths of the NMR spectrometers are limited by the FLL electronics. In both spectrometers the noise is dominated by the ¯ux noise in the SQUID and the observed signal-to-noise ratio (SNR) is as calculated for spin systems where the radio frequency (rf) excitation ®eld can penetrate the entire sample. The ®rst spectrometer is based on a commercial SQUID system and operates out to 50 kHz; the second uses a SQUID with additional positive feedback (APF) [6] and has a 3 dB bandwidth of 3.4 MHz. APF SQUID systems designed for NMR applications have been also developed by other authors [7]. Some of our earlier broadband work has been reported elsewhere [8]. In addition we report data obtained from another spectrometer which uses a tuned input circuit and the SQUID operating open-loop as a small-signal rf ampli®er at 1 MHz. The tuned setup, which has been used by a few authors [9, 10], is particularly suitable for the study of samples with low spin density and with T2 greater than several hundred microseconds, especially if the NMR pick-up coil can be cooled. GENERAL REQUIREMENTS FOR A PULSED NMR SPECTROMETER USING DC SQUIDS
In a pulsed NMR experiment a homogeneous static magnetic ®eld is applied to the spin system under investigation resulting in a net magnetization along the static ®eld direction. A large rf tipping pulse is then applied to the spin system for a short time (typically tens to hundreds of microseconds). The pulses should be large enough that the rf magnetic ®eld is much greater than any local magnetic ®elds in the sample. The maximum signal will be obtained if the pulse tips the magnetization through an angle of 908 into the transverse plane. Following removal of
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the pulse the magnetization, which precesses about the static magnetic ®eld direction, is the quantity detected. In SQUID NMR it is essential to have orthogonal transmitter and receiver coils because of the potentially harmful eect of the high rf ®elds from the transmitter. Even with a cross coupling of less than 1% the required pulsed ®elds correspond to of order 104 f0 coupled to the SQUID. This level can be reduced during the pulse by placing a series array of hysteretic SQUIDs known as a Q-spoiler [11] into the input circuit. The initial signals can be of order 1 m f0 or below and they decay to zero exponentially with the relaxation time T*2. It is important that the recovery time of the spectrometer is short compared with T*2. Without taking precautions the SQUID electronics would loose lock during application of the transmitter pulse. In general a reset pulse which opens the feedback loop has to be applied during the pulse and the loop is closed immediately after the pulse. The system recovery time is therefore related directly to the FLL bandwidth with large bandwidths being required for short recovery times. When studying metallic systems eddy currents which decay following the transmitter pulse can cause additional magnetic ®elds comparable in size to the expected signals. We have studied these eects in bulk metallic samples.
CALCULATION OF SIGNAL SIZE
The expected signal size in a pulsed NMR experiment is best calculated using the principle of reciprocity [12]. The total magnetic ¯ux, fp, threading the pick-up coil is given by 1 fp B1 :Mdvs ; s
where B1 is the ®eld created by unit current in the coil, M is the sample magnetization, and the integral is over the sample volume vs. Following a perfect 908 pulse the amplitude of the transverse magnetization is the same as that along the static ®eld direction before the pulse and is given by M = M0 = w0B0/(mrm0), where B0 = o0/g is the amplitude of the static NMR ®eld, w0 is the static magnetic susceptibility of the sample and g is the gyromagnetic ratio. The peak amplitude of the ¯ux coupled to the SQUID fsq is [13] fsq
fp Mi ; Li Lp
2
where Lp and Li are the self inductances of the pick-up coil and the SQUID input coil respectively, assumed to be much greater than any stray lead inductance, and Mi is the mutual inductance between the input coil and the SQUID. Using Eqs. (1) and (2) and integrating over the sample volume we obtain for the initial amplitude of the ¯ux coupled to the SQUID following a 908 pulse Mi w0 o 0 3 vs B1c k; fsq Li Lp mr m0 g where B1c is the ®eld at the centre of the receiver coil and the inhomogeneity factor, k, allows for the variation in B1 across the sample. Since the observable signal is limited by ¯ux noise in the SQUID, which is white above the 1/f knee at 01 Hz, the signal-to-noise ratio (SNR) in the NMR experiment is proportional to frequency. This should be contrasted with the situation in conventional pulsed NMR systems which have the pick-up coil as part of a tuned tank circuit which is followed by a voltage ampli®er. In that case the precessing magnetization induces a voltage across the coil which is proportional to the rate of change of ¯ux and the SNR goes faster than linear in frequency. Therefore at high enough frequencies the latter input con®guration gives a higher SNR but the former wins out at low frequencies. This is discussed in more detail in Section 8 (Comparison of tuned and untuned input con®gurations). In some cases at higher frequencies it is advantageous to use a tuned tank circuit input in order to get improved signalto-noise.
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THE 50 kHz BROADBAND DC SQUID NMR SPECTROMETER
The ®rst broadband spectrometer we constructed is similar to that described in [2] and [3]. We have used a Quantum Design DC SQUID system with a model 5000 controller which we modi®ed for the purpose. The static B0 ®eld is trapped in a niobium cylinder. The transmitter and receiver coils could be adjusted at room temperature to minimize the cross coupling. Both coils were wound from single ®lamentary Nb±Ti superconducting wire, CuNi clad with a 75 mm superconducting core. The cladding was removed with an HNO3 etch where necessary. The transmitter was a saddle coil designed for optimum homogeneity [12]. Receiver coils of both saddle and solenoidal form were used and in each case the inductance was made close to that of the SQUID input coil in order to optimize the coupled ¯ux [13]. All the low temperature input circuitry and transmitter circuitry are contained in niobium shields, in order to reduce noise from extraneous magnetic ®elds. We use four pairs of crossed diodes in series with the transmitter coil, placed at room temperature, in order to prevent noise from the transmitter electronics coupling to the SQUID when the pulses are o. The spectrometer is controlled by a pulse generator which, as well as gating the transmitter, activates both the reset circuitry (opening the feedback loop when the pulse is on) and the sample and hold (which removes the DC oset that arises as the SQUID regains lock when the feedback loop is closed [2, 3]). The purpose built cryostat has a pumped helium pot with a continuous ®ll line and can achieve temperatures down to 1.4 K. Plenty of space is available for housing SQUID sensors. Generally the SQUIDs are operated at 1.4 K although the SQUID temperature can be controlled independently of that of the sample. PERFORMANCE OF THE 50 kHz SPECTROMETER
The spectrometer has been tested on a variety of samples. As reported in [8], we have measured signals from 1021 spins of 195Pt in platinum powder at temperatures between 1.4 K and mf0/ p 4.2 K with the initial signal size being given by Equation (3) and with the noise of 05 Hz, dominated by ¯ux noise in the SQUID. Figure 1 shows an NMR signal from 1020 spins in 3He gas at around 40 kHz and 4.2 K. The narrow width of the line shows that the static ®eld homogeneity is not seriously aected by the superconducting rf coils. With this sample the recovery time of the spectrometer following a 1 mT pulse is less than 50 ms. We have used the
Fig. 1. Signal from 1020 spins in 3He gas at 4.2 K. Shown here is the Fourier transform of a free induction decay signal following a 908 pulse averaged 512 times. Signal has been mixed down using a lockin detector to get adequate spectral resolution in a 200 ms sample window.
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spectrometer to obtain signals from bulk Pt samples (here the signals are obtained only from spins within a skin depth) and from a single monolayer of 3He adsorbed on graphite. In the latter experiment silver foils were diusion bonded to the graphite for thermal contact. In both experiments the presence of the normal metal in the sample cell gave rise to an additional recovery signal following removal of the pulse due to eddy current decay. However we found that it was possible to ®t to the background and extract the NMR signal from the response. NMR MEASUREMENTS WITH AN APF SYSTEM
A broadband spectrometer is essential for measurements on samples with short T2 (for example most solid state samples) at low frequencies since the recovery time constant of a tuned circuit of quality factor Q, given by t = (2Q/o0), is excessive. One situation where low frequencies are essential is in ultralow temperature measurements on conducting materials where the eddy current heating eects are prohibitive at high frequencies. In order to obtain a short recovery time a large bandwidth is necessary. We have performed initial tests of a new broadband spectrometer based on a DC SQUID which uses APF to allow direct coupling between the SQUID and the read-out electronics, thus enabling a large bandwidth. A small cryostat was built for these tests which could be inserted into a helium transport dewar. The experiment was a proof-of-principle and was performed under non-optimized conditions. A large area (7.2 7.2 mm) multiloop DC SQUID fabricated at PTB was used. The SQUID has an inductance of Lsq = 400 pH and is equipped with an integrated APF coil, resulting in a transfer coecient p of 250 mV/f0 at the operating point. This particular SQUID has a ¯ux noise of 3 mf / Hz. The noise of the room temperature ampli®er is 0 p 0.74 nV/ Hz, and is thus comparable with the SQUID voltage noise. The ¯ux-locked loop operates with a measured bandwidth of 3.4 MHz. With the present shielding arrangement the maximum slew rate is 8 105 f0sÿ1. The SQUID is con®gured as an integrated magnetometer without an input coil. Since the NMR measurements require the use of a remote pick-up coil and a ¯ux transformer input circuit, an input coil is essential. An input coil was wound from 20 turns of niobium wire and was mounted under the SQUID carrier. Its self inductance Li = 4.8 mH. Due to the large thickness of the chip carrier (which was originally used for testing the SQUID), the measured coupling constant a was only 0.125. This results in a ¯ux sensitivity of 0.39 mA/f0, equivalent to a mutual inductance Mi = a(LiLsq)1/2 of 5.3 nH. The sample pick-up coil of inductance Lp = 2 mH was wound from NbTi wire and formed a superconducting ¯ux transformer with the input coil. We used a coil set that had previously been used on the 50 kHz spectrometer. The pick-up coil and NMR transmitter coil were both wound in saddle geometry on separate cylindrical formers fabricated from a machinable ceramic (MACOR) to close tolerances. Vacuum grease was smeared between the coil formers before assembly, which allowed easy adjustment of orthogonality at room temperature (by minimizing cross-talk) but solidi®ed to form a rigid structure when cold [14]. About 1% orthogonality was achieved. The sample consisted of platinum powder containing 1021 spins of 195Pt. Figure 2 shows a schematic of the APF SQUID based spectrometer. TTL pulses are provided by a computer controlled pulse generator. They are used for timing the rf transmitter pulse, resetting the FLL during the pulse (i.e., the output is forced close to zero voltage), and for triggering a sample and hold circuit which sets a zero starting point for data acquisition. Signals are captured and averaged on an oscilloscope (Tektronix TDS 410A), which is controlled by a PC with LabView software. A home built persistent NMR magnet in a superconducting shield is used to provide the static magnetic ®eld. The coil is wound from 0.1 mm diameter Nb±Ti wire on an annealed copper cylinder. It consists of 9 layers and 5310 turns, and is of inner diameter 16 mm and 80 mm long. It is placed in a 35 mm diameter niobium cylinder 100 mm long. The measured ®eld is 62.7 mTAÿ1, which is within 1% of the calculated value, and the predicted homogeneity is 10ÿ4 over 1 cm on-axis. In persistent mode at 10 mT the ®eld is stable after a few minutes and at 60 mT it is stable at a drift rate of a few f0 per day after about half an hour. It is possible to make NMR measurements essentially immediately after persisting the magnet. Since the static
Fig. 2. Schematic diagram of 3.4 MHz bandwidth NMR spectrometer based on a DC SQUID with APF.
596 C. P. LUSHER et al.
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®eld can be readily changed in this way, making NMR measurements as a function of frequency is convenient. PERFORMANCE OF THE APF SQUID NMR SPECTROMETER
Figure 3 shows some examples of NMR signals. They are Fourier transforms of free induction decays (FIDs) of platinum spins following a 908 transmitter pulse. The measurements below 100 kHz were made using a slightly dierent set-up from those at higher frequencies. At all frequencies a 908 pulse of length 294 ms and amplitude 0.1 mT was used. The recovery time after the transmitter pulse is about 10 ms; i.e. when the reset pulse is timed to come o 10 ms after the end of the transmitter pulse an NMR signal can be captured immediately. In order to study samples with larger linewidths the transmitter pulses must be made both shorter and larger. For these situations it is planned to include a Q-spoiler in the ¯ux transformer to limit the induced current during a transmitter pulse. With the ¯ux transformer input circuit the initial amplitude of the FID is expected to be proportional to the magnetization i.e. proportional to the NMR frequency (Equation (3)). Although the amplitude of the FFT is not linear in frequency, the inset to Fig. 3 shows that when corrected for an increase in linewidth a good linear dependence of signal size is observed. In fact 1/T*2 is linear in frequency, which would be expected if there is an increase in the static ®eld gradient due to ®eld inhomogeneities. The presence of a superconducting pick-up coil is clearly not a serious problem. The signal size is as predicted using the principle of reciprocity
Fig. 3. Fourier transform of free induction decays from 1021 platinum spins at 4.2 K; performed at Larmor frequencies of 38, 65, 85, 240 and 513 kHz. Inset: linear dependence of signal size (amplitude of FFT linewidth) on Larmor frequency.
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and the known circuit parameters. The signals shown here correspond to 105 averages. With the present set-up an rms signal-to-noise ratio of 10 is achievable with about 40 averages at 513 kHz. With a lower-noise SQUID and optimized input circuit an improvement of a factor of 6 in SNR is possible, equivalent to an rms SNR (rms amplitude to rms noise) of 10 in a single shot. At lower frequencies the measurements suer from a signi®cant amount of ®eld dependent noise, possibly attributable to magneto-acoustic resonances in the coil formers. In early experiments with the 50 kHz spectrometer the problem was particularly serious when using coil formers fabricated from stycast 1266 and perspex but they were signi®cantly improved by using MACOR. With the new set-up there are still a number of such resonances below 100 kHz, but it is possible to ®nd ``quiet'' regions within the spectrum. At higher frequencies these problems are not so serious. We have recently made modi®cations to increase coupling of the input coil to the SQUID and to achieve better inductance matching (Lp = Li). A hole was cut in the chip carrier to take a new input coil which was mounted directly under the chip. The new parameters were Li = 2.4 mH, Mi = 9.6 nH and a = 0.31. Preliminary NMR spectra have been obtained with an amplitude SNR a factor of three larger than that in Fig. 3. The coupling coecient cannot be increased to more than 00.4 with the octagonal SQUID geometry used, however by using a lower-noise SQUID with a higher transfer coecient another factor of two improvement in SNR is possible.
COMPARISON OF TUNED AND UNTUNED INPUT CONFIGURATIONS
The rms SNR for the untuned system with an input ¯ux transformer can be written as ! ! ! fsq B1cu ku vs w0 o0 Mi 1 p ; fN 2 2Lpu mr m0 g hf2N i1=2 Dfu 1=2
4
where we have neglected any stray inductance in the leads and have chosen optimal matching conditions (Lp = Li). The subscript u is used to denote the untuned system. hf2Ni1/2 is the rms ¯ux noise in the SQUID per unit bandwidth when optimally matched and Dfu is the measurement bandwidth. We assume SQUID noise dominates. Since the ¯ux noise in the SQUID is independent of frequency above the 1/f knee (01 Hz) then the SNR is proportional to frequency. We now calculate the SNR for the case of a tuned series tank input circuit. In this case the rate of change of magnetic ¯ux is detected and the SQUID acts as an rf ampli®er. The peak amplitude of the initial signal voltage induced across the terminals of the NMR pick-up coil is given by jVs j o0 fp o0 B1ct kt M0 vs
5
and the signal current is given on resonance by is = Vs/rp, where rp is the total series resistance in the input circuit which for a high-Q series tank circuit on resonance is given by rp = o0LT/Q, where LT = (Li + Ls + Lp) is the total inductance in the input circuit and Ls is any stray inductance associated with the leads. The subscript t denotes the tuned system. For the SQUID ampli®er operating under optimized conditions, which results in a minimum noise temperature, we have Qr 1 1, where r is the fractional inductance contributed by the SQUID input coil [9]. Therefore Lpt 1 LT 1QLi and any stray inductance can be ignored. The signal current on resonance is given by is
B1ct kt M0 vs Q : Lpt
6
The noise current on resonance has a contribution from both Johnson noise in the coil (or the resistive part of the input circuit) at temperature Tcoil and from noise in the SQUID ampli®er. The mean square current noise on resonance is given by
Broadband nuclear magnetic resonance using DC SQUID ampli®ers
hi2N i
4kB Tcoil TN Dft ; rp
599
7
where Dft is the measurement bandwidth for the tuned system and TN is the noise temperature of the SQUID ampli®er, which is a function of both the SQUID and the receiver coil parameters, including the tank circuit quality factor. Substituting for M0 gives the rms SNR with the tuned input con®guration ! ! ! 1=2 is B1ct kt vs w0 o3=2 QLpt 1 0 p ; 8 mr m0 g kB Tcoil TN hi2N i1=2 Dft 1=2 2 2Lpt which should be compared with Equation (4) derived p above for the untuned set-up. We note that if TcoilwTN then the SNR is proportional to Q. In order to obtain the maximum SNR the Q must be made as large as possible. The maximum allowed Q is constrained by T*2, since t = (2Q/o0), and will have a frequency dependence Q A o0T*2. However, in general the SNR increases faster than linear with o0. One can compare the two set-ups under various conditions. As an example we consider a sample of ®xed geometry and volume, vs, containing N spins and ask at what frequency the SNR is equal in the two cases. The SQUID and the receiver coil geometry are taken to be the same but the number of turns on the receiver coil Nc is dierent since the optimization conditions are dierent for the two con®gurations. In general we have (B1ck/Lp) = g/Nc where g is a purely geometrical factor. The ratio of the SNRs for the untuned and tuned con®gurations is given by ! !1=2 SNRu Nct Mi kB Tcoil TN Dft 1=2 : 9 SNRt Ncu hf2N i1=2 Dfu Qo0 Lpt We note that for a given sample at ®xed frequency the required measurement bandwidth is independent of the input coupling scheme and therefore Dfu = Dft. Also in the situation p where both pick-up coil inductances are optimized Lpt = QLi = QLpu so that (Nct/Ncu) = Q. Writing the SQUID parameters in terms of the coupled energy sensitivity ec = Lihf2Ni/(2 M2i) (i.e., the energy equivalent of the minimum detectable current in the input coil) we have !1=2 1=2 M2i SNRu kB Tcoil TN 1=2 1 kB Tcoil TN 1=2 : 10 SNRt opt Qo0 2ec Qo0 Li hf2N i We should point out that when a load is connected to the input coil of a DC SQUID the SQUID parameters can be signi®cantly altered, especially if the input coil is strongly coupled to the SQUID. In particular, the SQUID behaves as an isolated SQUID but with a reduced inductance Lesq = (1ÿa2e)Lsq, where a2e = a2 (Li/LT) [15]. For the tuned set-up under optimal conditions Lp wLi and the aect of loading is negligible, however it can be signi®cant in the untuned case if a2 is close to unity. Therefore the coupled energy sensitivity in Equation (10) is that which corresponds to the SQUID connected to an optimally matched ¯ux transformer. The crossover frequency oc at which the tuned circuit becomes the more favourable for obtaining maximum SNR is given by the frequency at which the ratio in Equation (10) is unity. It is a function of SQUID parameters, pick-up coil parameters and sample parameters (since the maximum usable quality factor in the tuned case is determined by the NMR properties of the sample). There are two factors constraining the quality factor. Firstly the electronic bandwidth must be wide enough to avoid distortion of the NMR line i.e., Do = o/Q w1/T*2 which implies that high-Q tuned systems are more useful for systems with long T*2s. Also the system recovery time following removal of the transmitter pulse must be fast compared with T*2. In practice a Qspoiler is incorporated into the SQUID input circuit in order to limit the maximum current in the input circuit during the pulse to of order the minimum Q-spoiler critical current, i1. Once the transmitter pulse is removed the current in the input circuit will decay exponentially from i1 with the natural time constant t and the spectrometer recovery time TR will be given by TR = (2Q/o0) 1n(i1/is) where is is a typical signal amplitude. As an example we take i1 to be the
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current which couples 5 f0 to the SQUID (as in our set-up) and choose an initial signal amplitude equivalent to 1 mf0. If we stipulate for example that we need TR ET*2/10 then this condition limits Qmax to be less than (o0T*2/200). Using this expression for Q we obtain !1=2 1=2 1 200kB Tcoil TN : 11 oc 2ec T2 The value of the crossover frequency clearly depends on the frequency dependence of T*2. However in order to get a feeling for its size we take a SQUID with ec = 200 h (where h is Planck's constant), a coil temperature of 4.2 K with TN negligible, and a frequency independent T*2 of 1 ms. This results in a value of 1.05 MHz for oc/(2p). The choice of the value of Q to insert into Equation (10) is subtle. For example if Tcoil wTN it is possible to choose a Q higher than one might naively expect based on the sample T*2 and then use feedback to increase the bandwidth but without signi®cantly reducing the SNR [10]. The eect of this is to lower oc. This method will not work, however, if Tcoil ETN. Note that for T*2 = 50 ms, typical of solid state samples, then oc/(2p) = 4.5 MHz with Tcoil = 4.2 K. In the above analysis we have assumed that other sources of noise for example additional noise from room temperature ampli®ers or sample noise are negligible. USE OF A SPECTROMETER WITH A TUNED TANK INPUT CIRCUIT
State-of-the-art SQUID ampli®ers with optimum pick-up coils are predicted to have noise temperatures of around a few tens of millikelvin at 1 MHz with TN A o0 [9]. Tuned systems therefore oer unprecedented sensitivity for systems with receiver coils at low millikelvin temperatures. At frequencies below 1 MHz this makes the system suitable for samples with a long T2, for example 3He. In this set-up a Q-spoiler is incorporated into the input tank circuit in order to reduce its ringdown time. This has a normal resistance of approximately 1 kO during the large transmitter pulses, thereby reducing the Q, but is superconducting at low currents. Samples with low spin density and with T2 of more than a few hundred ms can be studied in this way. We have used a spectrometer with a tuned input tank circuit (Q 0 30) in order to measure NMR signals at a frequency of 01 MHz from platinum powder and from 3He ®lms at 4.2 K and below. The commercial DC SQUID chip (a Conductus SQD1002 with a quoted ¯ux noise p hf2Ni1/2 of 3 mf0/ Hz) was operated open loop as a small signal rf ampli®er. The signal was coupled out via a tank circuit with a Q of 010 in order to ensure that the noise of the room temperature preampli®er was insigni®cant. The Q-spoiler was a Conductus SQD 1060 chip with a minimum critical current of 2 mA, which corresponds to 5 f0 coupled to the SQUID. In the 3 He experiment the sample cell was 1 cm diameter and 1 cm long and contained 760 Mylar discs. The total surface area for adsorption was 0.11 m2. The sample was 02 layers of 3He at 1.7 K. Two layers contain 1.5 1018 spins. Some of the signal from the ®rst layer is expected to be lost since those spins have a short T2. Here the signal-to-noise ratio was limited by Johnson noise in the pick-up coil. The pick-up coil had an inductance of 5.8 mH. A short length of constantan wire was placed in the input circuit (in the niobium shield of the SQUID) in order to limit the input tank circuit Q to 30. The pick-up coil inductance was chosen to be optimum given the quoted SQUID input coil inductance of 170 nH. The noise temperature of the SQUID ampli®er was estimated to be 60 mK, taking the contribution from the SQUID to the total outp put noise to be 2 hf2Ni1/2, as expected for the optimal source impedance [16]. Noise measurements at 4.2 and 1.7 K were consistent with this but the value of TN was impossible to measure with adequate precision given the minimum coil temperature of 1.7 K. In Fig. 4 we show a free induction decay following a 908 pulse averaged 1024 times. The Larmor frequency is 946 kHz. The measured signal size is of order 6 mf0 peak-to-peak at the SQUID. The signal capture startS 100 ms after the end of the 50 ms transmitter pulse. This recovery time is determined by the Q of the input tank circuit. The system 3 dB bandwidth is 20 kHz, corresponding to a noise bandwidth of 31.4 kHz. The rms SNR is 19.pIn this case the signal has been mixed down resulting in a decrease in the SNR by a factor of 2 due to noise contributions from both sidebands.
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Fig. 4. Mixed down signal from 02 layers of 3He adsorbed on Mylar at 946 kHz and 1.7 K averaged 1024 times. Sample contains 01018 spins. Signal capture starts 100 ms after end of 50 ms 908 transmitter pulse. This recovery time is determined by the Q of the input tank circuit. The noise bandwidth is 31.4 kHz.
Taking this into account we would expect a value of 13 for the SNR from Equation (8) with 1018 spins, in good agreement with the measured value considering the uncertainty in the number of observable spins. It should be pointed out that this sample has a very low spin density (a factor of 13,000 smaller than that of liquid 3He at this temperature and atmospheric pressure). AcknowledgementsÐWe thank Alan Betts, Francis Greenough, John Taylor and Tony Wilkinson for technical assistance. This work was supported by EPSRC (UK).
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
R.A. Webb, Rev. Sci. Instrum 48, 1585 (1977). L.J. Friedman, A.K.M. Wennberg, S.N. Ytterboe and H.M. Bozler, Rev. Sci. Instrum 57, 410 (1986). N.Q. Fan and J. Clarke, Rev. Sci. Instrum 62, 1453 (1991). D.M. TonThat and J. Clarke, Rev. Sci. Instrum 67, 2890 (1996). S. Kumar, W.F. Avrin and B.R. Whitecotton, IEEE Trans. Magnetics 32, 5261 (1996). D. Drung, H. Matz and H. Koch, Rev. Sci. Instrum 66, 3008 (1995). S.L. Thomasson and C.M. Gould, Czech. J. Phys 46, 2849 (1996). C.P. Lusher, M.R. Dann, R.P. Reed, M.E. Digby, B. Cowan and J. Saunders, Czech. J. Phys 46, 2831 (1996). M.R. Freeman, R.S. Germain, R.C. Richardson, M.L. Roukes, W.J. Gallagher and M.B. Ketchen, Appl. Phys. Lett 48, 300 (1986). H.C. Seton, J.M.S. Hutchison and D.M. Bussell, IEEE Trans. Appl. Supercond 7, 3213 (1997). See, for example, C. Hilbert, J. Clarke, T. Sleator, and E.L. Hahn, Appl. Phys. Lett. 47, 637 (1985). D.I. Hoult and R.E. Richards, J. Mag. Res 24, 71 (1976). J.H. Claassen, J. Appl. Phys 46, 2268 (1975). C. Connor, J. Chang and A. Pines, Rev. Sci. Instrum 61, 1059 (1990). J.M. Martinis and J. Clarke, J. Low Temp. Phys 61, 227 (1985). C. Hilbert and J. Clarke, J. Low Temp. Phys 61, 263 (1985).
Applied Superconductivity Vol. 6, Nos 10±12, pp. 591±601, 1998 # 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0964-1807(99)00016-2 0964-1807/99 $ - see front matter
BROADBAND NUCLEAR MAGNETIC RESONANCE USING DC SQUID AMPLIFIERS C.P. LUSHER*{, JUNYUN LI$, M.E. DIGBY$, R.P. REED%}, B. COWAN$, J. SAUNDERS$, D. DRUNG% and T. SCHURIG% $Department of Physics, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK %Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, D-10587 Berlin, Germany AbstractÐWe have constructed two pulsed NMR spectrometers in which the signal is coupled to the input coil of a low Tc DC SQUID using a superconducting ¯ux transformer, yielding broadband response, with bandwidth determined by the SQUID electronics. A 50 kHz bandwidth commercial system has been used to observe free induction decay signals from platinum powder, bulk platinum, 3He gas and surface monolayers of 3He in the temperature range from 1.4 to 4.2 K and at frequencies from 5 to 40 kHz. The observed signal-to-noise ratio is as calculated with the noise dominated by ¯ux noise in the SQUID in all samples but the bulk metal. A second system, which operates in ¯ux-locked loop mode with bandwidth of 3.4 MHz using a SQUID with additional positive feedback, has been used to observe NMR signals from platinum powder at frequencies from 38 to 513 kHz and at a temperature of 4.2 K. The advantage of this technique in the study of systems with short T2 at frequencies below 1 MHz is discussed. In addition we discuss the bene®ts of both broadband and tuned input circuits for NMR detection and we describe the performance of a spectrometer with a tuned input circuit which has been used to obtain signals at 1 MHz from platinum powder at 4.2 K and from 02 layers of 3He absorbed on a surface area of 0.11 m2 at 1.7 K. The ampli®er noise temperature is predicted to be 60 mK in the 3He experiment. This demonstrates the potential of the tuned set-up for measurements at low millikelvin temperatures on systems with low spin density and with T2 greater than several hundred microseconds. # 1999 Elsevier Science Ltd. All rights reserved
NOMENCLATURE T1 T2 T*2 fp f0 fsq fN hf2Ni1/2 Df B1 M M0 mr m0 vs B0 g o0 w0 Li Lp Lsq Mi B1c k Q Vs is rp
NMR Longitudinal Relaxation Time NMR Transverse Relaxation Time Free Induction Decay Time Constant Total magnetic ¯ux threading NMR pick-up coil Flux Quantum ¯ux coupled to SQUID ¯ux noise in the SQUID rms ¯ux noise in SQUID per unit bandwidth measurement bandwidth Magnetic ®eld produced by unit current in the pick-up coil Sample magnetization Magnitude of static magnetization Relative permeability permeability of vacuum sample volume Static NMR ®eld Gyromagnetic ratio Angular Larmor frequency = 2pv0 Static magnetic susceptibility SQUID input coil inductance NMR receiver coil inductance (also called pick-up coil) SQUID inductance Mutual inductance between input coil and SQUID ®eld at centre of receiver coil due to unit current inhomogeneity factor allowing for ®eld variation across the coil extra subscripts t and u denote tuned and untuned systems quality factor of tuned circuit initial signal voltage initial signal current total series resistance in input circuit
*Corresponding author. }Present address: Centre for Quantum Metrology, National Physical Laboratory, Teddington TW11 0LW, UK 591
592 LT Ls r Tcoil TN kB Nc SNR ec a ae Lesq TR t i1 oc
C. P. LUSHER et al. total inductance in input circuit stray lead inductance fractional inductance contributed to total input inductance by Li Temperature of resistive part of NMR pick-up coil noise temperature of SQUID ampli®er connected to NMR pick-up coil Boltzmann's constant number of turns on NMR pick-up coil rms signal-to-noise ratio (inital rms amplitude divided by rms noise) Coupled energy sensitivity of SQUID coupling constant between input coil and SQUID eective coupling constant when pick-up coil is connected eective SQUID inductance with pick-up coil connected spectrometer recovery time natural time constant of tuned circuit maximum value of current in input circuit (limited by Q-spoiler) crossover frequency at which tuned circuit more favourable than untuned for NMR
INTRODUCTION
Nuclear magnetic resonance (NMR) is a widely used technique for the study of magnetic systems. Measurements of the relaxation times T1 and T2 give information on atomic and molecular motions in physical systems. NMR response can also be used to give information on the state of materials and structural information can be obtained from frequency shifts. In many situations the magnetic response is small and with conventional spectrometers much averaging is necessary in order to obtain usable information. By utilizing the high magnetic ®eld sensitivity of the DC SQUID one can study these systems with NMR within a reasonable period of time. We have built two broadband pulsed NMR spectrometers which use a DC SQUID as the front end ampli®er and have demonstrated their performance on several physical systems. The spectrometers measure the precessing magnetization directly, rather than the more conventional method [1] of using a SQUID to monitor the longitudinal magnetization. The superconducting NMR pick-up coil forms a ¯ux transformer with the SQUID input coil. The pick-up coil is orthogonal to the applied static magnetic ®eld and the SQUID detects the precessing magnetization. There are a number of previous reports of DC SQUIDs being used to detect NMR in this way [2±5]. These systems have a distinct advantage over conventional NMR spectrometers in that they are broadband. This makes measurements as a function of frequency relatively simple and also results in short recovery times. The thin ®lm DC SQUIDs are operated in a ¯ux-locked loop (FLL) mode. The bandwidths of the NMR spectrometers are limited by the FLL electronics. In both spectrometers the noise is dominated by the ¯ux noise in the SQUID and the observed signal-to-noise ratio (SNR) is as calculated for spin systems where the radio frequency (rf) excitation ®eld can penetrate the entire sample. The ®rst spectrometer is based on a commercial SQUID system and operates out to 50 kHz; the second uses a SQUID with additional positive feedback (APF) [6] and has a 3 dB bandwidth of 3.4 MHz. APF SQUID systems designed for NMR applications have been also developed by other authors [7]. Some of our earlier broadband work has been reported elsewhere [8]. In addition we report data obtained from another spectrometer which uses a tuned input circuit and the SQUID operating open-loop as a small-signal rf ampli®er at 1 MHz. The tuned setup, which has been used by a few authors [9, 10], is particularly suitable for the study of samples with low spin density and with T2 greater than several hundred microseconds, especially if the NMR pick-up coil can be cooled. GENERAL REQUIREMENTS FOR A PULSED NMR SPECTROMETER USING DC SQUIDS
In a pulsed NMR experiment a homogeneous static magnetic ®eld is applied to the spin system under investigation resulting in a net magnetization along the static ®eld direction. A large rf tipping pulse is then applied to the spin system for a short time (typically tens to hundreds of microseconds). The pulses should be large enough that the rf magnetic ®eld is much greater than any local magnetic ®elds in the sample. The maximum signal will be obtained if the pulse tips the magnetization through an angle of 908 into the transverse plane. Following removal of
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the pulse the magnetization, which precesses about the static magnetic ®eld direction, is the quantity detected. In SQUID NMR it is essential to have orthogonal transmitter and receiver coils because of the potentially harmful eect of the high rf ®elds from the transmitter. Even with a cross coupling of less than 1% the required pulsed ®elds correspond to of order 104 f0 coupled to the SQUID. This level can be reduced during the pulse by placing a series array of hysteretic SQUIDs known as a Q-spoiler [11] into the input circuit. The initial signals can be of order 1 m f0 or below and they decay to zero exponentially with the relaxation time T*2. It is important that the recovery time of the spectrometer is short compared with T*2. Without taking precautions the SQUID electronics would loose lock during application of the transmitter pulse. In general a reset pulse which opens the feedback loop has to be applied during the pulse and the loop is closed immediately after the pulse. The system recovery time is therefore related directly to the FLL bandwidth with large bandwidths being required for short recovery times. When studying metallic systems eddy currents which decay following the transmitter pulse can cause additional magnetic ®elds comparable in size to the expected signals. We have studied these eects in bulk metallic samples.
CALCULATION OF SIGNAL SIZE
The expected signal size in a pulsed NMR experiment is best calculated using the principle of reciprocity [12]. The total magnetic ¯ux, fp, threading the pick-up coil is given by 1 fp B1 :Mdvs ; s
where B1 is the ®eld created by unit current in the coil, M is the sample magnetization, and the integral is over the sample volume vs. Following a perfect 908 pulse the amplitude of the transverse magnetization is the same as that along the static ®eld direction before the pulse and is given by M = M0 = w0B0/(mrm0), where B0 = o0/g is the amplitude of the static NMR ®eld, w0 is the static magnetic susceptibility of the sample and g is the gyromagnetic ratio. The peak amplitude of the ¯ux coupled to the SQUID fsq is [13] fsq
fp Mi ; Li Lp
2
where Lp and Li are the self inductances of the pick-up coil and the SQUID input coil respectively, assumed to be much greater than any stray lead inductance, and Mi is the mutual inductance between the input coil and the SQUID. Using Eqs. (1) and (2) and integrating over the sample volume we obtain for the initial amplitude of the ¯ux coupled to the SQUID following a 908 pulse Mi w0 o 0 3 vs B1c k; fsq Li Lp mr m0 g where B1c is the ®eld at the centre of the receiver coil and the inhomogeneity factor, k, allows for the variation in B1 across the sample. Since the observable signal is limited by ¯ux noise in the SQUID, which is white above the 1/f knee at 01 Hz, the signal-to-noise ratio (SNR) in the NMR experiment is proportional to frequency. This should be contrasted with the situation in conventional pulsed NMR systems which have the pick-up coil as part of a tuned tank circuit which is followed by a voltage ampli®er. In that case the precessing magnetization induces a voltage across the coil which is proportional to the rate of change of ¯ux and the SNR goes faster than linear in frequency. Therefore at high enough frequencies the latter input con®guration gives a higher SNR but the former wins out at low frequencies. This is discussed in more detail in Section 8 (Comparison of tuned and untuned input con®gurations). In some cases at higher frequencies it is advantageous to use a tuned tank circuit input in order to get improved signalto-noise.
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THE 50 kHz BROADBAND DC SQUID NMR SPECTROMETER
The ®rst broadband spectrometer we constructed is similar to that described in [2] and [3]. We have used a Quantum Design DC SQUID system with a model 5000 controller which we modi®ed for the purpose. The static B0 ®eld is trapped in a niobium cylinder. The transmitter and receiver coils could be adjusted at room temperature to minimize the cross coupling. Both coils were wound from single ®lamentary Nb±Ti superconducting wire, CuNi clad with a 75 mm superconducting core. The cladding was removed with an HNO3 etch where necessary. The transmitter was a saddle coil designed for optimum homogeneity [12]. Receiver coils of both saddle and solenoidal form were used and in each case the inductance was made close to that of the SQUID input coil in order to optimize the coupled ¯ux [13]. All the low temperature input circuitry and transmitter circuitry are contained in niobium shields, in order to reduce noise from extraneous magnetic ®elds. We use four pairs of crossed diodes in series with the transmitter coil, placed at room temperature, in order to prevent noise from the transmitter electronics coupling to the SQUID when the pulses are o. The spectrometer is controlled by a pulse generator which, as well as gating the transmitter, activates both the reset circuitry (opening the feedback loop when the pulse is on) and the sample and hold (which removes the DC oset that arises as the SQUID regains lock when the feedback loop is closed [2, 3]). The purpose built cryostat has a pumped helium pot with a continuous ®ll line and can achieve temperatures down to 1.4 K. Plenty of space is available for housing SQUID sensors. Generally the SQUIDs are operated at 1.4 K although the SQUID temperature can be controlled independently of that of the sample. PERFORMANCE OF THE 50 kHz SPECTROMETER
The spectrometer has been tested on a variety of samples. As reported in [8], we have measured signals from 1021 spins of 195Pt in platinum powder at temperatures between 1.4 K and mf0/ p 4.2 K with the initial signal size being given by Equation (3) and with the noise of 05 Hz, dominated by ¯ux noise in the SQUID. Figure 1 shows an NMR signal from 1020 spins in 3He gas at around 40 kHz and 4.2 K. The narrow width of the line shows that the static ®eld homogeneity is not seriously aected by the superconducting rf coils. With this sample the recovery time of the spectrometer following a 1 mT pulse is less than 50 ms. We have used the
Fig. 1. Signal from 1020 spins in 3He gas at 4.2 K. Shown here is the Fourier transform of a free induction decay signal following a 908 pulse averaged 512 times. Signal has been mixed down using a lockin detector to get adequate spectral resolution in a 200 ms sample window.
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spectrometer to obtain signals from bulk Pt samples (here the signals are obtained only from spins within a skin depth) and from a single monolayer of 3He adsorbed on graphite. In the latter experiment silver foils were diusion bonded to the graphite for thermal contact. In both experiments the presence of the normal metal in the sample cell gave rise to an additional recovery signal following removal of the pulse due to eddy current decay. However we found that it was possible to ®t to the background and extract the NMR signal from the response. NMR MEASUREMENTS WITH AN APF SYSTEM
A broadband spectrometer is essential for measurements on samples with short T2 (for example most solid state samples) at low frequencies since the recovery time constant of a tuned circuit of quality factor Q, given by t = (2Q/o0), is excessive. One situation where low frequencies are essential is in ultralow temperature measurements on conducting materials where the eddy current heating eects are prohibitive at high frequencies. In order to obtain a short recovery time a large bandwidth is necessary. We have performed initial tests of a new broadband spectrometer based on a DC SQUID which uses APF to allow direct coupling between the SQUID and the read-out electronics, thus enabling a large bandwidth. A small cryostat was built for these tests which could be inserted into a helium transport dewar. The experiment was a proof-of-principle and was performed under non-optimized conditions. A large area (7.2 7.2 mm) multiloop DC SQUID fabricated at PTB was used. The SQUID has an inductance of Lsq = 400 pH and is equipped with an integrated APF coil, resulting in a transfer coecient p of 250 mV/f0 at the operating point. This particular SQUID has a ¯ux noise of 3 mf / Hz. The noise of the room temperature ampli®er is 0 p 0.74 nV/ Hz, and is thus comparable with the SQUID voltage noise. The ¯ux-locked loop operates with a measured bandwidth of 3.4 MHz. With the present shielding arrangement the maximum slew rate is 8 105 f0sÿ1. The SQUID is con®gured as an integrated magnetometer without an input coil. Since the NMR measurements require the use of a remote pick-up coil and a ¯ux transformer input circuit, an input coil is essential. An input coil was wound from 20 turns of niobium wire and was mounted under the SQUID carrier. Its self inductance Li = 4.8 mH. Due to the large thickness of the chip carrier (which was originally used for testing the SQUID), the measured coupling constant a was only 0.125. This results in a ¯ux sensitivity of 0.39 mA/f0, equivalent to a mutual inductance Mi = a(LiLsq)1/2 of 5.3 nH. The sample pick-up coil of inductance Lp = 2 mH was wound from NbTi wire and formed a superconducting ¯ux transformer with the input coil. We used a coil set that had previously been used on the 50 kHz spectrometer. The pick-up coil and NMR transmitter coil were both wound in saddle geometry on separate cylindrical formers fabricated from a machinable ceramic (MACOR) to close tolerances. Vacuum grease was smeared between the coil formers before assembly, which allowed easy adjustment of orthogonality at room temperature (by minimizing cross-talk) but solidi®ed to form a rigid structure when cold [14]. About 1% orthogonality was achieved. The sample consisted of platinum powder containing 1021 spins of 195Pt. Figure 2 shows a schematic of the APF SQUID based spectrometer. TTL pulses are provided by a computer controlled pulse generator. They are used for timing the rf transmitter pulse, resetting the FLL during the pulse (i.e., the output is forced close to zero voltage), and for triggering a sample and hold circuit which sets a zero starting point for data acquisition. Signals are captured and averaged on an oscilloscope (Tektronix TDS 410A), which is controlled by a PC with LabView software. A home built persistent NMR magnet in a superconducting shield is used to provide the static magnetic ®eld. The coil is wound from 0.1 mm diameter Nb±Ti wire on an annealed copper cylinder. It consists of 9 layers and 5310 turns, and is of inner diameter 16 mm and 80 mm long. It is placed in a 35 mm diameter niobium cylinder 100 mm long. The measured ®eld is 62.7 mTAÿ1, which is within 1% of the calculated value, and the predicted homogeneity is 10ÿ4 over 1 cm on-axis. In persistent mode at 10 mT the ®eld is stable after a few minutes and at 60 mT it is stable at a drift rate of a few f0 per day after about half an hour. It is possible to make NMR measurements essentially immediately after persisting the magnet. Since the static
Fig. 2. Schematic diagram of 3.4 MHz bandwidth NMR spectrometer based on a DC SQUID with APF.
596 C. P. LUSHER et al.
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597
®eld can be readily changed in this way, making NMR measurements as a function of frequency is convenient. PERFORMANCE OF THE APF SQUID NMR SPECTROMETER
Figure 3 shows some examples of NMR signals. They are Fourier transforms of free induction decays (FIDs) of platinum spins following a 908 transmitter pulse. The measurements below 100 kHz were made using a slightly dierent set-up from those at higher frequencies. At all frequencies a 908 pulse of length 294 ms and amplitude 0.1 mT was used. The recovery time after the transmitter pulse is about 10 ms; i.e. when the reset pulse is timed to come o 10 ms after the end of the transmitter pulse an NMR signal can be captured immediately. In order to study samples with larger linewidths the transmitter pulses must be made both shorter and larger. For these situations it is planned to include a Q-spoiler in the ¯ux transformer to limit the induced current during a transmitter pulse. With the ¯ux transformer input circuit the initial amplitude of the FID is expected to be proportional to the magnetization i.e. proportional to the NMR frequency (Equation (3)). Although the amplitude of the FFT is not linear in frequency, the inset to Fig. 3 shows that when corrected for an increase in linewidth a good linear dependence of signal size is observed. In fact 1/T*2 is linear in frequency, which would be expected if there is an increase in the static ®eld gradient due to ®eld inhomogeneities. The presence of a superconducting pick-up coil is clearly not a serious problem. The signal size is as predicted using the principle of reciprocity
Fig. 3. Fourier transform of free induction decays from 1021 platinum spins at 4.2 K; performed at Larmor frequencies of 38, 65, 85, 240 and 513 kHz. Inset: linear dependence of signal size (amplitude of FFT linewidth) on Larmor frequency.
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and the known circuit parameters. The signals shown here correspond to 105 averages. With the present set-up an rms signal-to-noise ratio of 10 is achievable with about 40 averages at 513 kHz. With a lower-noise SQUID and optimized input circuit an improvement of a factor of 6 in SNR is possible, equivalent to an rms SNR (rms amplitude to rms noise) of 10 in a single shot. At lower frequencies the measurements suer from a signi®cant amount of ®eld dependent noise, possibly attributable to magneto-acoustic resonances in the coil formers. In early experiments with the 50 kHz spectrometer the problem was particularly serious when using coil formers fabricated from stycast 1266 and perspex but they were signi®cantly improved by using MACOR. With the new set-up there are still a number of such resonances below 100 kHz, but it is possible to ®nd ``quiet'' regions within the spectrum. At higher frequencies these problems are not so serious. We have recently made modi®cations to increase coupling of the input coil to the SQUID and to achieve better inductance matching (Lp = Li). A hole was cut in the chip carrier to take a new input coil which was mounted directly under the chip. The new parameters were Li = 2.4 mH, Mi = 9.6 nH and a = 0.31. Preliminary NMR spectra have been obtained with an amplitude SNR a factor of three larger than that in Fig. 3. The coupling coecient cannot be increased to more than 00.4 with the octagonal SQUID geometry used, however by using a lower-noise SQUID with a higher transfer coecient another factor of two improvement in SNR is possible.
COMPARISON OF TUNED AND UNTUNED INPUT CONFIGURATIONS
The rms SNR for the untuned system with an input ¯ux transformer can be written as ! ! ! fsq B1cu ku vs w0 o0 Mi 1 p ; fN 2 2Lpu mr m0 g hf2N i1=2 Dfu 1=2
4
where we have neglected any stray inductance in the leads and have chosen optimal matching conditions (Lp = Li). The subscript u is used to denote the untuned system. hf2Ni1/2 is the rms ¯ux noise in the SQUID per unit bandwidth when optimally matched and Dfu is the measurement bandwidth. We assume SQUID noise dominates. Since the ¯ux noise in the SQUID is independent of frequency above the 1/f knee (01 Hz) then the SNR is proportional to frequency. We now calculate the SNR for the case of a tuned series tank input circuit. In this case the rate of change of magnetic ¯ux is detected and the SQUID acts as an rf ampli®er. The peak amplitude of the initial signal voltage induced across the terminals of the NMR pick-up coil is given by jVs j o0 fp o0 B1ct kt M0 vs
5
and the signal current is given on resonance by is = Vs/rp, where rp is the total series resistance in the input circuit which for a high-Q series tank circuit on resonance is given by rp = o0LT/Q, where LT = (Li + Ls + Lp) is the total inductance in the input circuit and Ls is any stray inductance associated with the leads. The subscript t denotes the tuned system. For the SQUID ampli®er operating under optimized conditions, which results in a minimum noise temperature, we have Qr 1 1, where r is the fractional inductance contributed by the SQUID input coil [9]. Therefore Lpt 1 LT 1QLi and any stray inductance can be ignored. The signal current on resonance is given by is
B1ct kt M0 vs Q : Lpt
6
The noise current on resonance has a contribution from both Johnson noise in the coil (or the resistive part of the input circuit) at temperature Tcoil and from noise in the SQUID ampli®er. The mean square current noise on resonance is given by
Broadband nuclear magnetic resonance using DC SQUID ampli®ers
hi2N i
4kB Tcoil TN Dft ; rp
599
7
where Dft is the measurement bandwidth for the tuned system and TN is the noise temperature of the SQUID ampli®er, which is a function of both the SQUID and the receiver coil parameters, including the tank circuit quality factor. Substituting for M0 gives the rms SNR with the tuned input con®guration ! ! ! 1=2 is B1ct kt vs w0 o3=2 QLpt 1 0 p ; 8 mr m0 g kB Tcoil TN hi2N i1=2 Dft 1=2 2 2Lpt which should be compared with Equation (4) derived p above for the untuned set-up. We note that if TcoilwTN then the SNR is proportional to Q. In order to obtain the maximum SNR the Q must be made as large as possible. The maximum allowed Q is constrained by T*2, since t = (2Q/o0), and will have a frequency dependence Q A o0T*2. However, in general the SNR increases faster than linear with o0. One can compare the two set-ups under various conditions. As an example we consider a sample of ®xed geometry and volume, vs, containing N spins and ask at what frequency the SNR is equal in the two cases. The SQUID and the receiver coil geometry are taken to be the same but the number of turns on the receiver coil Nc is dierent since the optimization conditions are dierent for the two con®gurations. In general we have (B1ck/Lp) = g/Nc where g is a purely geometrical factor. The ratio of the SNRs for the untuned and tuned con®gurations is given by ! !1=2 SNRu Nct Mi kB Tcoil TN Dft 1=2 : 9 SNRt Ncu hf2N i1=2 Dfu Qo0 Lpt We note that for a given sample at ®xed frequency the required measurement bandwidth is independent of the input coupling scheme and therefore Dfu = Dft. Also in the situation p where both pick-up coil inductances are optimized Lpt = QLi = QLpu so that (Nct/Ncu) = Q. Writing the SQUID parameters in terms of the coupled energy sensitivity ec = Lihf2Ni/(2 M2i) (i.e., the energy equivalent of the minimum detectable current in the input coil) we have !1=2 1=2 M2i SNRu kB Tcoil TN 1=2 1 kB Tcoil TN 1=2 : 10 SNRt opt Qo0 2ec Qo0 Li hf2N i We should point out that when a load is connected to the input coil of a DC SQUID the SQUID parameters can be signi®cantly altered, especially if the input coil is strongly coupled to the SQUID. In particular, the SQUID behaves as an isolated SQUID but with a reduced inductance Lesq = (1ÿa2e)Lsq, where a2e = a2 (Li/LT) [15]. For the tuned set-up under optimal conditions Lp wLi and the aect of loading is negligible, however it can be signi®cant in the untuned case if a2 is close to unity. Therefore the coupled energy sensitivity in Equation (10) is that which corresponds to the SQUID connected to an optimally matched ¯ux transformer. The crossover frequency oc at which the tuned circuit becomes the more favourable for obtaining maximum SNR is given by the frequency at which the ratio in Equation (10) is unity. It is a function of SQUID parameters, pick-up coil parameters and sample parameters (since the maximum usable quality factor in the tuned case is determined by the NMR properties of the sample). There are two factors constraining the quality factor. Firstly the electronic bandwidth must be wide enough to avoid distortion of the NMR line i.e., Do = o/Q w1/T*2 which implies that high-Q tuned systems are more useful for systems with long T*2s. Also the system recovery time following removal of the transmitter pulse must be fast compared with T*2. In practice a Qspoiler is incorporated into the SQUID input circuit in order to limit the maximum current in the input circuit during the pulse to of order the minimum Q-spoiler critical current, i1. Once the transmitter pulse is removed the current in the input circuit will decay exponentially from i1 with the natural time constant t and the spectrometer recovery time TR will be given by TR = (2Q/o0) 1n(i1/is) where is is a typical signal amplitude. As an example we take i1 to be the
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C. P. LUSHER et al.
current which couples 5 f0 to the SQUID (as in our set-up) and choose an initial signal amplitude equivalent to 1 mf0. If we stipulate for example that we need TR ET*2/10 then this condition limits Qmax to be less than (o0T*2/200). Using this expression for Q we obtain !1=2 1=2 1 200kB Tcoil TN : 11 oc 2ec T2 The value of the crossover frequency clearly depends on the frequency dependence of T*2. However in order to get a feeling for its size we take a SQUID with ec = 200 h (where h is Planck's constant), a coil temperature of 4.2 K with TN negligible, and a frequency independent T*2 of 1 ms. This results in a value of 1.05 MHz for oc/(2p). The choice of the value of Q to insert into Equation (10) is subtle. For example if Tcoil wTN it is possible to choose a Q higher than one might naively expect based on the sample T*2 and then use feedback to increase the bandwidth but without signi®cantly reducing the SNR [10]. The eect of this is to lower oc. This method will not work, however, if Tcoil ETN. Note that for T*2 = 50 ms, typical of solid state samples, then oc/(2p) = 4.5 MHz with Tcoil = 4.2 K. In the above analysis we have assumed that other sources of noise for example additional noise from room temperature ampli®ers or sample noise are negligible. USE OF A SPECTROMETER WITH A TUNED TANK INPUT CIRCUIT
State-of-the-art SQUID ampli®ers with optimum pick-up coils are predicted to have noise temperatures of around a few tens of millikelvin at 1 MHz with TN A o0 [9]. Tuned systems therefore oer unprecedented sensitivity for systems with receiver coils at low millikelvin temperatures. At frequencies below 1 MHz this makes the system suitable for samples with a long T2, for example 3He. In this set-up a Q-spoiler is incorporated into the input tank circuit in order to reduce its ringdown time. This has a normal resistance of approximately 1 kO during the large transmitter pulses, thereby reducing the Q, but is superconducting at low currents. Samples with low spin density and with T2 of more than a few hundred ms can be studied in this way. We have used a spectrometer with a tuned input tank circuit (Q 0 30) in order to measure NMR signals at a frequency of 01 MHz from platinum powder and from 3He ®lms at 4.2 K and below. The commercial DC SQUID chip (a Conductus SQD1002 with a quoted ¯ux noise p hf2Ni1/2 of 3 mf0/ Hz) was operated open loop as a small signal rf ampli®er. The signal was coupled out via a tank circuit with a Q of 010 in order to ensure that the noise of the room temperature preampli®er was insigni®cant. The Q-spoiler was a Conductus SQD 1060 chip with a minimum critical current of 2 mA, which corresponds to 5 f0 coupled to the SQUID. In the 3 He experiment the sample cell was 1 cm diameter and 1 cm long and contained 760 Mylar discs. The total surface area for adsorption was 0.11 m2. The sample was 02 layers of 3He at 1.7 K. Two layers contain 1.5 1018 spins. Some of the signal from the ®rst layer is expected to be lost since those spins have a short T2. Here the signal-to-noise ratio was limited by Johnson noise in the pick-up coil. The pick-up coil had an inductance of 5.8 mH. A short length of constantan wire was placed in the input circuit (in the niobium shield of the SQUID) in order to limit the input tank circuit Q to 30. The pick-up coil inductance was chosen to be optimum given the quoted SQUID input coil inductance of 170 nH. The noise temperature of the SQUID ampli®er was estimated to be 60 mK, taking the contribution from the SQUID to the total outp put noise to be 2 hf2Ni1/2, as expected for the optimal source impedance [16]. Noise measurements at 4.2 and 1.7 K were consistent with this but the value of TN was impossible to measure with adequate precision given the minimum coil temperature of 1.7 K. In Fig. 4 we show a free induction decay following a 908 pulse averaged 1024 times. The Larmor frequency is 946 kHz. The measured signal size is of order 6 mf0 peak-to-peak at the SQUID. The signal capture startS 100 ms after the end of the 50 ms transmitter pulse. This recovery time is determined by the Q of the input tank circuit. The system 3 dB bandwidth is 20 kHz, corresponding to a noise bandwidth of 31.4 kHz. The rms SNR is 19.pIn this case the signal has been mixed down resulting in a decrease in the SNR by a factor of 2 due to noise contributions from both sidebands.
Broadband nuclear magnetic resonance using DC SQUID ampli®ers
601
Fig. 4. Mixed down signal from 02 layers of 3He adsorbed on Mylar at 946 kHz and 1.7 K averaged 1024 times. Sample contains 01018 spins. Signal capture starts 100 ms after end of 50 ms 908 transmitter pulse. This recovery time is determined by the Q of the input tank circuit. The noise bandwidth is 31.4 kHz.
Taking this into account we would expect a value of 13 for the SNR from Equation (8) with 1018 spins, in good agreement with the measured value considering the uncertainty in the number of observable spins. It should be pointed out that this sample has a very low spin density (a factor of 13,000 smaller than that of liquid 3He at this temperature and atmospheric pressure). AcknowledgementsÐWe thank Alan Betts, Francis Greenough, John Taylor and Tony Wilkinson for technical assistance. This work was supported by EPSRC (UK).
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