Ill

100 kHz

100 Hz

100 Hz

The soft pulse is 1000 times lower in amplitude compared to the hard pulse, but it is 1000 times longer than the hard pulse. Although both pulses deliver a 90° rotation when on-resonance, they have very different behavior off-resonance. The excitation profile of both pulses is a "sinc" function:

sinc(x) = sin (x)/x but the width of the main "hump" of the sinc function is different:

Excitation bandwidth = 1/tp = 1/10 ^s = 100 kHz for the hard pulse

Excitation bandwidth = 1/tp = 1/10 ms = 100 Hz for the soft pulse

Pulse power is the square of the pulse amplitude:

In the laboratory we can measure RF power in watts, but when we set up NMR experiments we use a relative power scale that is logarithmic: the decibel scale. For comparison of power levels, we compare to a standard power level Po that corresponds to zero on the decibel scale:

Power in decibels = 10log(P/Po)

Every time the power is increased by a factor of 2, we are adding 3 dB to the power level in decibels: log(2) = 0.301, 10 log(2) = 3.01. Because the decibel scale is logarithmic, multiplying power by a factor corresponds to adding to or subtracting from the power level in decibels.

We can also compare pulse amplitudes using the decibel scale, as we know that power is the square of pulse amplitude:

where B\ is the B 1 amplitude at one power setting and Bb is the amplitude at another setting, AdB decibel units lower in power. As the 90° pulse width is inversely proportional to B1 amplitude, AdB = 20 log(tJb0/t9J0), where t9J0 is the 90° pulse width at one power level and tJb0 is the 90° pulse width at a power level AdB decibel units lower in power. Thus to cut the 90° pulse width in half, we need to double the B1 amplitude (quadruple the pulse power), which requires a 6 dB increase in power: 20 log (2) = 6.021 dB. Likewise, to double the 90° pulse width would require a 6 dB decrease in pulse power. This "6 dB rule" is very useful to keep in mind.

Bruker and Varian not only use different zero points for their decibel scales, but also use the opposite sign: Varian considers decibel to be a power level as described above, but Bruker sees the decibel setting as an attenuation—higher decibel values correspond to lower power. As long as you know this, you will not have any problem, but be very careful because setting the wrong power level can fry equipment!

Vendor

Parameters

Minimum

Maximum

Definition

Bruker

pll, pl2

120 dB

—6 dB

"dB of attenuation"

Varian

tpwr, dpwr

0 dB

63 dB

"dB of power"

When you calibrate a 90° "hard" *H pulse, you can estimate the power levels for other uses. The most convenient way to express a power level is by the duration of the 90° pulse at that power level:

Application

90°

Y B1/2n

TOCSY mixing (MLEV-17)

30 xs

8333 Hz

ROESY mixing (cw)

75 xs

3333 Hz

1H decoupling (waltz-16)

90 [xs

2778 Hz

You can then use the decibel scale to estimate power settings. For example, suppose you calibrated the 90° pulse on a Bruker 500 to be 17.6 ^s for *H at a power setting of 3 dB, and you want to know the power setting that will give a 30 ^s 90° pulse (yB1/2n = 1/(4 x 30 ^s) = 8333 Hz). Just plug in the ratio of pulse widths:

As our point of comparison (i9J0) was at 3 dB, we add this number AdB to 3 to get the correct power setting: 7.6 dB. This calculation gives us an estimate of the power setting; to get an accurate value you would have to calibrate the 90° pulse (on resonance) using this value as a starting point. Because in this case we want a 90° pulse of 30 ^s, you would start with a 60 ^s pulse and adjust the pulse power (Bruker parameter pll) until you get a null (180° pulse). When you are calibrating pulse widths and pulse power, at low power, it is extremely important to be on-resonance for the peak you are observing during the calibration. When yB\/2n is small, the effect of being off-resonance by even a small amount can be dramatic. For example, for a 75-^s 90° pulse, yB\/2n is 3333 Hz and on a 600-MHz instrument you would tilt the Beff vector out of the x'-y' plane by 45° if you are off-resonance by the same amount (3333 Hz = 5.56 ppm for protons). Also, keep in mind that near the maximum setting of pulse power, the dB settings do not give as much power as you expect: they begin to "droop" in a process called "amplifier compression." This occurs in the top 6 dB or so of available pulse power. The dB calculations work much better below this range.

Exercise: Estimate the Varian power level settings for TOCSY mixing (8333 Hz), for ROESY mixing (3333 Hz), and for 1H decoupling (2778 Hz) if the 90° pulse is 21.3 ^s at a power setting of 59 dB.

8.13.1 Calibrating a Shaped Pulse

Starting from a "first guess" of power level, a shaped pulse should be calibrated to get exactly the correct pulse rotation. Calibration of a rectangular pulse involves changing the pulse duration (pulse width) while maintaining the power level (pulse height) constant with a peak on-resonance. We look for a null in the spectrum at the 180° or 360° pulse width. For a shaped pulse, the selectivity depends on the pulse width, so we keep that constant and adjust the pulse power, increasing or decreasing the vertical scale of the pulse shape to change the area under the curve. When the maximum amplitude is changed, the shape of the

Figure 8.49

pulse is maintained so that all the short pulses that make up the shaped pulse are adjusted in amplitude according to the same ratio. Because pulse power is set using a logarithmic scale (dB), the envelope of a pulse calibration will not be a simple sine wave as it is for varying the duration of rectangular pulses. For example, for a Gaussian pulse we see a maximum for the 90o pulse power and then decreasing intensity to a null for the 180° pulse power (Fig. 8.49, top). When the null point is located (180° pulse at 61.5 dB), we can decrease the maximum power by 6 dB (67.5 dB on Bruker) to get a 90° pulse, rather than dividing the pulse duration by two as we do for hard pulses. These power levels represent the power of the highest amplitude in the shape, which is at the center of a Gaussian pulse. Often it is better to calibrate a shaped pulse in the context of how it is used in the pulse sequence. For example, a 180° Gaussian pulse used in a PFGSE is calibrated for the strongest signal of the selected peak using a PFGSE sequence (Fig. 8.49, bottom). Even though we think of a 180° pulse producing a null, in the context of a PFGSE it produces the maximum refocusing and allows the second gradient to perfectly unscramble the twisted coherence produced by the first.

To estimate a starting point for the maximum power of a shaped pulse, we need to come up with a rectangular pulse that has the same "area" as the shaped pulse. This can be done by mathematically integrating the function used for the pulse shape, for example, the Gaussian function. The Bruker software does this in the PulseTool program, and Varian does it in Pandora's Box (PBox). For example, a 35-ms Gaussian 180° pulse that is truncated at 5% of the maximum amplitude takes up an area that is 50.4556% of the corresponding rectangular pulse with the same duration and the same maximum amplitude as the shaped pulse (Fig. 8.50). Thus, a rectangular pulse of duration 17.66 ms (0.504556 x 35 ms) with the same amplitude as the maximum of the shaped pulse would rotate the sample magnetization the same amount (180°) as the Gaussian pulse, if both pulses are on-resonance. This corresponds to a full rotation in 2 x 17.66 = 35.32 ms, and a yB 1/2n of 1/(35.32 ms) = 28.31 Hz. This is a very weak pulse! Suppose we have calibrated the 90° hard pulse at 3 dB

35 ms

0 0

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