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Design and construction of a high homogeneity rf coil for solid-state multiple-pulse NMR
JOIJRNAL
OF MAGNETIC
RESONANCE
50,
28 l-288 (1982)
Design and Construction of a High Homogeneity rf Coil for Solid-State Multiple-Pulse NMR S. IDZIAK* Max-Planck-Institut
AND U. HAEBERLEN
ftir Medizinische Forschung, Abteilung fir Molekuiare Physik, Jahnstrasse 29, 6900 Heidelberg, Germany Received June 1. 1982
A prescription is given for the construction of a coil having an optimal d homogeneity desired, especially, in multiplepulse solid-state NMR. The approach is to use a coil with variable pitch. The field in a helical coil and in the coil with variable pitch are compared. The field distribution across the sample and some experimental aspects associated with the nonuniformity of the rf field are also discussed.
The design of the transmitter-receiver coil of a multiple-pulse take into account the following considerations:
spectrometer must
(1) Duration tx,* of u/2 pulses; our design goal is tr12 G 0.7 ~.~secfor protons and a transmitter power P = 800 watts, spectrometer frequency v = 270 MHz. (2) Ring-down time t, of the pulses as limited by the quality factor Q of the probe circuit; our design goal is tr < 100 nsec (90% - 10%). (3) Optimum sensitivity compatible with tr. (4) Inductance L of coil; it is limited by the minimum capacitance Cmin of the tuning capacitor. (5) Homogeneity of the rf field inside the coil. (6) Sparking. The first four points in this list have led us to use until now, in our multiplepulse spectrometer operating at 270 MHz (I), a self-supporting helical coil with eight turns, an inner diameter of 5.2 mm, and a length of 7.6 mm. It is wound of silver-plated copper wire with a diameter of 0.5 mm. It is satisfactory with the exception of the homogeneity of the rf field. Since rf inhomogeneity is a factor limiting the resolution in multiple-pulse experiments (2-6) we undertook efforts to design a new coil with improved rf homogeneity. A boundary condition was that the performance of the new coil with regard to considerations (1) to (4) should not be degraded in comparison to the former design. The idea for improving the rf homogeneity is to use a coil with continuously vuving pitch (4). The problem is twofold: first we must determine by calculation how the pitch should vary along the coil, second, we must find a way to constructsuch a coil in practice. We start with some theoretical considerations. For a helical coil with constant pitch the current * On leave from the Institute of Molecular Physics of the Polish Academy of Sciences, Poznaft, Poland. 281
0022-2364182 J 14028 l-08$02.00/0 Copyright g3 1982 by Academic Press, Inc. All rights of reproduction in any form rosened.
282
IDZIAK
AND HAEBERLEN
path is described by x= rcosQ, y = r sin Q,
z = aQ, a = Q.s/23r, where r and s are, respectively, the radius and pitch of the coil, see Fig. 1. By using Biot-Savart’s formula the field B,(z) along the coil axis produced by a current I can easily be calculated, 27rIn l/2 - z l/2 + z B,(z) = c ( ((l/2 - z)~ + r2)‘j2 + ((l/2 + z)~ + r2)‘j2 ) ’
PI
where n is the number of turns per unit length. The field distribution on the axis of a helical coil with constant pitch is identical to the field distribution on the axis of a coil consisting of a dense stack of parallel rings. This becomes immediately evident by considering the field contributions originating in the currents in infinitesimally short pieces of the helical coil but can also be seen by comparing Eq. [2] with the formula for the field in the stack of parallel rings derived in, e.g., Purcell’s well-known textbook (7). In both cases, the field distribution depends only on r and k hence increasing the density of turns in a helical coil does not lead to an increase of the B1 homogeneity on the coil axis. For the same reason we do not expect any advantage with respect to rf homogeneity (on the coil axis) by using a coil wound of flat tape (2). One way to obtain a coil with continuously varying pitch is to replace Eq. [ lc] by z = a(Q + ulQp(“sign Q). 131 For u < 0 and k > 0, k # 1 we obtain a coil having a higher density of turns at the ends of the coil than toward its center. For a single-layer coil Eq. [3] makes sense only in that range of Q for which dz/dQ > 0. We proceeded by integrating BiotSavart’s formula numerically over the path given by Eqs. [ 1a], [ 1b], and [3], varying u and k while adjusting a in such a way as to keep the length of the coil fixed. The
FIG.
1. Coil geometry and definition of coil parameters.
RADIOFREQUENCY
283
COIL FOR NMR
Z---c 1
2
0
-2
-L
C v
FIG. 2. Top: The B,, field on the coil axis normalized to B p,, for constant pitch (c) and variable pitch (u) coils. The dotted and dashed lines indicate, respectively, +2 and +lO% deviations with respect to the field at the center of the coil. Bottom: Cross sections through the wire (4 = 0.5 mm) at 'p" = ?r+ n * 2n, II = 0, 1, 2, 3.
best homogeneity is obtained for values of u and k which give such small turn spacings at the end of the coil that we must expect sparking between adjacent turns. Thus a compromise between rf homogeneity and minimum turn spacing had to be made. For judging which distance between adjacent turns is still acceptable we relied on our experience with rf coils in pulsed NMR. The parameters for what we considered to be the best compromise are a = 5.46 X low5 mm, 2) = -0.24 X 10P3, k = 2, coil length I= 7.6 mm, radius r = 2.8 mm; (0 is to be inserted in radians. Figure 2 shows &AZ) for such a coil (top) and a cross section through the wires, I#J= 0.5 mm at 9, = ?r + n. 27r, n = 0, . . . , 3 (bottom). For comparison we also show B&) and a cross section through the wires for a coil with constant pitch. So far we have considered only the field on the axis of a coil, B,AO, 0, z). In order to learn how B,, varies along the x and y axes we calculated &Ax, 0, zJ,
I
-2
0
2
-2
0
2
FIG.
x.y[mml
3. Radial part of B,, for three cross sections of the coil along the x (continuous curve) and y axis
curve). The dashed and dotted lines show, as in Fig 2, +2 and +lO% deviations with respect to the field at the center of the coil. The left part presents results for the constant pitch coil, the right part for the variable pitch coil, given by Eqs. [la], [lb], and 131. (dotted
284
IDZIAK
CROSS-SECTION
C-D
AND HAEBERLEN
CROSS-SECTION
A-B
FOG.4. Design of the variable pitch coil; parts 1 and 2 (actually four pieces each) are made of boron nitride, part 3 of KEL-F.
B,dO, y, zi), zi = 0; 1 and 2 mm for both the constant and variable pitch coils for which B,, is displayed in Fig. 2. The calculation was carried out by numerically integrating the equation,
s +8r
r2 - r(y.sin Cp+ x.cos P) c -& [(x - r’ cos s)2 + (y - r sin ‘P)* + (z - &)2]3’2 #Y
[41
where the limits of integration are suitable for an eight-turn coil. The results are displayed in Fig. 3. The diagrams show, on the one hand, that the radial inhomogeneity in the region of interest is comparable to or less than the axial inhomogeneity, and, on the other hand, that it is automatically improved by improving the axial homogeneity of the field, which justifies our procedure of considering only B,dO, 0, z) when optimizing the geometry of the coil. Practical Construction of a Coil with a Continuously
Varying Pitch
In spite of spending much effort to make a self-supporting coil with varying pitch we did not succeed in getting sufficient precision and stability. Therefore, we had to devise a means somehow to fix the turns of the coil in the desired locations. Any support of the turns from inside the coil was considered intolerable because of inevitable loss of volume for the NMR sample. In the design which we eventually adopted the turns of the coil are held in their positions by combs with precisely machined grooves. There are four such combs on the outer circumference of the coil. Figure 4 shows the combs and the entire coil assembly. It is made of boron nitride which is free of protons, has very low dielectric losses, and is easily machinable. The end pieces are made of KEL-F. The (almost) cylindrical filling of space around the sample with material of nonzero magnetic susceptibility is essential for avoiding interference of the coil assembly with the B0 homogeneity. The gain in rf homogeneity realized with such a coil design is demonstrated in Fig. 5. The upper part shows the scope trace of the NMR signal from a water sample in a helical coil irradiated on resonance by a string of equally spaced 7r/2 pulses. The decay rate of this signal is a direct measure of the homogeneity of the rf field (2). The lower part shows the same type of signal in the newly designed coil. It is evident from these traces that the gain in homogeneity is about twofold. Calculations of the type that follow indicate that the homogeneity achieved is close to the theo-
RADIOFREQUENCY
COIL FOR NMR
285
retical expectation. We take this as evidence that the desired variation of the pitch of the coil has been realized with sufficient precision. The traces shown in Fig. 5 are probably familiar to most people working in highresolution solid-state NMR. Nevertheless, their shape resembling a pair of tongs is probably annoying to most of them. What they expect to see in this type of experiment are three traces, one upper and one lower “envelope” and a single flat trace in the center. However, the center trace never remains flat and single, it always develops the shape of a pair of tongs. One might suspect that this peculiar shape results from a ringing of the rf power after turn on of the pulse sequence. However, this is not so and is, as we shall show now, an almost inevitable consequence of the B, inhomogeneity. The signal S, after the Nth pulse of the string of nominally
FIG. 5. Scope traces of the NMR signal of a water sample irradiated on resonance by a string of equally spaced x/2 pulses for a helical coil (top) and a variable-pitch coil (bottom). The spacing of the pulses is :25 rsec. The radius of the spherical sample equals 2 mm.
286
IDZIAK
AND HAEBERLEN
r/2 pulses, applied on resonance to, say, a sample of water is given by S, = kA&
p(B,,) - sin (rB,,Nt,) - dB1, . [51 1 Note, that yB,,Nt, is the total phase acquired after Npulses by a spin packet “seeing” the field B,,. The value M0 is the equilibrium magnetization and k a constant of proportionality; p(Bl,)dBI, is the fraction of spins “seeing” an rf field in the range of B,, - - . B,, + dB1,. If p(B,J is symmetric with respect to By, = 7r/2ytP the same fractions of spins are “late” and “early” in the course of the u/2 pulse sequence by any given amount with respect to the central spin packet seeing the field By,. In this case Eq. [5] predicts two decaying sets of points for N odd and a single set of points SN = 0 for N even. Symmetric distribution functions p(B,,) have been assumed ad hoc in recent papers dealing with rf inhomogeneity (3, 4, 6). In most practical cases, however, p(Blz) is not at all symmetric about any value of B,,. This leads, as we shall see, to the peculiar shape of the center trace in Fig. 5. Consider, e.g., a spherical sample in the center of a helical coil. For this case we have calculated p(B,J neglecting radial field gradients. The result, shown in Fig. 6b, clearly indicates that p(B,,) is strongly asymmetric. It is also evident from physical reasons that p(B,,) should be peaked at the maximum value of B1,. Inserting p(B,J from Fig. 6b into Eq. [5] we have calculated S,. For an asymmetric distribution function p(B,,) it is not evident how we should adjust the width of a nominal 7r/2 pulse. Two reasonable choices are tp such that either y&By, = 7r/2 or rtpBE = r/2, where BtZ is the center of gravity of p(B& see Fig. 6. The question now is how to recognize such adjustments from traces such as those shown in Fig. 5. In order to answer this question we simulated such traces for p(B,,) appropriate to a sphere (radius 2 mm) in the center of a helical coil with parameters as described above, neglecting radial inhomogeneity. The results are displayed in Fig. 7 for three values of tp, chosen such that (a) yByztp = 7r/2; (b) IBM&, = 7r/2 (the geometry chosen implies BQB y, = 0.958) and (c) yB\& = 7r/2 with B\,/B?, = 0.993 chosen because the trace obtained for this condition resembles most closely the experimental one in the top part of Fig. 5.
FIG. 6. Volume distribution P along the z axis of the spherical sample of radius 2 mm (top) and the B,, field distribution p for this sample in a helical coil (bottom) with parameters as in Fig. 2. The dashed tine indicates the center of gravity of p(&).
RADIOFREQUENCY
287
COIL FOR NMR
_-
J
FIG. 7. Calculated NMR responses of a spherical sample of water in a helical coil with parameters as in Fig. 2 irradiated on resonance by a string of equally spaced pulses of three differing widths tp; top: y.BC& = r/2, where BC, = 0.958*B~z is the center of gravity of g(B& see Fig. 6b; center: rB’,& = u/2, B’;, = 0.993 *By, (closest resemblance to Fig. 5); bottom: rBy& = r/2.
From these simulations we learn, on the one hand, that the peculiar “tongs-like” &ape of the traces in Fig. 5 does indeed originate in the B,, inhomogeneity and the asymmetric distribution function p(Br,). They show, on the other hand, that, when starting such experiments with small values of the pulsewidth and then increasing it, the traces still contain a substantial amount of beats when the situation yBP,t, = 7r/2 is reached, whereas the slowest variation of S, (even) and S, (odd) corresponds closely to the situation rB?,t, = 7r/2. ACKNOWLEDGMENTS Manfred Hauswirth made the coil assembly and the delicate combs, we are aware that the success of this undertaking rests on the conscientious care and precision of his work. S.I. acknowledges hospitality and a stipend from the Max-Planck-Gesellschaft.
288
IDZIAK
AND
HAEBERLEN
REFERENCES 1. H. POST AND U. HAEBERLEN, J. Magn. Reson. 40, 17 (1980). 2. (a) W-K. RHIM, D. D. ELLEMAN, AND R. W. VAUGHAN, J. Chem. Phys. 59, 3740 (1973); (b) R. W. VAUGHAN, D. D. ELLEMAN, L. M. STACEY, W-K. RHIM, AND J. W. LEE, Rev. Sci. Instrum. 43, 1356 (1972). 3. A. N. GARROWAY, P. MANSFIELD, AND D. C. STALKER, Phys. Rev. B 11, 121 (1975). 4. N. Q. LAN, H. PFEIFER, AND H. SCHMIEDEL, Wiss. Z. Karl Marx Univ. Leipzig Math. Naturwiss.
Reihe 23, 498 ( 1974). 5. U. HAEBERLEN, “High Resolution NMR in Solids: Selective Averaging,” Suppl. I, “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Academic Press, New York/London, 1976. 6. M. MEHRING, “NMR, Basic Principles and Progress,” Vol. 11, pp. 88-101, Springer-Verlag, Berlin, 1976. 7. E. M. PURCELL, “Berkeley Physics Course,” Vol. 2, pp. 201-203, McGraw-Hill, New York, 1963.
OF MAGNETIC
RESONANCE
50,
28 l-288 (1982)
Design and Construction of a High Homogeneity rf Coil for Solid-State Multiple-Pulse NMR S. IDZIAK* Max-Planck-Institut
AND U. HAEBERLEN
ftir Medizinische Forschung, Abteilung fir Molekuiare Physik, Jahnstrasse 29, 6900 Heidelberg, Germany Received June 1. 1982
A prescription is given for the construction of a coil having an optimal d homogeneity desired, especially, in multiplepulse solid-state NMR. The approach is to use a coil with variable pitch. The field in a helical coil and in the coil with variable pitch are compared. The field distribution across the sample and some experimental aspects associated with the nonuniformity of the rf field are also discussed.
The design of the transmitter-receiver coil of a multiple-pulse take into account the following considerations:
spectrometer must
(1) Duration tx,* of u/2 pulses; our design goal is tr12 G 0.7 ~.~secfor protons and a transmitter power P = 800 watts, spectrometer frequency v = 270 MHz. (2) Ring-down time t, of the pulses as limited by the quality factor Q of the probe circuit; our design goal is tr < 100 nsec (90% - 10%). (3) Optimum sensitivity compatible with tr. (4) Inductance L of coil; it is limited by the minimum capacitance Cmin of the tuning capacitor. (5) Homogeneity of the rf field inside the coil. (6) Sparking. The first four points in this list have led us to use until now, in our multiplepulse spectrometer operating at 270 MHz (I), a self-supporting helical coil with eight turns, an inner diameter of 5.2 mm, and a length of 7.6 mm. It is wound of silver-plated copper wire with a diameter of 0.5 mm. It is satisfactory with the exception of the homogeneity of the rf field. Since rf inhomogeneity is a factor limiting the resolution in multiple-pulse experiments (2-6) we undertook efforts to design a new coil with improved rf homogeneity. A boundary condition was that the performance of the new coil with regard to considerations (1) to (4) should not be degraded in comparison to the former design. The idea for improving the rf homogeneity is to use a coil with continuously vuving pitch (4). The problem is twofold: first we must determine by calculation how the pitch should vary along the coil, second, we must find a way to constructsuch a coil in practice. We start with some theoretical considerations. For a helical coil with constant pitch the current * On leave from the Institute of Molecular Physics of the Polish Academy of Sciences, Poznaft, Poland. 281
0022-2364182 J 14028 l-08$02.00/0 Copyright g3 1982 by Academic Press, Inc. All rights of reproduction in any form rosened.
282
IDZIAK
AND HAEBERLEN
path is described by x= rcosQ, y = r sin Q,
z = aQ, a = Q.s/23r, where r and s are, respectively, the radius and pitch of the coil, see Fig. 1. By using Biot-Savart’s formula the field B,(z) along the coil axis produced by a current I can easily be calculated, 27rIn l/2 - z l/2 + z B,(z) = c ( ((l/2 - z)~ + r2)‘j2 + ((l/2 + z)~ + r2)‘j2 ) ’
PI
where n is the number of turns per unit length. The field distribution on the axis of a helical coil with constant pitch is identical to the field distribution on the axis of a coil consisting of a dense stack of parallel rings. This becomes immediately evident by considering the field contributions originating in the currents in infinitesimally short pieces of the helical coil but can also be seen by comparing Eq. [2] with the formula for the field in the stack of parallel rings derived in, e.g., Purcell’s well-known textbook (7). In both cases, the field distribution depends only on r and k hence increasing the density of turns in a helical coil does not lead to an increase of the B1 homogeneity on the coil axis. For the same reason we do not expect any advantage with respect to rf homogeneity (on the coil axis) by using a coil wound of flat tape (2). One way to obtain a coil with continuously varying pitch is to replace Eq. [ lc] by z = a(Q + ulQp(“sign Q). 131 For u < 0 and k > 0, k # 1 we obtain a coil having a higher density of turns at the ends of the coil than toward its center. For a single-layer coil Eq. [3] makes sense only in that range of Q for which dz/dQ > 0. We proceeded by integrating BiotSavart’s formula numerically over the path given by Eqs. [ 1a], [ 1b], and [3], varying u and k while adjusting a in such a way as to keep the length of the coil fixed. The
FIG.
1. Coil geometry and definition of coil parameters.
RADIOFREQUENCY
283
COIL FOR NMR
Z---c 1
2
0
-2
-L
C v
FIG. 2. Top: The B,, field on the coil axis normalized to B p,, for constant pitch (c) and variable pitch (u) coils. The dotted and dashed lines indicate, respectively, +2 and +lO% deviations with respect to the field at the center of the coil. Bottom: Cross sections through the wire (4 = 0.5 mm) at 'p" = ?r+ n * 2n, II = 0, 1, 2, 3.
best homogeneity is obtained for values of u and k which give such small turn spacings at the end of the coil that we must expect sparking between adjacent turns. Thus a compromise between rf homogeneity and minimum turn spacing had to be made. For judging which distance between adjacent turns is still acceptable we relied on our experience with rf coils in pulsed NMR. The parameters for what we considered to be the best compromise are a = 5.46 X low5 mm, 2) = -0.24 X 10P3, k = 2, coil length I= 7.6 mm, radius r = 2.8 mm; (0 is to be inserted in radians. Figure 2 shows &AZ) for such a coil (top) and a cross section through the wires, I#J= 0.5 mm at 9, = ?r + n. 27r, n = 0, . . . , 3 (bottom). For comparison we also show B&) and a cross section through the wires for a coil with constant pitch. So far we have considered only the field on the axis of a coil, B,AO, 0, z). In order to learn how B,, varies along the x and y axes we calculated &Ax, 0, zJ,
I
-2
0
2
-2
0
2
FIG.
x.y[mml
3. Radial part of B,, for three cross sections of the coil along the x (continuous curve) and y axis
curve). The dashed and dotted lines show, as in Fig 2, +2 and +lO% deviations with respect to the field at the center of the coil. The left part presents results for the constant pitch coil, the right part for the variable pitch coil, given by Eqs. [la], [lb], and 131. (dotted
284
IDZIAK
CROSS-SECTION
C-D
AND HAEBERLEN
CROSS-SECTION
A-B
FOG.4. Design of the variable pitch coil; parts 1 and 2 (actually four pieces each) are made of boron nitride, part 3 of KEL-F.
B,dO, y, zi), zi = 0; 1 and 2 mm for both the constant and variable pitch coils for which B,, is displayed in Fig. 2. The calculation was carried out by numerically integrating the equation,
s +8r
r2 - r(y.sin Cp+ x.cos P) c -& [(x - r’ cos s)2 + (y - r sin ‘P)* + (z - &)2]3’2 #Y
[41
where the limits of integration are suitable for an eight-turn coil. The results are displayed in Fig. 3. The diagrams show, on the one hand, that the radial inhomogeneity in the region of interest is comparable to or less than the axial inhomogeneity, and, on the other hand, that it is automatically improved by improving the axial homogeneity of the field, which justifies our procedure of considering only B,dO, 0, z) when optimizing the geometry of the coil. Practical Construction of a Coil with a Continuously
Varying Pitch
In spite of spending much effort to make a self-supporting coil with varying pitch we did not succeed in getting sufficient precision and stability. Therefore, we had to devise a means somehow to fix the turns of the coil in the desired locations. Any support of the turns from inside the coil was considered intolerable because of inevitable loss of volume for the NMR sample. In the design which we eventually adopted the turns of the coil are held in their positions by combs with precisely machined grooves. There are four such combs on the outer circumference of the coil. Figure 4 shows the combs and the entire coil assembly. It is made of boron nitride which is free of protons, has very low dielectric losses, and is easily machinable. The end pieces are made of KEL-F. The (almost) cylindrical filling of space around the sample with material of nonzero magnetic susceptibility is essential for avoiding interference of the coil assembly with the B0 homogeneity. The gain in rf homogeneity realized with such a coil design is demonstrated in Fig. 5. The upper part shows the scope trace of the NMR signal from a water sample in a helical coil irradiated on resonance by a string of equally spaced 7r/2 pulses. The decay rate of this signal is a direct measure of the homogeneity of the rf field (2). The lower part shows the same type of signal in the newly designed coil. It is evident from these traces that the gain in homogeneity is about twofold. Calculations of the type that follow indicate that the homogeneity achieved is close to the theo-
RADIOFREQUENCY
COIL FOR NMR
285
retical expectation. We take this as evidence that the desired variation of the pitch of the coil has been realized with sufficient precision. The traces shown in Fig. 5 are probably familiar to most people working in highresolution solid-state NMR. Nevertheless, their shape resembling a pair of tongs is probably annoying to most of them. What they expect to see in this type of experiment are three traces, one upper and one lower “envelope” and a single flat trace in the center. However, the center trace never remains flat and single, it always develops the shape of a pair of tongs. One might suspect that this peculiar shape results from a ringing of the rf power after turn on of the pulse sequence. However, this is not so and is, as we shall show now, an almost inevitable consequence of the B, inhomogeneity. The signal S, after the Nth pulse of the string of nominally
FIG. 5. Scope traces of the NMR signal of a water sample irradiated on resonance by a string of equally spaced x/2 pulses for a helical coil (top) and a variable-pitch coil (bottom). The spacing of the pulses is :25 rsec. The radius of the spherical sample equals 2 mm.
286
IDZIAK
AND HAEBERLEN
r/2 pulses, applied on resonance to, say, a sample of water is given by S, = kA&
p(B,,) - sin (rB,,Nt,) - dB1, . [51 1 Note, that yB,,Nt, is the total phase acquired after Npulses by a spin packet “seeing” the field B,,. The value M0 is the equilibrium magnetization and k a constant of proportionality; p(Bl,)dBI, is the fraction of spins “seeing” an rf field in the range of B,, - - . B,, + dB1,. If p(B,J is symmetric with respect to By, = 7r/2ytP the same fractions of spins are “late” and “early” in the course of the u/2 pulse sequence by any given amount with respect to the central spin packet seeing the field By,. In this case Eq. [5] predicts two decaying sets of points for N odd and a single set of points SN = 0 for N even. Symmetric distribution functions p(B,,) have been assumed ad hoc in recent papers dealing with rf inhomogeneity (3, 4, 6). In most practical cases, however, p(Blz) is not at all symmetric about any value of B,,. This leads, as we shall see, to the peculiar shape of the center trace in Fig. 5. Consider, e.g., a spherical sample in the center of a helical coil. For this case we have calculated p(B,J neglecting radial field gradients. The result, shown in Fig. 6b, clearly indicates that p(B,,) is strongly asymmetric. It is also evident from physical reasons that p(B,,) should be peaked at the maximum value of B1,. Inserting p(B,J from Fig. 6b into Eq. [5] we have calculated S,. For an asymmetric distribution function p(B,,) it is not evident how we should adjust the width of a nominal 7r/2 pulse. Two reasonable choices are tp such that either y&By, = 7r/2 or rtpBE = r/2, where BtZ is the center of gravity of p(B& see Fig. 6. The question now is how to recognize such adjustments from traces such as those shown in Fig. 5. In order to answer this question we simulated such traces for p(B,,) appropriate to a sphere (radius 2 mm) in the center of a helical coil with parameters as described above, neglecting radial inhomogeneity. The results are displayed in Fig. 7 for three values of tp, chosen such that (a) yByztp = 7r/2; (b) IBM&, = 7r/2 (the geometry chosen implies BQB y, = 0.958) and (c) yB\& = 7r/2 with B\,/B?, = 0.993 chosen because the trace obtained for this condition resembles most closely the experimental one in the top part of Fig. 5.
FIG. 6. Volume distribution P along the z axis of the spherical sample of radius 2 mm (top) and the B,, field distribution p for this sample in a helical coil (bottom) with parameters as in Fig. 2. The dashed tine indicates the center of gravity of p(&).
RADIOFREQUENCY
287
COIL FOR NMR
_-
J
FIG. 7. Calculated NMR responses of a spherical sample of water in a helical coil with parameters as in Fig. 2 irradiated on resonance by a string of equally spaced pulses of three differing widths tp; top: y.BC& = r/2, where BC, = 0.958*B~z is the center of gravity of g(B& see Fig. 6b; center: rB’,& = u/2, B’;, = 0.993 *By, (closest resemblance to Fig. 5); bottom: rBy& = r/2.
From these simulations we learn, on the one hand, that the peculiar “tongs-like” &ape of the traces in Fig. 5 does indeed originate in the B,, inhomogeneity and the asymmetric distribution function p(Br,). They show, on the other hand, that, when starting such experiments with small values of the pulsewidth and then increasing it, the traces still contain a substantial amount of beats when the situation yBP,t, = 7r/2 is reached, whereas the slowest variation of S, (even) and S, (odd) corresponds closely to the situation rB?,t, = 7r/2. ACKNOWLEDGMENTS Manfred Hauswirth made the coil assembly and the delicate combs, we are aware that the success of this undertaking rests on the conscientious care and precision of his work. S.I. acknowledges hospitality and a stipend from the Max-Planck-Gesellschaft.
288
IDZIAK
AND
HAEBERLEN
REFERENCES 1. H. POST AND U. HAEBERLEN, J. Magn. Reson. 40, 17 (1980). 2. (a) W-K. RHIM, D. D. ELLEMAN, AND R. W. VAUGHAN, J. Chem. Phys. 59, 3740 (1973); (b) R. W. VAUGHAN, D. D. ELLEMAN, L. M. STACEY, W-K. RHIM, AND J. W. LEE, Rev. Sci. Instrum. 43, 1356 (1972). 3. A. N. GARROWAY, P. MANSFIELD, AND D. C. STALKER, Phys. Rev. B 11, 121 (1975). 4. N. Q. LAN, H. PFEIFER, AND H. SCHMIEDEL, Wiss. Z. Karl Marx Univ. Leipzig Math. Naturwiss.
Reihe 23, 498 ( 1974). 5. U. HAEBERLEN, “High Resolution NMR in Solids: Selective Averaging,” Suppl. I, “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Academic Press, New York/London, 1976. 6. M. MEHRING, “NMR, Basic Principles and Progress,” Vol. 11, pp. 88-101, Springer-Verlag, Berlin, 1976. 7. E. M. PURCELL, “Berkeley Physics Course,” Vol. 2, pp. 201-203, McGraw-Hill, New York, 1963.