## The Shim System

Shimming is the process of adjusting the magnetic field to achieve the best possible homogeneity. By homogeneous we mean that the magnetic field strength does not vary significantly from one part of the sample to another. Because the resonance frequency (chemical shift) is directly proportional to magnetic field strength, a variation of 1 ppm in field strength from one location within the sample volume to another would lead to a peak with a 1 ppm linewidth (or 200 Hz on a 200-MHz instrument!). Since linewidths of 1.0 Hz can be routinely obtained, the magnetic field when well shimmed does not vary more than 5 ppb within the sample volume. The dictionary meaning of a "shim" is a thin piece of wood or metal placed within a gap to make two parts fits snugly. The NMR shim plays the same role—it increases the magnetic field strength in certain volumes of space so that it is uniform throughout. The inhomogeneity of the field is a complex function in three dimensions; the goal of the shim system is to exactly match that function with a function of opposite sign so that all of the inhomogeneity cancels out (Fig. 3.4). To match that function, a number of simple 3D functions (Z, Z2, Z3, XY, XZ2, etc.) are summed with different coefficients. For example, the Z (or Z1) shim coil creates a linear gradient of magnetic field along the vertical (Z) axis, which is the axis of the NMR sample tube. When the Z shim is set to zero, there is no effect; when it is set to a positive value, the field strength is slightly increased in the upper part of the sample and slightly decreased in the lower part of the sample, to an extent proportional to the vertical distance from the center of the sample. Increasing the Z shim setting increases the current to the Z shim coil and makes the gradient "steeper" so that the field varies more for a given vertical distance. Changing the Z shim setting to a negative value reverses the sense of the gradient so that the field is decreased in the upper part of the sample and increased in the lower part. Likewise, the other shim coils control other simple gradient functions, and the current put through each coil controls its contribution to the field correction (or, mathematically speaking, its coefficient). These shim currents are set by computer and can be saved and recalled as files. Your job as operator is to search the

n-dimensional space (where n is the number of shims available) to find the global optimum of homogeneity. To get instant feedback on homogeneity, you have the lock level as a guide. As the field becomes more homogeneous, the 2H peak of your deuterated solvent becomes sharper, and its peak height (which is equivalent to the lock level) becomes higher as the same peak area gets squeezed into a narrower and narrower peak. The newer and higher field instruments have a large number of shim functions (or shim "gradients") available, so that you may be trying to find an optimum in a 28-dimensional space! This can be a daunting task, and only those with experience and great patience attempt to adjust any but the simple low-order shims.

### 3.4.1 Shimming for Beginners

Routine users usually adjust only Z and Z2, whereas experienced users might attempt to adjust Z1, Z2, Z3, Z4, Z5, X, Y, XZ, YZ, XZ2, YZ2, XY, and X2-Y2. For beginners, the first thing to learn is to move the shim far enough past the optimum so that there is a significant and observable drop in lock level (e.g., 10%), and then move back to the setting that gives the highest lock level. The territory on both sides of the optimal setting needs to be explored, and the optimal setting is near the center of the two settings that degrade the lock level by 10% on either side. The other problem for beginners is understanding that you are not finished shimming until you cannot improve the lock level any more with the shims you are adjusting (e.g., Z1 and Z2). There is no universal "good" lock level at which you can stop shimming, only the "best" lock level for your sample achieved after going over the shims several times (e.g., Z1, Z2, Z1, Z2, etc.). Although you should not be shy to make large changes, if you change any shim too rapidly there may be a transient response that obscures the true effect of the shim. For example, increasing Z2 might give an increase in lock level, but when you stop changing the shim this effect goes away. A rapid change in shim setting in the opposite direction will cause a transient decrease in lock level. In this situation you have to make changes slowly and be sure to wait for a moment after making a change to see if the lock level has really changed. As the shim settings are changed, the lock phase may be affected. Improperly set lock phase can lead to a situation where the optimal lock level does not correspond to the best peak shape in your spectrum. Since you need to use the lock level as a criterion for shimming, it is important to reoptimize the lock phase from time to time as you change the shims. This can be done simply by adjusting the lock phase for maximum lock level, just as you would do with a shim.

### 3.4.2 Shimming and Peak Shape

The effect of large errors in Z shim settings on peak shape is simulated in Figure 3.5 for a singlet NMR peak with a natural linewidth of 1.0 Hz. Note that a linear Z gradient (bad Z shim setting) simply "stretches" the peak horizontally. This peak is now a map of the sample molecules, with the molecules at the top of the sample having a resonant frequency at the left edge of the peak and the molecules at the bottom giving rise to the right edge of the peak (actually the limits of the peak are the top and bottom of the probe coil, since only that part of the sample within the coil is "seen"). This is an NMR imaging experiment (MRI) and illustrates the principle of making images by NMR using a linear gradient of magnetic field. For example, if there were bubbles in the sample these would show up as dips in the peak at the point corresponding to the position of the bubble along the vertical axis, since sample molecules would be missing at these points. Note that the higher order "odd" Z gradients (Z3 and Z5, Fig. 3.5, left) have symmetrical "pedestals" at the base of the peak. These pedestals are lower relative to the top of the peak as the shim order increases from Z3 to Z5. The "even" Z gradients (Z2, Z4, and Z6, Fig. 3.5, right) have "porches" or "verandas" at the base of the peak on one side only. If the shim error is reversed (e.g., from Z2 too high to Z2 too low) the porch will move to the opposite side of the peak. Each instrument will usually have a different polarity of the even shim gradients, so you will

have to experiment to see which way to move the shim in order to move the porch into the peak. As with the odd Z gradients, as we move to higher order even Z gradients, the wide portion appears lower down at the base of the peak. Figure 3.6 shows the effect on lineshape of smaller errors in shim settings on an ideal peak with 1.0 Hz linewidth. A Z1 error just broadens the peak evenly from top to bottom, while a Z2 error leads to a bulge or shoulder on one side of the peak. Similar effects are seen with higher order Z shims, with odd shims leading to symmetrical bulges and even shims leading to bulges on one side. The bulges move down lower in the peaks as we go to higher order shims. All of this assumes that the shim coils (Z1, Z2, Z3, etc.) actually deliver pure mathematical changes in magnetic field (z, z2, z3, etc.) as a function of position in the sample (z coordinate). In reality, there is a good deal of mixing of these pure functions, such that the Z1 shim knob actually changes all the other shims a little bit as well. This means that in practice you may not see these ideal changes in lineshape, and it is more difficult to diagnose which shim needs to be adjusted just by looking at peak shape. Furthermore, when you make a change in a higher order shim, such as Z4, you will need to readjust the lower order shims, especially Z2, because the Z4 shim "contains" a bit of Z2. Likewise, changes in Z3 and Z5 will require readjustment of Z1 and Z2.

In general, high-order shim errors do not affect the linewidth at half-height, but they do affect the linewidth at the base of the peak and reduce the peak height by spreading the peak intensity into the pedestals and porches at the base. To assess the overall quality of shimming, it is best to measure the peak width in several places; typically for a singlet peak such as chloroform (CHCl3), the width in hertz is measured at 50% of peak height, 0.55% of peak height, and 0.11% of peak height. The 0.55% is conveniently determined by the

height of the doublet produced by the 1.1% 13CHCl3 present at natural abundance. These 13C "satellite" peaks are found at 105 Hz (1JCH/2) downfield and upfield from the main 12CHCl3 peak. For example, the quality of shimming could be reported as 0.5/8/11 Hz at 50/0.55/0.11% of peak height. If linewidth is only measured at the half-height level, you will be ignoring the higher order shim errors.

Often the effect of two shims is interactive, such that changing one shim setting affects the optimal setting of the other. This is commonly observed for Z and Z2 with short (low volume) samples. For example, you might visualize the effect of Z and Z2 on the lock level as a two-dimensional plot, like a topographic map (Fig. 3.7). Climbing a simple round peak (Fig. 3.7, left) is easy: just optimize Z (the east-west direction) by moving to the right (segment a) as long as the lock level continues to increase. The lock level reaches a maximum and begins to go down again (segment b), so you turn around and move left to return to the maximum. It is important to really see the lock level go down significantly, so you know you have reached the maximum for the noisy lock level. Next optimize Z2 (the north-south direction) by finding the direction (up or down on the map) that increases the lock level and then moving straight to the top of the peak (segment c). Again, to be sure you are at the maximum you need to go significantly beyond and return to the peak (segment d). But what if the surface is more like a ridge that runs from the southwest to the northeast (Fig. 3.7, right)? You might use Z1 to climb to the top of the ridge (segment a), but Z2 would not give any further improvement, even though the peak is a long way up the ridge. What you need to do is simultaneously adjust Z1 and Z2 so that you can move diagonally, like trying to draw a diagonal line with an Etch-a-Sketch. To do this on a spectrometer, note the lock level and then arbitrarily move Z2 away from the maximum in one direction to degrade the lock level by a certain amount, like 10% (segment b). Then use Z1 to optimize again (segment c). If the optimized lock level is better than where you started, you have made progress up the ridge. Now you only need to continue making small changes in Z2 in the same direction away from the Z2 optimum, followed by optimization of Z1. If the optimized lock level is worse, try an arbitrary change in Z2 in the opposite direction. The process is a zigzag approach to the peak, alternately going downhill and then uphill to the top of the ridge. Many shim pairs behave in this way, so you can see how shimming is an art requiring a lot of patience.

Another way to shim is to use the FID as a criterion for homogeneity instead of the lock level. The goal is to get a smooth exponential decay curve with the longest time constant (slowest decay) possible for the FID. Bruker uses the command GS to enter an interactive mode where the FID is acquired over and over again, displaying it each time

Figure 3.8

Figure 3.8

without summing in the sum to memory. On the Varian you can do the same thing by selecting FID instead of SHIM in the acqi window. A simple Z1 shim error (Fig. 3.8, bottom) will give an FID that not only decays faster but also goes though a series of evenly spaced nulls ("pinches") in the FID as it decays. At each null the FID signal reverses phase. Mathematically this can be described as a "sinc" function (y = sin(x)/x), which gives a rectangle in frequency domain when you do the Fourier transform. The rectangle is the peak shape you get from a linear Z gradient, the MRI experiment (Fig. 3.5). The sinc function and the rectangle can be viewed as a "Fourier pair" since Fourier transformation of one shape gives the other and vice versa. We will encounter this and other Fourier pairs later in the course of this book. As you improve the Z shim you will see the "pinches" move to the right, to longer times, corresponding to a narrower rectangle in frequency domain, and eventually off the "end" of the FID. After the last "pinch" is moved off the end, you should be able to maximize the signal at the end of the FID to get the slowest decay. The lock level may actually be going down as you do this! The reason is that either the lock phase is not adjusted right or there are other shims, especially higher order shims, which are interacting with Z1. If Z1 were the only misadjusted shim, the lock signal would go up as you improve the FID. You may be able to alternate between optimizing other shims (Z2, Z3, etc.) using the lock level and optimizing Z1 using the FID. You may also find that the Z2 shim moves the pinches, contrary to theory. Go with it! Shimming is an art, not a science. Sometimes if the FID has no "pinches" but the shape is not exponential, you can adjust higher order shims (Z3, Z4, Z5, Z6) to try to get an exponential FID shape. Then you can go back to the lower order shims to get a slower exponential decay. Shimming on the FID is easiest if you have a single peak dominating the spectrum so that the FID is a simple decay of one sine wave. For example, a small amount of CHCl3 in CDCl3 or d6-acetone, or of H2O in D2O is good for this kind of shimming.

The "pure" Z shims are called "axial" shims. Other shims contain X or Y in their names; for example, X, Y, XZ, YZ2, and so on. Spinning does not correct these "off-axial" errors; it simply moves the intensity to spinning sidebands, which are satellite peaks separated on either side of the main peak. If the sample is spinning at 20 Hz (20 revolutions per second), spinning sidebands will appear at 20, 40, 60 Hz, and so on, away from the main peak on either side. If there are large errors in the off-axial shims there will be large spinning sidebands. Another indication of poor off-axial shims is the increase in lock level observed when the spinning is turned on. If this is more than about 15% of the nonspinning lock level, the X and Y contributions to the shims are not optimal. The off-axial shims must be adjusted without spinning the sample. In general, low-order shims (Z, Z2, X, and Y) should be adjusted with small changes ("fine" setting), and all higher order shims can be adjusted using large changes in the shim value ("coarse" setting). A typical approach for off-axis shims is to adjust them in the order X, Y, XZ, YZ, XZ2, YZ2, XY, and X2-Y2, and then reoptimize X and Y.

If shims are really bad, you may be able to recall a recent shim file from the disk. Ideally, a spectrometer should be shimmed regularly (daily or weekly) by an expert, and these "current" shims should be saved. Rather than waste a lot of time trying to get home from someone else's «-dimensional wanderings away from the optimum, you can just read the latest shim file and start from there. Of course, you will still need to adjust Z1 and Z2 because these change a great deal based on the sample volume, position, and polarity of the solvent.