Spin Dynamics. Basics of Nuclear Magnetic Resonance.

Supplementary Notes

Updated Dec 14, 2005

Topics

Intrinsic angular momentum of macroscopic objects (page9)

Some large objects do "have angular momentum `by themselves`, without rotating", although the consequences are subtle and require fine experimental observations. For example, the Einstein-de Haas effect shows that the act of magnetizing a substance imparts angular momentum to it (for example, an object suspended by a fine thread starts rotating slowly as it is magnetized). A magnetized substance, like a lump of iron, does have angular momentum, without rotating! See, for example, L. D. Landau and E. M. Lifschitz, Course of Theoretical Physics, Vol.8, Pergamon Press, page 145.

Nuclear contribution to the susceptibility (pages 34)

The expression on page 34 for the nuclear contribution to the susceptibility may be derived as follows: According to Atkins, Physical Chemistry, 7th edition, page 802, the susceptibility is defined through the linear relation M=cH, where M is the magnetization and H is the applied magnetic field (in oersteds). If the susceptibility of the sample is low, so that it's own influence on the flux density may be ignored (a good approximation except for ferromagnets, superconductors, etc), we have B0=m0H and hence c=Mm0/B, where B0 is the applied field (flux density) in Tesla. The magnetization M is average magnetic moment per unit volume. For spins-1/2, the nuclear spin states have energy ▒(1/2)hbar g B0 and hence their populations in the field B0 are (1/2)(1▒ hbar g B0/2kBT). The magnetic moments along the z-axis for the two states are ▒(1/2)g hbar. The average magnetization per nuclear spin is therefore (1/2)(1+hbar g B0/2kBT)((1/2)hbar g )+(1/2)(1-hbar g B0/2kBT)(-(1/2)hbar g )=hbar2g2B0/4kBT. The magnetization is therefore M=hbar2g2B0c/4kBT, where c is the number of spins per unit volume. Hence the susceptibility is c=m0hbar2g2c/4kBT as given in the text.

Signs of frequencies (pages 48-53)

See the following exchange between myself and Stefan Berger (Leipzig):

SB: Take a concentrated solution of TMS and measure in the same sample without lock first the carbon resonance and afterwards the silicon resonance (negative gamma). Then change the magnetic field, let's say by increasing the zí coil by some units. Remeasure the two spectra. According to the diagrams in your book the signals of carbon and silicon should be displaced in different direction compared with their original position. Experimentally this is not the case, regardless which detection scheme is used (quad or quad off) and regardless, whether the instrument knows, that it is 29Si what it measures or just a "carbon" at 79 MHz. How would you comment this experiment?

MHL: With respect to your experiment, it's true that in the "thought experiment" spectrum of Fig.3.10 that the 13C and 29Si peaks move in opposite directions when the field is increased. However, in the real spectrometer, the accessible frequency windows are plotted using relative frequencies, as in sections 3.4 and 3.5. Since the 29Si spectrum gets turned around (Fig.3.15), the effect is that both peaks move in the same sense, which is as observed on the spectrometer. So I don't see a problem there.

The point about the instrument "not knowing" about the sign of gamma is an interesting one. This problem has two related parts: to do with the excitation of the spins, and to do with the detection of the NMR signals. On a spectrometer in which quadrature coils are used (as is the case in some imaging machines), then the instrument would really have to know about the sense of the precession in order to work at all, since the excitation circuitry would have to make a field rotating the correct sense. So there would be no paradox on such spectrometers.

In practice, on most spectrometers, the excitation and detection circuitry are linearly polarized (only one coil is used), in which case excitation and detection is simply implemented using both senses of rotation at the same time, and one simply throws away the counter-rotating components, both of the signal, and of the excitation field. Because of this, it is not necessary to program the sign of gamma into the instrument. In M. H. Levitt, J. Magn. Reson. 126, 164-182 (1997), I run through the situation quite carefully, using a model of the typical spectrometer electronics, and come to the conclusion that what actually happens is indeed equivalent to the conclusions of sections 3.4 and 3.5, even if the manufacturers do not implement this consciously.

See also my research group's page on the signs of frequencies and phases.

States and TPPI (pages 114-121)

The States and TPPI methods both lead to pure absorption 2D spectra, providing the appropriate conditions are met (see page 115), and both require the same total number of acquired transients. Nevertheless, they are not totally equivalent, and in some circumstances, the TPPI method is slightly superior. As described in the text, the States method combines two data sets which differ only in the phase of the excited coherences. The evolution frequency of the excited coherences is unchanged. In the TPPI method, on the other hand, the phase of the excited coherences is linked to incrementation of the evolution interval, generating an effective frequency change of the coherences. This turns out to be useful. For example, consider the case where longitudinal magnetization is present during the evolution interval t1. This magnetization does not evolve (neglecting relaxation), and so generates "axial peaks" on the w1=0 axis. In the States method, these axial peaks sit in the middle of the single-quantum spectrum, and must usually be removed by a further stage of phase cycling. In the TPPI method, on the other hand, the effective frequency shift of the coherences displaces the interesting 2D peaks from the vicinity of the w1=0 axis, so that in many circumstances, a further level of phase cycling is unnecessary. In such cases, the TPPI method is more economical of instrument time than the States method.

Sign of rotations (page 153)

The sign of the rotation operators introduced on page 153 are consistent with positive rotations about a right-handed axis system. Some NMR researchers and books use a different sign convention (for example, Hennel and Klinowski). See also signs.

Orbitals (page 157)

The orbital pictures in Fig.6.16 represent eigenfunctions of the angular momentum operators. However, they are not all eigenfunctions of the same angular momentum operator. For example, the pz orbital is an eigenfunction of the lz operator, with eigenvalue 0, and may be represented by the ket |1,0>. The px orbital, on the other hand is an eigenfunction of the lx operator, with eigenvalue 0. It is a superposition of two eigenfunctions of the lz operator, namely |1,+1> and |1,-1>.

It is difficult to draw the eigenfunctions |1,+1> and |1,-1> directly, since they are complex functions. The functions sketched in Figure 6.16 are all real functions, and only the s-orbital, one of the p-orbitals, and one of the d-orbitals are eigenfunctions of the z-component of angular momentum. The other orbitals are not eigenfunctions of the z-angular momentum but they are eigenfunctions of the angular momentum operators in other directions.

Do spins-1/2 have a quadrupole moment? (pages 171-174)

Nuclear physicists distinguish between the intrinsic quadrupolar moment of a nucleus, usually denoted Q0, and the effective quadrupolar moment, usually denoted Q. Roughly speaking, Q0 may be regarded as the "true" quadrupolar moment, while Q is the result of averaging the quadrupole moment over the "spinning motion" of the nucleus. Q0 may be finite for nuclei with spin < 1, while Q vanishes for such nuclei. The relevant quantity in NMR is the effective quadrupole moment Q. The discussion in the text is therefore valid, providing one realizes that the term "quadrupole moment" refers to the spin-averaged effective moment Q, and not to the intrinsic quadrupolar moment Q0.

Isotope Shifts (page 198)

One should distinguish between primary isotope shifts (for example, the difference in chemical shift for 1H nuclei in CHCl3 and for 2H nuclei in CDCl3) and secondary isotope shifts (for example, the difference in 13C chemical shift for 13CHCl3 and 13CDCl3). The effect discussed on page 198 is a secondary isotope shift. There are also primary and secondary isotope effects on J-couplings. See, for example N. D. Sergeyeva, N. M. Sergeyev and W. T. Raynes, Magn. Reson. Chem. 36, 255-260 (1998)).

Euler angles (page 220)

Follow this link for animations illustrating the operation of Euler angles.

Pulse bandwidths (page 269)

The relationship of the pulse bandwidth to the nutation frequency is more complicated than indicated on page 269. In the limit of a very short pulse (with a small flip angle), there is a Fourier relationship between the shape of a pulse in time, and its frequency response. In this regime, short pulses have wider frequency bandwidths than long pulses. However, most of the pulses used in NMR (with flip angles greater or equal to around 45 degrees) are outside the short-pulse regime. For pulses with commonly-encountered flip angles, the rf field (i.e. nutation frequency) determines the frequency bandwidth of the pulse, not the pulse duration.

Commutation Relationships (page 379)

The last two commutation relationships in Eq.12.26 only apply for spins-1/2, and are not trivial to derive. The easiest way is probably to multiply out the matrices for the operators, constructed as in section 12.9.1. Those who find this argument unsatisfying may prefer the following reasoning:
Take the commutator
[2I1z.I2z, 2I1x.I2x] = 2I1z.I2z.2I1x.I2x - 2I1x.I2x. 2I1z.I2z

Start with the first term

2I1z.I2z.2I1x.I2x = (2I1z.I1x).(2I2z.I2x)

Use the expressions given in Eq.6.81 (which only apply for spins-1/2) for each of the bracketed terms. One gets

(2I1z.I1x) = (1/2)( |a><a| - |b><b| )( |a><b| + |b><a| ) where all the bras and kets refer to spin number 1.

If one uses <a|a>=1, <a|b>=0, etc., this reduces to

(2I1z.I1x) = (1/2)( |a><b| - |b><a| ).

Putting the two operators together, we get

(2I1z.I1x). (2I2z.I2x) = (1/2)( |a><b| - |b><a| )1 * ( |a><b| - |b><a| )2

where the subscripts indicate which spin we are referring to. If we use the product basis for the two spins, this can be written as

(2I1z.I1x). (2I2z.I2x) =(1/2)( |aa><bb| - |ab><ba| - |ba><ab| + |bb><aa| )

If one repeats for the second term in the commutator, one gets

(2I1x.I1z). (2I2x.I2z) = (1/2)( -|a><b| + |b><a| )1 * ( -|a><b| + |b><a| )2

which is also

(2I1z.I1x). (2I2z.I2x) = (1/2)( |aa><bb| - |ab><ba| - |ba><ab| + |bb><aa| )

Hence the commutator vanishes.

Sign of nutation (page 259)

I define the initial phase of the rotating frame in such a way (Equation 9.17) as to impose a positive sense of nutation on the nuclear spins, independent of the sign of the magnetogyric ratio. With this convention, an ideal p/2 pulse always rotates the nuclear spins through an angle of +p/2 radians, in the rotating frame. This convention was established by Ernst and co-workers and is employed in Spin Dynamics. Although it is messy to define, it is convenient to use.

An alternative is to set the initial phase of the rotating frame to zero, in which case, an ideal p/2 pulse rotates positive-g spins through an angle -p/2 radians, while an ideal p/2 pulse rotates negative-g spins through an angle +p/2 radians, in the rotating frame. This convention is physically more natural but is more messy to use, since different diagrams are required for spins with positive and negative magnetogyric ratios.

A third possibility is to impose negative nutation frequencies on all spins, through an appropriate choice of rotating frame. This convention is implied in many diagrams in the NMR literature, and has been used effectively by many research groups, for example that of Ray Freeman.

All of these conventions lead to equivalent results, providing that they are used consistently.

See also my research group's page on the signs of frequencies and phases.

Are bicelles really bicelles? (pages 427-429)

Pages 427-429: It is possible that lipids do not form disk-like bicelles under the conditions typically used for biochemical NMR. There is recent evidence (Ad Bax, ENC, 2002) that the lipid bilayers are organized in highly perforated sheet-like structures (the "swiss cheese model"), rather than disks. Nevertheless, the precise nature of the lipid phase does not have appreciable consequences for the NMR properties of the partially-oriented molecules outside the bilayers.

Signal strengths of coherences in strongly-coupled systems (pages 393 and 590-591).

The signal induced by a coherence depends upon the amplitude of that coherence (as determined by the excitation efficiency and the history of that coherence) multiplied by a factor representing the coupling of the coherence to the quadrature NMR signal. In general, the detection factor is given by the matrix element of the I+ operator across the corresponding transition. In weakly-coupled systems, this factor is equal to unity for all 1-spin-(-1)-quantum coherences, and zero otherwise. In general, the detection factor is zero for all coherences which do not have order -1, but varies between the different (-1)-quantum coherences (see pages 590-591).

Simplifying the real part of Eq.15.4 (page 492)

The working used to go from Eq.15.4 to the equation at the end of page 492 was performed with the help of Mathematica. The intermediate steps may be viewed here.

Phases of rf pulses in NOESY, and sign of cross-relaxation (pages 501-509 and 543-548).

The rf phases assumed for 2D exchange spectroscopy and NOESY are given in Table 15.1 on page 502. In the first rown of this table, the second pulse is shifted in phase by p with respect to the first pulse. This choice of rf phases generates 2D spectra in which the diagonal peaks are always positive, and in which the cross peaks are positive for magnetization exchange experiments, and for NOESY experiments on slowly-tumbling molecules. Note however, that 2D experiments are often performed using the same rf phases for all three p/2 pulses, in the first step of the phase cycle. With this choice of rf phases, all peaks in the spectra are inverted in sign, compared to the analysis given in the text (for example, the diagonal and all cross peaks become negative for 2D magnetization exchange experiments). Personally, I find the choice of phases given in the text leads to a more transparent interpretation, in which the diagonal is always positive, and in which the sign of the cross peaks corresponds to the sign of the magnetization transfer. Nevertheless, these rf phase issues further complicate the controversy around the sign of the cross-relaxation rate constant (see note 3 on page 568). My personal position is that a mistake was originally made in choosing the definition of the cross-relaxation constant (probably because the first experiments were performed on small molecules), and the experiments have been adjusted afterwards in a misguided attempt to force the spectra to correspond to this erroneous choice of sign. In my view, it is now time to correct these historical errors.

Cogwheel phase cycling (page 625).

New ways have been discovered for constructing phase cycles, which are sometimes much more economical than the "nested" cycles described in section 17.10. See, for example, the cogwheel phase cycles developed in our research group.

Thanks

Contributions were made in part by:

David Siminovitch, Young Lee, Stefan Berger, Gareth Morris and Chunpen Thomas.

Please let me know by email if you have any more observations or discussion points.

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