3 Molecules in Electric and Magnetic Fields


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1 Chapte, page Molecules in Electic and Magnetic Fields. Basic Equations fom Electodynamics The basis of the desciption of the behaviou of molecules in electic and magnetic fields ae the mateial equations of Maxwell s theoy. The equation fo the magnetic induction B is B μ (H + M ) μ μ H μ ( + χ) H. (.) μ 4π 7 V s m (H m ) μ is the pemeability of vacuum, μ is the elative pemeability, which is nomally taken as a scala to descibe the isotopic popeties of a medium. χ is the dimensionless magnetic susceptibility. Magnetization M in equ. (.) has the dimensions of a magnetic field stength, and adds to the magnetic field stength H. In some textbooks you may find the magnetic polaisation J defined in analogy to electic polaisation P. It is used in B μ H + J, whee J has the same dimension as B. Fo the dielectic displacement D and the induced electic polaisation P we have D ε E + P ε ε E ε (+ χ e ) E and P χ e ε E (ε )ε E. (.) ε /μ c 8, s V m (F m ) ε is called the pemittivity of vacuum, ε is the elative pemittivity, and χ e is the electic susceptibility. The last two quantities ae eally tensos, and ae used to descibe anisotopic popeties. Fom the ight hand side of equ. (.) it follows that a static polaization of gases o liquids needs the pesence of an electic field. Even if the molecules have a dipole moment, it is aveaged to zeo in gases o liquids by themal motion.. The Electic Popeties of Molecules Definition of an electic pole: monopole (n ) is composed of a chage, which is descibed by a scala. dipole (n ) is epesented by a vecto. It is composed of two opposed chages which compensate each othe. It is in this way vey diffeent than a monopole. quadupole contains fou chages (n 4), which also add to zeo, have no dipole moment, and is descibed by a tenso. n octopole contains eight chages (n 8), that has neithe a net chage, no dipole moment o quadupole moment. It equies a fou dimensional desciption. CO can be descibed as a quadupole, and CH 4 as an octopole. The Figue above is taken fom tkins: Physical Chemisty, CD vesion, Fig. (.). The inteaction potential between an npole and n'pole is V n n'. It theefoe deceases as the numbe of involved chages inceases, since as the distances incease; the effects of the individual chages ae compensated. Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
2 Chapte, page Multipole expansion of chages q n with z y ' ' x obevation point q n n To claify a few tems, such as dipole moment and polaizability, we use the so called multipole expansion, which descibes the electic potential V () of a chage distibution. The chages q n ae located at ' n, and the oigin of the coodinate system is located within (o not fa fom) the chage distibution. The left figue of the wate molecule shows the chage of the oxygen nucleus located at ' and the obsevation point at. Electon chages ae dawn in oange (gey). q n V () 4π ε n n qn ( xn ) + ( x ) ( x ) i n i n j n xi! xi xj φ () + φ () + φ () q n + µ + 5 θijxx i j. (.) n The facto 4πε is intoduced on the left so that the potential has the SI unit of volts. The potential is given fo lage distances between the point of obsevation and the chage, i.e.» ' n. It is expanded by powes of / by taking the deivatives with espect to, the point unde consideation. The seies development shows that the potential of any chage distibution can be epesented by a sum of multiples. Let us now conside an electically neutal molecule, in which the positive nuclea chage and the negative electon chage compensate each othe. In this case, the fist tem of the expansion φ () is zeo. φ () is the dipole moment and φ () the quadupole moment. Hee we stop the expansion. φ () can be witten as μ / o μe /, whee e is the unit vecto in the diection. μ q n n (.4) n μ is defined as the dipole moment of a chage distibution. It does not depend on the location of the oigin in the case of neutality of the chage cloud of atoms o molecules. The unit of the dipole moment is sm. The old cgs unit named afte Pete Debye, in which D,564 sm, is still in use because the dipole moments of small molecules ae in the ange of D (H O,85 D, HCl,8 D). Be caeful not to confuse the Debye with atomic unit ea 8,478 sm, which efes to the elementay chage e,6 9 s and the Boh adius a 5,9 m. The H O molecule has ten positive elementay chages (q 8e) in the nuclei, and equally many negative elementay chages in the electon shell. displacement though the distance d of the cente of chage of the nuclei elative to the cente of chage of the electons leads to a dipole moment μ qd of,85 D. i, j Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
3 Chapte, page The vecto chaacte of µ allows the vecto addition of dipole moments of diffeent segments of a molecule. Examples ae the halogen substituted benzene molecules. We use the expeimentally detemined dipole moment of monohalogen benzene and use that to calculate the polysubstituted benzene, unde the assumption of a egula hexagon. See at left Fig.. taken fom tkins. ow we come to the last tem φ () in equ. (.), which descibes the potential of a quadupole. The quadupole moment is [ ( ) ( ) ] θ ij q n x n x i n j n ij n δ (.5) at the oigin. δ ij is the Konecke symbol. lthough we will not discuss molecula quadupoles and multipoles of highe ode, nuclea quadupoles play an impotant ole in spectoscopy. Fom equ. (.5) we can see that the quadupole tenso has no tace. Even though single magnetic chages do not exist, we can wite a elationship fo magnetic potential analogous to equ. (.). It is impotant that we teat the magnetic moment, which is also epesented by μ, since, togethe with the electic dipole moment, it will be used in the electomagnetic dipole adiation in the chapte.5 concening magnetic popeties. Statements about whethe a molecule contains a pemanent dipole moment will be done using symmety consideation in chapte 4. E.g., cubic symmety goups and the goups D indicate non pola molecules. ow we conside dielectic mateial consisting of paticles without pemanent dipole moments, e.g. CO. moment μ ind can be induced though the electic polaizability α (units: sm /V) unde the influence of an extenal electic field E. The coesponding polaizability tenso α is defined by μ ind α E (.6) This linea effect is sufficient fo the consideation of weak fields. But if we ae dealing with stong fields, fo example in lases, the equation μ ind α E + β E + γ E +... (.7) is the basis of the consideation of effect of nonlinea optics (LO). The β tem is called hypepolaizability. onlinea effects play an impotant ole in the lase spectoscopy. α has the units sm/(v/m), which is sm V. Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
4 Chapte, page 4 This SI unit has not succeeded in diving out a modified cgs unit, the socalled polaizability volume α', which has the unit 4 cm Å what is simila in magnitude to the mola volume. The convesion is: α α (.8) 4πε Some isotopic values fo α'/å ae CCl 4 :,5; H :,8; H O:,48; HCl:,6. vey anisotopic molecule is benzene with the values: isotop:,, pependicula: 6,7, paallel:,8. The elationship of the size of the induced to the pemanent dipole moment is clealy seen in wate: It has a pemanent dipole moment of,8 D o 6, sm. The polaizability of wate is α',48 Å. This value has to be multiplied by 4πε, 4 and becomes,64 4 s m V. If we multiply that with a elatively high field stength of 7 V m, we get,64 sm, which gives us an induced dipole moment moe than thee odes of magnitude smalle than the pemanent one. Back to the induced effects: s a elationship of the induced polaisation P to the induced dipole moment μ I of paticles pe unit volume, the fist pat of the equation P μ i α E, (.9) V V i is valid fo any paticles, wheeas the second equals sign is valid only fo identical paticles that do not influence each othe. onpola molecules only have displacement polaizability. In gases thee is no mutual influence of the induced dipole moment μ ind between the paticles and we have P ρ α E (.) M with the vogado numbe, the density ρ and the mol mass M. Fom equ. (.) and equ. (.) we get ρ ε + α. (.) Mε With this and the equation below we can detemine the polaizability a of the molecules, if we know the density ρ, the mola mass M, the capacitances of an empty capacito (C ) and one filled with the gas unde study (C), by means of the equation C ε. (.) C Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
5 Chapte, page 5 E extenal E additional In mateials of highe density, the dipoles ceate an additional field which affect thei neighbous. This field has the same diection as the extenal field and depends on the geomety. Loenz calculated fo a spheical hollow space in a dielectic with the isotopic polaization P, that the field local to the hollow space E local is inceased elative to the extenal field E by the exta field P/ε. See the figue on the left. + With that, equ. (.) becomes ρ P P α E + (.) M ε With equ. (.), we can eplace E: PM ( ε + ) ( ε ) α P P P α ρ + ε ( ε ) ε ε With some eaangement, we get the ClausiusMosotti equation ε ε + M α P ρ ε m 4π. (.4) α, (.5) which connects the macoscopic paametes ε, ρ, and M with the molecula quantity α, and also defines the mola polaizability P m. Ou pevious consideation have assumed static fields, which ae caused by a diect cuent (DC) souce applied to a capacito If we eplace DC by C (altenating cuent) with the fequency ν, we get the coesponding magnetization of the field only in the case, if the chages can change thei oientation quickly enough. The electonic polaizability poduced by shifting the positively chaged nucleus with espect to the negative electon shell (in molecules and in spheically symmetic atoms) takes place in less than 4 s. The polaisation by the shifting o vibation of the ions in a molecule o lattice (ion polaization, distotion polaization) happens a thousand times moe slowly, on the ode of s. Both types of polaization ae united unde the tem of displacement polaization. The oientation polaization is much slowe and theefoe plays no ole in the index of efaction in the optical ange. It is caused by the lining up of pemanent molecula dipoles which ae pesent even in the absence of an extenal field. The dielectic elaxation of the oientation polaization can be expeimentally examined with high fequencies and helps detemine the dynamics of the system. The figue on the left, Fig..4 taken fom tkins, shows fequency anges of the vaious types of polaization. It also demonstates that the polaization can incease with fequency. Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
6 Chapte, page 6 In chapte. we will show that fo the efactive index n c /c, the Maxwell elation n ε applies. Using that elationship, we can wite the ClausiusMossotti equation (.5) as n n M + ρ P m. (.6) ccoding to ou ealie assessments, as the fequency inceases, optical excitement should have a lowe polaization and accoding to equ. (.6) a lowe index of efaction as well. This should happen because the highe fequencies, the less of a ole the distotion polaization plays. Expeiment, howeve, shows in a few cases exactly the opposite elationship, as shown to the left in table. taken fom tkins. This dispesion (inceasing polaization o inceasing index of efaction with inceasing fequency) is due to the fequency of the incoming light being close to the esonance fequency of the system. This will be moe deeply discussed in chapte.. We now etun to pola molecules. lthough the displacement polaization of dielectic substances is hadly dependent on the tempeatue, the oientation polaization of pola molecules in paaelectic substances has a stong tempeatue dependence. The calculation of the tempeatue dependency of the polaization was fist calculated by Paul Langevin in 9 in the following way: molecule at an angle θ to the applied extenal electic field E, and with a pemanent dipole moment µ, contibutes μ cosθ to the macoscopic dipole moment and μ E μ E cosθ to the total enegy (Enegy μ E). Using Boltzmann statistics, we get the pobability P Boltzmann that the molecule will have the angle θ: ( xcosθ ) μe x ( xcosθ ) sinθ dθ kt exp PBoltzmann mit. (.7) π exp The integal in the denominato nomalizes the pobability to one ove all possible oientations. The aveage dipole moment in the diection of the field is detemined using equ. (.7) μ π π ( xcosθ ) μ exp cosθ sinθ dθ μ P Boltzmann cosθ sinθ dθ. (.8) π exp sinθ dθ ( xcosθ ) Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
7 Chapte, page 7 By substituting y cosθ and dy sinθ dθ, we get ( xy) μ y exp dy μ. (.9) exp dy ( xy) Fom integal tables, we use y e xy x x x x e + e e e dy und x x e xy e dy x e x x. (.) Fom equ. (.9) and equ. (.) we get x x e + e μe μ μ L mit L coth x und x, (.) x x e e x x kt whee L is the Langevin function. If we use the dipole moment of wate, which is 6, sm, and the elatively lage field stength of 7 V m, we get 6, Vs, which is compaable to kt 4,4 Vs at oom tempeatue. Paamete x in equ. (.) is a vey small numbe. Tems past the linea tem (second tem) can be neglected in the seies expansion of the coth x in equ. (.). The Langevin function can be eplaced by L x/ μe/kt (high tempeatue appoximation). It follows that μ E μ, (.) kt and with the paticle numbe pe volume ρ/m, the oientation polaization is P Oientation ρ μ E. (.) M kt n simila equation was fist developed as a Cuie law fo tempeatue dependent paamagnetism. pplied to paaelectic substances, we can wite P Oientation+ Displacement ρ μ α + E and ε M kt ρ μ + α + ε M kt (.4) fo the sum of dielectic and paaelectic polaisation instead of equ. (.). The same ovelapping of both effects changes the ClausiusMosotti equation, equ. (.5), to ε ε + M μ Pm α + kt. (.5) ρ ε Equation (.5) is known as the Debye equation. If we measue the tempeatue dependency of the elative dielectic constant, and wite down the mola polaisation against /T, we get μ as the slope accoding to equ. (.5), and α as the yintesect. Fo gases, ε is in the magnitude, fo wate at K is ε 78,5. Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
8 Chapte, page 8. bsoption, Dispesion, Refaction The popeties of absoption, dispesion, and efaction have been long known due to optical examinations. In the yea 886, Heinich Hetz expeimentally demonstated the existence of electomagnetic waves and thei equivalence to light waves. fte Hetz s achievement, the electomagnetic theoy of James Clek Maxwell, developed fom 86 to 864, became the basis of examining optical absoption and dispesion phenomena. s the name indicates, electomagnetic waves have two components. The figue below shows a linealy polaised wave tavelling in the xdiection though a homogenous isotopic medium. The electic field E oscillates in the xyplane and the magnetic field stength H oscillates in the pependicula xzplane. Both oscillations have the same fequency ν and the same wave vecto k, which in this case only has the xcomponent k x π/λ. The amplitude vectos E and H also only have one component each. The following equations descibe the popagation of the linea polaized wave: E y H z y E z H cos(k x x πν t), cos(k x x πν t). (.6) zchse, H ychse, E xchse usbeitungsichtung The diection of popagation and the diection of the electic field detemine by definition the plane of polaisation of the light. In this case, it is the xyplane. The figue on the left shows a linealy polaized electomagnetic wave tavelling in the xdiection. Fo the detemination of the popagation speed c of the wave, we put into equ. (.6) in the agument of the cosine function a constant phase, fo example zeo. We then detemine dx/dt and obtain c λν εεμμ c ε μ. (.7) Fom that we get the speed of light c as the speed of popagation of an electomagnetic wave in the vacuum, in which ε μ. The enegy density w (enegy pe unit volume) is ED fo an electic field o BH fo a magnetic field. The time aveage of cos ωt is ½. Fom that we obtain fo the enegy density of a linealy polaised electomagnetic wave w ½ ε ε E y + ½ μ μ H z. (.8) The PoyntingVecto S, which efes to the enegy flux density though a unit aea, points along the xdiection and is the poduct of the enegy density and the speed of light: S x wc. (.9) Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
9 Chapte, page 9 Fom that we aive at the impotant conclusion, that the enegy flux of the adiation in the diection of popagation depends on the squaes of the amplitudes of the field stengths. cδt x, k, S The figue on the left shows the enegy flux of an electomagnetic wave, popagating in the xdiection. The wave vecto k and the Poynting vecto S also point in the xdiection. It takes the enegy enclosed in the box the time Δt to flow though the suface. If we set Δt to one second and set equal to the unit aea, we get fo the enegy density a powe density of the same numeical value. The phase speed c λν, which is defined as the poduct of wavelength and fequency, is educed in compaison to speed of light in a vacuum, c, when the electomagnetic wave tavels though a medium with an index of efaction n >. The educed value is c c /n. We will show that the fequency dependency of n leads to a dispesion, which can be descibed using a classical model. It will be also shown that the imaginay pat of a complex index of efaction descibes the damping of an electomagnetic wave. Fo this pesentation we conside an electic field with the amplitude vecto E (, E, ), which has a complex time dependency exp(iωt) instead of the cos ωt of equ. (.6). The diffeential equation of a damped oscillation foced by an extenal field is m d dt + mγ d y dt + mω y q E exp(iωt), (.) y whee the mass of the oscillato is m, the chage q, and the chaacteistic fequency ω. γ is the damping constant. With an exponential tial solution of y y exp(iωt) we obtain y m qe ( ω ω + iγω) (.) as the complex amplitude of the oscillation. n induced electic dipole moment μ ind appeas in the y diection: q E μ y q y exp(iωt). (.) m ω ω + iγω ( ) With oscillatos pe unit volume, we obtain P ind χ e ε E μ (.) as the induced electic polaisation, and with that a complex susceptibility χ e q. (.5) ε m ω ω γω ( + i ) The eal and imaginay pats of χ e ae not independent, as can be shown by multiplying numeato and denominato of equ. (.5) by the conjugated complex of the paenthesis Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
10 Chapte, page Fom the definition n c /c, μ ε fo vacuum and equ. (.7), it follows that n c c με. (.6) Since we ae not consideing feomagnetic mateials, we can set μ with sufficient accuacy, and we obtain the Maxwell elation n ε + χ e. (.7) We should note that these quantities ae fequency dependent. Fo example, the oientation polaization mentioned ealie has no effect on the susceptibility in the optical fequency ange. Fom equations (.5) and (.7) it follows that the index of efaction epesents the complex quantity n + q. (.8) ε m ω ω γω ( + i ) To sepaate this into eal and imaginay components, diffeent conventions ae in use. We wite n n' i n". (.9) When n, which is the case in gaseous media, we can make the appoximation n (n + ) ( n ) (n ). ea the esonant fequency we find ω ω «ω o ω + ω ω ω. With that we have and n' + n" q ω ω 4εmω ( ω ω) + ( γ / ) q γ 8ε mω ( ω ω) + ( γ / ) (.4). (.4) The fequency dependent quotient in equ. (.4) is descibed by a function in the fom of y /( + x ), which is commonly called Loentz cuve. The paamete γ is the fullwidthathalfmaximum of the cuve. f Loentz On the left see a Loentz cuve and its halfwidths Δω / and δω / (full width at half maximum). / Δω / δω / γ ω ω Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
11 Chapte, page The meaning of the eal and imaginay components can be claified by the following consideations: nalogous to equ. (.6), we have fo a wave popagating in the xdiection E y y E exp [i(ωt k x x)]. (.4) The wave vecto k can be eplaced by nk, whee k with k ω/c is the wave vecto in the vacuum. Fom equation (.9) and (.4) it follows that E E E y y exp[i (ωt k x {n' in"} x)] y exp[ n"xω/c ] exp[ik x (c t n'x)]. (.4) The fist exponent on the ight side of equ. (.4) descibes a damping of the wave. Late we will show how the absoption, descibed by the imaginay pat of the index of efaction n", is elated to expeimentally measuable extinction coefficient. The second exponent descibes the dispesion. In connection with equ. (.4), we get fom that the dependency of the phase speed on the fequency..4 Intemolecula Inteactions The electical popeties of molecules ae as impotant fo the intemolecula inteactions as the absoption and dispesion. lthough the epulsive foces ae not entiely descibable using electostatic inteactions, the van de Waals foces whose basis is the Coulomb inteaction is descibable. The potential enegy of two point chages q and q sepaated by the distance is 4πε qq V. (.44) d q q q ow we will calculate the inteaction of a fixed dipole μ d q with a chage q. Let us conside a simple model with a chage on the axis of the dipole moment, see left. Fo d «and x d/ «we obtain fom supeimposition the attaction and epulsion of the individual chages V q q 4πε + q q q q + + d 4πε + x 4πε 4πε q q x μ q. (.45) We boke off the seies m x + x m x +... (.46) ± x afte the linea tem. It is appaent fom the diagam and equ. (.45) that the potential is zeo fo a pependicula oientation of the dipole moment and the distance vecto. Fo all othe geometies we get a potential popotional to. Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
12 Chapte, page This is changed if the dipole can feely otate, fo example aound an axis pependicula to. We label the nuclea chages with q and let the cente of the electon chages q feely otate at a distance d fom the fixed cente of nuclea chage. We eplace the otation by the two exteme points. One of these points is shown in the figue above. t the othe point, the chage q is at a distance d fom the chage q on the dipole axis. We add the potentials of these two oientations (and divide the esult by two to get an aveage potential). In doing this, we have to expand the paentheses of equ. (.45): (.47) + x + x x In the expansion of the factions in equ. (.47) the tems of zeo and fistode cancel out, see equ. (.46), and the even tems stating at the second ode emain, i.e. d /. If we put this back in equ. (.45), we get a dependency of. Rotations of the inteaction patne aise the negative powe of the distance dependency of the potential. The fact that the potential fom equ. (.45) by the otation gets multiplied by the facto d/ «, indicates a eduction of the inteaction potential as the negative powe inceases. d q q μμ V. 4πε d q q s a futhe example, let us calculate the potential enegy of two fixed dipoles on a common axis, as shown on the left. It is easy to see that the sum of the chages gives the expession that we wote on the ight side of equ. (.47) fo a diffeent eason. The quadatic tem emains and we obtain (.48) In feely movable dipoles the negative powe of the distance dependency is inceased to six. The eason fo this is not analogous to the example above (point chage and movable dipole), but athe though the consideation of the pobability fo diffeent oientations of two dipoles given by the Boltzmann distibution, which was neglected in the examples above. The inteaction between a pemanent dipole moment and the dipole moment of a neighbouing molecule induced by the pemanent dipole moment can be descibed by equ. (.48) if we use μ to epesent the pemanent dipole moment, and μ the induced one. ow we have the geneal elationship fo the induced dipole moment μ ind αe and thus μ α E, (.49) whee E is the field poduced by the pemanent dipole moment of molecule one. Using the same method as fo the addition of the potentials in equ. (.45), but note that (+x) x + x..., it is possible to calculate the electic field stength of a dipole using the wellknown equation fo the electic field stength of a point chage q E 4πε We obtain μ E. 4πε. (.5) (.5) Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
13 Chapte, page Equation (.5) with equ. (.49) put into equ. (.48) gives V μ α 4πε μ μ α 4πε πε 6, (.5) whee α' is the cgs unit of the polaizability intoduced in equ. (.8). Since the induced dipole moment always has the same diection as the pemanent dipole, the mobility of the molecules has no effect on V(). The electic inteaction of nonpola molecules occus due to fluctuations in the electon density distibution of molecules o noble gases. Let us conside a molecule, which has a time dependent dipole moment due to fluctuations of the electon density. Even though this dipole moment aveages out in shot time fames, it is always inducing a codiectional dipole moment in the neighbouing molecule. Due to this induced dipole moment, thee is always an attactive inteaction with a 6 dependency. The stength of the inteaction also depends on the polaizability of molecule, since this affects the ceation of the fluctuating dipole moment. This dispesion inteaction, fist descibed by Fitz London, is descibed by the empiically discoveed London fomula: I I α V 6 α. (.5) I + I I and I ae the ionisation enegies of the two molecules. To sum up, we have the following distance dependencies f() and enegy contibutions E fo inteactions between molecules: Inteaction f() E/kJ mol Comments IonIon / 5 only fo ions IonDipole / 5 DipoleDipole / fixed pola molecules / 6,6 otating pola molecules London (Dispesion) / 6 all molecules The epulsive contibutions of intemolecula inteaction is often dealt with by intoducing a tem with a lage (>6) negative powe of the distance. The LennadJones (n,6) potential as the supeposition of epulsive and attactive contibution has the fom V C C6 n, (.54) n 6 whee is often given fo n, though values up to 8 have been used. n exponential function exp( C/) descibes bette the decay of obitals functions o thei ovelap, which is the eason fo the epulsion. Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
14 Chapte, page 4.5 Magnetic Popeties The basis is the Maxwell equation given aleady at the beginning of chapte : B μ (H + M ) μ μ H μ (+χ) H. (.) The magnetic susceptibility χ is often witten as κ. The mola magnetic susceptibility χ m should not be fused with the magnetic susceptibility. We have χ m V m χ. Since the magnetization M is, in analogy to the electic popeties, the sum of the magnetic moments pe unit volume, the units ae m fo micoscopic magnetic moments. The magnetic moments connected to electon o nuclea popeties ae usually labelled with μ, and the magnetic moments of molecules ae usually labelled with m. s a natual micoscopic magnetic moment we use the Boh (electon) magneton μ B 9,74 4 m (magnetic moment of an electon with the obital quantum numbe m l and spin quantum numbe m s ½), and the nuclea magneton μ μ B m e /m n 5,5 7 m. In diamagnetic mateials, χ is negative, and in paamagnetic mateials it is positive. few values (.4 HWM) in ppm (multiply values by 6 ) at K (9 K fo liquid O ): Diamagnetic Mateials Paamagnetic Mateials H (gas), O (gas),86 H O 9, O (liquid) 6 acl,9 Dy (SO 4 ) 8H O 6 Cu 7,4 l, Bi 5 CuSO 4 5H O 76 Most molecules ae not paamagnetic, since they only have full electon shells. Diamagnetism occus due to the magnetic moment of the electon induced by an extenal field B, which is oppositely diected to the extenal field accoding to Lenz s ule. The magnetic polaizability, defined in analogy to electic polaizability, cf. equ. (.6), is m ind ξ B (.55) with the magnetizability ξ (zeta), sometimes labelled β. We also have m ind /V mol M χ H χ B/μ μ and with μ ξ /V mol χ /μ. (.56) Fo hydogen gas, we get m ind m with χ 9 (see table above) in a field of T. We see with that the induced magnetizability is moe than six odes of magnitude smalle than the Boh magneton. futhe analogy to the electic popeties of equ. (.4) is to be found in the following elationship, which adds the induced contibutions of molecules without pemanent magnetic moment and a tempeatue dependent contibution fom molecules with pemanent magnetic dipole moments: m χ + m μ ξ. (.57) kt In equ. (.57) we see the empiical Cuie law, with χ m + C/T. Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
15 Chapte, page 5 ll molecules have a diamagnetic contibution to thei susceptibility. This is due to the ciculating cuents in the gound state of the molecule s occupied obitals. If the ciculation is foced to occu in nomally unoccupied higheenegy obitals, we get a tempeatue independent paamagnetism (TIP). If the molecule contains unpaied electons, the tempeatue dependent spin paamagnetism dominates. This is the basis of electon spin esonance (ESR, EPR) and will be explained hee in moe detail. mateial is called paamagnetic if it has no macoscopic magnetic moment in the absence of an extenal magnetic field, but has a magnetic moment in the diection of the field, if an extenal magnetic is applied. We can imagine that the andomly oiented micoscopic magnetic dipole moments of unpaied electons in a paamagnetic mateial ae oiented by the application of an extenal field. The existence of such magnetic moments is equal to the existence of electon shells not completely closed, o the existence of unpaied electons. Paied electons have the same quantum numbes n, l, m, but diffeent spin quantum numbes s +½ and s ½. In atoms and molecules with closed electon shells, all electons ae paied, i.e. the esulting obital and spin moments ae zeo. To examine such paticles with EPR, they have to be moved into a paamagnetic gound state, fo example with adiation (fo example, the constuction of fee adicals o tiplet states). Electon spin paamagnetism allows EPR spectoscopy on the following substances: a) Fee adicals in solids, liquids, o gases, which accoding to definition epesent an atom, molecule o ion with an unpaied electon, fo example CH. Fee adicals ae instable. They have a finite lifetime and can be ceated by adiation. b) The natually existing ions of the tansition metals, which belong to the d, 4d, 5d, 4f and 5f goups of the peiodic table, compose moe than half of the elements of the peiodic table. The palette of the positively to negatively chaged ions contains up to seven unpaied electons. To descibe the obital and spin magnetism, we will begin with a classical pictue and then conside the quantum mechanical aspects: The electon in a cicula obit with adius and angula fequency ω ceates a cuent I eω/π, whee e,6 9 C denotes the elementay chage. In geneal, the magnetic moment of a cuent I, which encicles the suface, is µ I. With π follows µ ½ e ω. Using the angula momentum L m e ω and electon mass m e 9,9 kg, we get fo the obital magnetism a magnetic moment which depends on L: µ L e L. (.58) m e Fo spin magnetism, we stat with a otating sphee (the axis of otation goes though the cente of mass) of mass m e and chage e. We then divide this sphee into small volume elements, in which the elationship of the segment chage though the segment mass is independent of the segment size. We then follow the same steps fo each segment that we followed fo the electon in a cicula obit. In the final summation, the individual contibutions do not depend on the distance of the segment of chage fom the midpoint. We aive fo the dipole moment of spin magnetism at µ S e S, (.59) m e Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
16 Chapte, page 6 what is analogous to equ. (.58), but S is the spin of the electon. In quantum mechanics, it is well known that because of the quantization of the angula momentum, the absolute value is: L h l( l+ ) and S h ss ( + ). (.6) hee l is the obital angula momentum quantum numbe and s is the spin quantum numbe. The components in the diection of an extenal magnetic field in the z diection ae: L z l z h m l h m h bzw. S z m s h s h. (.6) Thee ae l+ diectional quantum numbes (magnetic quantum numbes) fo the obital magnetism: m l m l, l +,..., l, l (.6) and only two diectional quantum numbes fo electon spin magnetism m s s ½, +½, (.6) the use of s fo the electon spin quantum numbe +½ and also fo the diectional quantum numbes ±½ could be a souce of confusion. If we conside the zcomponent of the magnetic moment in equ. (.58) and set the angula momentum L z with h to the smallest value othe than zeo fom equ. (.6), we get as an elementay unit of the magnetic obital moment in an extenal magnetic field the Boh magneton: µ B e m e Jm h 9, (.64) Vs Based on the Boh magneton, we intoduce the gfacto (which is impotant in EPR) fo an abitay magnetic quantum numbe m: μ gμ B m( m+ ). (.65) We will subsequently use small lettes fo the quantum numbes of individual electons and capitals fo the total quantum numbes of seveal electons in a shell, as explained in chapte 4. By compaing equ. (.65) to equ. (.58) and equ. (.6) it immediately follows that fo the obital magnetism g L. This has been expeimentally shown with a pecision of 4. Fo spin magnetism of the fee electon, we had a gfacto in equ. (.59), (.6), and (.65) which is not in ageement with expeimental esults. leady in equ. (.59) if we wee to intoduce the quantity ½h fo the electon spin S with equ. (.6) and equ. (.6), we would have a contadiction with the expeimentally poven esult, that we get a whole Boh magneton fo the magnetic spin moment of the electon with the spin quantum numbe s ½. In othe wods, it is appoximately equal to the that belonging to the obital angula momentum quantum numbe l. Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
17 Chapte, page 7 ppaently the classical consideations fail fo spin magnetism, although they wee successful in the case of obital magnetism. This failing of the classical model is called the magnetomechanical anomaly of the fee electon. The coespondence between theoy and expeiment was estoed by Paul dien Mauice Diac in 98 with elativistic quantum mechanics. The most accuate cuent value fo the gfacto of the fee electon is g e,9486(). The magnetic esonance is based on the socalled the Zeeman esonance effect, i.e. the tansitions between states which come to being though the magnetic splitting of a state, nomally the gound state. Fo the sake of completeness, we should explain hee the nomal and anomalous Zeeman effect, which was found in the optical spectoscopy. The nomal Zeeman effect occus in singulet states fo which the total spin S. It holds that J L, all states ae split into L+ levels in which the enegy sepaation between neighbouing levels only depends on the extenal magnetic field. Fo the change of the diectional quantum numbe M in a tansition, we have ΔM, ±. Fom that we get fo optical tansitions with ΔL thee lines, the nomal Zeeman tiplet. Fo the histoically named anomalous Zeeman effect of non singulet atoms, the RusselSaundes coupling fo the addition of total obital angula momentum and total spin is valid: J L + S. The magnetic moments µ L and µ S point in the diection of L o S, accoding to equ. (.58) and (.59). Because of the diffeing values of g L and g e, µ J does not point along the diection of J. S J µ S M J The diagam on the left shows the vecto composition in the anomalous Zeeman effect. Because of the LScoupling we get J L + S. The magnetic moments µ L and µ S ae paallel to thei moments of otation and ae added to M J, which is not paallel to J. The quantity μ J consideed in equations (.66) to (.69) is not the absolute value of M J but athe its pojection onto the Jaxis. µ L L Fo the absolute values of the magnetic moments, it follows fom equ. (.65): μ g μ L L+, μ μ L L B g μ S e B J J B ( ) ( ), ( ). g μ J J + S S+ (.66) The esulting total angula momentum J is constant in time. L and S pecess aound J, so that only the components of thei magnetic moments paallel to J have a measuable effect. Fom that it follows that: ( ) cos( L,J ) μ ( ) cos( S,J ) μ J L μ B e B g L L+ + g S S+ (.67) With the help of the cosine fomula, we get cos L + J S LJ ( L,J ) bzw. cos( S,J ) fo the angle between L and S and S and J. S + J L SJ. (.68) Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
18 Chapte, page 8 With L h LL ( + ), compae equ. (.6) and the elevant equations fo S and J, ( + )( + ) + { ( + ) ( + ) }( ) μ J J g g S S L L g g J J μ J B e L e L ( + ). (.69) If we connect equ. (.69) with the lowe line in equ. (.66) and futhe sets g L g e, we get the facto: J( J + ) + S( S + ) L( L+ ) gj, (.7) J( J + ) which was named afte lfed Landé. He calculated this facto using the BohSommefeldquantum mechanics with the quantum numbes J instead of J(J+) etc. g J is the gfacto fo the anomalous Zeeman effect. When stong magnetic fields distub the LScoupling, L and S pecess diectly aound the extenal magnetic field. s a consequence of this effect which has been named the PaschenBack effect in honou of Fiedich Paschen and Enst Back, the optical tansitions again assume the simple splitting obseved in the nomal Zeeman effect..6 Inteactions with Electomagnetic Radiation, Spontaneous and Induced Tansitions, Radiation Laws spontaneous event needs no extenal influence to occu. The light of a themal adiato, which we can visually see, occus when a substance at high tempeatue spontaneously emits quanta of light. n induced o stimulated event only occus with extenal influence. ccodingly, absoption is always induced (stimulated). But emission can be induced, if a fequency equal to that of the light to be emitted is extenally input. Let us now conside two enegy levels of an isolated paticle, see below. Since the following consideations ae applicable to any states, we will label them with i and j. Hee, and in the next two sections, we will set i and j. Let E > E and E E hν, whee h 6,66 4 Js (the Planck constant). The occupation numbes of the states ae and. Enegy E hν B ρ ν The numbe of paticles which go fom state to state is, d B w ν dt, (.7) E B ρ ν bsoption Induced emission Spontaneous emission whee B w ν is the absoption pobability with the spectal enegy density w ν. The enegy absobed by the paticles fo the tansition is given by dw abs hν d. (.7) The enegy emitted in the fom of adiation by the tansition fom to is dw em hν d. (.7) Fo the balance of paticles that go fom to, we need to conside a spontaneous tansition pobability in addition to the tansition pobability B w ν : d (B w ν + ) dt. (.74) Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
19 Chapte, page 9 The pobability does not depend on extenal fields. The pobability of an induced tansition, howeve, does depend the extenal field. It is the poduct of the B coefficients with the spectal enegy density w ν of the extenal fields in the fequency ange fom ν to ν + dν. The spectal enegy density w ν has the units of an enegy pe volume and fequency. Instead of this quantity, the spectal beam density L ν is often used. L ν is the powe in the fequency ange ν to ν + dν that is emitted pe unit aea in a cone of solid angle Ω. solid angle Ω would mean that m is cut out of the total suface aea of 4π m of a sphee with a adius of m. The apetue angle of the cone is about 66. In a vacuum, whee the speed of light is c it holds that L ν w ν c /4π. (.75) B and B ae the Einstein coefficients fo absoption and induced emission. With the help of these coefficients lbet Einstein could find a simple and secue poof of the adiation law in 97. The adiation law was discoveed at the end of 9 by Max Planck though an intepolation (of the behaviou of the second deivative of the entopy with espect to the enegy) between Wien s adiation law and Rayleigh Jeans adiation law. Einstein s deivation stats with a closed cavity in a heat bath at the tempeatue T. Because of equilibium, we have fo two abitay states between which tansitions occu that the numbe of absobed and emitted quanta of enegy must be equal. w ν is in this case the spectal enegy density of a black body, labelled with ρ ν. Fom ( + B ρ ν ) B ρ ν it follows that B ρ + B ν. (.76) ρ ν On the othe hand, Boltzmann statistics can be applied to this system: g E exp g E kt g g exp hν kt. (.77) k efes to the Boltzmann constant, and h is Planck s elementay quantum of action. The statistical weights, g, ae fom now on set to g g, i.e. a degeneation of the enegy levels will not be consideed. With equations (.76) and (.77) we aive at: ρ ν B h ν kt e B. (.78) In equ. (.78), no statement about the elationship between B and B is made. If we make the plausible assumption that as T, ρ ν must also hold, we get fom equ. (.78) the elation that B B. Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
20 Chapte, page Fo the detemination of the elationship between and B the adiation law fom 9 stated by Lod Rayleigh and James Hopwood Jeans is used. In the low fequency ange (hν «kt), equ. (.78) should fulfill the RayleighJeans law ρ ν 8 πν c kt, (.79) which we will deive late using classical statistics. With exp (hν/kt) + hν/kt and hν «kt, we get fom equ. (.78) by setting B B ρ ν kt Bhν. (.8) Fom equations (.79) and (.8) it follows fo abitay elationships between hν and kt that the valid elationship between the spontaneous and induced tansition coefficients is B 8 h πν. (.8) c Equation (.8) put into (.78) leads us to the famous Planck adiation law: π ν ρ 8 h ν c e hν kt. (.8) If we use the wavelength dependent enegy density ρ λ dλ instead of the fequency dependent enegy density ρ ν dν, we conclude that in a vacuum: π ρ 8 hc λ 5 hc. (.8) λ λkt e using the elationships ν c /λ and dν c /λ dλ. The RayleighJeans law, which applies when hν «kt, is used in Einstein s deivation of the Planck adiation law. Othe adiation laws wee not used, but can be pesented as esults of the Planck adiation law in the fame of the Einstein deivation: Fo hν» kt it holds that exp (hν/kt)», and we get fom equ. (.8) as a special case the Wien adiation law, deived by Wilhelm Wien in 896 (up to the factos late detemined to be 8πh/c and h/k): π ν ρ 8 h ν c e h ν kt. (.84) Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
21 Chapte, page If we constuct the fist deivative of equ. (.8) with espect to the wavelength and set this to zeo, we get the a maximum of the spectal enegy density of the black body at λ max. The wavelength follows the elationship λ max T const. hc,8978 mm K (.85) k 4, 965 and descibes a displacement of the maximum of the intensity distibution to shote wavelengths as the tempeatue inceases. (The numbe 4,965 is the zeo point of the deivative, ounded to the neaest decimal. Because of this, the numbe,8978 is also ounded). This law, deived by Wien in 89, is known as Wien s displacement law. It was the basis of his thoughts fo the fist fom of his adiation law. t K, the maximum of the adiation of a black body is in the infaed at appox. μm. Only at about 4 K does it move into the visible spectum. Fom equations (.8) and (.85) we get the law ρ λ max const. T 5 (.86) fo the enegy density in the ange of the maximum. Fo completeness, mention also Josef Stephan s empiical law of 878, late claified with themodynamics by Ludwig Eduad Boltzmann. It is known as the StefanBoltzmann law, and is aived at by integating equ. (.8): ρλdλ T π k 5 ch σ T4. (.87) The total adiation of the black body is popotional to the fouth powe of the tempeatue. We stess again that in the above equations, enegy densities ae used. To convet to the often used beam density, use equ. (.75). Fo example, the facto σ in equ. (.87) is changed into π 5 k 4 /(5c h ) 5,67 8 W m K 4, if we use L λ instead of ρ λ. Js m ρ ν/ RayleighJeans 5 K Planck 5 K Wien 5 K Planck K 4 wave numbes / cm On the left we see the fequency dependency of the spectal enegy density of the black body at a tempeatue of 5 K accoding to the laws of Wien, Rayleigh Jeans, and Planck. On the hoizontal axis, instead of the fequency ν, the wave numbe ~ ν ν/c is used. Using Planck s law, the cuve fo K is also shown. The visible ange of the spectum lies between cm and 6 cm. Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
22 Chapte, page If we use the Einstein coefficients, we get the elationship between spontaneous and induced emission pobabilities by ewiting equ. (.8): B ρν hν. (.88) kt t a tempeatue of K, the equilibium between both pobabilities is at ν k K / h 6,5 Hz, and ~ ν 8 cm o λ 48 μm, in the fa infa ed. That is tue fo black body adiatos, which ae best made using a tempeed cavity whose adiation escapes though a small hole. In a lase, much highe beam densities escape than in a black body. By concentating the beam density in an extemely small fequency spectum, the induced emission in a lase dominates, even in highe fequency anges. To expand upon the elationship of spontaneous to induced emission, we intoduce chaacteistic vibations, o modes. Theeby we can use eithe the photon pictue o the wave pictue in a closed cubic cavity with paallel mios. In the photon pictue, a photon is eflected back and foth between the mios. In the wave pictue, the field stength of a standing wave disappeas at the edge of the cavity. Fo that eason, we have to use a whole numbe multiple of λ/ fo the distance between the mios L. You can find in othe textbooks a futhe wave pictue which uses a wave moving back and foth instead of a standing wave. In that case, the distance between the mios L has to be a whole numbe multiple of λ, and the wave vecto k detemined fom the diffeent positive and negative diections of popagation whee k (π/l) (n x, n y, n z ) fo positive and negative values of n i. In ou futhe consideations, we use the pictue of a standing wave in a vacuum. The wave vecto fo an abitay standing wave in a cube of edge length L is: k π L (n x, n y, n z ) (.89) whee n i is a positive whole numbe. It is valid with k π/λ ν ω π c λ k c π c L n + n + n z. (.9) vecto potential can be deived fom the sum of all modes whee a j x y sin (k j ω j t). (.9) j The vecto amplitudes a j epesent time dependent vectos and evey index j and evey wave vecto k j stand fo a cetain combination of (n x, n y, n z ). We assume that is the vecto potential of the electomagnetic field and set div. With that it holds fo evey value of j the scala poduct k j a j. The wave vecto is theefoe pependicula to the amplitude vecto. The wave is tansvesal and can be epesented as a linea combination of two linealy polaized waves. Fo this eason evey vecto k j has two chaacteistic vibations, o modes (o states). Due to the wave vecto definition in equ. (.89), the k vecto can be epesented by a point in a thee dimensional kspace. The diffeence between this space and ou nomal D space is that it only contains points fo whole numbe values of n x, n y and n z. The numbe Δn of possible values of k in the intevals Δk x, Δk y and Δk z is equal to the poduct of Δn x Δn y Δn z. Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
23 Chapte, page With of k i (π/l) n i we get Δn L π Δk x Δk y Δk z. (.9) The numbe of points between k and k + Δ k is equal to the volume of a spheical shell. Since only positive values of n i ae consideed, only the elevant octant must be consideed (/8 of the total volume of the spheical shell): Δn L π 4π k Δ k. (.9) 8 If we additionally take into account that fo evey vecto, the two polaization possibilities of the wave give two modes, then the numbe of diffeent modes pe unit volume is Δn L π k Δ k. (.94) The tansition fom diffeences (Δ) to diffeential quantities (d) occus when we eplace Δn/L with n(ν) dν (the numbe of modes pe unit volume in the diffeential fequency ange) and Δ k by d k having consideed k πν/c. With that we get n(ν) dν 8 πν dν. (.95) c Hee we make an insetion, in ode to deive the adiation law afte Rayleigh. The enegy of a classical oscillato is the sum of potential and kinetic enegy which amounts kt fo each oscillation. This gives ρ ν n(ν) kt and we obtain the RayleighJeans law ρ ν 8 πν kt. (.79) c Coming back afte the (Rayleigh)insetion we we put equ. (.8) into (.95). Then the elationship of the emission coefficients is B n(ν) hν. (.96) By expanding this elationship with the spectal enegy density w ν the elationship of the induced to spontaneous emission pobability follows Bwν wν n ( ) ν h ν enegy of photons volume fequency volume fequency numbe of modes numbe of the photons. numbe of modes enegy of a photon Related to a single mode, this means that: the elationship of the induced to the spontaneous emission pobability is equal to the numbe of photons in this mode fo an abitay mode. With that the epesentation of the induced emission in the diagam at the beginning of this chapte has the following explanation: Induced emission occus when a photon with the appopiate enegy meets a mode containing many photons. (.97) Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
24 Chapte, page 4 With the peviously deived elationship, an easy consideation of the inteaction of electomagnetic adiation with atoms, molecules, and solid bodies is possible. lthough we will use quantum theoetical esults in ou consideations, we will not get into the deivations of these esults. In the case at hand, we can eplace the exact quantum theoetical desciptions with semiclassical deivations. Let us conside a dipole, fo example an antenna whose chage distibution changes with the cicula fequency ω. The time dependent electic dipole moment is μ(t) μ cosωt. (.98) The adiant powe of a classical (spontaneous) adiating dipole follows fom electodynamics as the aveage adiated powe. The time aveage of a peiodic function f is epesented by f. P em 4πε ( t) d μ 4 ω μ c dt πε c, (.99) cf. e. g. Landau/Lifschitz II, p. 5. In the tansition fom the middle to the ight pat of the equation, the aveage cos ωt ½ was used. The coespondence pinciple touches on the fact that quantum mechanical systems fo high quantum numbes obey the laws of classical physics. Though that it was possible to detemine selection ules and make statements about intensity and polaisation of spectal lines. By using this pinciple, we can take the following path: put into equ. (.99) the opeato fo the dipole moment μ q, in which q epesents the absolute value of the chages sepaated by the distance. The vecto µ is eplaced by vecto opeato $μ o q $ which is multiplied by two. The facto two is intoduced because of the two possibilities of the electon spin. With this opeato, it follows that the dipole moment of a tansition fom state to state is: M q ψ $ ψ dτ, (.) whee ψ is the wave function of state and ψ * is the complex conjugated function of state. The integation is done ove all vaiables of the functions (in this case ove space). Fom that we get the expectation value of the powe fom equ. (.99): P 4 ω πε c M. (.) Since is a vecto opeato, M is also a vecto: M $ M + M + M. x y Fo a spontaneously adiating dipole, we get the tansition pobability z P hν ω πε c M h 6π ν ε c h M. (.) Using the elation /B 8 hν /c, equ. (.8), and equ. (.) the Einstein coefficient of the induced emission can be calculated: B π M. (.) ε h Molecula Physics D. Feude Chapte Fields, vesion ovembe 5
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