Radiation Damping and Pulse Calibration

Radiation damping causes broadening in the NMR resonances of very strong signals (such as the ¹H signal of pure water) as a result of currents induced in the coil from the strong transverse magnetization.  Radiation damping can also produce asymmetry and phase irregularities in the affected resonances.  These problems make pulse calibration by the standard nutation curve problematic when very strong signals are used for the calibration.  The left-hand panel (black) of the figure below shows the standard ¹H nutation curves for 0.1% H₂O in D₂O (bottom) and 80% H₂O in D₂O (top).  In both cases, single-scan spectra with a recycle delay of 30 sec were collected and plotted horizontally.  The pulse was varied from 1 µsec to 24 µsec in steps of 1 µsec.  In the case of 0.1% H₂O in D₂O, the nutation curve is well behaved and one is easily able to read off the 90°, 180°, 270° and 360° pulse durations.  In the case of 80% H₂O in D₂O, where radiation damping is a problem, the nutation curve is not well behaved.  There are asymmetry and phase distortion problems which make it impossible to determine the 90° pulse, based on maximum signal height, with any accuracy.  Nor is it possible to determine a reliable 180° based on the first minimum.  The spectra show very little distortion in the vicinity of the second minimum so the 360° pulse can be used reliably to determine the 90° pulse.  The right-hand panel of the figure (red) shows the integrals of the corresponding nutation spectra.  The integrals for both samples behave similarly.  It is clear that even in the case of severe radiation damping, one is able to determine a well behaved nutation curve from the integrals.

Determining 90° and 180° Soft Pulses

Many modern NMR pulse sequences (e.g. the 1D gradient selective NOESY experiment) depend on shaped pulses for selective excitation or inversion of specific resonances.  Since the width of the excitation profile of a shaped pulse is determined by its duration, the pulse duration is chosen by the user for the selectivity needed.  The longer the pulse, the higher the degree of selectivity.   Some spectrometer software will calculate the pulse duration based on a selected region in a spectrum containing the desired resonance for excitation.  The calculation depends on an initial pulse calibration.  The standard calibration method for hard pulses involves incrementing the pulse duration at a fixed power level.  The 90° pulse is at the first maximum and the 180° pulse is at the first null.  Since the duration of a the selective pulse is fixed by the desired selectivity, the 90° and 180° pulses must be found by varying the pulse power rather than the pulse duration.  The figure below shows the calibration for three different 50 msec shaped pulses on a Bruker AVANCE spectrometer.

An on-resonance water signal was observed as a function of pulse power using a selective one-pulse sequence.  The scale is in units of decibels of attenuation.  Maximum power is at -6 dB so the scale goes from low power on the left to higher power on the right.  The 90° and 180° pulses are indicated with arrows.  The intensity profiles are not sinusoidal due to the logarithmic dB scale.  The 90° and 180° pulses are separated by 6 dB of attenuation as expected.

Measurement of 13C 90° Pulses in Solids via Cross Polarization

The direct measurement of ¹³C 90° pulses in solids under MAS conditions by the conventional method suffers from the very low inherent sensitivity of ¹³C and is very time consuming due to the typically long ¹³C T₁'s.  These problems can be at least partially overcome by using ¹H - ¹³C cross polarization which has a potential four-fold sensitivity gain and also a time advantage as the repetition rate depends on the ¹H T₁ rather than the ¹³C T₁, the former typically being less than the latter by a factor of ten.  The 90° pulses are measured by carrying out the usual cross polarization contact which leaves the ¹³C magnetization along the -y axis.  The contact is followed by a ¹³C -x phased pulse (φ_-x) which rotates the magnetization towards the z axis in the -yz plane.  The acquisition follows with high power ¹H decoupling.  The sequence is illustrated in the figure below.

When φ_-x = 0°, one observes the usual positively phased CP spectrum.  As φ_-x is increased the signal decreases until φ_-x = 90° at which point the ¹³C magnetization is on the z axis and a null signal is observed.  As φ_-x is increased further, +y magnetization is created and a negative signal is observed until φ_-x = 180° at which point the magnetization is on the y axis and a maximum negative signal is observed, etc..... The 90° pulse can be read directly from the first null or 1/3 of the second null at 270°.  The vector diagrams and a typical measurement (where φ_-x was increased from 0.5 µsec to 20 µsec in 0.5 µsec steps) are illustrated in the figure below.

Fast 90 Degree Pulse Determination

Almost all NMR measurements rely on the correct calibration of 90° pulses. This is traditionally done by collecting a series of spectra as a function of pulse duration, finding a null for the 180° or 360° pulse and calculating the 90° pulse by simple division by 2 or 4 in the case of the 180° and 360° nulls, respectively. This determination, although trivial, can be very time consuming. Wu and Otting* have presented a much faster method of determining a 90° pulse based on measuring the nutation of a magnetization vector directly. Continuous nutation is depicted in the figure below. Here, the sample is subjected to continuous irradiation about the x axis. While being irradiated, the magnetization vector rotates in the z-y plane at a nutation frequency proportional to the pulse power. The magnetization on the -y axis is defined by a sine function. Fourier transformation of this magnetization gives an antiphase doublet centered at zero whose splitting Δν is twice the nutation frequency. The reciprocal of the nutation frequency is the time it takes the magnetization vector to rotate one complete cycle in the z-y plane and therefore the time it takes to rotate by one quarter of a cycle (i.e. the 90° pulse duration) is defined as 1/(2 Δν). The problem with continuous irradiation is that the sample must be irradiated at the same time magnetization is being detected. To eliminate this problem, a scheme similar to homonuclear decoupling is used where the radiation is turned off long enough to sample a data point. This is depicted in the figure below.Here each dwell period is divided up into a period for irradiation and a period for detection. The duty cycle for the irradiation is the fraction of time for which the sample is being irradiated. The magnetization is sampled when the power is off. As in the case for continuous irradiation, the magnetization vector still rotates in the z-y plane however, the rotation is slower as it is scaled according to the duty cycle. The duration of the 90° pulse is d/(2 Δν), where d is the duty cycle for irradiation. An example of this is shown in the figure below. The nutation spectrum was measured for HDO using a duty cycle, d = 0.10 and a power level of 12 dB (Bruker). Since the response of the amplifiers is linear, the 90° pulses at higher power levels can be calculated. Each decrease by 6 dB cuts the duration of the 90° pulse in half. In this case the 90° pulse at 0 dB was calculated to be 10.93 µsec at 0 dB based on the measured 90° pulse of 43.71 µsec at 12 dB. This pulse agrees to within a couple of percent of that measured by the more traditional method however, the measurement took only a few seconds. You can use a program called "pulsecal" on newer Bruker spectrometers to do this in complete automation.

--

* Peter S.C. Wu and Gottfried Otting J. Mag. Res. 176, 115 (2005).

90 Degree Pulse Determinations

In the routine procedure of R.F. pulse optimization, distorted nutation curves can sometimes be observed, as can be seen in the calculated graph below. This is usually due to the short recycle delay time (D1) not sufficient for complete relaxation. With increasingly shorter delay time, the maximum of the nutation curve shifts to lower flip angle, which makes the curve asymmetric. Under this circumstance, the 90 degree pulse determined from the maximum of the intensity will be inaccurate. On the other hand, if the recycle delay time is not too short, say 1 to 2 times of T1, the length of 180 degree pulse can still be determined with reasonable accuracy.

Many thanks to Eric Ye of the National Ultra-high Field NMR Facility for Solids for contributing this post.

90 Degree Pulses for I = n/2 Quadrupolar Nuclei in the Solid State

The 90 degree pulse for an I = n/2 quadrupolar nucleus in the solid state depends on the strength of the rf pulse with respect to the quadrupolar frequency. If the strength of the pulse is much greater than the quadrupolar frequency, the pulse is non-selective and excites all transitions equally. If however it is much less than the quadrupolar frequency, then the pulse is selective to the central (m = 1/2 - m = -1/2) transition. The duration of the pulse producing a maximum signal is shorter for selective vs. non-selective pulses at a similar power level. In solution, where the quadrupolar interactions is averaged by random isotropic molecular motion or in the solid state, if the symmetry around the I = n/2 nucleus is cubic, the quadrupolar frequency is small with respect to the strength of the rf pulses and the pulses are non-selective. When the symmetry around the I = n/2 nucleus in the solid state is non-cubic, the quadrupolar frequency is significant and the pulses are very often selective to the central transition. This is illustrated in the figures below for the ²³Na MAS spectrum of a mixture of NaCl (cubic) and Na₂SO₄ (non-cubic). The first figure shows the ²³Na MAS spectrum labelling each component of the mixture. The second figure shows the effect of increasing the pulse duration. One can clearly see that the 90 degree pulse for NaCl is close to twice that of Na₂SO₄.