Townes & Schawlow Chs 5,12

Townes & Schawlow Chs 5,12 - SPECTROSCOPY C.H.Townes 3...

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Unformatted text preview: SPECTROSCOPY C.H.Townes 3 A.L.Schawlow 114 MICROW’AVE SPECTROSCOPY TABLE 4-8. STRUCTURES OF ASYMMHTRIC ROTORS WHICH IIAVE BEEN OBTAINED FROM MICROWAVE SPECTRA (Concluded) Molecule Structure ltcfm'lmcc l H\ 3:“39' . 11—. us}. was“) my 7,, lid] 1.203 "a a i ‘h' . 3 ms 5?: 178 l . , ., ozs (son 5 "W. [em [5m] .‘e O [875] p m/ 03 O n °a9' advent of high-resolution microwave spectroscopy. A large number have now been worked out by microwave techniques, and it seems feasible to solve the rotational spectrum of any asymmetric rotor which does not have serious complications due to internal motions (of. Chap. 12) or an exceptionally complex hyperfine structure (of. Chap. 6). The Stark effect has been extremely valuable in identifying and working out this type 0f SDGCl-I‘Um (of. Chap. 10). The structures of asymnmtric rotors which have been obtained from microwave results are given in Table 4—8. CHAPTER 5 ATOMIC SPECTRA While most microwave spectra have their origin in absorption by molecules, certain types of atomic spectra may fall in the microwave region. Atomic theory is important even for molecular spectroscopy because it is often convenitnit to consider a molecule as being composed of atoms “’llUSB properties are not too greatly different from their proper- ties in the free state. Moreover, many molecular phenomena. are sufficiently analogous to phenomena in atoms so that it is worthwhile to study first the simpler atomic case. This chapter will present a summary of those parts of the theory of atomic spectra which are needed for microwave spectroscopy. More extensive treatments are given in the several books devoted to the subject (c.g., A. C. Candler |79], G. llerzberg [[24], L. Pauling and S. Goudsmit [23], H. E. “iliite [53], and for a more advanced, quantum—mecha-nical treatment, R. U. (I‘ondon and G. .H. Shortley [56]). 5—1. The Hydrogen Atom. The. simplest atom is that of hydrogen, consisting of a. single proton and an electron. It is described by the wave equation 8 Q , r We + L,“ (or e W = 0 or. in spherical coordinates, 10 ,at 1 are lift/.614!» 735 (r 01') + sin! 56¢? + 7'2 sin 6 <36 \sm 9 06 + 8—}? (W ~ w = 0 (5-1) where the nucleus, or more exactly the center of mass of the electron and nucleus, is taken as the origin of coordinates, and ,u = mill/(M + m.) is the reduced mass of the atom. W is the total energy of the atoma V = ~Z63/r is the potential energy, Z is the nuclear charge in units of the proton charge. and c is the proton charge. By EL process of separating the variables similar to that used for the dint-omit}. molecule (Chap. 1), the wave equation may be solved [62. giving 4’ : Ri")®(6l¢m(¢l (5'2) 115 116 MICROWAVE SPECTROSCOPY where 1 r . CIJ_ = —-— 6”“ (5—3) H ‘\/er (21+ no — |-m!)1 * , , = —. m , ~ ,4 GM: [ 2(£+ lm‘)! _ Pi (00:. 0) () ) _ [IL—.13 27’"?- t Aer'nug 2H1 EZT r i“ R”: — l [(-n + l)!]3n"a‘,‘ (no.0) (’ L”! man (1)“) and where n = l, 2, . . . is the principal quantum number I = U, 1, 2, . . . , (n — 1) is the orbital quantum number m : ——l, ml + 1, . . . , Z? '1, l is the magnetic quantum number and should not be confused with the same sym- bol used for the mass of the electron Pi“' = associated Legendre polynomial L35} = associated Laguerre polynomial on = h2/4r2pez is the radius of the tiret orbit of the hydrogen atom in the Bohr theory It may be observed that Eq. (5-1) for the hydrogen atom is exactly the same as the wave equation (1-11) for a diatomic molecule with the potential V(r) replacing the molecular potential (Tr). The hydrogen atom may, in fact, be regarded as a diatomic molecule with the proton and electron as the two atoms. The parts of the wave function (5-3) and (5-4) which depend on angle are identical with those for a diatomic or linear molecule (1-5) and (1-6). They are the some for any spherically symmetric potential, since these functions represent conservation of total angular momentum (Z or J) and of the projection of the angular momen- tum (m or ill) on a chosen axis. Unlike the diatomic molecule, the dependence of the potential on r in this atomic case is very simple and the radial wave function (5-5) can he determined. In more complex atoms, the potcnticl for a. single electron may often be considered spher- ically symmetric, so that the angular parts of the wave function are unchanged. However, the dependence of the potential on r is usually very difficult to determine, so that the radial wave function and energy cannot be exactly obtained. Figures 5-1, 5-2, and 5-3 show the radial and angular distribution of the electrons. The S electron wave function is seen to he the only one which does not vanish at the center of the nucleus. The s electron is also the only one which has a spherical charge distribution. The allowed energy levels for the hydrogen atom are given by 2 2 — L2“? = —Rhc a (5-6) nigh? n? W: where R : 21r3pc4/ch3 is the Rydberg constant in cm“. W is in ergs; t0 conx'ert to cm", divide by he. On this model of the hydrogen atom, ATOMIC srscrna 117 0 5 1000 0 5 10 i5'a —r—> FIG. :5-1. The radial part of the hydroe‘ function of the distance between the on electronic wave function Rm: plotted as a electron and the nucleus. 15 25 m L_ J— O 1 ‘6‘" r ‘4 _l_ l 900 O 6 12 18 a0 2p L41: 4p 70 2 4 6 800 O 6 12 1890 O 8 16 24 3200 1500 O 9 18 2300 O 12 24 36 4800 _I_J ‘ F0“ 7l0 20 3300 O 15 30 4500 O 12 24 36 48 60 7200 I'm. .‘i—J. the electronic density distribution HINHJF - 1-H I’lflllf‘d :15 ll. fimc- .c . (flé‘Cf-rournucleal' distance for Sllx'o Um 0f the rui status of the hvdrom‘n atom Tl ' ‘ 7‘ a ‘ I _ ‘ r _ h . . ie ordinate shin: thL piohnlnhty of finding the electron between spherical shells of radii r and r , (T. 118 MICROWAVE SPECTROSCOPY which neglects electron spin and relativistic effects, the energy is inde- pendent of l and m, and depends only on the principal quantum number it. Since the angular dependence of the wave functions is the same as for the diatomic molecule, the selection rules and intensity relations which depend on angular momentum are identical. As in Chap. 1, it can be shown that transitions occur between energy levels such that l’=li1 and m’:m,mi1 A state with l = 0 is called an S state; states for which I a 1, 2, 3, . . . are, respectively, P, D, F, G, H, . . . states. a? 3 ll?" :7 "efecirens E 3);, 9 _ '69— electrons I=O m=0 1:1 m=+1 m:0 m=—I 99 ' l d l 7 I —‘ k _ electrons l l I=2 m=+2 m=+i m=0 m=-1 m:—2 l $0 I f ‘ I "' electrons [=3 m=+3 m=+2 1713+: m=0 m=—l m=—2 =-3 FIG. 5—3. The probability- density distribution factor (4)2 plotted as a function of 9 for s, p, 05, f electrons. For states with m = 0 the scale is approximately 1(3- + 1) times that. of the other states having the same value of 1. (After White [53].) 5-2. Atoms with More Than One Electron. While a wave equation can be written for many-electron atoms, it is not possible to solve it exactly. To a good approximation, however, each electron may be considered to move in a spherically symmetrical field produced by the nucleus and all the other electrons. Then the angular part of the solution will be just the same as for the hydrogen atom, and the electron can be characterized by quantum numbers I and m- as before. The radial part of the wave function can only be approximated. Some useful methods of making the approximation and obtaining numerical or sometimes analytical wave functions have been discussed by Hartree, Fock, Fermi, Thomas, Slater, and others {305}. Particularly simple are the alkali—type atoms which have one electron outside a spherically symmetrical core of “closed electron shells.” As ATOMIC SPECTRA 119 long as the valence electron stays out beyond all other electrons in the atom, it moves in a Coulomb field with Zen = 1. To this extent the wave functions and energy levels resemble those of the hydrogen atom, and each state is characterized by quantum numbers n, l, m. The energy is still independent of m and for large values of 1 does not depend greatly on I. But for small values of I, especially for 8 states where l = 0, the wave functions (Figs. 5-1 and 5—2) show a relatively large probability of the electron’s being near the nucleus. In these states, the electron —55—5p——_50’—5f——59 0 ~45 —4p—- qd—M 10,000 10000 —35 —3p—- 30' ' 20,000 zopoo 30.000 —2s —-—2; 30,000 _2" 7 40.000 40,000 E E —23 U U a; 50,000 — 50,000 a 3 C} _ ’ so 000 3 g » E 60,000 .2 r, 70,000 '— r0000 30.000 80,000 90,000 90,000 100,000 100,000 110,000 13 110,000 1" LF FIG. 5-4. Energy levels of hydrogen and an alkali atom. The term value, or energy, of the atom is referred to a zero which represents the energy after the valence electron has been completely removed or the atom ionized. spends considerable time inside the closed electron shells, in a region of larger 2,“, so that its binding energy is increased. Figure 5-4 shows the energy levels of a hydrogen atom and those of an alkali atom. The energies for levels in a complex atom are often given by a formula analogous to Eq. (5-6) for the hydrogenic case. However, since Eq. (5-6) no longer applies exactly, it can be made to agree with the observed energies only by modifying the value of n or the value of Z, or both. Thus for the first case, _ Hthfi _ i RthEL W = n*2 _ (n — a}? (5-7) 120 MICROWAVE. SPECTROSCOPY where R = Rydberg constant _ Z5, = effective nuclear charge in the outer region of the atom. Zo = 1 for alkali atoms, but may be 2, 3, 4, . . . for ionized alkalilike atoms such as Be+, B”, Cttt, etc. it effective principal quantum number; not an integer a principal quantum number, and is an integer a' : quantum defect From Eq. 5—7) it is seen that :k I. *=n—a' (0—8) fl If n. is retained and Z is modified, the term value or energy depends on an effective Z, Zn”, such that ' __ li’nchZE,f (54)) W = n, Equations (5-7) and (5—9) are written down by purely formal aniilligy with the expression which applies to a hydrogenic atom, Since n in Eq. (5-7) or Z,” in Eq. (5-9) are not integers but are empirical constants. these equations are useful only when the same constant can be used for several purposes. Equation (5-?) is quite useful for describing the ten: values of alkalilike atoms because the quantum defect, or : n i n , varies only slowly with n. and 1. Thus if 71* is known for one level, its value, and hence the term value for another level, can be found at least approximately. ' I An effective value of Z is often used in connection \vith atomic fine structure, or for the energy levels of dilferent atoms in an isoelectronic sequence. it is then sometimes convenient to express Z.“ in terms of ZN,” and anr, which are effective nuclear charges in the inner and outer region of the atom. 7 5-3. Fine Structure, Electron Spin, and the Vector Model. When spectral lines are examined with instruments of moderate resolving power, thev are found to have a structure. 219., to consist of several components. This “fine structure" is explained by attributing to the electron a spin angular momentum and a magnetic moment. The angular momentum is related to a spin quantum number s, and its magnitude is given by in units h.,/'2-7r. For a single electron .9 always has the value 57‘ This 5' should not be confused with the same letter used to denote state with t : 0. The corresponding spin magnetic moment is l + (cg/210, in units of the. Bohr magneton* lieflarmr, where m is the electron mass. The fine structure constant a 27H“?th and approximately equals T37. A further quantum number no : i7} is rerouted to describe the projec- MTl‘fl' “alll' niagncton is usually taken us a posilive quantity even though the nicetron (barge is negative. ir'gurdlt'ss of this convention, it should be remembered “mi “1“ "lf‘t'lron magnetic morncnt is opposite to the direction 0! its spin. ATOMIC SPECTRA 121 tion of s on some fixed direction, just as 7m gives the projection of the orbital angular momentum on a fixed direction. The electron’s orbital angular momentum and spin angular momentum are vectors and will therefore add in some way similar to the ordinary rules of vector addition to form a resultant denoted by j. The magnitude of j is also quantized. It is specified by the total—angular-momcntum quantum numborj which has the valuesj = l + s andj : ll — 8' (when .9 = é). The corresponding vector equation is l=1+s ' (5.10) where the two possible values of j correspond to l and 5 being nearly parallel or nearly antiparallel. The magnitudes of j, 1, and s are, respectively, m, +71), and Vets- Howcver, sometimes equations are written as if the magnitudes were 3', l, and s. This convention is frequently employed because of its conciseness (cf. [23]). Using it, one may speak of 1 and 5 being parallel or antiparallel for then the magnitude of their resultant would just be I + s or ll fl The method leads to equations which must finally be corrected by replacing 3'2 by j(j + 1), l2 by I“ + l), and 32 by 3(3 + 1). The two orientations of 8 relative to I produce levels with slightly different energies. They are distinguished by appending the value of j as a subscript to the term symbol, for example, Si', Pi, Pg, 1),, I);. Since there are in general two possible orientations of 5 relative to l, for any given 1 there are two possible j values. The multiplicity of the term is therefore said to be 2, and this is denoted by a superscript 2 to the left of the term symbol. The doublet symbol is used even for 8 states where only one value of j is possible. Some typical states in a one-electron (hydrogenic or alkalilike) spectrum are 98;, 210;, 31);, QDé, EDS, . . . . The two levels 2P; and 2P; form a fine—structure doublet term. The splitting between the two levels with different values of j which occurs when l is not zero is caused primarily by the magnetic interaction between the electron spin and electron orbital magnetic moments. This interaction can be derived from the magnitude of the magnetic field at the electron caused by the relative motion of the nucleus with respect to the electron. The splitting, however, is reduced by another contribu— tion to the energy exactly half as large and in the opposite direction which is connected with a precession of the electron axis due to relativistic effects [7]. When both are taken into aceount, the energy level for a hydrogenic atom with nuclear charge Z is displaced by an amount 22 W‘_l ehZ (1 _ 4r2m3c2 _ SI COS (5,1) (5-11) GHAP'J‘ER 12 THE AMRIONIA SPECTRUIVI AND IIINDERED h'IOTIONS 12—1. Introduction. Because of the intensity and richness of its spectrum. :uninonin has played 3, major role in the development.- of micro— wave speetroneopy. It has provided :1. lurge number of msin observable lilittr-i on whieh to try both experimental tei.-.l1nirgue>; and theory. NHa also provides the Simplest and most thoroughly works-d out. (txnmple of .‘L clans of Spt‘t'll‘fl which will continue to om-upy and puzzle inirirmrave spawhti'oscopists for many years The moat. important hindered motions all involve the quantum- Him-hanit-ul tunneling eil'eet. That- is. the“! are mot-ions Wllit'll (:unnot oreur in chin: cal mechanics because of energy («insidomtiona but are allowed by the wave nature of quantum meehnnim. For example. in the ground vilimtionnl state of Nlla, the molei'ule does not. have enough energy to allow the. nitri'ig‘en to be found in the plune oi the hydrogens because of the large pot-oiu-ial-euerg}: hump in this pmiiion. as indicated in Fig: lB-l. its nl‘tunl penetration of the. plnnn ol' the liyclroueus and rapid vihrntion from one bide of it. to the other is (filled the tunnel ell'eet heunwe, if the nitrogen cannot (limb lll(:}')ill'(‘11i'-l£1llllll. it. mi.th "tunnel " in Ol‘th‘I‘ to get through. In principle, the .\'ll;‘ inversion is n vilirutionnl motion. Although vibrations normnlly give frequencies in the iufl‘z'u‘m'i region. the .\f ll. invernion so much slowed down by the hindering: poten- tinl thnt its frequent-y lies in themicrowaveregion. 'l‘he(lunlitntivei‘liniige from ordinary viln‘ntionnl levels where there is no hump in the potential to spectra involving; hindered n'iotions. vilu‘utionul levels owurring,r in pairs when the. hindering liunip orctll‘s‘ iF< diwiiussed in (hip. When the. Vibrntionnl enemy in so grout that the NH; moles-tile has Hilflltf'lt‘lll'. energy to invert ('lfl:~1>lt‘:!ll_\’1 the vilarntionnl levels no longer ()(‘tilll‘ in Close palm but. tl-l't.‘ similar to more normal equally Hpaeed \‘il.i1‘e.i.tit.innl levels. lloii'ever. tlngv are Mill markedly inlluent'cd by the preHenee of the potential hump. and hem-e in this (use of no “tunneling.” one may still spent; of some hindering; of the ITlOilOl'l. .Xnut'llm' type of liind<_-i'e(l motion is the. rotntion of one part of a. mole- cule with rr-spm‘d. to the remnimler. ’l‘hun the hydrogen nltnvhed to oxym'iu in (."ll,,i Jli (Fig. 12-2) has three pwwihle ponitions of equal energy lint must tunnel through n. 1)::t':‘1‘.ti;il hump l-vtwwu‘: them :‘is innlirntod in Fig. 12-3. llL‘IlLiC the rotation (if this hydrogen LLJ‘UUIlti Ll'n.‘ C l) liond Ron THE AMMONIA SPECTRITM AND HENDERJ‘JD MOTIONS 301 is hindered. Similarly, the methyl group in CHI/T‘s (Fig. 12-4) has three positions of equal energy separated by potential humps. Usually hindered motions do involve two or more positions of equal energy. If these positions are only Ininimn, which are not equal in energy. many of O «0.000 T E 5 —20,000 3 a; E, .73 E 450000 *5 D. 440,000 i l '50‘0003 a t 0 1 2 3 S ——3' i (a) __. 40,000 E § 2 745,000 ., E I? 7500001 . s_> (bl Fro. 12-1. Potential curve of NHL The variable 3 is a measure of the distance between the nitrogen and the plane of the hydrogens. (b) shows the. lower part of the. potential curve in more detail, and the. energy levels. the characteristic properties and interesting features of the hindered motions Considered here do not occur. When hindered motions have frequencies in or near the microwave region. the symctrn are oft-en quite diliieult to interpret l.)(‘f‘2111Fi(2 the num— ber of lines falling in the, rniz-rou'ave region is greatly inereahed. and he muse there in no exact solution for the energy levels in these cases. In 304 MICROWAVE SPECTROSCOPY barrier in terms of the frequency of inversion. This rate of penetration may also be calculated from the “tunnel effect.” Consider now the nitrogen striking the barrier at s = —so from the left. Its wave function will partly penetrate the barrier and extend over into the right-hand potential minimum. The amount of penetration can be roughly evaluated by examining Schrodinger’s equation in the classically forbidden region. dhp_2u E h, (V — WM» = 0 (12-5) An approximate solution to (125) is a = exp [a % f more) — News: (12—6) if V does not vary too rapidly with 5'. Since V — W is positive in the classically forbidden region, this corresponds to an exponentially decreas- ing function. which has unity value at the boundary where the nitrogen strikes (the increasing exponential which is also a solution is omitted, since it would give the N the largest probability of being found on the right). The amplitude of the wave function which penetrates the boundary is then abs-n. = exp {— [2ttv — W)]ids} —80 u = exp [— if)" [2M s words} = 313 (12-7) Now in a time t which is large enough for the nitrogen to oscillate on the left side many times, but small compared with the time required for inversion, the nitrogen will strike the left side of the potential barrier vat times, where :20 is the vibrational frequency. Each time it may be considered to partially penetrate and transmit an amplitude 1/ A? to the right side. The transmitted amplitudes add up to give a total amplitude after time t of pot/A2, so that the probability of penetration in time t is (yet/A2)? This is similar to, but differs slightly from, the usual expres- sion for radioactive decay or other types of barrier penetration by tunneling ([305], p. 22), because the transmitted wave on the right-hand side of the barrier is trapped in an identical potential minimum and is added to by successive transmitted waves. Equating the amplitude pot/A2 to the amplitude for in; in (12—4), we have rot V 1rvt 2 s or V : —U «A2 A2 (12-8) which is the result obtained by Dennison and Uhlenbeck. In the ground state Of N113, vfivo 9: 1200, so that. :19 ’m' 400 or c". THE AMMONIA SPECTRUM AND HINDERED MOTIONS 305 Because A.2 is large and varies exponentially with air or (V — WV, changes in either of these quantities can very drastically affect the inversion freQLIency v. For example, if the reduced mass a is increased by a factor of 2, such as would be roughly done by changing from NH3 to IlD3, v decreases by efi‘ifi—l) or a factor of 1]. The inversion frequency is cor- respondingly sensitive to the potential, and most molecules have such high potentials and heavy masses that the inversion frequencies are less than 1 cycle/sec. hIany molecules invert so slowly that they have not succeeded in inverting during the few billion years’ life of our planet. N113 in the first excited level of this vibrational mode has a much higher inversion frequency because of the increased value of W. Dennison and Uhlenbeck L33] were able to assume a simple shape for the potential barrier and obtain values for the inversion frequencies in the ground and excited states which agreed roughly with experimental observations. The exact form of the potential barrier assumed is not critical, since A depends only on an integral of this energy, and not on details of shape. However, hianning [61] found a potential function which has the general shape to be expected for NHa, and for which the wave equation could be more easily solved and accurate values for the energies obtained. Blan- ning’s potential function is V . _ _ —— = 6(3,ool seeh4 he i_ 29 Ct: 109,619 sech2 2;: (12—9) where V/hc = potential in units of cm—1 3: = a coordinate which is dependent on the distance of the nitrogen from the plane of the hydrogens 6.98 X libs/pi Where ,u is the reduced mass in atomic mass units This potential is zero for large 5 (or r), is symmetric about 3 = 0, and has a peak at s : 0, where the potential is 743,068 cm—l. It has two minima where the potential is — 45,140 cm‘l. rl‘he constants in (12-9) are also chosen to give close to the correct equilibrium configuration for the molecule, the vibration frequency, and the correct inversion frequency for one level. If the distance between hydrogens is assumed to stay constant during the motion so that they move together as a rigid triangle, then .r is taken as equal to s, the distance between the nitrogen and the plane of this triangle. If m is the mass of the hydrogens and flI that of nitrogen, then the reduced mass for this case is simply ,u = BrmM/(Sm —}— M). However, probably a better approximation to the motion is to assume the N7H distance remains constant, and only the HiNiH angles change with vibration [110]. In this case a: is taken as the distance along an are moved by the hydrogens from the median plane, i.e., a plane through the nitrogen and perpendicular to the molecular axis. If the angle between ,0 306 MICROWAVE SPECTROSCOPY this plane and the NfiH bond is taken as a, the proper reduced mass can be shown to be ,u = 377101! + 3m sin2 a)/’(3m + .11). This reduced mass varies only a small amount with 0:, and its value may be assumed to be that at the equilibrium angle of a0 = 21°49. Different reduced masses in hianning’s potential will give different equilibrium heights for the NH;; pyramid. I'loweverJ these heights do not deviate significantly from the observed value. The values of inversion frequency for various isotopic species of ammonia calculated from Manning’s potential are shown in Table 12-1 TABLE i2-1. vaEnsioN Momennems or AMMONIA IN MEGACYCLEs The “first excited vibrational state" (a; = 1) corresponds to approximately 050 cm—1 excitation of the vibration of the nitrogen against the plane of the. liydrogens. The calculated values are given by Manning [61] and Newton and Thomas [306] except for the calculation of NUD; with the Manning potential, which was done by A. Javan and J. Lutspeich. ————._*__—_—Jfi* ll“ 9 ext-‘t d (iI‘UllIld state . mi \ I P Vlbralmnnl state. Nun. Nuna mun Nun. Nuin Observed value, Mc. . . , _ . . . . . . , , 23,780 1000 22,705 1.095.000 Calculated from Manning poten— tial . . . . . . . . . . . . . . . . . . . . . . . . . 25.000 1250 Calculated from Newton-Thomas potential..................... 1 17 , 000 . . . . . . 780 . 000 83 _, 000 23 , 800 690, 000 and compared with experimental results. The potential constants have been chosen to fit some of the data, including the energy of the first vibrational level. The excited vil.)rat-ional levels are fitted rather satis- factorily by this potential [61]. Although the inversion frequencies are given approximately correctly, it does not seem possible to obtain a really satisfactory fit of all the data with a potential of D'Iaiining’s type. This will be seen again when fine structure of the inversion spectrum is discussed. Newton and Thomas [306] used a potential of the form V i [[(0.377)2 — 82 2 i i - -— — ” 4 . —1 _ he i 0.536 + i 1i X 3'1“ X 10 L111 (12 10) Where 3 is measured in angstrom unit-s. Here the hydrogens are con— Sidured to move as a rigid triangle. Values for some of the inversion ff‘3‘ltleiicies calculated by an approximate method [3013] from this poten— tial are given in Table 13-1. Form (12-10) for the potential appears to SW0 results which are comparable in accuracy with those from the, pomp tial (12-9}, THE AMMONIA SPECTRUM AND HINDERED MOTIONS 307 12—3. Inversion of Other Symmetric Hydrides. A very simple form of potential has been found by Costain and Sutherland [678] to give fairly accurate values for the inversion frequency of N113, and has been used to estimate the inversion frequencies of RH; and AsH3. This is a poten- tial of the form V = Skim)? + Sic-AM)? (12—11) where Ar = change in the N—II bond length A6 : change in the H- --N- H bond angle The force constants k, and 15.5, as well as the ratio of Ar to A8, can be obtained by a normal coordinate treatment of the observed vibrational frequencies of N113. Their evaluation gives V = 3.89 X 10“(A6)2 cm—1 (12—12) where 5 is measured in radians. This gives a potential hill of 2077 cm—I between the two minima in good agreement with Manning’s value of 2072 cm”. In fact, the entire potential curve given by (12-12) is very close to that given by (12-9). A similar evaluation of the potential hill from vibrational constants and molecular geometry for PH3 and Asli3 gives the following results [678]. For PHa: V = 5.3 X 104035)” cnr1 (12—13) Height of potential hill abdve minimum = 6085 end—1 Inversion frequencies: Ground state = 0.14 Me First excited state 2 7.2 hie For Asllaz V = 41.55 X 1.040382 cm—1 (12—14) 50 = 0.585 radian Height of potential hill above minimum Inversion frequencies: Ground state : 1/2 cycle/year First excited state = 1 cycle/day 11,220 em—1 Thns the inversion splitting of 13113 is probably large enough to give a splitting in microwave measurements of the rotational spectrum, whereas Asl—Iz would take two years to go through a cycle of inversion and should hence have no observable splitting. These numbers illustrate the very rapid variation in frequency as the potential barrier height 13 changed. . 12-4. Fine Structure of the Ammonia Inversion Spectrum Rotation- Vibration Interactions. Discussion of the inversion spectrum has so far ...
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Townes & Schawlow Chs 5,12 - SPECTROSCOPY C.H.Townes 3...

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