4 SPECTROSCOPY: SOME PRELIMINARY CONSIDERATIONS EXAMPLE 1.2 The C=O bond in formaldehyde vi- certain nuclei to move with uniform periodic motion of a brates( stretches, then contracts)with a frequency of 5.13 x different type 10>Hz. (a)What frequency of radiation could be absorbed by this vibrating bond?(b) How much energy would each 1.4 UNCERTAINTY AND THE QUESTION OF photon deliver? (c) To which region of the electromagnetic TIME SCALE spectrum does this radiation belong?(d)Are photons of this region capable of breaking bonds? If you have ever tried to take a photograph of a moving object, you know that the shutter speed of the camera must be D Solution:(a)From Eq (1.5)we know the frequencies adjusted to avoid blurring the image. And, of course, the faster must match; therefore, vohoton =5.13 x 10 Hz. (b)From the object is moving, the shorter must be the exposure time to Eq.(1.3) "freeze"the motion. We have very similar considerations in photon=hv=(663x10-34Js(513×103s Suppose you owned a collection of very extraordinary chameleons that were able to change colors instantaneously =340×10-20J=205 k mol-l from white to black or black to white every I s. If you took a picture of them with a shutter speed of 10 s, each of the little (c)From Table 1. I we see that radiation of this frequency critters would appear to be gray. But if you decreased the and energy falls in the infrared region.(d)No. This amount of energy is less than half that required to break ones and white ones in roughly equal numbers but no gray even the weakest chemical bond. However, absorption of ones. Thus, to capture the individual colors, your exposure such a photon does create a vibrationally excited bond time must be significantly shorter than the lifetimes of the which is more likely to undergo certain chemical reactions pecies, in this case the l-s lifetime of each colored form an is the same bond in its ground state 口 There are many types of molecular chameleons, that is olecules that constantly undergo some sort of reversible At this point you might think that the frequency-matching reorganization of their structures. If absorption of the photon requirement places a heavy constraint on the types of absorp- is fast enough, we will detect both the"black"and"white" tion processes that can occur. After all, how many kinds of orms of the molecule. But if the absorption process is slower periodic motion can a particle have? The answer is that even than the interconversion, we will detect only some sort of a small molecule is constantly undergoing many types of periodic motion. Each of its bonds is constantly vibrating; the to the question: How long does it take for a particle to absorb molecule as a whole and some of its individual parts are a photon? Unfortunately, such a question is impossible to rotating in all three dimensions: the electrons are circulating answer with complete precision through their orbitals. And each of these processes has its own In 1927, w. Heisenberg, a pioneer of quantum mechanics. characteristic frequency and its own set of selection rules stated his uncertainty principle: There will al ways be a limit governing absorption! to the precision with which we can simultaneously determine All of the above forms of microscopic motion are what we the energy and time scale of an event. Stated mathematically, might describe as intrinsic. That is, the motion takes place all the product of the uncertainties of energy (AE)and time(an) by itself, without intervention by any external agent. How ever, it is possible under certain circumstances to induce particles to engage in additional forms of periodic motion △E△t≥h (16) Still, to achieve resonance, we need to match the frequency of this induced motion with that of the incident radiation([Eq (1.5)). Thus, if we know the energy of a given photon to a high order For example, an ion (or any charged particle, for that of precision, we would be unable to measure precisely how matter)follows a curved path as it moves through a magnetic long it takes for the photon to be absorbed. Nonetheless, there field. If we carefully adjust the strength of the magnetic field, is a useful generalization we can make Using Eq (1.3),we the ion will follow a perfectly circular path, with a charac- can substitute h Av for the AE in Eq(1.6), giving teristic fixed frequency that depends on its mass, charge velocity, and strength of the magnetic field. Matching this characteristic cyclotron frequency with incident electromag netic radiation of the same frequency can lead to absorption, and this is the basis of a technique known as ion cyclotron where Av is the uncertainty in frequency. As a result the time resonance(ICR)spectroscopy, We will discover in Chapter required for a photon to be absorbed(Ar) must be approxi 2 that a strong magnetic field can also be used to induce mately as long as it takes one "cycle"of the wave to pass the
REVIEW PROBLEMS(Answers in Appendix I particle. That length of time, fo in Figure I 1, is nothing more 2. Electromagnetic radiation is characterized by its fre- than 1/v. This result stands to reason if we consider that the quency(v) or wavelength(A), which are inversely particle would have to wait through at least one cycle before proportional [Eq.(1. 2)]. Radio-frequency radiati it could sense what the radiation frequency was. At least we sed in NMR spectroscopy typically has frequencies now have an order-of-magnitude idea of how fast our shutter on the order of 200-750 MHz speed must be in order to"freeze"a given molecular event 3. The energy of a photon(a quantum of radiation)is We will encounter the uncertainty principle at several points directly proportional to its frequency [Eq. (1.3)] along our voyage through NMR spectroscopy 4. For radiation to be absorbed by a particle, the fre- quency of the radiation must equal the frequency of the EXAMPLE 1.3 Suppose our NMR experiment re- particle's periodic motion quired the use of rf radiation with a frequency of 250 MHz to 5. The Heisenberg uncertainty principle [Eq.(1. 6)] de- examine formaldehyde(see Example 1. 2). will this NMR fines the time scale of radiation absorption event as experiment enable us to see the various individual lengths of inversely proportional to the radiation,s frequency eC=o bond as it vibrates, or will we detect only a time-av Processes that occur faster than the spectroscopic time eraged bond length? scale are time averaged during the absorption proce B Solution: The vibrational time scale(1/v=1/(5. 13 x 10 REVIEW PROBLEMS(Answers in Appendix 1) Hz)=1.9x 10-14 s)is much shorter(faster)than the NMR 1.1. The linear HCN molecule rotates around an imaginary time scale [I/v= 1/(2.5 x 108 Hz)=4 x 10-9sI axis through its center of mass and perpendicular to the Therefore, NMR can only detect a time-averaged C=o molecular axis. The frequency of this rotation is bond length 4.431598x 10 0 Hz.(a)What frequency of radiation could be absorbed by this rotating molecule?(b) To Equipped with this knowledge about electromagnetic ra- which region of the electromagnetic spectrum does diation, periodic motion, resonance, and time scale, we are such radiation belong? now ready to enter the intriguing world of the atomic nucleus. 1.2. When laser light with A= 1064 nm impinges on a CHAPTER SUMMARY sample of formaldehyde(Example 1. 2), most of the light is scattered elastically. But a small number of scattered photons emerge with A= 1301 nm.Account Nuclear magnetic resonance spectroscopy involves for this exact wavelength. (This is an example of Raman the interaction of certain nuclei with radio-frequency spectroscopy. (rf) electromagnetic radiation when the nuclei are im- 1.3. What is the shortest lifetime a species could have and mersed in a strong magnetic field still be detectable with visible light having 2= 500 nm?
MAGNETIC PROPERTIES OF NUCLEI 2. 1. THE STRUCTURE OF AN ATOM A=Z+N (2.1) The compounds we examine by nmr are composed of mole The isotope 2H is usually referred to as deuterium(D), or cules, which are themselves aggregates of atoms. Each atom heavy hydrogen, but most isotopes of other elements are has some number of negatively charged electrons whizzing identified simply by their mass number. The atomic mass around a tiny, dense bit of positively charged matter called the listed for each element in the periodic table is a weighted nucleus. The size of an atom is the volume of space that the average, the fractional abundance of each isotope times its electron cloud occupies. However, >99.9% of the mass of an exact mass, summed over all naturally occurring isotopes atom is concentrated in its nucleus, though the nucleus occu- pies only one trillionth(10-12)of the atom Is volume. Even the EXAMPLE 2.1 Tritium, H, is a radioactive(unstable nucleus can be further dissected into other fundamental par- isotope of hydrogen. What is the composition of its nucleus? ticles, including protons and neutrons, not to mention a host of other subnuclear particles that help hold the nucleus to O Solution: Since the atom is an isotope of hydrogen, Z gether and give nuclear physicists something to wonder about I. The mass number A is 3 and therefore, from Eq. (2.1), N=2. Thus, the nucleus consists of one proton and two 2.1.1 The Composition of the nucleus neutrons. It is the number of protons in an atoms nucleus(Z, the atomic EXAMPLE 2.2 Natural chlorine(Z= 17)is composed number) that determines both the atoms identity and the of two isotopes, 35Cl and 37Cl. The atomic mass listed for charge on its nucleus. In the periodic table of the elements chlorine in the periodic table is 35.5. (a)What is the compo- (Appendix 2)the atomic number of each element is shown to sition of each nucleus?(b) What is the natural abundance of the right of its chemical symbol. Every nucleus with just one each isotope?(You may assume for the purposes of this proton is a hydrogen nucleus, every nucleus with six protons question that the exact mass of each isotope is exactly equal is a carbon nucleus, and so on. Yet, if we carefully examine a to its mass number, though in general this is not the case. large sample of hydrogen atoms, we find that not all their nuclei are identical. It is true that all have just one proton, but 0 Solution:(a)Chlorine-35 has Z= 17(17 protons), A they differ in the number of neutrons. Most hydrogen atoms 35, and N= 18(18 neutrons) 3'CI has Z=17, A=37,and in nature(99.985%, to be exact) have no neutrons(N=O), but N=20.(b)Since the atomic mass of 35.5 is a weighted a small fraction(0.015%)have one neutron(N=I )in average of a mixture of CI and 37Cl, we can use a little to the proton. These two forms are the naturally oc algebra to calculate the fraction(f)of each isotope stable isotopes of hydrogen, and they are given the H andH, respectively. The leading superscript is the mass (5·35)+(/·37)=35.5 number(A)of the isotope, which is the integer sum of z and N: and since only the two isotopes are present
2.1, THE STRUCTURE OF AN ATOM 7 s+/37=1.00 the extemal field)or"down"(in the opposite direction to the external field). These two spin states are degenerate(i.e Therefore have the same energy) in the absence of an external magnetic field. Moreover, if all the electrons in an atom are paired (i.e (/5·35)+(1.00-/5)(37)=355 each orbital contains two electrons), all up spins are canceled by down spins, so the atom as a whole has zero magnetic 35/35-37/3=355-37 moment However, when unpaired electrons are immersed in an external magnetic field, the two states are no longer degener s=0.75(75%) and /37=0.25(25%) ate. An electron oriented opposite to the field(s=-y in Figure 2. 1)has lower energy(and greater stability)than an electron 2.1.2 Electron Spin oriented with the field(s =+2). It is the interconversion of these two spin states that is centrally important to the tech- Before we delve further into the properties of the nucleus, let nique known as electron paramagnetic resonance spectros us momentarily shift our attention back to one of the electrons copy( Chapter I 1). But for now, we return to the nucleus zooming around the nucleus. Just like photons, electrons exhibit both wave and particle properties. Each electron wave 2.1.3 Nuclear Spin in an atom is characterized by four quantum numbers. The first three of these numbers can be taken as the electrons The proton is a spinning charged(Z= 1) particle too, so it address and describe the energy, shape, and orientation of the should not surprise us to learn that it also exhibits a magnetic spin quantum number s, which can assume only two values, es, absence of an external magnetic field. To differentiate nuclear +or-.(Why #, was selected rather than, say, #I will be spin states from electronic spin states, we will adopt the described a little later. )The Pauli exclusion principle tells us convention of labeling nuclear spin states with the nuclear that no two electrons in an atom can have exactly the same set spin quantum number m. Thus, for a proton, m can assume of four quantum numbers. Therefore, if two electrons occupy values of only +i or-1. We describe such a nucleus as having the same orbital (and thus possess the same first three quan- a nuclear spin())of Because nuclear charge is the opposite tum numbers ), they must have different spin quantum num- of electron charge, a nucleus whose magnetic moment is bers. Therefore, no orbital can possess more than two aligned with the magnetic field (m =+2) has the lower energy electrons, and then only if their spins are paired (opposite) ( Figure 2.2).(If an isotope has a negative H, the lowest energy Is there any other significance to the spin quantum num- state is the one with the most negative m value. ber? Yes, indeed! Because the electron can be regarded as a Perhaps surprisingly, neutrons also exhibit a magnetic particle spinning on its axis, it has a property called spin moment and a nuclear spin of I=i, even though they are angular momentum. Further, because the electron is a uncharged. Therefore, they too can adopt two different orien- charged particle(Z=-1), this spinning gives rise to a mag. tations in a magnetic field. But because the sign of u for a netic moment(symbol u) represented by the boldface vector neutron is negative (Table 2.1). the more stable orientation arrows in Figure 2. 1, We describe such a species as having a corresponds to m= magnetic dipole. The two possible values of s correspond to So, we have established that 'H nuclei (i.e, protons)e the two possible orientations of the magnetic moment vector hibit two possible magnetic spin orientations. What about in an external magnetic field, up"(in the same direction as other isotopes? From Chapter I you might remember that b m=+1/2 m=-1/2 (higher energy) (lower energy) (lower energy) (higher energy) Figure 2. 1. Two possible orientations of the magnetic moment (H) Figure 2.2. Two possible orientations of the magnetic moment(u) of a spinning electron in an extemal magnetic field (B of a spinning proton in an external magnetic field (Bo)
8 MAGNETIC PROPERTIES OF NUCLEI TABLE 2.1 Magnetic Properties of Selected Particles" Isotope % N A Y 0 -183.2629.167-1913150 0.322 HHL 2675124257592792680 0015 2 141.06486.535660.857387273×103965×10-3 103.9616.5463.25603×10 0.293 BB℃ 19.58 328748457541.800774×1021.99×10 5567 11 85.82813.66026880 3.55×10 0.165 67264010.70540.7021990 1.59×10 9963 19325307560403471.6×10 1. 01 x 10 -271074.31420.282980 1.04×10-3 0037 1.8930 291×10-2 25166740412627270 0.833 70.76l11.262 2.216l 0.14 25×10 697061L.0943.6385 0.149 53.1428.45780554770 784×10-3 15 6.63×10 33s 0.7616 20.5173.26540.6425764×1022.26×10 26,2124.17170.8209|-7.89×10 4.70×10- CI 24.47 3.472 0.683 6.2l×10 271×103 "Abstracted in part from The 64th CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton FL. 1984 Magnetogyric ratio in units of 10radT-s- " Resonance frequency in megahertz in a l-Tfield. magnetic moment in nuclear magnetons ' Electric quadrupole moment in barns sEnsitivity (relative to proton)for equal numbers of nuclei at constant field: S=7.652 x 10-H(+Iyr. Zeeman found only certain isotopes give rise to multiple Group 2: Nuclei with Both Z and N Odd (and Therefore nuclear spin states when immersed in an external magnetic A Even). field. This is because only isotopes with an odd number of protons (odd Z)and/or an odd number of neutrons(odd N) Such nuclei must have an odd number of unpaired proton possess nonzero nuclear spin Nuclei with zero nuclear spin (I=))spins, and an odd number of unpaired neutron(=2 (those with an even Zandeven N) have zero nuclear magnetic spins, so the net magnetic spin must be a nonzero integer Ii.e moment and cannot be detected by nmr methods an integer multiple of 2)]. Such nuclei are detectable by Here is the reason that the parity( odd or even number)of NMR. A few common examples are 2H(I= 1), 0B(=3) protons and neutrons is so important: A proton spin can only N(=I), and 50V(1=6) pair with(cancel) another proton spin, but not a ne and vice versa. This rule allows us to assign every isotope to one of thr Group 3: Nuclei with Even Z and odd N, or Odd Z and Even N(n Either Case odd A). Group 1: Nuclei with Both Z and N even (and Therefore These nuclei must have an even number of proton spins(all A Even). paired) and an odd number of unpaired neutron spins, or vice In such nuclei all proton spins are paired and all neutron spins versa. Therefore, the net magnetic spin is an odd integer are paired, resulting in a net nuclear spin of zero(I=O). Such multiple of =, and these nuclei can be detected by NMR. Here nuclei are invisible to NMR. Some examples include the are some examples: H(=D),B(=D),C(=D,N abundant isotopes 2C, 6O, 80, and 32s (=0=吓F(=i(=3P(=分