Spectrometers for Multiple-Pulse NMR

Spectrometers for Multiple-Pulse NMR J. D . E L L E T T , J R . , M. G. GIBBY, U . H A E B E R L E N , * L. M. H U B E R , f M. M E H R I N G , A. P I N E S , AND J. S. W A U G H DEPARTMENT OF CHEMISTRY AND RESEARCH LABORATORY OF ELECTRONICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS

I. Introduction II. Spectrometer Design and Construction A. Pulse Programmer B. rf Source . C. Four-Phase Generator D. Transmitter E. Duplexer . F. Probes G. Receivers . H. Phase Detection I. Signal Processing J. Video Field Pulsing K. Field Stabilization Operation III. A. Introduction B. Pulse Program C. Tuning D. Results Note Added in Proof

117 122 122 134 136 139 147 150 153 153 157 162 163 168 168 168 169 172 176

I. Introduction The repertoire of transient N M R has been considerably enlarged in the past few years by a number of new experiments. We have in mind particularly, but not exclusively, a variety of "coherent averaging" techniques 1 - 3 with which we have ourselves been heavily involved. In these experiments one applies to the sample a long burst of strong magnetic excitations in which the rf am­ plitude and phase, and sometimes the Zeeman field H0, are programmed in periodic but often complicated fashion. One selects from the complex transient * Present address: Max-Planck-Institute, Heidelberg, Federal Republic of Germany. t Present address: Dow Chemical Company, Midland, Michigan. 1 J. S. Waugh, C. H. Wang, L. M. Huber, and R. L. Void, /. Chem. Phys. 48, 662 (1968). 2 J. S. Waugh, L. M. Huber, and U. Haeberlen, Phys. Rev. Lett. 20, 180 (1968). 3 U. Haeberlen and J. S. Waugh, Phys. Rev. 175, 453 (1968). 117

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nuclear response a set of sample points separated in time by one cycle1 of the excitation: These points trace out the free induction decay (FID) which the system would execute under the influence of an "average Hamiltonian" differ­ ing from the actual spin Hamiltonian in a way which is characteristic of the excitation chosen. In this chapter we discuss the design of spectrometers which are capable of performing coherent averaging experiments (as well as most other transient NMR experiments). As a framework we shall use the generalized block diagram of Fig. 1, where we have labeled the blocks to correspond with parts of Section II of this chapter. At the level of this figure, one coherent spec­ trometer looks pretty much like another: it is in the contents of the blocks that any special requirements will become evident. Roughly speaking, they include in our case: (a) high rf power and fast response, (b) complex programming capability, (c) high stability of many parameters during long bursts of excitation. Requirement (a) is a familiar one in connection with studies of FID's and 7\ processes in solids, and we shall discuss only certain special aspects of it in this chapter. Unfortunately it has appeared to some, including commercial suppliers of spectrometers, that a conventional "high powered" pulsed NMR system could be adapted to coherent averaging experiments by adding a programmer satisfying (b) above. This is typically not so, as a result of requirement (c) being far more stringent for coherent averaging experiments than for the more C

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usual pulsed NMR methods. The reason for this is partly that the nuclear precession transient may be lengthened by a very large factor over the normal FID by applying many cycles of a very strong excitation. The longer the decay, the more disastrous become the errors which accumulate from very small perturbations of each cycle of excitation : to make the case more concrete, consider the relatively simple Lee-Goldburg experiment,4 in which one applies a rotating rf field of amplitude Hl, with the Zeeman field offset from exact resonance by AH0. For large H1 and AH0, the dipole-dipole broadening of the spectrum behaves as though it were reduced by a factor. a = i(3cos20-l);

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It is desirable and within the realm of possibility to make a as small as, say, 10" 3 . According to (1), this establishes a tolerance of the order of 0.1% on the adjustment and stability of the amplitude Hl of the rf field during the burst. While there are various ingenious ways of cancelling some parts of such errors, it is clear that averaging experiments often do place unaccustomed require­ ments on amplitude stabilities and other parameters. Pulse spectrometers are nearly as old as magnetic resonance itself. Cor­ responding to the development of the art, there is a substantial literature 5-10 dealing with the subject. We will discuss various experimental problems and their solutions with specific reference to two spectrometers built in our laboratory. Spectrometer A (Fig. 2) operates at a fixed frequency of 54 MHz (chosen for its suitability to *H and 1 9 F resonance in a rather ancient Varian electro­ magnet which was originally part of a high resolution 60 MHz proton spec­ trometer). The pulse programmer for this spectrometer is hard-wired using DTL integrated circuits, and programs are changed by changing plug-in program cards. The last two stages of the transmitter are tuned (accounting for our tendency not to leave 54 MHz) and deliver more than 4kW to a matched load. Data are recorded on a standard signal averager: they can be extracted on punched paper tape for further processing (Fourier transformation, etc.) on a remote computer. The necessity for signal averaging forced us to develop a NMR field lock for this spectrometer, based in our case on a 7 Li resonance from an external sample to avoid interference between the field lock and the pulse spectrometer. 4 5 6 7 8 9 10

M. Lee and W. I. Goldburg, Phys. Rev. 140, AÌ26Ì (1965). W. G. Clark, Rev. Sci. Instr. 35, 316 (1964). P. Mansfield and J. G. Powles, / . Sci. Instr. 40, 232 (1963). J. C. Buchta, H. S. Gutowsky, and D. E. Woessner, Rev. Sci. Instr. 29, 55 (1958). K. Luszczynski and J. G. Powles, / . Sci. Instr. 36, 57 (1959). J. Schwartz, Rev. Sci. Instr. 28, 780 (1957). A. G. Redfield, Rev. Sci. Instr. 27, 230 (1956).

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Spectrometer B (Fig. 3) was designed with the benefit of experience gained with spectrometer A. An important element of B is a small computer (Digital Equipment Co. PDP-12A). The computer allows us to realize a pulse pro­ grammer of very great versatility—a necessity on the basis of our discovery that it is possible to invent new pulse experiments at the rate of one a week or greater. Moreover we can now process the results on line: we had found that the usual conclusion drawn after waiting 24 hr for a Fourier transform from the remote computer was that the spectrometer had been adjusted incorrectly. In order to obtain greater mobility in frequency, the possibility of multiple resonance experiments, and freedom from various transient phase behavior to be discussed, we designed this spectrometer essentially without tuned circuits except for the probe itself. It operates over the range 20-250 MHz, the upper limit corresponding to proton resonance in a 60kG superconducting magnet (Oxford Instrument Co.), which is the only part of the spectrometer not operating at the time of this writing. Because of its wide band design, the transmitter output power of spectrometer B is limited to ~ 1.5kW pulsed or 700 W average. II. Spectrometer Design and Construction A. PULSE PROGRAMMER

In this section we describe two pulse programmers. The first, programmer A, is a simple clock-controlled device with pulse sequence timing determined by hardwired logic circuits situated on plug-in cards. This allows for convenient transfer from one pulse sequence to another. Construction is relatively simple and inexpensive and so is the interface with the rest of the spectrometer. The second programmer, B, is a more versatile computer controlled device constructed for our more sophisticated pulse sequences and elaborate data processing. The versatility is gained at the expense of simplicity of design, construction, and interfacing. The description of each programmer is written as a complete self-contained unit as far as possible and each may be read separately. However, certain units common to both devices will be found in more detail in the description of B. 1. Pulse Programmer A The plug-in pulse programmer (see Fig. 4) was designed to provide the timing for our earlier multiple pulse experiments. The pulse sequence it produces is determined by a single card of integrated circuits, which can be replaced by different cards to change the pulse sequence. Our present library of program cards enables us to use this programmer to perform the Carr-

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Purcell, Meiboom-Gill, phase-alternated, three-pulse, four-pulse, six-pulse, HW eight-pulse, and Evans eight-pulse experiments.3 These experiments require that the programmer be able to control the following events in the spectrometer. (i) Prior to the beginning of the burst of rf pulses, dummy loads must be switched on to draw current from the transmitter power supplies so that they will attain a steady state before the loading by the burst begins. The ac input power to the plate supply of the final amplifier must be increased during this preloading and also during the rf burst to offset the voltage droop caused by the loading. (ii) At the beginning of the burst, scope displays must be triggered and signal digitizing must be initiated. (iii) A burst of rf pulses made up of repeated groups of "cycles" of pulses must be produced. The length of this burst and the width of the pulses must be adjustable. In some experiments the pulses within a cycle have different phases so they must be routed through the different phase channels of the spectrometer. Also video current pulses coincident with the rf pulses may be required. (iv) Once during every cycle of pulses the signal digitizer must be triggered. (v) After the burst the spectrometer must wait a variable delay time to allow the sample to remagnetize before repeating the experiment. This is called the recycle delay. The plug-in pulse programmer is built with "830 series" DTL integrated circuits. All timing within the programmer except for the widths of the rf pulses and the duration of preloading is derived from a single clock square wave. This clock signal may be generated internally with a simple RC astable multivibrator or it may be externally supplied. The timing of both the leading and trailing edges of the rf pulses are independently adjustable with respect to the clock time. The trailing edge of each pulse is set by a single one-shot, which is triggered on every clock signal to provide a stable delay. Thus, we have one control which varies the widths of all the pulses in a burst. The leading edges of the pulses in each phase channel are individually determined by in­ dependent one-shots which provide fine adjustments of the pulse widths. There is also a provision for generating an rf pulse at the beginning of the burst whose width is set independently of the others. This pulse may be used in the CarrPurcell and other experiments where the first pulse width differs from the rest. The integrated one-shots available at the time this programmer was con­ structed (e.g., type 851) generate pulse widths which are somewhat dependent on the time which has elapsed since the previous pulse—a situation which cannot be tolerated in many of our experiments. Accordingly we used home­ made one-shots built with discrete transistors, designed to permit rapid

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restoration of circuit equilibrium. At the present writing, much improved TTL one-shots are available (e.g., type 9601), and were used in programmer B. The proper sequence of phases and spacings for a given multiple pulse experiment is obtained by selecting various phase channels with "enabling" pulses generated by the logic circuits mounted on the plug-in card. These logic circuits are usually ring counters or related circuits which are advanced at the clock times. They run continuously, stepping through their cycle of pulses and in addition generating a "cycle count" pulse once each cycle. This pulse triggers the signal digitizer and advances a ripple counter which de­ termines both the length of the burst and the delay between the bursts. Both lengths are adjustable. The program cards are usually designed so that when the burst length is set to its shortest value only a single rf pulse is produced. This makes it easy to obtain a simple Bloch decay to check spectrometer settings. A typical multiple pulse experiment begins when the one-shot which gen­ erates the preloading time receives a trigger pulse from either the manual start switch or from the ripple counter. This circuit holds the transmitter power supply dummy loads on for 80 msec and then waits until a synchronizing pulse is received before turning them off and initiating the rf burst. The synchroniz­ ing pulse is usually the cycle count pulse, although it can be supplied externally to synchronize the burst with an external event. For example, it is helpful in tracking down hum problems to be able to synchronize the burst to the 60 Hz line frequency. The burst begins on the first cycle count pulse after receipt of the synchronizing pulse. At that time the enabling pulses of the program card are themselves enabled and a "burst trigger" pulse is generated which clears the ripple counter, initializes the program card circuits, triggers the initial pulse, and is available externally to trigger scope displays and initialize data collection. The burst continues until the ripple counter exceeds a preset power of 2, at which time the enabling pulses are disabled, shutting off the burst. The logic circuits on the program card continue to produce cycle counts and thus advance the ripple counter while the sample (and the transmitting tubes) recover from the burst. The experimental sequence begins again when the ripple counter exceeds a second preset value which determines the recycle delay. 2. Pulse Programmer B a. Introduction. The complexity of pulse experiments in our laboratory has increased appreciably in recent years. Previously, the implementation of new pulse experiments invariably necessitated the time consuming design and con­ struction of specific logic circuits. To overcome this obstacle we therefore set out to produce a more versatile and generally powerful pulse programmer, and the end result of our endeavor is described in the following sections. The

FIG. 5. Functional schematic of the computer-controlled pulse programmer of spectrometer B.

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device contains an independent memory unit of its own which carries all the information necessary for the execution of an arbitrary pulse sequence, and is preloaded and otherwise serviced by an on-line computer. The realization of new pulse experiments is thus essentially reduced to a computer software problem—a considerable advance over previous methods—and recent ex­ periments have already utilized this facility with promising efficiency.11 A schematic diagram of the programmer is shown in Fig. 5. Components include mostly homemade circuits and currently available TTL and DTL logic. Section II, A, 2, b describes the basic operation including the memory, cycling, and addressing. Sections II, A, 2, c-g describe the different units involved in the production of a pulse sequence, and Section II, A, 2, h elab­ orates on the points of access and function of the computer in servicing the instrument. To summarize, Section III shows a sample pulse program and the result of one computer controlled experiment. b. Basic Operation. The heart of the programmer is an extendable 384 bit flip-flop memory organized as sixteen words of 24 bits each. Each word is assigned an address in the memory and contains an instruction to be executed by the programmer. The assignment of bits within each word is depicted in the schematic diagram. Here we give a brief description of the function of these bits, the basic cycling of the instrument and addressing of the memory. More specific details on the different units involved are presented in later sections. The cycling of the instrument is achieved very simply. Each instruction contains several basic pieces of information : (i) the function to be performed (pulse output, sampling, counting, etc.), (ii) address of the next instruction, and (iii) the delay until execution of the next instruction. The main operating unit is the major state generator. This emits 2 pulses called TP1 and TP2 about O.l/zsec apart. TP1 triggers the execution of the current instruction and the specified delay and TP2 then selects the next word specified by the next address (NA) bits in the memory. The major state generator is then retriggered at the end of the delay through the end-delay gate and the next instruction is executed. In this way cycling is maintained and we see that the cycle time is redetermined at each instruction and is normally not clock-controlled as in pulse programmer A. Summarizing, the basic chain of events is : (i) end of previous delay triggers major state generator, (ii) TP1 triggers execution of instruction and next delay, and (iii) TP2 selects next instruction. The function of the memory bits is as follows : 11

W-K. Riiim, A. Pines, and J. S. Waugh, Phys. Rev. Lett. 25, 218 (1970); Phys. Rev. 3B, 684 (1971).

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(i) NI N2 N3 N4—Next address. There are sixteen words coded in binary with addresses 0000-1 111. The addressing of a selected word is done in "x-y" fashion. One of four "*" lines and one of four " / ' lines specify which one of the sixteen words is to be accessed. This memory location may be written into by the computer in loading or read by the programmer during operation. To understand the addressing more explicitly let us assume that the programmer is just executing the instruction in word 0010 and the NA bits N1-N4 of this word contain 0011 as the next address. (This next address may or may not be modified at the adder as explained later in this section.) In any case, TP1 transfers the number 0011 or the modified address to the NA register whence TP2, after execution of instruction 0010, then transfers it to the current address register where it becomes the new current address. The decoders use the 4 bits from the current address register as two sets of two, each of which is decoded to 1 of 4 lines, the x and y sets mentioned above. These are used, by driving them with current-sinking transistors, to select the corresponding word 0011 (or modification) in the memory. (ii) X, X, F, F, V, V and H. These bits specify whether or not a pulse is to be produced in any of the six output channels. For every bit that is set to logic 1 a pulse will be produced in the corresponding output channel. X9 X, Y9 Y designate the 4 channels used to gate the 4-phase box described in Section IT, C. VV are auxiliary outputs which may be used for producing video pulses,3 double resonance pulses, or for general triggering purposes. Bit H, the hold bit, extends the capabilities of the pulse output. If H is set to 0 we obtain normal pulsed output. If H is set to 1 the specified output channels are held open for the duration of the present delay, i.e., until the execution of the next instruction where they may be closed or, of course, held open. All this is discussed further in Section II, A, 2, f. (iii) C, Dl, D2, D3. These bits specify the delay. Dl D2 D3 specify one of 8 available codes for analog delays (Section II, A, 2, d) and a recycle delay (Section II, A, 2, d) and C specifies a clocked delay using a crystal clock with computer-controlled timing (Section II, A, 2, d). (iv) S-Sample bit. When set to 1 the sample-and-hold is triggered and the magnetization sampled. (See Section II, A, 2, g). (v) SB, BC, DC. Counter bits: These control three counters called the subburst, burst, and decade counters. When any of these bits is set to 1 the corresponding counter is incremented until overflow occurs at some specified count (either set manually or under computer control) and the advance decoder then modifies the next address by adding to it some number dependent on which counters have overflowed. This allows us to accomplish skipping and branching in the pulse program after a specific preset number of instruc­ tions or groups of instructions. A more complete discussion of this facility appears in Section II, A, 2, e.

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It is often useful to step through a pulse program one instruction at a time in order to examine the contents of the programmer memory or to check the program's operation. This can be done using the single instruction switch. In normal operation this is set to allow triggering of the major state generator through the end-delay gate. In single-step operation the end of a delay will not trigger the next instruction. Instead each instruction is triggered manually through the execute switch which produces a pulse from the corresponding pulse generator. All pushbutton switches of this sort are equipped with circuits to prevent multiple triggering due to contact bounce. A little thought, together with the example at the end, will convince the reader that most feasible pulse sequences require very few of the sixteen available memory locations. Thus another switch, the memory bank switch, enables us to store and use two pulse programs each of less than eight words. The switch allows the programmer to operate in either the lower memory bank, locations 0-7, or in the upper bank, locations 8-15. For longer pulse sequences the whole memory may of course be used as one unit. c. Recycling and Initialization. During an experiment one usually wishes to repeat the pulse sequence periodically; for example, in order to make adjust­ ments or perform some averaging. The experiment is thus composed of a major cycle consisting of what we call a "burst" during which the pulses are emitted and the signal observed, and a "recycle delay" during which the spin system returns to full equilibrium before initiation of the next burst. During the recycle delay the data may also be manipulated and the programmer serviced if necessary. The recycle delay is one of the 8 possible delays selected with each instruc­ tion. When its code is encountered by TP1, a 1 KHz astable multivibrator is activated. The output of this is fed into two channels, in one of which the frequency is divided by 1000. One or the other channel, selected by a manual switch, is fed to a 12 bit ripple counter. The overflow bit of the counter is selected on the front panel giving us recycle times of 1-211, in powers of 2, with msec or sec units. The end of the recycle delay does not trigger the execu­ tion of the next instruction as with other delays. Instead it is sensed by the initializer (consisting of three monostables) which prepares for initialization and triggering of the next burst. First, a clear pulse is emitted (output 1 of initializer), which after current amplification by a driver clears all the registers in the programmer. This includes the various counter registers, the current address register, the clock counter, and all skip and interrupt flags connected to the computer. Clearing may also be accomplished manually using the clear pushbutton. About l/isec later a second pulse "BG" (output 2) is sent to the mode select switch from which it may be directed to one of several places depending on the mode selected. The available modes and subsequent opera­ tion are:

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(i) Normal Operation. In this mode the retriggering is automatic. The pulse BG (output 1 of mode select) enters the end delay gate which then triggers the first instruction through the major state generator. (ii) Computer Control. Here the pulse BG (output 2) is channeled to the computer and continuation is thus computer controlled, the triggering of a new burst being initiated through input 2 of the end delay. This is discussed further in Section II, A, 2, h. (iii) Bloch Decay. This mode is used if one wishes at any time to look at a single pulse, or a Bloch decay of the sample, independent of the contents of the programmer memory. There are six positions in this mode, X, X, ..., V, and for the one selected a single corresponding output pulse is produced by hardwiring of the appropriate bit and disabling of the memory (symbolized by output 4). BG (output 3) is channeled to trigger the main pulse generators (see Section II, A, 2, f) and retrigger the recycle delay. Thus recycling here is automatic and a single pulse is repeated with a recycle delay period. (iv) Off. The pulse BG is unused, there is no retriggering of the burst, and programmer operation ceases. The initial cycling of the computer is activated by the start pushbutton switch, after loading a new pulse program or after cessation of operation for any number of reasons. This activates the pulse generator and simulates the end of a recycle delay, thus performing the necessary initialization and starting operation in the selected mode. d. Delays. The 8 available delay codes are used as follows: (i) Four Variable Analog Delays. Each analog delay consists of two identical monostable multivibrators (type 9601) with RC time constants controllable manually. The first monostable is triggered by a pulse from the delay decoder when its code is encountered by TP1. The end of the pulse triggers the second monostable and the end of the pulse from this triggers the major state generator through the end-delay gate and the next instruction is then executed. The use of two cascaded monostables is an artifice for insuring that the same delay can be immediately retriggered if this is called for by the next instruction. These delays cover different ranges. Two produce delays varying from 0.2 to 8/isec and two from 1 //sec to 1 msec. These may also be multiplied by one of the counters available (Section II, A, 2, e) so that any practically useful delay is accessible. (ii) One Fixed Analog Delay. This is the same as above except not con­ trollable on the front panel. Normally it is set at about 1 //sec and is useful when one wishes to generate an instruction immediately on termination of a 90° pulse without using one of the variable delays. (iii) Recycle Delay. This was described in detail in Section II, A, 2, c. (iv) Two are unused. One is available for extension and one remains open

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to be used when the crystal clock is requested and we want no extra delay triggered simultaneously. In addition to these delays we have available a crystal clock. When bit C is set to 1 a gate is opened and a 1 MHz crystal clock counts into a series of 6 decade counters providing us with 7 selectable channels with counting units of 1 ^sec to 1 sec. The desired channel is selected manually or by the computer and the clock register then fills at the corresponding frequency. The latter is a 12 bit ripple counter which can be preloaded and read by the computer. This opens up two modes of operation for the clock: (i) To produce a clocked delay the clock counter is preloaded by the computer with the complement of the desired delay. This is useful for experiments where computer-controlled timing of pulses is necessary; for example, in TY measurements, (ii) In order to measure the length of a burst the clock counter is preloaded with 0 by the computer. The clock gate is then opened at the beginning of the burst and is automatically shut when a recycle delay is requested. The real time of the burst may then be read by the computer. e. Counters. Clearly, for most conceivable pulse experiments it is necessary to be able to count—we may need to produce a certain number of pulses or cycles of pulses, or to sample a fixed number of times, and so on. Three counters in the programmer provide this facility; called the subburst counter, burst counter, and decade counter they count respectively from 1 to 2 7 in powers of 2, from 2 5 to 2 11 in powers of 2, and from 1 to 1000 continuously, and all have either manual or computer-controlled selection. When a counter memory bit is encountered by TP1 the corresponding counter control looks at the counter register to see if it has overflowed at the preselected bit. If not, the register will then be incremented by TP2. If it has overflowed, i.e., the count is over, the register will be cleared by TP2 in prep­ aration for recounting. When overflow has occurred at some instruction, the next instruction is modified and branching of the pulse program can occur. Modification of the address is simple—the next address (see Section II, A, 2, b) enters a 4 bit binary adder as one addend. The other addend is a decoded 4 bit binary number with 3 bits from a decoder and 1 from the memory bank switch (Section II, A, 2, b). The decoder will present a number different from 0 in the selected memory bank only if an overflow has occurred in one or more of the counter registers. The number there presented will add to the next address 1 for a subburst overflow, 2 for a burst overflow, and 4 for a decade overflow, thus permitting separate branching for each of these counters inside the selected memory bank. Obviously, the counters may also be used as mul­ tipliers, specifically to create long pulses or delays and provide a convenient extension of their ranges. The sample pulse program in Section III demonstrates some uses of these counters.

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f. Pulse Output. A major feature of most pulse experiments is the necessity for very sensitive pulse width adjustments. There are six pulse output channels in the programmer and these are divided into 2 groups labeled XXYY and V V. The first four provide gating for the 4 phase box described later. The other two provide separate gating, for example, for a second frequency in double resonance experiments or video current pulses in tilted coil experiments.3 The pulse output is produced quite simply. TP1 triggers two pulse generators— the rf and F main pulse generators. These are monostable multivibrators with RC time constants selectable manually, from 0 to 4/isec. They are fed re­ spectively into 4 and 2 channels which are then gated by 6 pulse selection gates. We now differentiate again between the modes of operation described in Section II, A, 2, c. (i) Normal Operation or Computer Control. The gates are controlled by the 6 appropriate bits in the memory. (ii) Bloch Decay. The gates are controlled by bits set from the mode select switch and the memory is disabled totally. For every bit that is 1 the corresponding gate is open and the main pulse passes through to the pulse generators and triggers a pulse from one of 6 monostables with RC constants again determined manually and with sensitive continuous selection of 0-4/isec. The output from the pulse generators is thus a pulse for each pulse bit set to 1 either by the memory or the mode select switch, with each pulse separately controllable. Now we introduce the possi­ bility for an overall pulse width control—this is done separately for the two groups of pulses at the pulse trimmer gates as described for pulse pro­ grammer A. The last manipulation that the pulses undergo occurs at the final pulse gates. These provide another useful facility described in Section II, A, 2, b, namely, the possibility of holding open a pulse gate until the next instruction occurs. This is necessary when the desired output is not short pulses but long ones or continuous rf bursts with or without phase switching as employed in some recent experiments.11 This is determined by bit H in the memory. If set to 0, the trimmed pulses pass straight through the final gates and form the final output. If set to 1, the selected pulse channels are not closed at the end of the trimmed pulse and are kept open until the end of the current delay or more if necessary (by keeping H and the pulse bit set to 1). Clearly this also extends the range of accessible pulse widths by having them set with the delays. If the mode of operation is Bloch decay then the hold function is suspended and only pulsed output is possible. A switch which opens the final driver gates (disable spectrometer switch) enables us to disconnect the logic from the rf circuitry, thus avoiding burning out the transmitter with excessive duty cycles when new pulse sequences are being tested.

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g. Sampling. At any instruction where we wish to sample the magnetization the S bit is set to 1, opening the sample gate. TPl then triggers the integrate and hold described in Section I and the magnetization is sampled a manually variable time later. In addition to this a sample register is incremented. The output of this 10 bit ripple counter is connected to the computer direct address lines and one higher order bit is provided from the sample and hold unit. The latter bit selects one of two buffers in computer core for the deposit of real and imaginary digitized data from the dual phase detector. The sample register increments the address within each buffer thus providing sequential storage of digitized data in computer core whence it may be displayed and otherwise manipulated by the computer. h. Computer Access. The great versatility in programming experiments and in signal processing is provided by the computer and this has several points of access to the programmer, all provided by standard interface techniques. The more important of these access points are symbolized in the schematic diagram by a "12" (for PDP 12). The computer has two main functions, namely, servicing the pulse programmer, and storing and processing data. Here we describe the more important aspects of these two functions. (i) Servicing the Pulse Programmer. This includes loading in pulse programs, setting the counter registers for preset counts, selecting the clock decade, preloading and reading the clock counter, triggering and controlling the operation of some auxiliary external devices, and triggering the bursts when in computer control mode. All of these are carried out either when the pro­ grammer is at a standstill or just at the end of a recycle delay during operation. Loading is carried out simply: Assuming the program is in computer core, either from the teletype or from tape, the computer steps through the pro­ grammer memory by incrementing the current address register and reads in the pulse program through the I/O bus. Programs may be read in and changed at any time during operation—this is entirely determined by the available computer software. Our library of pulse programs currently contains about 50 entries. Servicing is initiated when the computer is interrupted by the BG pulse at the end of a recycle delay (Section II, A, 2, c). The service functions mentioned above are then performed and the computer then triggers a new burst through input 2 of the end-delay gate. (ii) During the burst data are stored in computer core. The end of the burst interrupts the computer through the recycle delay request. During the recycle delay, processing of the data occurs. This may include, for example, displaying and plotting, storing on tape, averaging, fourier transformation, counter and clock time readout, etc. The end of the recycle delay is sensed by the computer which then enters a service routine as in (i). If signal processing has not been completed the computer may choose, under software control, to lengthen the

134

J. D. ELLETT ET AL.

recycle delay. All in all, operation is extremely flexible and a variety of service and signal processing routines have been used for different experiments. The last section shows an example of one such simple experiment. B. rf SOURCE

Spectromer A is currently being used only to study *H and 1 9 F resonances, both at a frequency of 54 MHz. (This value was chosen to be consistent with the field-strength capabilities of an elderly Varian 12 in. electromagnet.) The field is locked to a 7Li resonance from an external sample, as described in Section II, K below. Therefore we need not only 54 MHz but also signals at 20.987 and 22.308 MHz, which correspond to the Li frequency when the Larmor frequencies of *H and 1 9 F, respectively, are 54.000 MHz. Signals very close to the three frequencies are generated from a single thermostatted

r-\ Crystal oscillator Bulova PC 0X0-103 18.000 MHz

W> Avantel» UA-301 ΌΑ-302

54.000

Tripler

Frequency synthesizer

MHz

A

Loche

y_y

oscillate

«

Relcom •mixer

Frequency synthesizer

<ΜμΑ: or

22.320 20.970

are

SN7

MHz MHz

^OM

1000 pF

Input: i a 0 0 0 MHz Output. 4:3 20 MHz or 2.970 MHz plus sidebands separated by 0 . 7 2 0 MHz

NAND

gates

FIG. 6. Frequency synthesizer used to generate the frequencies needed to lock the magnet of spectrometer A to a 7Li resonance while observing either *H or 19 F at 54.0MHz. Resistance in ohms.

SPECTROMETERS FOR MULTIPLE-PULSE NMR

135

18MHz crystal oscillator as follows: the oscillator signal is tripled to obtain 54 MHz. The other two frequencies are derived in a homemade frequency synthesizer (Fig. 6) made with TTL integrated logic circuits. Frequency divi­ sion by factors of 2 and 5 is easily accomplished with flip-flops and SN7490 decade counters. Frequencies can be added in NAND gates, although because logic signals are rectangular, harmonics in addition to the desired one are produced. The logic circuit generates the two frequencies: and

20.970MHz = 1 8 ( l + i + 2 r ) 22.320MHz = 1 8 ( 1 + Η Τ Ϊ Γ )

The proper carrier is selected from the comb of frequencies separated by 18/25 = 0.72 MHz by a phase-locked oscillator (Fig. 7). The remaining fre­ quency error lies in the audio range and depends on placement of the two samples in the magnet, on chemical shifts, etc. It is removed by use of a variablefrequency first-sideband detection of the 7 Li field-lock signal, as described in Section II, K.

FIG. 7. Details of the frequency tripler and locked oscillator appearing in Fig. 6. Tripler circuits tuned to 54 MHz. Locked oscillator circuits tuned to 20.970 MHz or 22.320 MHz.

136

J. D . ELLETT ET AL.

Spectrometer B obtains its master frequency from a General Radio Model 1164A frequency synthesizer. As we shall see later, the Larmor frequency is the lower sideband obtained by mixing a high frequency vs with a standard 30 MHz carrier. The spectrometer is designed to operate over the range 20-250MHz, so vs must be variable between 50 and 280MHz. To do this we generate harmonics of the main synthesizer output (vmax = 70MHz) with a broadband frequency doubler. The desired harmonic is selected with a Boonton Model 203 tuned rf power amplifier (Hewlett-Packard Co.). The 30 MHz is available at a low level as one of the standard reference signals of the syn­ thesizer. It is amplified and buffered by a simple two-stage tuned transistor amplifier. No additional signal is needed for field-locking purposes in this spectrometer, since it is intended to be used with a persistent superconducting magnet whose field stability is adequate for our needs. A point has been made in some commercial pulse spectrometers of making the rf source coherent with the clock which generates the pulse timing. It is easily verified that such coherence would be without any advantage if all the rf circuits following the gating were linear. Of course they often are not, and it is therefore possible to envision situations in which the pulsed rotation of magnetization in the rotating frame might differ appreciably from one sup­ posedly identical pulse to the next if the number of rf cycles contained in the pulse is small. At high operating frequencies these effects are expected to be small : in spectrometer A a 90° pulse typically contains ~40 cycles of the rf excitation, and small variations in the manner in which the first and last cycle are gated should not be critical. To test this assumption, spectrometer A was for a time converted to "fully coherent" operation by deriving its programmer clock from a suitable point in its rf frequency synthesizer. No important changes in the performance of any experiments were noted. For spectrometers employing low rf frequencies and short pulses, or more generally for esthetic reasons, complete synchronization is probably none­ theless desirable. C. FOUR-PHASE GENERATOR

The more elaborate transient NMR experiments require pulses or bursts of rf power having a variety of different carrier phases. A typical example is the WAHUHA experiment2 which requires 90° pulses along the x, —x, >>, and —y directions in the rotating frame, corresponding to four different carriers with relative phases 0, 90°, 180°, and 270°. Other experiments may require intermediate phases. The four mentioned make a convenient and easily generated set, from which any others can be derived by linear combination of signals.

SPECTROMETERS FOR MULTIPLE-PULSE NMR

137

To insure purity and constancy of phase from beginning to end of a pulse, it is desirable to generate the four signals as separated cw carriers, which may then be individually gated and combined. A simple way to generate them is by dividing a standard signal into four branches, each feeding a suitable length of delaying coaxial cable. In spectrometer A we have done this. Since fine trimming of phases during an experiment is necessary, we have inserted into three of these lines General-Radio air-dielectric coaxial line stretchers, which provide a very smooth and precise phase controllability. (Commercial variable delay lines having a sliding contact to a delay helix are not satisfactory in this regard, and in addition have typically inadequate bandwidth.) The four cw lines are fed to gates which operate under control of the programmer. Broadband double balanced mixers, now inexpensively available from many sources, are convenient for this purpose. They are made conducting to rf by feeding a current of ~10mA into the local oscillator port. Since the im­ pedances presented to rf source and load change somewhat when the gates are switched, it is desirable to buffer both. We have found Avantek (Santa Clara, California) modular wideband amplifiers convenient for this and many other purposes. These one stage amplifiers are built on chips of microstrip which can be mounted and combined with other wideband components, such as microstrip-mounted mixers, in standard shielded boxes provided by the same manufacturer. In spectrometer B, we have used a similar system, somewhat different in detail. This spectrometer is designed to operate from 20 to 250MHz, and it would be most awkward and expensive to make the phase generator operate satisfactorily over this range. We have therefore made a single-frequency (30MHz) device, and translated its pulsed rf output to the desired operating frequency by mixing with another carrier of variable frequency. The four cw phases are generated and trimmed by means of a "quadripole network," built at our request by Merrimac Research and Development, Inc. at a fabulous cost, and now more cheaply available as a standard item. This consists of a 90° hybrid junction and two 180° hybrids, plus voltage-controlled varactor phase trimmers. It has a nominal 10% bandwidth centered at 30 MHz, and thus has somewhat more stable phase characteristics than phase generators produced with resonant elements such as coaxial lines. A higher frequency (50-280 MHz) is combined with the output of the pulsed 30 MHz four-phase generator in a single-sideband mixer to obtain the finally desired pulsed rf at frequencies between 20 and 250 M Hz. (See Section II, B for a discussion of rf sources.) The mixer, shown in Fig. 3, is of conventional design and uses one narrowband (30MHz) and one broadband (40-300MHz) quadrature hybrid. The reason for employing a SSB suppressed carrier mixer instead of a much less expensive conventional mixer is that the ensuing power amplifiers are all of broadband design. It is important not to waste their power output capability

138

J. D. ELLETT ET AL.

on unwanted sidebands which are of no interest to the nuclear spins. The SSB mixer has a carrier and unwanted sideband rejection of 30dB, which is entirely adequate for this purpose. When constructing a transistor driver for the rf switches, it was discovered that turning the gate on with a sharp current pulse (risetime < lOnsec) pro­ duced an rf pulse modulated with a decaying transient of the same frequency as the carrier. By rounding the corners of the current pulse slightly (risetime ~ 30-40nsec), this effect was significantly suppressed. Figure 8 is a diagram of the switch and driver.

LOGIC INPUT

Ι.5ΚΛ

o

·—VWV—

FIG. 8. rf switch and driver de­ signed to suppress switching transients.

>300

+ 5V

27Λ

»-5V

Much has been made 5 ' 1 2 ' 1 3 of the problem of obtaining adequate on-off ratio in the gating systems of coherent pulsed NMR spectrometers. In fact the rf switches in the four arms of the phase generator are not by themselves adequate to this purpose, even though the last two stages of the transmitter are also gated. We find that the leakage with gate closed can be lowered below detectability by adding one additional master rf switch following the singlesideband generator. Although the placement of this additional gate gives adequate on-off ratio, we have probably not made most efficient use of the device. Placing the gate before the SSB mixer on either the 30MHz or high frequency channel would have utilized the nonlinear properties of the mixer to high level signals only. In principle, this would yield a better on-off ratio. Due to greater capacitive leakage of the rf switches at higher frequencies, it behooves one to place the gate on the 30MHz channel. 12 13

K. R. Jeffrey and R. L. Armstrong, Rev. Sci. Instr. 38, 634 (1967). I. J. Lowe and D. E. Barnaal, Rev. Sci. Instr. 34, 143 (1963).

SPECTROMETERS FOR MULTIPLE-PULSE NMR

139

D. TRANSMITTER

1. Amplitude of the Hl Field Multiple pulse line narrowing experiments require that yHi>

ll^intll

(2)

where || Jf int || is the "magnitude" of the interaction Hamiltonian of the nuclei, y the gyromagnetic ratio, and Hl is the rf magnetic field strength in the rotating frame. Since the local dipolar fields of protons in solids is of the order of 20 G, Hx fields of greater than 100 G should be applied to the sample. According to Clark 5 the H1 field in a simple solenoid is given by H, = 3(PQ/v0V)i/2 = 3(P/AvV)*9

(3)

where Hl is the rotating field in Oe, P the rf power in watts, Q = v0/Av the quality factor of the coil, v0 the resonance frequency of the circuit in MHz, and V is the volume of the coil in cc. The bandwidth Δν of an RLC circuit is related to the rise time TR of the field pulse (10-90%) as follows: TR = 2/πΔν = 2β/πν 0 .

(4)

In order to reduce transient phase effects (see Section II, D, 2) the rise and fall times of the pulsed Hl field have to be very short. TR should be <200nsec, i.e., Δ ν ~ 3.3 MHz. Equation (3) indicates that under these conditions, even with such small volumes of the sample coil as 0.25 cm 3 , an rf power of at least 1 kW is needed to produce fields greater than 100G. The transmitter of spectrometer A consists of several stages, as shown in Fig. 2. The final amplifier can produce 4kW rms of pulse power at duty cycles of up to 20% for bursts of several seconds. This homemade tuned class C amplifier uses two 4CX250B tubes operating in parallel, similar to one de­ scribed in ref. 14 (see Fig. 9). It is driven by a 300 W tuned class C stage using a 3E29 tube in a push-pull configuration. This driver stage is preceded by three wideband chain amplifiers in series. As will be shown below, transient phase effects occur in such a multistage transmitter with several tuned circuits due to coupled circuit ringing and grid current operation. In order to reduce these effects, the transmitter of spec­ trometer B is a broadband distributed amplifier (Model M404P/Mod 1, In­ struments for Industry, Inc., Farmingdale, N.Y.) which was built for us to the following specifications : 14

M. Mehring and O. Kanert, Proc. 14th Colloq. AMPERE {At. Mol Etud. Radio Elee), 1966 p. 988. North-Holland Pub., Amsterdam (1967).

140

J. D. ELLETT ET AL.

Frequency range: 20-250MHz Power gain: 50dB Power output: 700W in continuous mode 1300W in pulse mode ( < 20% duty cycle) Risetime: ^ 50nsec 2. Constancy ofHl During a Long Burst A typical burst for a 4-pulse experiment contains four pulses of less than ljwsec width with a cycle time of about 20/xsec. The complete burst length might range from 10msec to several seconds. Care must be taken to keep the rf power supplied to the probe constant over the burst. We find that a droop of less than 1% in power is sufficient to degrade the experiment very sub­ stantially in many cases. During the burst the two final stages of the trans­ mitter in spectrometer A can draw enough current to cause their plate voltages to drop by more than 20%. The rf power at the output of the transmitter droops with the plate voltage, making it impossible to adjust multiple pulse experiments properly over the entire length of a burst.

Connected via auarier

to
Intimai »créer i+CX250B Capacito«· ο·ί îoïkft

FIG. 9. 45 MHz tuned amplifier providing up to 4 kW of rf pulse power. All coil windings widely spaced to minimize interwinding capacitance. Typical operating voltages are Vp = 3.2kV; Vs = 500V; Vg= -135 V. LL cable refers to Lundy Lossy line cable type 009 N.

141

SPECTROMETERS FOR MULTIPLE-PULSE NMR

200

5 V logic pulse in

&.ZV

2N35ÓOKN550/

to grids

10 fe I0W

-I50V

FIG. 10. Grid driver used to preload the amplifier of Fig. 9 in anticipation of a pulse train. Resistance in ohms, capacitance in picofarads.

A voltage regulator with sufficient current-passing capacity and fast enough response for this application would be expensive. An unregulated supply sufficiently "stiff" to prevent droop would also be costly and large in size. Faced with this situation, we have adopted the inelegant but effective procedure of "preloading" the supply with a dummy load for a short time prior to the pulse burst. This allows the output voltage to reach its steady-state value before the dummy load is replaced by the rf amplifier. The dummy load must be adjustable to draw the same current as the transmitting tubes draw during the burst and must be left on long enough for the plate supplies to reach a steady state. Two parallel 3E29 tubes are used to preload the plate supply of the final amplifier, which is usually set at 3kV and supplies 0.6A during a burst. The plates of the preloading tubes are connected to the power supply output, and the tubes are biased off until approximately 100 msec prior to the start of a burst, at which time their grids are switched to zero with a high voltage transistor grid driver controlled by the pulse programmer as shown in Fig. 10. The plate current drawn by the preloading tubes is controlled by adjusting their screen voltage. The tubes are biased off again when the rf burst begins. It is not possible to set the unloaded output of our plate supply high enough for it to remain at the required 3 kV when loaded. For this reason it is necessary to increase the voltage setting of the final plate supply when it is loaded. This is accomplished by placing a rheostat in series with the high voltage transformer primary of the supply and then bypassing this with a Triac switch controlled by the pulse programmer. When the Triac is open the rheostat reduces the ac voltage across the primary keeping the output voltage below its maximum safe value. Closing the Triac places the full ac voltage across the primary and allows the power supply to supply current to the load at 3 kV.

142

J. D . ELLETT ET AL.

A single 3E29 tube is used to preload the 2kV plate supply of the driver amplifier. It is not necessary to switch the voltage setting of this power supply. The rf output voltage of the chain amplifiers of the transmitter also droops during a burst. This drooping can be reduced by preloading them with a burst of ~ 1 MHz rf during the time the plate supplies of the final stages are being preloaded. Since this frequency is far removed from the frequency to which the final stages are tuned, the final stages are not excited and the pre­ loading signal is blocked from reaching the probe and receiver. The amount of loading is adjusted by varying the amplitude of the 1 MHz signal. By carefully adjusting these three preloading mechanisms it is possible to achieve multiple pulse bursts whose amplitude is constant to <2%. 3. Phase Transients1** When a pulse of rf field Hx (t) is applied in the rotating frame, the effect on a spin system is described by the transformation3 h

L 1 (/) = r e x p [ - / J * ' j r i ( 0 ] , o

(5)

where Vindicates a time ordering and ^ ( 0 =

-yH^O-I.

(6)

It is usually assumed that Hi keeps a constant direction in the rotating frame during the pulse. If this is so, the time ordering in (5) can be omitted, and the angle of rotation ß depends only on the area under the pulse u

ß = yjdt'Hi(t') o

(7)

and not on its shape. Then there would be only an esthetic advantage to obtaining square pulse envelopes. It does not seem to be widely known that this assumption of constant direction of H t during a pulse is not valid in practice when, as is always the case, the probe circuit is tuned. Suppose the probe, represented by a series RLC circuit, is excited by a stimulus
0 ^ / ^ tp

= 0 otherwise. The current in the circuit, which determines the rf field, is given by the inverse 14a

See Note Added in Proof, page 176.

143

SPECTROMETERS FOR MULTIPLE-PULSE NMR

Laplace transform : /(/) = JSP" ' {I(s)} =<£-' {E(s)/Z(s)},

(8)

where Z(s) is the impedance of the RLC circuit and E(s) and I(s) are the Laplace transforms of e(t) and /(/). A straightforward calculation yields /(/) =

-,/τ[\ . 8εβ+1 "Il ^ο f 8εβ+1 1 Q smœt H cosœt — e τ/τ\ Q smœrt H — coscori >

4

œrL{

4

L

for

and, writing t =t — tp, i{t) = (V0/œrL) Qe~t/T sinœrt

for

0 ^ / < tp

JJ

(9)

t ^ tp .

Here Q = coL//? is the circuit quality factor, τ = 2/Δω is the risetime of the circuit, where Δω = bandwidth, ωΓ = {(1/LC) 2 —(Δω/2)2}1/ζ is the resonant frequency, and ε = (ωΓ — ω)/Δω is a dimensionless deviation of the exciting frequency from the resonant frequency of the circuit. A steady state is reached for t > 2τ. The approximations Q2 > 1, ε ^ 1 have been used above. Note that the steady-state current is slightly out of phase with the driving signal, as is well known. The phase deviation is, according to Eq. (9), tan^ = ( 8 e ß + l ) / 4 ß .

(10)

A phase detection of the rf signal /(/) in phase with the steady-state current gives results : i0(t) = (V0/œrL) £(1 - e~t/T cosAO, = (V0lcorL) Qe~t/r cosA / ,

0 < t < tip (11)

/ > tp

For the current in "quadrature" one obtains iq(t) = (V0/curL) Qe~th sin Ai, = (V0/œrL) Qe-tlx

0 < t < tp

[sinAf' - ( l / 4 ß ) cosAf'],

(12) / > tp.

where Δ = ωΓ — ω is the deviation from resonance frequency. Equations (11) and (12) show the well-known switching characteristic of an RLC circuit, with the switching time τ and oscillating parts due to the misadjustment A. Equation (12) indicates that there is in fact always a quadrature component of the current during rise- and fallarne of the rf pulse, which is nonzero even

144

J. D. ELLETT ET AL.

for Δ = 0. We call this effect "phase glitch." The ratio of the maximum quad­ rature component to the maximum in phase component of the current can be easily determined from Eqs. (11) and (12) to be δ = (2/ε)(Δ/Δω).

(13)

This phase glitch ratio δ can in practice easily reach the value of 0.2 or greater if the adjustment is made to maximize the pulse height on a scope. Figure 11 shows the phase detected rf pulse in phase and in quadrature. The upper trace has been attenuated by lOdB, i.e., δ = — lOdB approximately, which is fairly big, but the in-phase signal still does not show oscillating parts. The pulse shape and phase glitch in Fig. 11 shows the behavior as described by Eqs. (11) and (12). The actual phase glitch, as measured in the probe circuit, is always bigger than one would expect from a single RLC circuit, because of transmission cables, tube switching, and tuned amplifier stages, so that even with a most careful adjustment the phase glitch is a potential problem. The phase glitch affects the time evolution operator Ll(t) because ffî l(t) in the steady state of the pulse does not commute with Jf j (t) during the riseand falltimes, where the phase glitch occurs. Using the timing of the pulse as

FIG. 11. Phase-detected rf pulse which has passed through a tuned amplifier. The two traces are the components detected in phase and 90° out of phase, respectively, with the rf carrier. The lower trace has been magnified by a factor of 10 dB to show clearly the "phase glitch" discussed in the text.

145

SPECTROMETERS FOR MULTIPLE-PULSE NMR

given by Fig. 12 and Eqs. (11) and (12), the Hamiltonian of a pulse in the y direction of the rotating frame is approximately given by * Ί ( 0 = - ytfiiO - e~tlT)Iy - Δί£Γ ί / τ / χ }, and

jf^t)

0 < t < tp (14)

it t

= -yH1{e' ~ ^Iy

+ A(t- t^e'^'^W^

t> tp.

Using the Magnus expansion for Lx (t) we can express Lx{t) = e x p { - it{JP\0) + &[ι) + &[2) + ···)} with the average Hamiltonians ^[°\ evaluation of j f (10) and j?\l) leads to

(15)

JF\l) etc. as expressed in ref. 3. An

Ll (t -+ oo) = exp {i(aly + blz)}9

(16)

with a= -yHdtp

+ Te-'*'*)

and b = - fr2 ^ Δ τ 3 ! * ! - τ ^ " ί ρ / τ ( ^ Γ + ^ + 2^ + ^ " 2 ^ Ί . Under the assumption of small rise- and falltimes, i.e., e~tp/T-*0 one gets a = -γΗχ

tp

and b=

2

2

3

(17)

-& Ηχ Ατ .

Using the relative misadjustment parameter ε = Δ/Δω and τ = 2/Δω the z component of the time evolution operator Ll (t) can be written as

b=

i+

-tfHS^e.

(18)

i IN-PHASE SIGNAL

δ+ ^

JT"

QUADRATURE SIGNAL

FIG. 12. Definition of parameters used in the analysis of the "phase glitch" effect of Fig. 11.

146

J. D. ELLETT ET AL.

FIG. 13. Phase-detected receiver output during a phase-alternated multiple pulse experi­ ment on 19 F in liquid CF3COOH. Time scale: 2 msec/div. cycle time: 50//sec. The oscil­ lation apparent in the left-hand part of the trace corresponds to an apparent shift off exact resonance resulting from the phase glitch effect. The absence of beats in the free induction decay at the right, after the pulse train is turned off, shows that the spectrometer was actually set on resonance.

Thus the net effect of the phase glitch is a slight deviation of the rotation axis into the z direction, so that it is not possible for the spins to reach the — z direction with a single rf pulse. The z component of the rotation can be measured in a phase-alternated experiment in a liquid.15 Figure 13 shows the effect on resonance of a burst of phase-alternated 90° pulses on 1 9 F in CF 3 COOH with a pulse spacing of 25/isec. Instead of a straight line which would occur if b = 0, a beat frequency of 500Hz can be seen. As 80 pulses are applied during 2msec, b = 0.078 or each pulse includes a rotation of 4.5° about the z direction. In order to reduce these phase glitch effects we use a very strongly damped probe circuit for line narrowing experiments on solids, making δ small by making Δω large accord­ ing to (13). 15

J. D. Ellett, Jr., and J:S. Waugh, /. Chem. Phys. 51, 2851 (1969).

147

SPECTROMETERS FOR MULTIPLE-PULSE NMR

E. DUPLEXER

The ideal duplexer is a fast single pole double throw switch which connects the probe to the transmitter during transmission mode and disconnects the receiver, whereas during signal reception the probe is connected to the receiver and completely disconnected from the transmitter. The literature 5,16 " 20 son XMTRV

ΑΛ Α Α Α

>

-\Zl·

wwv— g o5 0WÎ K 2 W

50 Ώ

22 Ω —WWv

75 Λ

λ / 4 CABLE

50 n RCVR

ll.2uH

Jf

HIGH VOLTAGE CAPACITOR (a)

XMTR 50Ω >

91 n , λ / 4 CABLE

r-71 VLJ

50ßRCVR

-T"

HIGH VOLTAGE CAPACITOR

(b)

50 Ω

XMTR 5 0 SI

22 Λ WW^

90 W

]£

50 Ω 12 W

75 a

λ / 4 CABLE 50 a RCVR

HIGH VOLTAGE CAPACITOR 15 Λ

4;l

2.4/xh

reewMj

0

• TYPICALLY 6

IN9I4

FIG. 14. Probe circuits employing diodes for passive switching and resistors for damping. (a) has a simple series-tuned sample circuit, (b) employs a quarter-wave resonant line to bring the switching diodes and associated components outside the magnet, (c) is an adap­ tation of (a) to permit the introduction of video current pulses to the coil. 16 17 18 19 20

I. J. Lowe and C. E. Tarr, / . Sci. Instr. [2] 1, 320 (1968). K. W. Gray, W. N . Hardy, and J. D. Noble, Rev. Sci. Instr. 37, 587 (1966). R. A. McKay and D. E. Woessner, / . Sci. Instr. 43, 838 (1966). J. J. Spokas, Rev. Sci. Instr. 36, 1436 (1965). U. Haeberlen, Ph.D. Thesis, Tech. Hochshule, Stuttgart, 1967.

148

J. D . ELLETT ET AL.

contains numerous discussions on means of approximating this behavior. We have used passive diode duplexers of several types based on radar practice, as shown in Fig. 14. In Fig. 14a, the rf power is delivered by a 50 Ω coaxial cable to the 50Ω/90λΥ damping resistor of the probe circuit. During trans­ mission mode the rf voltage switches the pair of crossed diodes "on" with a low on-resistance, so that only a small amount of rf is dissipated in the diodes. The second pair of crossed diodes at the 200Ω side of the 1:4 rf transformer are also switched into the "on" position if the rf reaches 0.5 Vrms at that point. As the "on" resistance of these diodes is very low, the other side of the 32/4, 75 Ω cable has a very high input impedance so that only a small amount of rf power is incident on the receiver. If the rf power is turned off, the LC-circuit rings down with its natural decay time determined by the damping resistor until it reaches ~0.5 Vrms, the "cut on" voltage of the diodes. A further decrease in voltage across the diodes turns the diodes into the "off" position, where their resistance is very large and their capacitance is approximately 2pF. The transmitter is then isolated from the probe. The nuclear signal is transformed by the 75 Ω cable so that it matches the 200 Ω input impedance of the receiver transformer. Since a smaller damping resistor is used in the receiving mode, the quality factor Q of the coil is switched to a higher value. Several modifications of this circuit can be considered. Figures 14b and 14c show variations which we have employed on various occasions. The circuit of Fig. 14c is employed when we wish to apply simultaneous rf and video ex­ citation to a "tilted coil" probe (see below). It is a switched Q series RLC circuit for the rf similar to the first design. The sample coil, however, is con­ nected to rf ground in parallel with a second coil of the same inductance. The video current pulse is added to the coils through Cg9 whose value must be large enough to ground the 54 MHz rf, protecting the current puiser output, but not so large as to degrade the video pulse rise- and falltimes. The high voltage tuning capacitor must also block the video pulse from the damping circuit. When these conditions are met the video pulse circuit is effectively a parallel RLC circuit with a voltage generator in the inductive branch. The fastest risetime without ringing for this circuit driven by a voltage step occurs when ARC = L/R. The circuit in Fig. 14c has a video pulse rise- and falltime of 120nsec, which matches the switching time of the output resistors of the current pulse gen­ erator (see below). The rf pulse rise- and falltimes are the same as the ones for the first probe circuit. Note that all of the duplexers described here are passive in nature, i.e., switched by the rf excitation itself and not by some additional control signal. In principle one can obtain a valuable extra degree of freedom by using suit-

SPECTROMETERS FOR MULTIPLE-PULSE NMR

149

able active duplexers; 18 ' 19 it would be desirable to introduce heavy damping of the probe circuit until its rf voltage has fallen to the receiver noise thresholds not merely to the ~0.5V switching level of silicon diodes. While some success has been achieved by others in this way for spectrometers operating at rela­ tively low rf frequencies, our experience with frequencies of 50 MHz and higher has been discouraging—no doubt only because of the insufficiency of our expertise. We have tried a number of methods, with the invariable result that the overall blocking time of the spectrometer is not reduced by com­ parison with simpler passive techniques. The reasons for this failure are

FIG. 15. Detected receiver output (1 //sec/div, spectrometer A) showing overloadrecovery characteristics. A 1 //sec pulse begins 3 div from the beginning of the trace. The receiver is completely paralyzed for ~ 1 //sec after the end of the pulse, after which the beginning portion of a free induction decay can be seen.

difficult to determine experimentally, owing to the very large dynamic range involved (~ 180dB). We suspect that the general difficulty is connected with the reexcitation of an rf transient in the probe circuit when the active damper is switched off. Such transients can in principle be avoided by the use of switch configurations which are symmetrical with respect to rf ground, and by con­ trolling the rate at which the duplexer passes through its switching point. However, in practice it is difficult to satisfy these conditions to the extreme degree of accuracy necessary to avoid transients which appear large to a very sensitive receiver. In any case we have bypassed this interesting problem for

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J. D . ELLETT ET AL.

the present, since recovery times adequate to our purposes have proved ob­ tainable by passive techniques without an insufferable loss in sensitivity (see Fig. 15). F. PROBES

1. Single Coil vs. Crossed Coil The old question of single coil vs. crossed coil probe configurations sorts investigators into two logic-tight compartments.5 Nevertheless we feel im­ pelled to state some of the reasons for the obvious superiority of the single coil geometry for pulsed NMR experiments. The primary one is that it permits us to obtain the largest possible rf field Hl in a sample of given volume Vs for a given available rf power P: Hx cc(P/Vt)i/2 where Vt is the volume of the transmitter coil, so it behooves one to wind the coil as closely as possible around the sample. Since the same desideratum applies to the receiver coil, it pays to make them one and the same. Actually, the requirement for large Hi in many of our experiments is so paramount that we have often used rather small coils (i.d. ~ 5 mm) at the price of having to use small samples, so as to make the best use of the finite rf power available to us. Of course a single coil configuration is attractive from the viewpoint of mechanical simplicity, especially when one contemplates problems associated with limited space in the magnet, thermostating of samples, and the like. Tt is often said that this simplicity is bought at the price of complexity in the isolation of transmitter from receiver circuits. The latter problem really consists of two distinct parts: (a) In a pulse spectrometer the receiver input (designed to handle microvolt signals) must be protected from damage by the transmitter (which produces kilovolt signals). A crossed coil probe can solve this problem by arranging geometrically for ~60dB of isolation between the two coils. It is easy to obtain a comparable isolation in a single coil spectrometer by means of duplexers based on hybrid junctions or nonlinear elements (diodes), or com­ binations of the two (see Section II, E). Such techniques have been highly developed in radar technology. They permit one to realize quite independent equivalent circuits for transmission and reception modes. (b) A more difficult general problem is connected with the speedy recovery of the system from overloads caused by the rf pulses. Part of this is a question of receiver design (see Section II, G), but the major part arises simply because the sample circuit is tuned, and therefore rings down (if it is singly tuned) after its excitation is removed with a time constant 2Q/a>. Because of the enormous dynamic range (~180dB) over which the probe operates, one must allow for a recovery time ΖΓ~21τ before an undistorted nuclear signal can be

SPECTROMETERS FOR MULTIPLE-PULSE NMR

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observed. At first sight it might seem that a crossed coil probe avoids this dif­ ficulty : if the transmitter and receiver circuits communicate only through the agency of the nuclear spins, the receiver circuit can be given a high g, ad­ vantageous from the standpoint of sensitivity, since it is not excited by the transmitter transient. Similarly the transmitter circuit can be given a relatively high Q, thereby obtaining a large H1 from a modest input power, since its ringdown is not seen by the receiver. The trouble with this argument lies in the finite isolation between the two circuits, which cannot easily be made greater than, say, 60dB if the bandwidth is to be adequate for handling short pulses and if continual balancing adjustments are not to be made. Taking this figure, and a total dynamic range of 180dB extending from lkV to Ιμν, it is clear that the receiver circuit is excited at the 1V level and will require 14τΑ instead of 21τΛ to ring down. Similarly the transmitter circuit requires 14τΓ to ring down to the lmV which is equivalent to 1 μ\ at the receiver input. This shortening of tR by a factor off for given values of Q is measurable but clearly does not solve the fundamental problem. To obtain rapid recovery in either type of spectrometer requires the same fundamental attack on the damping of tuned circuits (see Section II, E). A final point concerns the homogeneity of the Hl field. The usual way to obtain high homogeneity is to make the sample occupy only a small fraction of the volume of the transmitter coil. This is routinely done in low-level cw crossed coil spectrometers (where in fact one does not need a homogeneous Hx) because it is mechanically convenient and only a very small H1 is needed. In a pulse spectrometer one cannot afford to pay the price of multiplying the transmitter power by the same factor as the increase in coil volume. When good Hl homogeneity is required, one must pay painful attention to the coil winding geometry in either a crossed coil or a single coil probe. 2. Tuning In order to obtain very large Hl with a feasible power input, as well as to recover a nuclear signal with acceptable sensitivity, it is (unfortunately) necessary to make the sample coil part of a tuned circuit. At least part of the tuning capacitance for this circuit should reside in the magnetic field near the sample coil, to confine as much as possible of the rf magnetic field to the coil itself where it is useful. This is particularly important at high frequencies, where the stray reactance of coil leads can become large compared to the inductive reactance of the coil. Thus we must have a nonferromagnetic variable capacitor, small enough to insert into the magnet gap, and capable of handling the large rf voltages (several kV) present during a high powered transmitter pulse. The probe arrangement of Fig. 16 is suited to this purpose. The tuning capacitance appears in series with the ground return of the coil, and is adjusted by means of the threaded copper sleeve which forms the probe housing. The

152

J. D. ELLETT ET AL. 5 mm TEFLON PLUG FUSED QUARTZ TUBING HIGH VOLTAGE CAPACITOR

3 3/4

3 13/16 CYLINDER THREADED INTERNALLY SO THAT IT CAN BE ADVANCED ALONG QUARTZ TUBE COPPER LOCKING NUT ALUMINUM SUPPORT BAR THREADED COPPER TUBING

TEFLON SPACER

FIG. 16. Construction of the high voltage probe. Tuning is accomplished by advancing the threaded outer shell with respect to the inner cylinder of a capacitor. The sample is thermostatted by passing dry N 2 through the center chamber. Dimensions in inches unless otherwise noted.

electrical contact across the threads need not be especially perfect, since the capacitance between the two surfaces already constitutes a much lower impedance than the tuning capacitor itself. We find that Pyrex glass is not a suitable substitute for the quartz insulating tube, and that even with quartz the Teflon tape wrapping on the inner electrode of the capacitor is necessary. The sample coil itself is a simple solenoid —1cm long and —6 mm in diameter, supported by its leads. It can be mounted vertically as shown to provide access through the top of the probe for changing samples, horizontally to produce an Ht of direction suitable for use in a superconducting solenoid, or diagonally for introducing video Zeeman pulses in addition to the rf (see Section II, J). Probes of this type lend themselves to variable temperature operation. One passes a stream of preheated or precooled gas up the quartz chimney in the center. This could even take the form of a double-walled vacuum jacket if one wished to be so elegant.

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G. RECEIVERS

The chief requirements of the receiver are that it have a low noise figure, sufficient bandwidth to pass the nuclear signals of interest, and the fastest possible recovery from the ~ 1V overload appearing at the output of the duplexer. In spectrometer A we employ a commercial i.f. strip (RHG Model EBT-101) which has a bandwidth of 40 MHz centered at 60 MHz. In order to obtain respectably rapid recovery it had to be modified in two respects: crossed diodes, slightly forward biased to reduce the additional signal needed to make them conduct heavily, were placed across the collector loads of the first three stages. Second, low-g electrolytic capacitors were placed between the collector supply line and ground of each stage to reduce power supply ringing after an overload. With these changes, as well as some rebuilding of the input matching circuit to obtain an impedance near 50Ω, the unit performs well (see Fig. 15). Spectrometer B employs a superheterodyne receiver, shown in Fig. 3. The preamplifier stage is given as large a bandwidth as possible, to cover the maximum part of the spectrometer frequency range without replacement. Since it needs only enough gain to raise the signal above the noise level of the ensuing mixer, one need not worry about nonlinearities induced by the relatively large total noise power present in a very wide bandwidth. After trying several units, we have found two types which combine low noise figure with rapid recovery from overload. One is the combination of Fairchild MHA 300-01 and MHA 70-01, which together cover the frequency range 40-500 MHz. These are small and intended for stripline mounting, and so are conveniently incorporated in a shielded case with the mixer circuitry. The other is MITEQ model MQ 20007, which covers the entire 20-250MHz band in one unit with a (claimed) noise figure of 3.5dB. The preamplifier is followed by a broadband double-balanced mixer, whose local oscillator port is driven by the frequency/ 0 + 30MHz available from the rf source, thus providing a 30MHz i.f. output. The local oscillator signal is gated so that the i.f. amplifier is not subjected to a signal within its passband until after the greater part of the overload is over. By this means a relatively conventional, moderately narrowbanded i.f. strip can be employed without modification. At present we ues an RHG model EVT 30V39 (30MHz center frequency, 10 MHz bandwidth). H. PHASE DETECTION

The precessing part of the nuclear magnetization is often represented as a complex function, which it is convenient to think of as a superposition of isochromats or quasi-isochromats, each displaced by a small amount Qt

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J. D . ELLETT ET AL.

from the spectrometer frequency ω, and damped according to a real envelope function fi(t): Ji = Mx + iMy = X / ( ( 0 exp [ι(ω + Ω,·) / - (/>,·] .

(19)

i

A receiver coil oriented along the x direction in the laboratory detects only the real part of Ji : Mx = Σ / ί ( 0

C0S

U( - Φι] ,

(20)

but no loss of information is entailed as long as one is not concerned with the sign of the nuclear gyromagnetic ratio. Now suppose the receiver coil signal is linearly detected, after amplification, by mixing with a cosinusoidal signal at the spectrometer frequency, cosco0/. After removal of frequencies near ω and 2ω, the remaining video signal from the /th isochromat is Set(t) = /,(/) cos(Qt.r -,·) = / , ( / ) Re exp [/(Ω,/ - φί)1 .

(21)

Notice that this signal is unchanged by reversing the signs of Ω, and φί9 i.e., one cannot tell whether a given isochromat lay above or below the spec­ trometer frequency. The ambiguity still exists if one uses a sinusoidal rather than a cosinusoidal reference in the mixer, inasmuch as that would only have the effect of changing all the (unknown) φί by π/2. This problem is a real one if one wishes to fourier transform Sf(t) to obtain the slow passage NMR spectrum: in such a spectrum every line at Ω, has a ghost at — Ω,. The real part of the (complex) Fourier transform is even and the imaginary part odd about Ω = 0. In high resolution NMR spectroscopy one often begs this question by offsetting the spectrometer from resonance sufficiently that one knows in advance that all Ω, will have the same sign. Such a procedure has certain disadvantages in principle: (a) To satisfy the sampling theorem, the sampling rate must be at least twice the largest value of | Ω,· |. This critical Nyquist rate is twice as high when the spectrometer is offset as it would be if the spectrometer could be set at the center of the spectrum; and any extra offset which one introduces to be sure of making all Ω(· positive adds to this problem. (b) More importantly, one likes to make Hi in a pulse experiment sufficiently large that the local fields can be neglected during a pulse—so that, for example, a 90° pulse is indeed a 90° pulse for all isochromats in the spectrum. This requires setting yH^ > | 0 | m a x . Clearly the spectrometer must be capable of twice the Hl9 i.e., four times the rf pulse power, as if the offset were not em-

SPECTROMETERS FOR MULTIPLE-PULSE NMR

155

ployed. While this may not be terribly important in high resolution proton NMR, it may be so in high resolution NMR of other species with larger chemical shifts and smaller y, and is crucial in pulse experiments on solids. The ambiguity in sign of Ω, is easily resolved without offsetting the spec­ trometer if one simultaneously records the output of two rf phase detectors whose reference signals are in phase quadrature: cos cot and — sinatf. The two outputs can now be treated as the real and imaginary parts of a complex video signal

which contains all the information picked up by the receiver coil. This signal makes an ideal input to a fast-Fourier transform computation, inasmuch as the usual algorithms work naturally with complex quantities. Of course the total signal sampling rate, counting both channels, must be the same as if the spectrometer were offset and a single channel used. A further advantage of this sort of single sideband detection is in signal-tonoise ratio (SNR). The signals in the two channels combine coherently, whereas the random noise components are statistically independent and add incoherently. Thus, for a general signal which appears in both channels, a SNR improvement of 3 dB is realized over a single phase detector. Double phase detection has also been employed by Redfield (see the pre­ ceding chapter in this volume). In our spectrometer B it takes the form shown in Fig. 3, which is of particularly simple construction. As first assembled, the tolerances on phase and amplitude unbalance of the components used are such that one can expect 1-2° deviation from perfect quadrature and a few percent deviation from equal gain in the two channels. If these errors are ignored they can introduce weak "ghost line" effects in Fourier-transformed spectra. If desired, the components can be trimmed with deliberately intro­ duced "stray" impedances. Alternatively, one can remove the errors later by computation. In our applications we find the latter advantageous, since the corrections required may in some cases differ substantially from the ones just envisioned. Consider, for example, any of the "magic angle" experiments, such as the Lee-Goldburg experiment,4 in which the magnetization appears to precess not about the z axis but about the (111) direction in the rotating frame. A single isochromat then gives a signal which differs in phase by 120° rather than 90° in the two-phase detector outputs—a difference which is more easily accommodated computationally than by adjustment of the phase detectors. An interesting and occasionally informative display can be obtained by plotting the outputs of the two phase detectors against one another, to get what amounts to a moving picture of the tip of the magnetization vector as viewed down the z axis toward the equatorial plane of the rotating frame.

156

J. D. ELLETT ET AL.

';.*;·. * ix /;"

.. »· :

.*

(b)

/ /

...

(e)

FIG. 17. Plots of the real vs. the imaginary part of the nuclear precession signal, repre­ senting the path of the transverse component of the magnetization vector as seen looking down the z axis of the rotating frame, (a) Hahn 90°-90° spin echo, slightly off resonance. This display shows the integrated-and-held signals presented to the ADC, and is useful in verifying that the latter's dynamic range is efficiently used but not exceeded, (b) CarrPurcell sequence slightly off resonance, (c) Same, but with pulse train poorly adjusted. Both bottom traces have been digitized, transferred to the computer, and read out on an x-y plotter.

SPECTROMETERS FOR MULTIPLE-PULSE NMR

157

We have incorporated such a display in spectrometer B at the outputs of the two integrate-and-hold channels (see below), as a quick means of verifying that the signal does not exceed the range of our analog-to-digital converter. In Fig. 17a we show a typical display obtained from the seminal 90°-90° spin echo experiment of Hahn. 21 An x-y plot may also be obtained after accumulation of the digitized data by the computer. Figures 17b and c show such plots from well adjusted (b) and poorly adjusted (c) Carr-Purcell trains in a liquid. I 23456789···

run run n n

J

TLTLJl· DIHD

I H D

(a)

JT_

I H · · ·

(b)

(c)

FIG. 18. (a) Schematic representation of the signals to be expected during a multiplepulse experiment having two pulses per cycle. Odd-numbered intervals represent receiver blocking and even-numbered intervals contain two interwoven nuclear signals, (b) Selection of one of the two nuclear signals by suitable gating, (c) Output of an integrate-hold-dump (IHD) circuit, which comprises an optimum correlation receiver for signals of this type. I. SIGNAL PROCESSING

1. Sampling The sampling of an analog signal at discrete points in the time domain is a necessary procedure for preparing the signal for digital processing. Multiple pulse NMR experiments require sampling for a more fundamental reason as evidenced by the special character of the signal S(t), shown in Fig. 18a, which approximates a time-domain-multiplexed pulse-amplitude-modulated signal. S(t) is a schematic representation of a typical video output from the spec­ trometer's phase detector. (Figure 19 is an actual phase detector output from the six pulse experiment.3 The odd-numbered intervals are of no interest as they depict the pulse overload and the subsequent recovery of the receiver. The useable nuclear signal and the accompanying noise is contained in the even-numbered intervals. For the example at hand (two pulses/cycle), the two sets of intervals {S2 + 4.K} and {S4 + 4K} K= 1, 2, ... separately trace out an 21

E. Hahn, Phys. Rev. 80, 580 (1950).

158

J. D. ELLETT ET AL.

FIG. 19. Actual receiver output during a multiple-pulse experiment having six pulses per cycle.

equivalent Bloch decay corresponding to the average Hamiltonian 1 of the spin system. For the moment consider the single set {S2 + 4.K} which is selected by an appropriate gating circuit to obtain pulse-amplitude-modulated signal plus noise shown in Fig. 18b. By implication, one sample is to be taken for each pulse (K) of {S2 + 4KÌ· According to the sampling theorem, this is just the Nyquist rate for preserving all the information contained in this signal. The noise, however, contains Fourier components at higher frequencies which should be removed to prevent their folding over and adding to the lower frequency noise as a result of the sampling process. For Gaussian white noise, it can be shown 22 that an optimum filter for this case has a frequency response which is the Fourier transform of a unit pulse of the same width as the signal pulses. Such a response is realized by the convolution of {Sn} with that unit pulse in the time domain. For our case, this simply amounts to integration of signal plus noise during each pulse and resetting (dumping) the integrator before each new pulse. This is what one expects intuitively: an integrator rejects noise optimally during the pulse since it behaves like a filter of very long time constant. Such a filter would 22

D. J. Sakrison, "Communication Theory", pp. 280ff. Wiley, New York, 1968.

■O

PULSE

WW

CAGIO

INTEGRATE

DELAY 2

ADDRESS ACCEPT PULSE

FIG. 20. Simplified diagram of the sampling circuits. The sequence is initiated by the pulse programmer. The samples are taken simultaneously and digitized sequentially.

o

OUTPUT OF DOUBLE PHASE DETECTOR

(PULSE PROGRAMMER)

SAMPLE

DELAY I

2

C r

i

H h3 r w

c r

S

O 50

5ö c« TI

S m H m

§

m O H

160

J. D . ELLETT ET AL.

distort the signal by virtue of its memory of previous pulses. This is prevented— at the unavoidable cost of preserving noise components below the Nyquist frequency—by the dumping process. The optimization of the filter is not usually of crucial importance. In spec­ trometer A, we have simply filtered the signal of Fig. 18b with a single section RC filter of time constant matched to the sampling rate, a procedure which is convenient and satisfactory. The filtered signal is sampled by a track-and-hold amplifier triggered at the proper instant by the pulse programmer. While in the hold mode, the result is digitized. In spectrometer B we have implemented the demultiplexing and optimum filtering by means of gated integrators, as shown in Fig. 18c. Under control of the pulse programmer the circuit is allowed to integrate (I) during the signal pulses. It then holds (H) the result while the analog-to-digital converter acquires the result and then dumps (D) until the beginning of the next signal pulse. An abbreviated diagram of the sampling circuit is given in Fig. 20 with timing sequence shown in Fig. 21. As remarked above, signals {S2 + ^Ki anc * {£4+4*} of our example contain equivalent information, and should perhaps both be sampled in the interests of maximum signal-to-noise performance. Often, signals of the {S4} type can be made to vanish and transfer their information to {S2} by suitable rf phasing adjustments. Moreover, the rules for combining the various informationbearing signals vary in a complicated way from one type of pulse experiment to another, and we have not felt that the gain in sensitivity was worth the complexity in signal processing which would be required. 2. Digitizing In spectrometer A, the output of the sampler is fed directly to a Fabri-Tek 1062 signal averager which digitizes in two modes. The high-speed, lowresolution mode yields data continuously with 5 bit accuracy at ljusec per point. The low-speed, high-resolution mode, synchronized to the pulse pro­ grammer, produces 10 bit data in 50/isec. The digital data are stored in the memory of the signal average to await further processing. In spectromer B, the outputs of the two phase detector channels are simul­ taneously integrated and held as described earlier, and then sequentially fed through an FET multiplexer to a Bunker Ramo Model 850 ADC. This device makes an 8 bit conversion in 1 /zsec, although the total time required for con­ verting each pair of values is ~3/^sec, allowing for the settling times of the operational amplifiers and the multiplexer. The gray-coded output of the ADC is converted to ones-complement binary in a parallel array of exclusiveor gates before being strobed into the computer through the one-cycle direct memory access facility. The minimum interval between commands to sample a pair of points is ~9/isec, including the time required for the computer to respond to a break request and accept the digitized data.

161

SPECTROMETERS FOR MULTIPLE-PULSE NMR

SAMPLE PULSE (PULSE PROGRAMMER)

|~|_

_TL

DELAY I (ADJUSTABLE) INTEGRATE (ADJUSTABLE) XY SCOPE

DELAY 2

ADC

-v -v

TRIGGER

BREAK REQUEST (CALLS COMPUTER) ADDRESS ACCEPT (COMPUTER PULSE) MEMORY ADRRESS BIT (DETERMINES MEMORY BANK OF DATA CHANNEL) BTS5 (COMPUTER PULSE)

V V

B-BREAK (COMPUTER PULSE)

-v

BTS5-B- BREAK (RELEASES BREAK REQUEST)

-V

_T~L

J~L

JT

1_Γ

IS

FIG. 21. Timing sequences for Fig. 20.

3. Processing of the Digital Signal The digitized signal from spectrometer A, stored in the core memory of the signal averager, is made available for computer processing (Fourier trans­ formation, etc.) by punching it on paper tape. The contents of the signal averager is also available in analog form for oscilloscope display as a repetitive scan of the memory. The power spectrum of the nuclear signal can be obtained by feeding this scan into a spectrum analyzer plug-in unit of a storage oscillo­ scope. While this spectrum contains no phase information, it is a useful device in adjusting the spectrometer. For spectrometer B, signal processing is substantially simplified. The digitized signal from either channel of the phase detector is displayed on the CRT of the PDP-12. In seconds, either mode of the Fourier transformed signal is ready for display. (Both the raw nuclear signal and its transform can be plotted by means of a Hewlett-Packard 7004A x-y recorder and point plotter

162

J. D . ELLETT ET AL.

which is activated by the computer under software control. Differing from the usual incremental plotter, the recorder plotter is driven by the two analog levels from the CRT display of computer which determine the xy coordinate of the point. The point plotter emits a pulse when its servos have reached null and a point has been plotted. This pulse is used to interrupt the computer, which then supplies the next point.) J. VIDEO FIELD PULSING

A complete arsenal of pulses ideally permits one to apply magnetic fields in arbitrary directions in the rotating frame. The rf pulses most commonly employed in NMR are confined to the x-y plane in this frame, their direction in this plane being defined by the phase of the rf carrier in the laboratory. A field of arbitrary direction is conveniently generated as the resultant of two fields simultaneously applied: an rf field defining the x and y components as just mentioned, and a dc (hereafter called video) field in the ±z (Zeeman) direction. The video field can be generated in a number of ways. An appealing possi­ bility is to make use of the field-frequency duality, ω = yHi characteristic of magnetic resonance. The equivalent of a field AH0 in the z direction can be obtained by stepping the spectrometer source off-resonance by Aœ = yAH0. This method has the evident advantages that a large AH0 can be generated without a large amount of power, and that the field AH0 is as homogeneous as the main Zeeman field. Quick switching of AH0 can be achieved without sacrifice of stability by single sideband mixing techniques, where Δω cor­ responds to a modulation of the main Larmor frequency carrier. The chief problem with this method arises when one tries to define the rotating frame, which conventionally rotates at the spectrometer frequency. If the frame is imagined to follow the switching by Δω, one must take into account the instantaneous and infinite accelerations experienced by the spins in this frame. Alternatively one can work in a constant interaction representation cor­ responding, for example, to a frame always rotating at ω 0 . Then the Hamiltonian during the periods when the carrier has been sidestepped has an explicit time dependence. While these problems are certainly not insuperable, they introduce some additional complication into "handwaving" pictures of the evolution of the spin system. A little reflection shows that it also becomes incumbent on the experimenter to synchronize the switching of sidebands with the phase of the modulations which produce them: in a rough sense one can say that the switching from one reference frame to another should be done at the special times when these mutually rotating frames "correspond." Most of our generation of video Zeeman fields has in fact not been done by the sideband technique, but by the actual generation of video magnetic fields.

SPECTROMETERS FOR MULTIPLE-PULSE NMR

163

This is partly so, somewhat perversely, because of our desire in certain experi­ ments to avoid excessive homogeneity of AH0. In the "magic angle" dipolar narrowing experiments3 one wishes to maintain ίαηξ = Hl/AH0 = ^/2 as accurately as possible over the sample. Hl is generated by a coil which is not large compared to the sample (see Section II, F) and so has some inevitable inhomogeneity. If AH0 could somehow be generated by a current in the same coil, the ratio of Hx to AH0 would remain constant over the sample. We have accomplished this by tilting the sample coil so that its axis makes an angle α ~ 45° with the direction of / / 0 , 3 and exciting it simultaneously with rf and video currents from separate and mutually isolated sources (see Section II, E). The spins then cleverly pick out of the total magnetic field Hamiltonian the two secular parts (in the rotating frame) J-frf = — yHv sinoc/^. — yAH0 cosa/ 2 , where Η± and AH0 are the rf and video fields referred to the axis of the coil. The exciting currents and/or a are adjusted to satisfy the magic angle condition. It is to be noted that the cancellation of inhomogeneity effects is successful only to the approximation that the field in the coil is everywhere parallel to its axis. The video current puiser shown in Fig. 22, when triggered by the pulse programmer, produces ~ 2 A current pulses of independently variable widths and of either polarity. The output transistors, a 2N 3720 and a 2N 3507, are high current core driver transistors. Their switching speeds limit the rise- and falltimes of the video pulse to ~ 120nsec. The rf rise- and falltimes should be degraded to the same value if the effective field in the rotating frame is to grow and shrink parallel to itself. The same circuit could be used for other purposes as well: by exciting a suitable arrangement of conductors one could for example apply pulsed field gradients, useful in diffusion studies 23 and in studying complex 7\ phenomena in liquids by Fourier transform techniques. 24 K. FIELD STABILIZATION

When signal averaging is convenient or necessary, time stability of the dc field is a major consideration. The dc field source for spectrometer B is a superconducting solenoid which has sufficient stability for most experiments of interest. Spectrometer A uses a Varian high resolution electromagnet, which is susceptible to significant field drifts. For this system, an NMR stabilizer was constructed which locks the magnetic field to a 7 Li resonance 23 24

E. O. Stejskal and J. E. Tanner, / . Chem. Phys. 42, 288 (1965). R. L. Void, J. S. Waugh, M. P. Klein, and D. E. Phelps, J. Chem. Phys. 48, 3831 (1968).

164

J. D . ELLETT ET AL.

i-ϊΐ

SPECTROMETERS FOR MULTIPLE-PULSE NMR

165

in an external sample. The spectrometer operates at 54 MHz, and so is suitable only for 1 9 F and *H studies using the electromagnet. Depending on which of these is being studied at the moment, one wishes to excite the 7 Li sample at 22.308 or 20.987 MHz, respectively. See Section II, B for a discussion of the production of these frequencies. Since the frequencies are not exactly commensurable, means have been provided for varying the effective 7Li frequency over a small range. This has been done by employing first-sideband detection in the control system, and deriving the modulation from a variable audio oscillator. Pulse modulation of the carrier frequency ("time-shared operation") 25 is convenient because the transmitter can be easily isolated from the receiver without recourse to a critical and frequency-sensitive bridge adjustment.5 The facts that much of the rf power is dissipated into a large number of sidebands, and the signal-to-noise ratio is somewhat degraded because the nuclear signal is not observed during the overload and recovery from the pulses, are not of great importance for this application. Figure 23 is a diagram of the field stabilizer, which has been designed as much as possible around broadband components to permit easy modification of operating frequency. The oscillator (cw) source is divided into two channels by means of a conservative power divider (A. R. Anzac Iso-T). One channel is gated by broadband double balanced mixers driven by the logic generator. The carrier pulse is amplified by two Avantek modules and transmitted to the probe through a broadband hybrid junction (A. R. Anzac Iso-T), which provides ~30dB of isolation between transmitter and receiver. The receiver consists of six stages of broadband amplification (using Avantek modules), providing a gain of 10dB each. The last stage is followed by another mixer switch which gates the receiver off during the overload and recovery. This gating enables one to employ high post-detection gain without severe blocking of the operational amplifiers employed. The output of the gate is then phase detected against the original source to obtain the dispersion mode of the particular sideband. The output of the phase detector is voltage-amplified with a type 709 operational amplifier, fed through a blocking capacitor, and gated off during the overload and recovery with an FET. This combination of capacitor and FET eliminates dc drift due to the first operational amplifier stage. At the 25

E. Grunwald, C. F. Jumper, and S. Meiboom, /. Amer. Chem. Soc. 84, 4664 (1962).

FIG. 22. Circuit for applying video current pulses of two opposite polarities (A and B) to the sample coil of Fig. 14c. The bottom part of the diagram is the circuit of the discretecomponent one shot employed. Resistance is in ohms, capacitance in picofarads unless otherwise specified. Diodes are 1N914. Positive one-shot identical to negative except that (a) supply voltage polarities are reversed; (b) all diodes are reversed; (e) 2N3906 transistors are used.

C W SOURCE

ISO-T

MIXER

COMPUTER DEVICES

RELCOM Ml

LINE

DELAY

A.R.ANZAC TV-50

r

!

UA 301

AVANTEK

LOGIC

+ 2x UA 103

704

.

UA 3 0 2

AVANTEK

GENERATOR

AVANTEK UAIOI + 3x UA 102

2N5033

"1

MIXER GATE

A.R.ANZAC MLF-3P

FIG. 23. Block diagram of the N M R field stabilizer.

704

A.R.ANZAC MLF-3P

MIXER GATE

J

MIXER GATE

A.R.ANZAC MLF-3P

I

A.R.ANZAC TV-50

ERROR -o OUTPUT

ISO-T

PROBE

m r r m H H m H

167

SPECTROMETERS FOR MULTIPLE-PULSE NMR

same time, the high dc gain of the stabilizer is maintained. The final operational amplifier stage determines the open loop transfer function of the field stabilizer. The choice of this transfer function is motivated by the open loop transfer function of the device receiving the error signal, i.e., magnet power supply, correction coils, frequency synthesizer, etc. Details concerning this matching problem can be found in any standard text on linear feedback systems. The logic generator is depicted in Fig. 24. The output of a variable frequency audio oscillator is clipped by silicon diodes and sharpened up with a Schmitt trigger. Each of the flip-flops divides the frequency of the oscillator by two. The outputs of the flip-flops are properly combined to give a pulsing frequency of one-fourth the oscillator frequency and pulse width of one-fourth of the cycle. Both the receiver and FET gates are activated for one-half of the cycle to allow for overload and recovery. This enables signal detection only during the last half of the period. The transistor switches bus the logic to the various gates of the stabilizer. This system typically holds the Varian magnet to within ±15 Hz of a 54 MHz resonance, a figure which could be improved if necessary by use of a control sample with a narrower resonance. Its bandwidth (20-70 MHz) allows obvious flexibility in application. Important also is the fact that it can be constructed with relative ease from available components, albeit not cheaply.

946-2P

D—=£>f TO TRANSMITTER GATE

ξ 500 il

+ 5V 2N3640 TO FET GATE

Ì

2N3646

T0

R E C E V ER

]

-ΛΛΛΛ

FIG. 24. Logic generator for Fig. 23.

G A TE

.

ft—I

/

1-3 kß

168

J. D. ELLETT ET AL.

m . Operation A. INTRODUCTION

This section presents a brief example of the operation of spectrometer B. We have chosen to demonstrate the application of the four-pulse sequence 1-3 to solid perfluorocyclohexane, previously described using spectrometer A. 26 This experiment exhibits the more important features of the spectrometer and demonstrates the power of these pulse trains in extracting the parameters of chemical shifts and electron-coupled spin-spin interactions from otherwise featureless dipolar-broadened NMR spectra in solids. B. PULSE PROGRAM

Pulse programs are written in a format closely related to the organization of the pulse programmer memory described in Section II, A. Table I shows TABLE I CA

X X

Y

Y V

0000 0001 0010 0011 0100 0101 0110

1 1 1 0 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

0 0 0 0 0 0 0

V H

e

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

Dl D2 D3

s

1 1 1 1 0 1 0

1 0 1 0 0 0 0

0 0 0 1 0 0 0

0 0 0 1 1 0 1

SB B DC NI N2 N3 N 0 0 0 0 0 0 1

0 0 0 0 0 0 1

0 0 0 0 0 0 0

0 0 0 0 0 0 0

1 1 1 0 1 1 0

0 0 0 0 0 1 0

0 0 0 0 1 0 1

the pulse program used for the present four-pulse experiment. Nl-DC rep­ resent the instruction bits described in detail in Section II, A and CA depicts the location of this instruction in the pulse programmer's memory. The program is entered in octal form into the computer (see Fig. 25) whence it may be stored on magnetic tape for future use or loaded into the pulse pro­ grammer. Operation can then begin under control of the computer. With reference to Section II, A it may easily be verified that this pulse program will produce the following pulse sequence : (*, τΐ9 -χ,τ2,γ9τί,

-y,

τ 2 )„, R D

where x denotes an x pulse, etc. and τχ and τ 2 are the analog delays called for by the delay codes (001) and (010). RD denotes a recycle delay called for by code (111). Sampling is initiated through the S bit between the x and — x 26

J. D. Ellett, Jr., U. Haeberlen, and J. S. Waugh, J. Amer. Chem. Soc. 92, 411 (1970).

SPECTROMETERS FOR MULTIPLE-PULSE NMR

169

FIG. 25. Pulse program for a four-pulse (WAHUHA) experiment as typed into the computer in an interactive programming routine. The author of the routine is a British subject.

pulses, and n, the number of four-pulse cycles, is determined by the burst counter. The sampling frequency is decided by the subburst counter; when the latter is set to w, the magnetization is sampled every m cycles. At the end of the recycle delay the pulse train is automatically reinitialized at address (0000) by the computer. During this delay, data sampled in the pulse train is displayed and the next sections demonstrate how it may be accumulated and averaged, transformed, plotted, or stored on tape. C. TUNING

The line-narrowing efficiency which the four-pulse experiment affords us is known to depend critically on the adjustment of rf phases and pulse widths. 3,27 27

U. Haeberlen, J. D. Ellett, Jr., and J. S. Waugh, to be published; J. D. Ellett, Jr., Ph.D. Thesis, M.I.T. 1970 (unpublished).

170

J. D . ELLETT ET AL.

If the pulse train consisted of ideal "(5-pulses," each one of a definite and constant rf phase, the adjustment of the experiment would be a relatively simple matter. We know, however, that severe distortions in the coherent averaging process are introduced both by the finite pulse widths and by the "phase glitches" (Section II, D) which occur during the pulse rise- and falltimes. It has been shown3 that these effects can be substantially reduced by deliberately adjusting the pulses away from their ideal 90° widths and quad­ rature phases. Since these width and phase connections are extremely small, making their direct measurement and adjustment a difficult task, we have preferred to develop an indirect criterion for adjustment of the pulse train. This proceeds in four main stages : 1. Employing conditions similar to those which are to be applied to the solid of interest, preliminary adjustments are made on a liquid sample. In the case of the present example, trifluoroacetic acid is placed in the probe and the spectrometer is set to operate exactly at its 1 9 F resonance frequency. Pulse widths are now brought to the 90° condition by adjusting the widths until a null is obtained in the signal following a pair of closely spaced pulses of the same phase. The phases are then set to quadrature by detecting the rf pulses or the transient signal following an rf pulse of each phase in the dual phase detector. Where the X, X pulses produce nulls in one channel of the phase detector, the Y, Y pulses should produce nulls in the secondary channel. We mention that all these adjustments are made with standard pulse programs which are loaded and interchanged readily by the computer. 2. A four-pulse program, for example that of Section III, B, is now entered into the pulse programmer memory under computer control. The cycle time is set to the desired value (in this case 21.5/xsec) with τί/τ2 set to 2 and the pulse train applied to the same liquid at resonance. That the rf pulses are not ideal is immediately obvious, since beats are obtained in the liquid transient signal (Section II, D). These are a direct manifestation of the spurious ac­ cumulative rotations of magnetization caused by the finite pulse widths and phase distortion in the pulse train. Adjustment is now carried out by a trial and error iterative procedure. The four rf phases and pulse widths are varied sequentially in cyclic fashion in an effort to minimize the above beat frequency. With careful adjustment it is possible to reduce this to one or two beats over the usual solid decay time of several milliseconds. The slowly decaying mag­ netization of the liquid thus acts as a sensitive probe, aiding us to compensate for the various nonidealities of the rf pulses and eliminating the need for their direct measurement. However, we stress that this approach is strictly empirical and our only assurance of its legitimacy is the fact that it does work quite well in practice. 3. A final pulse width adjustment is now made on a prototype solid (in this case CaF 2 ) to maximize the line-narrowing efficiency on a dipolar coupled

SPECTROMETERS FOR MULTIPLE-PULSE NMR

171

Observed frequency 4.0

2.0

4.0

6.0

Transmission offset frequency 8.0 ( hHz )

FIG. 26. Experimental determination of the effective field-scaling factor in a multiple pulse experiment.

system. Since this adjustment depends on the off-resonance frequency at which the spectrometer operates, 27 it is advisable to set the frequency to the range at which the experiment is to be carried out. The reason for working off resonance in the first place is that the line-narrowing efficiency is enhanced somewhat due to additional coherent averaging effects.27 We utilize this advantage in most of our line-narrowing experiments as well as in other experiments which depend on the averaging effects of pulse trains. A length­ ened decay of the 1 9 F CaF 2 signal is now observed and this is maximized using the overall pulse width adjustment (Section II, A). Normally, decays of several milliseconds are obtained in this way. 4. A final check on the adjustment is now possible. As shown previously 1-3 a reduction in chemical shifts (or off-resonance beat frequencies) of ^/3 is produced by the four-pulse sequence, and this scaling should be observed experimentally if the experiment is properly adjusted. It is very important to carry out this examination to ensure that we have not obtained spuriously lengthened decays in CaF 2 with a misadjusted pulse train. 28 This scale factor is also necessary to calculate the effective magnetic field for interpreting the spectra obtained. Figure 26 shows the scale factor obtained for the present experiment. This was measured by varying the spectrometer frequency and plotting it vs. the frequency of the peak obtained by Fourier transformation of the sampled CaF 2 signal. The slope of the line is 1.71 in close agreement with the expected value, indicating a good adjustment of the experiment over a frequency range 28

Lengthened decays can be obtained over small frequency ranges with misadjusted pulse trains having scale factors widely different from y/3. These ultimately yield very poor resolution.

172

J. D. ELLETT ET AL.

FIG. 27. Free induction decay of

19

F in solid C 6 F 12 .

of 16 kHz. At this stage, we are now in a position to proceed with the experi­ ment on perfluorocyclohexane. D. RESULTS

The line-narrowed NMR spectra of solid perfluorocyclohexane (C 6 F 12 ) exhibit marked changes in the temperature range 0°C down to —100° C. At the high temperature end a single narrow peak is obtained indicating rapid interconversion between conformations which renders the fluorine nuclei magnetically equivalent. In the intermediate range, the spectrum passes through a broad collapsed form to a resolved AB quartet 26 showing that conformation interconversion has slowed down, and at even lower tem­ peratures the lines broaden and a resolved chemical shift anisotropy is distinctly visible.29 As an example of the operation of the four-pulse experi­ ment we choose to show a line-narrowed spectrum taken in the intermediate range above. The tuning described in Section III, C above was carried out at — 69° C since we had in mind operation in this temperature range. Figure 27 shows an oscilloscope trace of the free induction decay of C 6 F 1 2 at this temperature. The signal is negative since we have photographed the output of the phase 29

R. G. Griffin, private communication.

SPECTROMETERS FOR MULTIPLE-PULSE NMR

173

detector channel 180° out of phase with the rf. The decay of the magnetization is rapid and characteristic of a solid with appreciable dipolar coupling. In order to obtain the absorption spectrum, the signals from both phase detectors are digitized with the help of an appropriate pulse program and then ac­ cumulated and averaged by the service routine. Figure 28a shows the signal of Fig. 27 processed in this way (note that all computer processed signals are inverted) with a sampling frequency of 87 kHz. These points together with those from the secondary channel of the dual phase detector form the complex input into the Fourier transform subroutine of the service program, and Fig. 28b shows the absorption half of the transformed data. The width of the line (~ 10 kHz) is of course dominated by the dipolar interactions between the 1 9 F nuclei. The small break in the curve comes at zero frequency of the Fourier transform and is caused by a slight misadjustment in the dual phase detector. Figure 29 shows the substantial change induced in the free induction decay upon application of the four-pulse sequence with the parameters given in Section III, C. Photographed are the outputs of the two channels of the phase detector. The two traces in each photograph depict the magnetization during the longer delays between the x and — x, and the y and — y pulses. Between the — x and y, and — y and x pulses the signals form traces too faint to be seen. The decay of the magnetization is much slower than that in Fig. 27 reflecting the effective reduction in dipolar broadening, and some beat structure due to resolved chemical shifts and scalar couplings is clearly discernible. Note that the magnetization does not decay to zero but reaches a finite value (from which it decays with a much larger time constant.) This phenomenon can be explained in terms of a resonance-offset averaging theory 27 and involves spin-lattice relaxation along the (111) direction of the rotating frame. (a)

(b)

5 KHz FIG. 28. (a) Signal of Fig. 27 after sampling at an 87 kHz rate, averaging several passes, and plotting, (b) Fourier transform of A.

174

J. D. ELLETT ET AL.

^ fe #. /K-fJ^|iaa|jOl J1_HrJjiiLiiLiiJ_UL

*?i^^^p;^^^^:i^^^^^^é|

FIG. 29. Raw phase detector outputs during a four-pulse experiment on solid C 6 F 1 2 . A and B are the two components detected in phase quadrature.

SPECTROMETERS FOR MULTIPLE-PULSE NMR

175

FIG. 30. Form taken by FIG. 29 after demultiplexing, integrate/ hold, and signal averaging.

In this experiment the magnetization was sampled every four cycles between the x and — x pulses (subburst counter set to 4 as explained in Section III, B) and Fig. 30 depicts the data plotted after digitizing and averaging of the sampled transients from both channels of the phase detector (Fig. 29). Finally the absorption and dispersion spectra obtained from these decays by Fourier transformation are shown in Figs. 31A and B. The line to the left of the AB quartet is due to the superimposed slow relaxation along the (111) direction mentioned above and always occurs at zero frequency of the Fourier trans­ formed spectra in these experiments.

I KHz

FIG. 31. Absorption and dispersion spectra obtained by Fourier transforming Fig. 30. The anomaly toward the left is at the frequency zero of the transform, and is associated with spin-lattice relaxation along the (111) direction of the rotating frame.

176

J. D. ELLETT ET AL. ACKNOWLEDGMENTS

The construction of these spectrometers was supported to various degrees by grants from the National Science Foundation and the National Institutes of Health, and also through the MIT Research Laboratory of Electronics by the Joint Services Electronics Program under Contract DA-28-043-AMC-02536(E). Drs. Ellett and Huber were NSF Predoctoral Fellows, Mr. Gibby is a Hertz Foundation Fellow. Dr. Haeberlen was a Stipendiat of the Deutsche Forschungsgemeinschaft. The authors thank their colleagues W. K. Rhim, R. G. Griffin and L. M. Goodman for much stimulation and technical aid, and Mrs. W. E. Jouris for project administration. NOTE ADDED IN PROOF

The "phase glitch" effect discussed on pp. 142-146 does not vanish for ε->0, as implied, if the turn-on and turn-off phases of the rf are arbitrary. A fuller discussion of this effect and its experimental consequences is given by M. Mehring and J. S. Waugh, Rev. Sci. Instr. (in press). We thank Prof. I. J. Lowe for stimulating the writing ofthat paper.