2.5 RELAXATION MECHANISMS AND CORRELATION TIMES I Bo Figure 2. 14. Laboratory frame diagram of"effective"spin-spin relaxation. Here, M is not shown the lattice nucleus (if the target nucleus drops to a lower tion time can be equated to the time required for a molecule energy level)or energy will be absorbed from the lattice to move a distance equal to one molecular diameter In both nucleus(if the target nucleus is promoted to a higher energy cases t is an average measure of how long the two nuclear level). These exchanges continue at a rate governed by Ti until magnetic dipoles remain in the appropriate relative orienta- equilibrium is reestablished tion to interact. Furthermore, it can be shown that The above dipole-dipole mechanism for spin-lattice re- laxation depends on the interaction of the target nucleus with the magnetic field B, of a lattice nucleus with magnetic IVOt ude of B, is governed by the equa (3cos2-1) where vo is the precessional frequency of the target nucleus (2.11) This equations tells us that for very fast molecular motion (i.e when I/t > 2vo), I/T, is proportional to t(T, is inversely where e is the angle between the external field Bo and a line (molecular motion slows), relaxation time decreases(the rate of length r connecting the two nuclei. This equation shows of relaxation increases). Conversely, for slow molecular mo- that the effectiveness of spin-lattice relaxation(as measured by how short T, is)is increased by the lattice nucleus having tion(i.e. when 1/t, < 2vo), T is directly proportional to a large u and being as close to the target nucleus as possible te: they both increase together. The minimum in T,(ca 10-3 to(i. e, in the same molecule or in high concentration in the s), and hence the most efficient spin-lattice relaxation, occurs bulk medium) when t=(2ivo-I In addition to the direct interaction of magnetic dipol As we will see later, the magnitudes of relaxation and spin-lattice relaxation can proceed by way of interactions correlation times are influenced by many factors, such as between the magnetic dipole of the target nucleus and fluctu- temperature, viscosity of the medium, and size of the mole. ating electric fields in the lattice. This is why neighboring cules involved. For example, in crystalline solids where all quadrupolar nuclei(those with 1>2, Section 2. 1)can bring translational and rotational motion has ceased, T values are about very efficient spin-lattice relaxation(short T values). exceptionally large while T2 values are exceptionally small As mentioned above, the frequency of rotational or trans- A major problem that results from inefficient spin-lattice lational motion of magnetic and electric fields in the lattice relaxation is that the target nuclei are much more easily nuclei is critical to the effectiveness of spin-lattice relaxation saturated(Section 2.3), making it difficult to obtain the de- It cannot be too fast or too slow. It is common to express the sired NMR signal. And, as we will see in Section 3.5,highly frequencies of these types of molecular motion in terms of a efficient spin-spin relaxation gives rise to very broad signal so-called correlation time te If the angular rotation fre- peaks. On the other hand, by measuring relaxation times and quency is o(in radians per second ), the rotational correlation correlation times, we are able to obtain detailed information time is 1/, the time required for a molecule (or part of a about how the giant molecules(e.g, polymers and proteins) molecule)to rotate I rad. Similarly, the translational correla tually move(see Chapter 14)
20 MAGNETIC PROPERTIES OF NUCLEI CHAPTER SUMMARY cally aligned along(for example)the x axis, and net nuclear magnetization M lies in the r,, a' plane. No 1. The nucleus of an atom consists of a number(2) of recession around Bo appears in the rotating frame protons and a number (N) of neutrons. The atomic Il. Spin-spin relaxation can be accomplished either by number Z determines the identity of the nucleus, while mutual energy exchange between two target nuclei or the sum Z +N determines the mass number(a)of the by inhomogeneities in the local magnetic fields. In nucleu either case, the relaxation is entropy driven and in- 2. Isotopes of a given element have the same value of Z volves no net change in energy of the system of target but different values of N and A spins. Spin-lattice relaxation involves energy ex 3. Nuclear spin(n)is a property characteristic of change between a target nuclear spin and fluctuating isotope and is a function of the parity of Z and N. magnetic or electric fields in the lattice(the collection values of I can only be zero, n(an integer), or n/2 of neighboring nonidentical magnetic nuclei). The (where n is an odd integer). Only if1+0 can the isotope efficiencies of both types of relaxation depend criti be studied by NMR methods. The most frequently ally on the similarity of the oscillation frequency (or studied nuclei are those with /=i(e. g, H and c) correlation time)of the interacting nucleus compared 4. Each isotope with I*0 has a characteristic magne to the precessional frequency of the target nucleus togyric ratio (Y, Table 2. 1)that determines the fre- quency of its precession in a magnetic field of strength ADDITIONAL RESOURCES Bo[Eq (2.6)]. It is this frequency that must be matched by the incident electromagnetic radiation(actually, the 1. There is an excellent computer tutorial entitled The oscillating magnetic field B)) for absorption to occur. Basics of NMR Spectroscopy, written by Joseph P. 5. When a collection of nuclei with 1# 0 is immersed in Hornak and available through him at the Department a strong magnetic field, the nuclei distribute them- of Chemistry, Rochester Institute of Technology, Rochester. NY 14623. This software uses realistic selves among 2/+I spin states (orientations), each with its own value of magnetic spin quantum number graphic animations to show such processes as spin m The quantity 21+ I is called the multiplicity of spin equilibration, absorption, and relaxation states. Nuclei in each spin state precess at the same 2. Becker, E. D. High Resolution NMR, 2nd ed, Aca- frequency demic. New York, 1980 6. The energy of the ith nuclear spin state is given by Eq REVIEW PROBLEMS(Answers in Appendix 1) 7. The relative population of each spin state is deter- 2. 1. How many protons and neutrons are there in a B mined by the Boltzmann distribution, Eq (2.8). Under nucleus? Hint: B has atomic number 5 conditions of a typical NMR experiment the rato or 2.2. To which of the three groups of nuclei does " B belong? spin state populations is near unity, differing only by Without looking at Table 2.1, what can you say about the I value for 0B? 8. If the two(or more)spin state populations become 23. Using the data in Table 2.1, determine the spin multi- equal, the system is said to be saturated and no net plicity of B and all possible values of m Then draw bsorption can occur. a diagram(resembling Figure 2.3)showing how the 9. After B, is turned off, nuclei can change their nuclear energy of each spin state of B varies with Bo spin orientations through two types of relaxation proc- 2.4. Calculate the precessional frequency of IoB in a5.87-T esses. Spin-lattice(longitudinal)relaxation(governed by relaxation time Ti)involves the return of the nuclei magnetic field to a Boltzmann distribution Spin-spin(transverse) 2.5. Calculate the energy gap between adjacent spin states relaxation(governed by relaxation time T2 or T2)in of B in a 5.87 T magnetic field. volves the dephasing of the bundled nuclear spins. 2.6. Calculate the population ratio of the m=-3 to the m Normally T:<T2<T +3 states of B in a 5.87-Tmagnetic field at 25oC 10.The rotating frame(of reference)is a Cartesian coor- 2.7. Suppose we are studying H nuclei at a field strength of dinate system where the x and y axes( designated x 5.87-T. Assume that the oscillating field B, is only 10-5 and y,)rotate around the z(z)axis at the precessional as strong as Bo.(a) How fast will M precess around frequency of the target nuclei. The rotating frame is B. (b) How much time is required for M to precess one drawn as it would appear to an observer precessing at full revolution around B ?(c)Calculate the flip angle the same frequency. In the rotating frame B, is stati- a at the following times after irradiation with B, begins
2.5 RELAXATION MECHANISMS AND CORRELATION TIMES 21 0. 0.10, and 0. 20 ms(I ms 10-3s)(d)Draw a magnitude(length)of the M vector, compared to its rotating-frame diagram that shows M at each of the initial equilibrium magnitude? (b)Use a rotating-frame above times. (e)Suppose B, is turned off 0. 20 ms after diagram to show what happens to M after B, is turned it was turned on. Describe what happens to M in terms off of T and T? using a rotating-frame diagram 2.9. Explain what effect dissolved oxygen(O,) might have 2.8. A collection of H nuclei is irradiated by B, to give a on longitudinal relaxation of H nuclei. Hint: The oxy Aip angle of 60, during which about a fourth of the gen molecule has two unpaired electrons with the same excess up spins absorb photons and flip to down spins Then B is turned off. (a)At this point, what is the
3 OBTAINING AN NMR SPECTRUM 3. 1 ELECTRICITY AND MAGNETISM Suppose that instead of a bar magnet we use sional motion of M(the net nuclear magnetizatic From a physics course in your past you are probably aware Section 2. 3)to induce an oscillating current li.e..a hat there is an intimate connection between electricity and quency alternating-current(ac)signal) in a coil of wire.We magnetism. Let us review a few of the relevant physical will orient the coil so that its axis lies anywhere in the x,y plane, for example, along the y axis and perpendicular to Bo, as in Figure 3.3a. Henceforth. any time there is a compo- 3.1.1 Faraday Induction in the Receiver Coil nent of M oscillating in the r, y plane(M Figure 2.7D), an alternating current of the same frequency will be induced in when a steady direct current of electricity(electrons) passes the coil's circuitry. We will label this loop the receiver coil through a loop of wire [by attaching the ends of the wire to a for it is here that the NMR signal is generated battery or other source of direct-current(dc)voltage], a steady magnetic field is established along the axis of the loop(a line perpendicular to the loop and through its center); see Figure 3.1. The higher the current (amperes), the greater is the strength of the resulting magnetic field. The field is also strengthened by using a coil of wire made up of several loops or by coiling the wire around an iron bar. These principles are used in the construction of all electromagnets and supercon ducting magnets (Section 3.2) If the direction of current flow in the coil is reversed, so is the direction of the magnetic field. And if the current is oscillating (i.e, alternating current). the resulting linearly polarized magnetic field will oscillate(change directions)at the same frequency(Figure 3.2) Now let us take the same loop of wire and replace the voltage source with an ammeter. Initially, of course, no cu rent registers in the ammeter. However, if we pass a bar magnet down into the coil, the needle of the ammeter deflects indicating a current as long as the bar magnet is moving Change the direction of the magnet's movement and you will see that the direction of the current reverses. This effect is Figure 3. 1. Magnetic field B along the axis of a loop of wire called Faraday induction: A current is induced in the wire carrying direct current. The dotted lines indicate a few of the lines by the movement of the magnetic field near the wire of magnetic nux
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