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CHAPTER 6
PRINCIPLES OF NUCLEAR MAGNETIC
RESONANCE AND ELECTRON
SPIN RESONANCE
INTRODUCTION
In this chapter, we consider the interaction of molecules with radiation when molecules are placed into a strong magnetic field. The fundamental properties of atoms that are important for this discussion are the nuclear spin, for nuclear magnetic resonance (NMR), and the electron spin, for electron spin resonance (ESR) or, equivalently, electron paramagnetic resonance (EPR). Strictly speaking, the concept of spin can be rigorously defined only by the use of quantum mechanics. However, we will use a semiclassical approach in which spin in the nucleus or in an electron can be represented as a charge moving in a circular path. This movement creates a magnetic dipole that can be thought of as a bar magnet. In the absence of a magnetic field, the magnetic dipole is oriented randomly, and only one energy level is associated with the electron or nucleus. But in the presence of a magnetic field, the magnetic dipoles (or bar magnets) tend to be oriented either in the direction of the field or opposed to it, thus creating multiple energy states. Application of quantum mechanics to this situation indicates that the orientation of the magnetic dipole and the energy states are quantized with characteristic quantum numbers. For electrons, the spin quantum number, S, is 1/2, and the two spin states are +1/2 and - 1/2, represented as the familiar arrows pointed up or down. Neu- trons and protons also have a spin quantum number of 1/2 so that the nucleus
103
Spectroscopy for the Biological Sciences , by Gordon G. Hammes Copyright © 2005 John Wiley & Sons, Inc.
104 PRINCIPLES OF NUCLEAR MAGNETIC RESONANCE AND ELECTRON SPIN RESONANCE
of an atom has a characteristic spin quantum number, I. Simple rules exist for determining the nuclear spin quantum number. Nuclei with an even mass number and even charge number have no nuclear spin (I = 0). Nuclei with an odd mass number have a half integral spin (I = 1/2, 3/2, 5/2 etc.). Finally, nuclei with an even mass number and odd charge number have integral spin (I = 1, 2, etc.). The number of energy levels associated with spin is determined by the spin quantum number. For an electron, the number of energy levels is 2S + 1, or 2. In the absence of a magnetic field, no distinction can be made between these two quantum states: They have the same energy, that is, they are degenerate. In the presence of a magnetic field, however, the alignment of the magnetic moments with and against the magnetic field creates two distinct energy levels. The hydrogen nucleus also has a spin quantum number, I, of 1/2, with two pos- sible orientations of its magnetic dipole (or magnetic moment) in a magnetic field and two energy levels (2I + 1). On the other hand, 23 Na has I = 3/2 and four energy levels in the presence of a magnetic field. In general, the quantum states in the presence of a magnetic field are characterized by quantum numbers ranging from I to - I in integral steps.Thus, for sodium, these quantum numbers are 3/2, 1/2, - 1/2, and - 3/2. Although the average orientation of the nuclear spin state of a proton is either aligned with the magnetic field or against it, the conservation of angular momentum requires the magnetic moment associated with the orientation to rotate about the direction of the field. This is analogous to a top spinning on its axis and rotating around a ver- tical line due to gravity. A quantum mechanical treatment of nuclear spins in a magnetic field of a strength H provides an explicit equation for the energy levels, E:
In this equation, bN is the nuclear magneton and is a universal constant cal- culated from the properties of nuclei: bN = 5.051 ¥ 10 -^27 Joules/Tesla. The nuclear g factor, g (^) N, is a constant, but it is different for each atom, and mI is the spin quantum number characterizing the orientation of the magnetic moment in the magnetic field (I to - I in integral steps). This equation also defines the gyromagnetic ratio, g, which is a frequently used constant. Values of the nuclear spin quantum mumber, I, and g for some nuclei of biological interest are presented in Table 6.1. The dependence of the energy on magnetic field for atoms with a total nuclear spin quantum number of 1/2 is shown in Figure 6-1. The difference in energy between the quantized levels increases as the magnetic field increases. Also shown is the precession of the magnetic moment about the field direc- tion for the two possible orientations. The energy difference between the energy levels can be calculated, again with the assistance of quantum mechanics, by the requirement that the change
E = - g (^) N b (^) N HM (^) I = -(h 2^ p g)HmI
106 PRINCIPLES OF NUCLEAR MAGNETIC RESONANCE AND ELECTRON SPIN RESONANCE
ference is small the actual population difference between energy levels is also very small at room temperature: the ratio of populations in the two states is exp(-E/kT) = 0.99993 for an 11.75 Tesla magnetic field at 37°C. For this same magnetic field, the resonant frequency for 13 C is 130 MHz. The quantum mechanical calculation of the energy levels for an unpaired electron in a magnetic field is quite similar to that for nuclear spin, except that the electron always has spin 1/2. The energy difference between levels is
(6-3)
where gS = 2.0023 and bS = 9.274 ¥ 10 -^24 Joules/Tesla for electrons. Because the Bohr magneton is much larger for an electron than a proton, the energy levels are further apart, and the resonance energy frequency is much larger. (bS and bN differ by the ratio of the mass of the proton to the mass of the electron, 1836.) For a 1 Tesla field, the resonant frequency is 28,000 MHz = 28 GHz. This frequency is in the microwave region, and quite different experimental tech- niques are required for ESR and NMR. Finally, we return to the precession of the magnetic moment about the direction of the magnetic field, as depicted in Figure 6-1. The angular preces- sion frequency, that is, how fast the magnetic moment is rotating about the vertical line, is w = 2 pu = gH. This is called the Larmor frequency. The visual- ization of rotating magnetic moments is useful when considering the effects of changing magnetic fields on the nuclear spins. We first discuss NMR in some detail because it is extensively used in biology. Although ESR has been used to obtain important information about biological systems, it is less extensively used and will receive relatively brief consideration.
NMR SPECTROMETERS
The first NMR spectrometers placed a sample in a fixed magnetic field and applied a fixed radiofrequency by means of a coil perpendicular to the field direction.The magnetic field was varied by a coil until resonance was achieved, with the absorption being detected by a third coil. This is analogous to the methods used in visible and ultraviolet spectroscopy. Although the magnetic field applied to the sample is uniform, the actual magnetic field at the nucleus is dependent on a number of environmental factors so that the field must be scanned to find the energy absorption (resonance) condition. We will return to this matter a bit later. A typical spectrum would display the absorption of energy versus frequency, as shown schematically in Figure 6-2. The absorption of radiation is difficult to detect because the populations of the energy states are very similar, as discussed above. In practice, this means that relatively high concentrations of the species being observed must be present. The sensitivity of detection depends on the characteristics of the
DE = g (^) S bSH
NMR SPECTROMETERS 107
nucleus being observed and its natural abundance. The sensitivity also is enhanced by increasing the energy difference between the two states, which is the reason that instruments with larger magnetic fields are being developed continuously.The proton, 1 H, provides the best sensitivity, and its natural abun- dance is 99.98%. Consequently, the most extensive measurements have been carried out with proton NMR. Although the natural abundance of 31 P is essen- tially 100%, the sensitivity of detection relative to protons is only about 6%. Probably the second most studied atom with NMR is 13 C. Its natural abundance is only about 1%, but its prevalence in biological compounds is very extensive. Furthermore, its abundance in molecules of interest can be enhanced through synthesis or bacterial growth with 13 C enriched compounds. Regrettably, the most abundant isotope of carbon, 12 C, does not have a nuclear spin. A variety of other isotopes have been studied with NMR. Most notable for biologists are 14 N, 15 N, and 19 F, which can often be substituted for H in sub- stances of biological interest. As discussed in Chapter 1, the frequency dependence of the absorption can be transformed into a time dependence and vice versa through Fourier trans- forms. The most common method of obtaining an NMR spectrum today is to apply a timed radio frequency pulse and then watch the nuclei return to their equilibrium configurations. The time dependence of the return to equilibrium or free induction decay (FID) can be transformed from a time-dependent signal into a frequency spectrum, as shown schematically in Figure 6-2. Multiple FIDs can be combined to produce an average FID that has less noise than a single FID. Consequently, Fourier transform instruments are con- siderably more sensitive than continuous wave instruments so that lower con- centrations can be used. The timing and nature of the pulses can be quite
Time (seconds)
Fourier transform
0.0 0.2 0.4 0.6 0. Frequency (Hz)
Free induction decay
Spectrum
800 600 400 200 0
Figure 6-2. The free induction decay of the signal from nuclei in a magnetic field is shown ( left ) after a short pulse of radiofreqency radiation is applied to the sample. A Fourier transform of the free induction decay gives the familiar nmr spectrum on the right where the absorption is plotted versus the frequency. Copyright by Professor T. G. Oas, Duke University. Reproduced with permission.
CHEMICAL SHIFTS 109
pounds are generally highly shielded. Conversely, a low electron density at the nucleus causes a downfield shift and increase in d. The range of chemical shifts for 1 H and 13 C in various compounds are shown in Figure 6-3. In principle, the chemical shifts of nuclei in proteins should provide infor- mation about the protein structure. However, chemical shifts alone are not suf- ficient to determine protein structure. The average chemical shifts of various nuclei for amino acids in denatured proteins are given in Table 6.2. Note that they are quite similar for all of the amino acids. When secondary structures such as a-helices or b-sheets are present, changes in chemical shifts occur so that the amounts of various secondary structures present can be inferred from the NMR spectra. Some of the largest changes in chemical shifts in proteins and nucleic acids are observed for aromatic rings of nucleotides, tyrosine, phenylalanine, and
dd / p.p.m.
14 12 10 8 6 4 2 0
—CO 2 H*
—CH*O
Ar—H*
ArOH*
—C=CH*
ROH*
—CO 2 CH* 3
ArOCH* 3
RCOCH* (^3)
—CH*
Ar—CH* 3
R—NH* 2
—CH* (^2)
RCH* 3 (a) 1 H resonances Si(CH*^3 )^4
d / p.p.m.
300 200 100 0
(R 3 C*)
R=C*=R'
R=C*=R'
R—C*HO
RC*OOH
RC* N
C*—X in ArX
C=C*<
C C*—
C=C*<x
(R 3 C*)-
(b) 13 C resonances R—*C—H
Figure 6-3. Range of typical chemical shifts for 1 H and 13 C resonances. P. W. Atkins, Physical Chemistry , 3rd edition, W. H. Freeman, New York, NY, 1986, p. 489. © 1978, 1982, 1986 by Peter W. Atkins. Used with permission of W. H. Freeman and Company.
110 PRINCIPLES OF NUCLEAR MAGNETIC RESONANCE AND ELECTRON SPIN RESONANCE
tryptophan. These shifts are due to the interaction of the external magnetic field with the delocalized electrons of the aromatic ring. Nuclei above or below the ring usually have decreased chemical shifts, whereas those near the edges have increased chemical shifts. These effects are called ring currents and can cause unusually large chemical shifts. These chemical shifts can provide infor- mation about the structure of macromolecules and about alterations in struc- ture due to changes in the environment such as temperature and the addition of various chemical agents.
SPIN-SPIN SPLITTING
The chemical shift is caused by interactions between the nucleus and nearby electrons. A conceptually different interaction is transmitted between nearby nuclei by intervening electrons participating in chemical bonds. Basically, the spin state of a neighboring nucleus alters the shielding a nucleus experiences. This effect is smaller than typical chemical shifts and is called spin-spin split- ting and is often referred to as scalar coupling. Unlike chemical shifts, the mag- nitude of spin-spin coupling is independent of the magnitude of the applied magnetic field.
TABLE 6-2. Average Chemical Shifts of Random Coil Amino Acids (ppm)
Amino Acid a-^1 H Amide- 1 H a-^13 C Carbonyl-^13 C Amide- 15 N
Ala 4.33 8.15 52.2 177.6 122. Cys 4.54 8.23 56.8 174.6 118. Asp 4.71 8.37 53.9 176.8 120. Glu 4.33 8.36 56.3 176.6 121. Phe 4.63 8.30 57.9 175.9 120. Gly 3.96 8.29 45.0 173.6 108. His 4.60 8.28 55.5 174.9 119. Ile 4.17 8.21 61.2 176.5 123. Lys 4.33 8.25 56.4 176.5 121. Leu 4.32 8.23 55.0 176.9 121. Met 4.48 8.29 55.2 176.3 120. Asn 4.74 8.38 52.7 175.6 119. Pro 4.42 — 63.0 176.0 128. Gln 4.33 8.27 56.0 175.6 120. Arg 4.35 8.27 56.0 176.6 120. Ser 4.47 8.31 58.1 174.4 116. Thr 4.35 8.24 62.0 174.8 114. Val 4.12 6.19 62.2 176.0 121. Trp 4.66 8.18 57.6 173.6 120. Tyr 4.55 8.28 58.0 175.9 122.
Reproduced with permission from D. S. Wishart and B. D. Sykes, Chemical Shifts as a Tool for Structure Determination, Meth. Enyzmol. 239 , 363 (1994). © 1994, with permission of Elsevier.
112 PRINCIPLES OF NUCLEAR MAGNETIC RESONANCE AND ELECTRON SPIN RESONANCE
amide bound occurs. Scalar coupling can also occur between different nuclei, for example, a proton and 15 N. The magnitude of the spin-spin splitting is called the coupling constant and is designated by J. The number of covalent bonds separating the nuclei in question is often appended as a prior superscript and a post subscript designates the atoms involved. For example, the coupling con- stant for the amide proton and the a-C proton would be written as 3 JHN-Ha. The spin-spin coupling constant for protons on adjacent atoms varies con- siderably, from about 0 to 10 Hz. This variation is due to different torsional angles between the protons as defined below:
If the torsional angle is 90°, the coupling is close to zero, whereas when it is 0° or 180°, it is about 10 Hz. If free rotation about the bond occurs, an average coupling constant is obtained. The dependence of the coupling constant on the torsional angle is given by the Karplus equation:
(6-6)
The constants A, B, and C can be calculated or established from measurements with molecules having known dihedral angles. If free rotation about the bond occurs, the coupling constant will be some average of the possible dihedral angles. However, if the bond is constrained, such as in a peptide linkage that has double bond character or in a folded macromolecular structure, the cal- culated dihedral angel can provide useful information about the structure. This is especially true when used in conjunction with other information. This simple picture of spin-spin splitting is not rigorous. It provides a useful and adequate explanation for relatively simple situations but breaks down when considering multidimensional NMR, which is discussed later.
RELAXATION TIMES
Thus far we have not dealt explicitly with the time scale for nuclear spin tran- sitions. If we think of a pulsed NMR experiment, we can envisage nuclear spins being oriented in a specific direction by the pulse, with the precession of spins about the direction of the field. When the pulse is turned off, the nuclei will come to equilibrium with regard to their environment. The rate at which a par- ticular nuclear spin returns to equilibrium depends on interactions with other nuclear spins, and this in turn will depend on the fluctuating fields experienced as the molecules tumble. This mode of relaxation involves a change of energy between the spin systems and their environment. The return of a spin popu-
J = A + B cos q +Ccos^2 q
H H
C C
q=0° q= 180 °
H
C C
H
RELAXATION TIMES 113
lation in a magnetic field to its equilibrium population follows first-order kinetics, and the reciprocal of the first-order rate constant is called the spin- lattice relaxation time , T 1. Typically it is in the range of tenths of a second to seconds for protons. Simply put, the value of T 1 depends on the interactions of a nuclear spin with its neighbors, including the solvent, and how fast the molecule rotates. The molecular rotation is characterized by a rotational correlation time that is basically a measure of the rotational diffusion constant of a molecule. Measurements of T 1 can, in fact, be used to determine rotational diffusion constants of rigid molecules. In some cases, chemical exchange of nuclei in different environments can contribute to T 1 although this is somewhat unusual. In general, anything that gives rise to magnetic fluctuations in the environment can contribute to this relaxation, for example, unpaired electrons or dissolved oxygen. A second mode of nuclear spin relaxation is possible that does not involve the exchange of energy of the magnetic moment with its environment. In terms of the picture of a nuclear spin being oriented in a field and precessing about the field direction, the spin-lattice relaxation time can be thought of as char- acterizing the return of the spin orientations to their equilibrium positions.The spin-spin relaxation time , T (^) 2, on the other hand, is associated with the rate of precession about the field direction. Basically, T 2 is a measure of alterations in the precession frequency during nuclear spin relaxation. This alteration is dif- ferent for different nuclear spins so that the rates of precession change with respect to each other. This is called a loss of phase coherence. This can be viewed as an exchange of energy within the spin system. It does not change the net population of the excited states. The dominant factor determining T 2 is the rate of molecular tumbling. The molecular tumbling effect is quantitatively different for T 2 and T (^) 1. Because T (^2) is generally much shorter than T 1 in liquids, the line widths of spectra are deter- mined by T 2 , 1/T 2 = pu1/2 where u1/2 is the peak width at one-half of its maximum value. As previously stated, rapidly tumbling molecules produce relatively sharp lines whereas slowly tumbling molecules have relatively broad lines. Chemical reactions provide one of the most interesting examples of how T (^2) can be altered. Consider the simple chemical reaction
(6-7)
Assume that a specific proton has a different chemical shift in A and B. (This means they have a different Larmor frequency.) If the chemical exchange rate is very slow compared to the difference in chemical shifts (i.e., smaller than the difference in Larmor frequencies), two distinct peaks will be seen in the NMR spectrum, as shown in Figure 6-5 (top). At the other extreme, if the chemical exchange rate is very fast relative to the difference in the chemical shift frequencies, states A and B will interconvert many times during the NMR experiment, and the two frequencies are effectively averaged. Consequently,
A WB
MULTIDIMENSIONAL NMR 115
reactions that can be studied is determined by the chemical shift difference between the two states. For example, if the difference is 100 Hz, then rates of chemical reactions in the range of 1/100 = 10 -^2 s can be studied. Reactions such as the rate of exchange of hydrogens between proteins or amino acids and water have been investigated with this method.
MULTIDIMENSIONAL NMR
Thus far we have considered what is now called one-dimensional NMR, namely determining the spectrum for a specific nucleus by scanning the mag- netic field or by analysis of the frequencies associated with the free induction decay following a frequency pulse. One-dimensional analysis has been invalu- able in determining the structures of small molecules and can also be used to obtain information about macromolecules. However, in order to get definitive structural information about macromolecules, multidimensional NMR is nec- essary. The genesis of this field was in the early 1970s and its vigorous evolu- tion continues to this day. The underlying principle of multidimensional NMR is to find “cross-peaks” that link two resonances. This linkage can be either through space or through a small number of chemical bonds. Finding these cross-peaks allows the spatial relationships to be determined between the nuclei responsible for the two resonances. These connections between reso- nances are sometimes called coherence pathways. A detailed presentation of multinuclear NMR is beyond the scope of this text. However, the concepts can be understood by considering two- dimensional NMR in a qualitative manner. Two-dimensional NMR can be viewed as the assembly of one-dimensional spectra in an array. The experi- ments consist of four stages, illustrated in Figure 6-6. In the preparation phase a frequency pulse is applied to the system. An evolution phase of length t 1 then
t 1 tm t 2
EVOLUTION MIXING DATA ACQUISITION
Figure 6-6. Schematic representation of a two-dimensional NMR experiment in which a radiofrequency pulse (black bar) is applied to the sample initially. After a period t 1 , a second pulse is applied, followed by a mixing time, tm. The data are acquired after a final pulse. The number of pulses and the various times depend on the type of experi- ment being carried out.
116 PRINCIPLES OF NUCLEAR MAGNETIC RESONANCE AND ELECTRON SPIN RESONANCE
occurs and is followed by a mixing period ending in a second pulse. (A pulse may also be applied between the evolution and mixing phases, depending on the specific two-dimensional experiment being carried out.) Finally, a free induction decay occurs for a time t 2 in the data acquisition phase. In the actual experiment, t 1 is varied incrementally and the signal is collected for the period t 2 , ultimately giving a large number of time points. This experiment is repeated many times and signal averaged. The nature of the pulses will depend on the specific experiment. Qualitatively, the pulses flip the spins (magnetic moments) in the field that is created, and the end of the mixing phase flips them again. In the evolution phase, the magnetic moments associated with the nuclear spin will partially return to equilibrium at a rate that depends on T 1 and T (^) 2. The return to equilibrium also occurs in the collection phase. In contrast, one- dimensional NMR uses a single pulse, followed by free induction decay and data collection. The design of these experiments can be quite tricky, as it depends on what connections between spin states is being probed and what the relaxation times are. The representation of the result is done by carrying out a Fourier transform on both the t 1 and t 2 data sets and converting them into a two- dimensional plot of u 1 versus u 2 with the third dimension being the amplitude of any resonance peaks that are observed. A very simple illustration is given in Figure 6-7 for two interacting spin systems. Peaks that occur on the diago- nal have the same frequency in both dimensions and correspond to the one- dimensional spectrum, whereas those occurring off the diagonal represent cases where different spin systems interact during the mixing period (coher- ence transfer). Contours are usually used to indicate the amplitudes of the res- onance peaks, rather than a third dimension. The interactions (coherence transfer) between nuclear spins can be either homonuclear (same nuclei) or heteronuclear (different nuclei). One of the first two-dimensional experiments carried out was COSY (COrrelated SpectroscopY). This experiment identifies pairs of nuclei that are linked by scalar coupling (spin-spin splitting connectivities). The interaction between these nuclei occurs during the relaxation taking place in the mixing phase and results in cross-peaks in the spectrum. Scalar coupling (off diago- nal peaks) between nuclei means they are within three covalent bonds. A typical COSY spectrum is shown in Figure 6-8 for l cro repressor, a DNA binding protein that regulates phage development. One of the most important two-dimensional NMR experiments is Nuclear Overhauser Effect SpectroscopY (NOESY). This is a through space interac- tion that takes place because of the interactions between the magnetic dipoles of two nuclear spins. In terms of our previous discussion, this primarily involves the T 1 mode of relaxation and is coupled with rotational motion. The NOE is the NMR equivalent of fluorescence resonance energy transfer dis- cussed in Chapter 3. In fact, an NOE can be observed in one dimension. If a sufficiently large magnetic field is applied at the resonance condition of a given nucleus, the spin system becomes saturated, that is, the ground and excited
118 PRINCIPLES OF NUCLEAR MAGNETIC RESONANCE AND ELECTRON SPIN RESONANCE
One of the most useful multidimensional spectra is HSQC (Heteronuclear Single Quantum Correlation), an example of which is shown in Figure 6- for a protein involved in proton transport across a membrane (subunit c of ATP synthase from E. coli ). This spectrum uses both 1 H and 15 N and selec- tively detects only pairs of covalently attached nuclei. Each spot in the contour map represents such a pair, with the position on the horizontal axis repre- senting the proton resonance frequency and the position on the vertical axis representing the resonance frequency of the nitrogen nucleus. Every amino acid, except for proline, has a backbone amide so essentially every residue is represented in this spectrum. Side chains containing amides will give rise to additional resonances. This gives direct information about which nitrogen is coupled to which hydrogen. Furthermore, the resonances are usually quite well
PPM
PPM
7.2 6.8 6.
7.2 6.8 6.
Y
Y
Y
F
F
F
Figure 6-8. Example of a COSY spectrum for the protons of the aromatic spin systems in l cro repressor protein. Tyrosine rings are connected with a dashed line and pheny- lalanine rings by a solid line.The diagonal and cross-peaks can be easily seen. Reprinted in part with permission from P. L. Weber, D. F. Wemmer, and B. R. Reid, Biochemistry 24 , 4553 (1985). © 1985 by American Chemical Society.
MULTIDIMENSIONAL NMR 119
resolved. The HSQC spectrum provides information that is useful for assign- ing the observed resonances to specific amino acid residues. Extending NMR to dimensions greater than two involves similar concepts. Multiple pulses and even field gradients are used. These dimensions can be in terms of different nuclei and/or combining two-dimensional experiments. The most common nuclei studied in addition to 1 H are 13 C and 15 N. In these cases,
6.5 6.
PPM
PPM
6
7
6
8
8 9
Figure 6-9. Example of a NOESY spectrum for a complex of distamycin A and DNA. Aromatic C6H resonances of adenine and guanine and C2H resonances of adenine are shown along the vertical axis. The Cl¢H resonances are shown along the horizon- tal axis. Sets of sequential connectivities are denoted by dotted, dashed, and solid lines. Reprinted in part with permission from J. G. Pelton and D. F. Wemmer, Biochemistry 27 , 8088 (1988). © 1988 by American Chemical Society.
MAGNETIC RESONANCE IMAGING 121
MAGNETIC RESONANCE IMAGING
One of the most remarkable advances in diagnostic medicine has been the evolution of magnetic resonance imaging (MRI). Although x rays readily dis- tinguish hard objects, such as bones, they do not distinguish soft tissue struc- ture. MRI, on the other hand provides excellent images of tissues and is able to distinguish between various types of tissue. Protons in water are the primary nucleus used for detection although applications with 13 C, 31 P, and 19 F have been developed. The principle underlying MRI is the use of a magnetic field gradient. Since the resonance frequency is directly proportional to the mag- netic field, the frequency of the resonance will depend on its location in the magnetic field. The intensity of the absorption is dependent on how many protons are present. If a linear magnetic field gradient is applied, the position of the resonance will change, also linearly, as the field is varied. Thus a plot of the amplitude of the resonance versus frequency is equivalent to a plot of the integrated number of protons versus distance. A series of cross-sections can be obtained by rotating the sample in the field, or by moving the field around the sample. These cross-sections can then be reconstructed to give a three- dimensional image. In soft tissue, the amount of water varies for different tissues, so the density of protons varies. In addition, the various tissues are characterized by signifi- cantly different T 1 values. The difference in proton density can be shown in reconstructions by varying the darkness of the shading. This can be seen in Figure 6-11 where the MRI image of an adult human brain is shown. In addi- tion to being able to distinguish various soft tissues, MRI is noninvasive, as contrasted to x rays or injections of foreign substances, including radioactive isotopes, required for other imaging techniques.
Figure 6-11. MRI of an adult human brain. Courtesy of Professor Scott Huettel, Duke University Brain Imaging and Analysis Center. Reproduced with permission.
122 PRINCIPLES OF NUCLEAR MAGNETIC RESONANCE AND ELECTRON SPIN RESONANCE
ELECTRON SPIN RESONANCE
We now return briefly to ESR. Unlike NMR, ESR is not observed for most materials. This is because electrons are usually paired and consequently have no net magnetic moment. However, free radicals and other paramagnetic sub- stances have unpaired electrons that give rise to ESR spectra. Because the frequencies associated with transitions between the energy levels of unpaired electrons in a magnetic field are in the microwave region, special techniques are required for placement of the sample in the magnetic field. Conceptually the experiment is the same as for NMR. The magnetic field is varied until resonance is found. The usefulness of ESR in biological systems arises from the interactions between nuclear spins and the electron spin. This gives rise to “hyperfine struc- ture” in the spectrum. For example, if a neighboring nucleus has a nuclear spin of 1/2, it will have two orientations in the field and will split each of the two energy states of the electron into two, as shown in Figure 6-12. Because of quantum mechanical selection rules, only two transitions between these four energy states are allowed. If the nuclear spin is 1, six energy levels are created and three transitions between the levels are allowed. The hyperfine structure can provide information about the environment of the paramagnetic species.
H
Energy
mI = 1 / 2 mS = 1 / 2 mI = -^1 / 2
mI = -^1 / 2 mS = -^1 / 2 mI = 1 / 2
Figure 6-12. Hyperfine splitting of the energy levels of an electron in a magnetic field, H, by a nuclear spin. The electron has a spin quantum number, m (^) S, of ±1/2, and the nucleus has a spin quantum number, m (^) I, of ±1/2.
