2 Absorption and Emission of Radiation

Transcription

1 Chapter, page Absorption and Emission of Radiation. Eletromagneti Radiation In the year 886, Heinrih Hertz experimentally demonstrated the existene of eletromagneti waves and their equivalene to light waves. After Hertz s ahievement, the eletromagneti theory of James Clerk Maxwell, developed from 86 to 864, beame the basis of examining optial absorption and dispersion phenomena. As the name indiates, eletromagneti waves have two omponents. Fig.. shows a linearly polarized wave traveling in the x-diretion through a homogenous isotropi medium. The eletri field strength E osillates in the x-y-plane and the magneti field strength H osillates in the perpendiular x-z-plane. Both osillations have the same frequeny ν and the same wave vetor k, whih in this ase only has the x-omponent k x π/λ. The amplitude vetors A E and A H also only have one omponent eah. The following equations desribe the propagation of the linear polarized wave: E y H z A y E A z H os(k x x πν t), os(k x x πν t). (.) z-ahse, A H y-ahse, A E x-ahse Ausbreitungsrihtung Fig.. A linear polarized eletromagneti wave traveling in the x-diretion. The diretion of propagation and the diretion of the eletri field determine by definition the plane of polarization of the light. In this ase, it is the x- y-plane. Isotropi homogenous mediums are desribed by the material equations of Maxwell s theory. For the dieletri displaement D and the indued eletri polarization P, it holds that D ε E + P ε r ε E ε (+ χ e ) E. (.) ε refers to the permittivity of a vauum, ε r is the relative dieletri onstant and χ e is the eletri suseptibility. The magneti indution B has a material equation analogous to equ. (.). It is B μ (H + M ) μ r μ H μ (+χ) H. (.) μ is the permeability of a vauum, and μ r is the relative permeability onstant. In ontrast to the polarization P in equ. (.), the magnetization M in equ. (.) has the same dimension as field strength, and adds to the magneti field strength H. The eletri ontribution to the eletromagneti radiation plays the most important role for the majority of spetrosopi tehniques, whereas the magneti ontribution and the units introdued in equ. (.) are essential for tehniques of the magneti resonane. For the determination of the propagation speed of the wave, we put the onstant phase zero into the argument of the osine funtion in equ. (.). We then determine dx/dt and obtain λν. (.4) εεμμ ε r μ r r r From that we get the speed of light as the speed of propagation of an eletromagneti wave in the vauum, in whih ε r μ r. Spetrosopy D. Freude Chapter "Radiation", version June 6

2 Chapter, page The energy density w (energy per unit volume) is ED for an eletri field or BH for a magneti field. The time average of os ωt is ½. From that we obtain for the energy density of a linearly polarised eletromagneti wave w ½ ε r ε E y + ½ μ r μ H z. (.5) The Poynting-vetor S, whih refers to the energy flux density through a unit area, points along the x-diretion and is the produt of the energy density and the speed of light: S x w. (.6) From that we ome to the important onlusion that the energy flux of the radiation in the diretion of propagation is proportional to the squares of the amplitudes of the field strengths. Δt A x, k, S Fig.. Energy flux of an eletromagneti wave propagating in the x diretion. The wave vetor k and the Poynting vetor S also point in the x diretion. In the time Δt, the energy in the ube flows through the surfae A. If the time Δt is hosen to be one seond and the area A is the unit area, energy density and power density have the same numerial value.. Dipole Moments and other Quantities from Eletrodynamis To larify a few terms, suh as dipole moment and y polarizability, we use the so alled multipole expansion, whih desribes the eletri potential V (r) of a harge r r' distribution. The harges q n are loated at r' n, and the r r' origin of the oordinate system is loated within (or not x far from) the harge distribution. The left figure of the z water moleule shows the harge of the oxygen nuleus loated at r' and the observation point at r. Eletron harges are drawn in orange (grey). q n q ( x ) ( x ) ( x ) obervation point V (r) 4π ε n r rn n n + i n i n j n xi! xi xj r φ () + φ () + φ () q r n + µr + 5 θijxx i j. (.7) n r r The fator 4πε is introdued on the left so that the potential has the SI unit of volts. The potential in equ. (.) is for large distanes between the point of observation and the harge, i.e. r r» r' n. It is expanded by powers of /r by taking the derivatives with respet to r, the point under onsideration. The series development shows that the potential of any harge distribution an be represented by a sum of multiples. i, j Spetrosopy D. Freude Chapter "Radiation", version June 6

3 Chapter, page Let us now onsider an eletrially neutral moleule, in whih the positive nulear harge and the negative eletron harge ompensate eah other. In this ase, the first term of the expansion φ () is zero. φ () is the dipole moment and φ () the quadrupole moment. Here we stop the expansion. φ () an be written as μr /r or μe r /r, where e r is the unit vetor in the r-diretion. μ q n r n (.8) n μ is defined as the dipole moment of a harge distribution. It does not depend on the loation of the origin in the ase of neutrality of the harge loud of atoms or moleules. The unit of the dipole moment is Asm. The old gs unit named after Peter Debye, in whih D,564 Asm, is still in use beause the dipole moments of small moleules are in the range of D (H O,85 D, HCl,8 D). Be areful not to onfuse the Debye with atomi unit ea 8,478 Asm, whih refers to the elementary harge e,6 9 As and the Bohr radius a 5,9 m. φ () in equ. () desribes the potential of a quadrupole. The quadrupole moment is [ ( ) ( ) ] θ ij q n x n x i n r j n ij n δ (.9) at the origin. δ ij is the Kroneker symbol. From equ. (.7) follows θ ij θ ji and from equ. (.9) we an see that the quadrupole tensor has no trae. Even though single magneti harges do not exist, we an write a relationship for magneti potential analogous to equ. (.7). The magneti moment, whih is also represented by μ, plays together with the eletri dipole moment an important role in the eletromagneti dipole radiation. ow we onsider dieletri material onsisting of partiles without permanent dipole moments, e.g. CO or CH 4. A moment μ ind an be indued through the eletri polarizability α (units: Asm V ) under the influene of an external eletri field E. The orresponding polarizability tensor α is defined by μ ind α E (.) This linear effet is suffiient for the onsideration of weak fields. The basis of the onsideration of the non-linear optis (LO) is the extended equation μ ind α E + β E + γ E +... on-linear effets play an important role in the laser spetrosopy. For weak fields we have the eletri polarization P as the indued dipolar moment per unit volume, f. equ. (.), P χ e ε E, (.) where the eletri suseptibility χ e is a salar dimension-less unit for isotropi material. The eletri field E is stati, if it is aused by a diret urrent (DC) soure applied to a apaitor. If we replae DC by AC (alternating urrent) with the frequeny ν, we get the orresponding magnetization of the field only in the ase, if the harges an hange their orientation quikly enough. The eletroni polarizability produed by shifting the positively harged nuleus with respet to the negative eletron shell takes plae in less than 4 s. The polarization by the shifting or vibration of the ions in a moleule or lattie (ion polarization, distortion polarization) happens a thousand times more slowly, on the order of s. Both types of polarization are united under the term of displaement polarization. The orientation polarization is muh slower and therefore plays no role for the index of refration (optial Spetrosopy D. Freude Chapter "Radiation", version June 6

4 Chapter, page 4 range). It is aused by the lining up of permanent moleular dipoles whih are present even in the absene of an external field. The dieletri relaxation of the orientation polarization an be experimentally examined with high frequenies and helps determine the dynamis of the system. DC spetrosopy is the frequeny dependent measurement of the relative dieletri onstant.. Absorption and Dispersion The phase speed λν, whih is defined as the produt of wavelength and frequeny, is redued in omparison to speed of light in a vauum,, when the eletromagneti wave travels through a medium with an index of refration n >. The redued value is /n. We will show that the frequeny dependeny of n leads to a dispersion, whih an be desribed using a lassial model. It will be also shown that the imaginary part of a omplex index of refration desribes the damping of an eletromagneti wave. For this presentation we onsider an eletri field with the amplitude vetor A E (, E, ), whih has a omplex time dependeny exp(iωt) instead of the os ωt of equ. (.). The differential equation of a damped osillation fored by an external field is m d dt + mγ d y dt + mω y q E exp(iωt), (.) y where the mass of the osillator is m, the harge q, and the harateristi frequeny ω. γ is the damping onstant. With an exponential trial solution of y y exp(iωt) we obtain qe y m ( ω ω + iγω) (.) as the omplex amplitude of the osillation. An indued eletri dipole moment μ ind appears in the y diretion: q E μ y q y exp(iωt). (.4) m ω ω + iγω ( ) With osillators per unit volume, we obtain P ind χ e ε E μ (.5) as the indued eletri polarization, and with that a omplex suseptibility χ e q. (.6) ε m ω ω γω ( + i ) The real and imaginary parts of χ e are not independent, as an be shown by multiplying numerator and denominator of equ. (.4) by the onjugated omplex of the parenthesis. In a vauum we have μ r ε r. From the definition n / and equ. (.4), it follows that n με r r. (.7) Spetrosopy D. Freude Chapter "Radiation", version June 6

5 Chapter, page 5 Sine we are not onsidering ferromagneti materials, we an set μ r with suffiient auray, and we obtain the Maxwell relation n ε r + χ e. (.8) We should note that these quantities are frequeny dependent. For example, the orientation polarization mentioned earlier has no effet on the suseptibility in the optial frequeny range. From equations (.6) and (.8) it follows that the index of refration represents the omplex quantity n + q. (.9) ε m ω ω γω ( + i ) To separate this into real and imaginary omponents, different onventions are in use. We write n n' i n". (.) When n, whih is the ase in gaseous media, we an make the approximation n (n + ) ( n ) (n ). ear the resonant frequeny ω ω «ω or 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 / ) (.). (.) The frequeny dependent quotient in equ. (.) is desribed by a funtion in the form of y /( + x ), whih is ommonly alled Lorentz urve. The parameter γ is the full-width-athalf-maximum of the urve. The Lorentz urve will be desribed in more detail in hapter.6. The meaning of the real and imaginary omponents an be larified by the following onsiderations: Analogous to equ. (.), we have for a wave propagating in the x-diretion E y A y E exp [i(ωt k x x)]. (.) The wave vetor k an be replaed by nk, where k with k ω/ is the wave vetor in the vauum. From equation (.) and (.) it follows that E E E y A y exp[i (ωt k x {n' in"} x)] A y exp[ n"xω/ ] exp[ik x ( t n'x)]. (.4) The first exponent on the right side of equ. (.4) desribes a damping of the wave. Later we will show how the absorption, desribed by the imaginary part of the index of refration n", is related to experimentally measurable extintion oeffiient. The seond exponent desribes the dispersion. In onnetion with equ. (.), we get from that the dependeny of the phase speed on the frequeny. Spetrosopy D. Freude Chapter "Radiation", version June 6

6 Chapter, page 6 If we set the harge of the osillator q to be the elementary harge e, equation (.) desribes the total absorption of atoms with a single valene eletron. The eletrons i in state i an, through absorption, move into new states k (inluding non-disrete states in the ontinuum). For this reason, only a portion f ik of the total absorption has to be onsidered for the transition from the state i to the state k. For these so-alled osillator strengths it holds: f ik. (.5) k With the osillator strengths f ik, the disrete transitions an be introdued into the lassially derived equation. The imaginary part of the index of refration then beomes: n" e i ε m ω f ik γ ik. (.6) + k ( ω ω ) ( γ ω) ik ik Here, the half-width of the absorption line for the transition from i k is γ ik, and it has to be summed over all possible exited levels k. Sine the frequenies ω ik streth over a wide range it is not possible to input a single frequeny that fulfils the ondition ω ω ik «ω ik for all values of k. For this reason, we did not make use of the approximation that ω ω ik «ω ik and ω + ω ω ω in the derivation of equ. (.6) in ontrast to the proedure followed in the derivation of equations (.) and (.). We annot, therefore, diretly ompare equ.(.6) with equ.(.4). We will return to an explanation of extintion oeffiients in hapter.8, equ.(.6)..4 Spontaneous and Indued Transitions, Radiation Laws A spontaneous event needs no external influene to our. The light of a thermal radiator, whih we an visually see, ours when a substane at high temperature spontaneously emits quanta of light. An indued or stimulated event only ours with external influene. Aordingly, absorption is always indued (stimulated). But emission an be indued, if a frequeny equal to that of the light to be emitted is externally input. Let us now onsider two energy levels of an isolated partile, see below. Sine the following onsiderations are appliable to any states, we will label them with i and j. Here, and in the next two setions, we will set i and j. Let E > E and E E hν, where h 6,66 4 Js denotes the Plank onstant. The oupation numbers of the states are and. Energy E hν B ρ ν The number of partiles whih go from state to state is, d B w ν dt, (.7) E B ρ ν A Absorption Indued emission Spontaneous emission Fig.. Absorption, indued and spontaneous emission. where B w ν is the absorption probability with the spetral energy density w ν. Spetrosopy D. Freude Chapter "Radiation", version June 6

7 Chapter, page 7 The energy absorbed by the partiles for the transition is given by dw abs hν d. (.8) The energy emitted in the form of radiation by the transition from to is dw em hν d. (.9) For the balane of partiles that go from to, we need to onsider a spontaneous transition probability A in addition to the transition probability B w ν : d (B w ν + A ) dt. (.) The probability A does not depend on external fields. The probability of an indued transition, however, does depend on the external field. It is the produt of the B oeffiients with the spetral energy density w ν of the external fields in the frequeny range from ν to ν + dν. The spetral energy density w ν has the units of energy per volume and frequeny. Instead of this quantity, the spetral beam density L ν is often used. L ν is the power in the frequeny range ν to ν + dν that is emitted per unit area in a one of solid angle Ω. A solid angle Ω would mean that m is ut out of the total surfae area of 4π m of a sphere with a radius of m. The aperture angle of the one is about 66. In a vauum, where the speed of light is it holds that L ν w ν /4π. (.) B and B are the Einstein oeffiients for absorption and indued emission. With the help of these oeffiients Albert Einstein ould find a simple and seure proof of the radiation law in 97. The radiation law was disovered at the end of 9 by Max Plank through an interpolation (of the behavior of the seond derivative of the entropy with respet to the energy) between Wien s radiation law and Rayleigh-Jeans radiation law. Einstein s derivation starts with a losed avity in a heat bath at the temperature T. Beause of equilibrium, we have for two arbitrary states between whih transitions our that the number of absorbed and emitted quanta of energy must be equal. w ν is in this ase the spetral energy density of a blak body, labeled with ρ ν. From (A + B ρ ν ) B ρ ν it follows that A B ρ + B ν. (.) ρ ν On the other hand, Boltzmann statistis an be applied to this system: g E exp g E kt g g exp hν kt. (.) k refers to the Boltzmann onstant, and h is Plank s elementary quantum of ation. The statistial weights, g, are from now on set to g g, i.e. a degeneration of the energy levels will not be onsidered. Spetrosopy D. Freude Chapter "Radiation", version June 6

8 Chapter, page 8 With equations (.) and (.) we arrive at: ρ ν B A h ν kt e B. (.4) In equ. (.4), no statement about the relationship between B and B is made. If we make the plausible assumption from T follows ρ ν, we get from equ.(.4) the relation B B. For the determination of the relationship between A and B the radiation law from 9 stated by Lord Rayleigh and James Hopwood Jeans is used. In the low frequeny range (hν «kt), equ. (.4) should fulfill the Rayleigh-Jeans law ρ ν 8 πν kt, (.5) whih we will derive later using lassial statistis. With exp (hν/kt) + hν/kt and hν «kt, we get from equ. (.4) by setting B B ρ ν A kt Bhν. (.6) From equations (.5) and (.6) it follows for arbitrary relationships between hν and kt that the valid relationship between the spontaneous and indued transition oeffiients is A B 8 h πν. (.7) Equation (.7) put into (.4) leads us to the famous Plank radiation law: π ν ρ 8 h ν e hν kt. (.8) If we use the wavelength dependent energy density ρ λ dλ instead of the frequeny dependent energy density ρ ν dν, we onlude that in a vauum: π ρλ 8 h 5 h. (.9) λ λkt e using the relationships ν /λ and dν /λ dλ. The Rayleigh-Jeans law, whih applies when hν «kt, is used in Einstein s derivation of the Plank radiation law. Other radiation laws were not used, but an be presented as results of the Plank radiation law in the frame of the Einstein derivation: For hν» kt it holds that exp (hν/kt)», and we get from equ. (.8) as a speial ase the Wien radiation law, derived by Wilhelm Wien in 896 (up to the fators later determined to be 8πh/ and h/k): π ν ρ 8 h ν e h ν kt. (.4) We use the first derivative of equ. (.9) with respet to the wavelength and set this to zero, we get the maximum of the spetral energy density of the blak body at λ max. The wavelength follows the relationship Spetrosopy D. Freude Chapter "Radiation", version June 6

9 Chapter, page 9 λ max T onst. h,8978 mm K (.4) k 4, 965 and desribes a displaement of the maximum of the intensity distribution to shorter wavelengths as the temperature inreases. (The number 4,965 is the zero point of the derivative, rounded to the nearest deimal. Beause of this, the number,8978 is also rounded). This law, derived by Wien in 89, is known as Wien s displaement law. It was the basis of his thoughts for the first form of his radiation law. At K, the maximum of the radiation of a blak body is in the infrared at approx. μm. Only at about 4 K does it move into the visible spetrum. From equations (.9) and (.4) we get the law ρ λ max onst. T 5 (.4) for the energy density in the range of the maximum. For ompleteness, mention also Josef Stephan s empirial law of 878, later larified with thermodynamis by Ludwig Eduard Boltzmann. It is known as the Stefan-Boltzmann law, and is arrived at by integrating equ. (.9): 6 4 ρλdλ T 4 8π k 5 h σ T4. (.4) The total radiation of the blak body is proportional to the fourth power of the temperature. We stress again that in the above equations, energy densities are used. To onvert to the often used beam density, use equ. (.). For example, the fator σ in equ. (.4) is hanged into π 5 k 4 /(5 h ) 5,67 8 W m K 4, if we use L λ instead of ρ λ. ρν /Jsm,E-5,E-5 8,E-6 6,E-6 4,E-6 Rayleigh-Jeans 5 K Plank 5 K Wien 5 K Fig..4 Frequeny dependeny of the spetral energy density of the blak body at a temperature of 5 K aording to the laws of Wien, Rayleigh Jeans, and Plank. On the horizontal axis, instead of the frequeny ν, the wave number ~ ν ν/ is used. Using Plank s law, the urve for K is also shown. The visible range of the spetrum lies between m and 6 m.,e-6 Plank K,E+ 4 wave numbers / m Spetrosopy D. Freude Chapter "Radiation", version June 6

10 Chapter, page If we use the Einstein oeffiients, we get the relationship between spontaneous and indued emission probabilities by rewriting equ. (.6): B A ρν hν. (.44) kt At a temperature of K, the equilibrium between both probabilities is at ν k K / h 6,5 Hz, and ~ ν 8 m or λ 48 μm, in the far infra red. That is true for blak body radiators, whih are best made using a tempered avity whose radiation esapes through a small hole. In a laser, muh higher beam densities esape than in a blak body. By onentrating the beam density in an extremely small frequeny spetrum, the indued emission in a laser dominates, even in higher frequeny ranges. To expand upon the relationship of spontaneous to indued emission, we introdue harateristi vibrations, or modes. Thereby we an use either the photon piture or the wave piture in a losed ubi avity with parallel mirrors. In the photon piture, a photon is refleted bak and forth between the mirrors. In the wave piture, the field strength of a standing wave disappears at the edge of the avity. For that reason, we have to use a whole number multiple of λ/ for the distane between the mirrors L. You an find in other textbooks a further wave piture whih uses a wave moving bak and forth instead of a standing wave. In that ase, the distane between the mirrors L has to be a whole number multiple of λ, and the wave vetor k determined from the different positive and negative diretions of propagation where k (π/l) (n x, n y, n z ) for positive and negative values of n i. In our further onsiderations, we use the piture of a standing wave in a vauum. The wave vetor for an arbitrary standing wave in a ube of edge length L is: k π L (n x, n y, n z ) (.45) where n i is a positive whole number. It is valid with k π/λ ν ω π λ k π L n + n + n z. (.46) A vetor potential A an be derived from the sum of all modes where x y A sin (k j r ω j t). (.47) j a j The vetor amplitudes a j represent time dependent vetors and every index j and every wave vetor k j stand for a ertain ombination of (n x, n y, n z ). We assume that A is the vetor potential of the eletromagneti field and set diva. With that it holds for every value of j the salar produt k j a j. The wave vetor is therefore perpendiular to the amplitude vetor. The wave is transversal and an be represented as a linear ombination of two linearly polarized waves. For this reason every vetor k j has two harateristi vibrations, or modes (or states). Due to the form of the wave vetor presented in equ. (.45), the k vetor an be represented by a point in a three dimensional k-spae. The differene between this spae and our normal -D spae is that it ontains only points for whole number values of n x, n y and n z. The number Δn of possible values of k in the intervals Δk x, Δk y and Δk z is equal to the produt of Δn x Δn y Δn z, i.e. it is valid beause of k i (π/l) n i Δn L π Δk x Δk y Δk z. (.48) Spetrosopy D. Freude Chapter "Radiation", version June 6

11 Chapter, page The number of points between k and k + Δ k is equal to the volume of a spherial shell. Sine only positive values of n i are onsidered, only the relevant otant must be onsidered (/8 of the total volume of the spherial shell): Δn L π 4π k Δ k. (.49) 8 If we additionally take into aount that for every vetor, the two polarization possibilities of the wave give two modes, then the number of different modes per unit volume is Δn L π k Δ k. (.5) The transition from differenes (Δ) to differential quantities (d) ours when we replae Δn/L with n(ν) dν (the number of modes per unit volume in the differential frequeny range) and Δ k by d k having onsidered k πν/. With that we get n(ν) dν 8 πν dν. (.5) Here we make an insertion, in order to derive the radiation law after Rayleigh. The energy of a lassial osillator is the sum of potential and kineti energy whih amounts kt for eah osillation. This gives ρ ν n(ν) kt and we obtain the Rayleigh-Jeans law ρ ν 8 πν kt. (.5) Coming bak after the (Rayleigh-)insertion we use (.7) and (.5). Then the relationship of the emission oeffiients is A B n(ν) hν. (.5) By expanding this relationship with the spetral energy density w ν the relationship of the indued to spontaneous emission probability follows Bwν wν A n ( ) ν h ν Energy of photons Volume Frequeny Volume Frequeny umber of Modes umber of the Photons. umber of Modes Energy of a photon (.5) Related to a single mode, this means that: the relationship of the indued to the spontaneous emission probability is equal to the number of photons in this mode for an arbitrary mode. With that the representation of the indued emission in Fig.. has the following explanation: Indued emission ours when a photon with the appropriate energy meets a mode ontaining many photons..5 Calulation of Trasition Probabilities The onsideration of the interation of eletromagneti radiation with atoms, moleules or solids requires a quantum mehanial treatment like in the text books of Haken and Wolf. But Spetrosopy D. Freude Chapter "Radiation", version June 6

12 Chapter, page here we will replae the exat quantum theoretial desriptions with semi-lassial derivations. Let us onsider a dipole, for example an antenna whose harge distribution hanges with the irular frequeny ω. The time dependent eletri dipole moment is μ(t) μ osωt. (.54) The radiant power of a lassial (spontaneous) radiating dipole follows from eletrodynamis as the average radiated power (Landau/Lifshitz II, p. 5). The time average of a periodi funtion f is represented by f. P em 4πε ( t) d μ 4 ω μ dt πε. (.55) In the transition from the middle to the right part of the equation, the average os ωt ½ was used. The orrespondene priniple touhes on the fat that quantum mehanial systems for high quantum numbers obey the laws of lassial physis. Though that it was possible to determine seletion rules and make statements about intensity and polarization of spetral lines. By using this priniple, we an take the following path: put into equ. (.55) the operator for the dipole moment μ qr, in whih q represents the absolute value of the harges separated by the distane r. The vetor µ is replaed by vetor operator $μ or q $r whih is multiplied by two. The fator two is introdued beause of the two possibilities of the eletron spin. With this operator, it follows that the dipole moment of a transition from state to state is: M q ψ r$ ψ dτ, (.56) where ψ is the wave funtion of state and ψ * is the omplex onjugated funtion of state. The integration is done over all variables of the funtions (in this ase over spae). From that we get the expetation value of the power from equ. (.55): P 4 ω πε M. (.57) Sine is a vetor operator, M is also a vetor: M $r M + M + M. x y For a spontaneously radiating dipole, we get the transition probability z A P hν ω M πε h 6π ν ε h M. (.58) With A /B 8 hν /, equ. (.7), and equ. (.58) the Einstein oeffiient of the indued emission an be alulated: B π M. (.59) ε h The equations (.58) and (.59) desribe the relationship of the emission oeffiients B and A to the absorption oeffiients B ( B ) with the dipole moment of the transition M, whih is onneted to the wave funtions of the states under onsideration by equ. (.56). Sine the partiles to be studied are haraterized by these wave funtions, equations (.58) and (.59) represent an important basis of the interation of partiles with eletromagneti Spetrosopy D. Freude Chapter "Radiation", version June 6

13 Chapter, page radiation. They are the basis of many spetrosopi experiments. The dipole moment M of the transition is rarely alulated. Even without alulation, we an determine whether M is zero or has a finite value from the symmetry onsiderations of hapter (forbidden and allowed transitions)..6 Lifetime and atural Line Width Let us onsider state in Fig.. to be the exited level and assume that it is not oupied at thermal equilibrium. An exitation at time t aused the oupation. The transition from state to an be aused by both spontaneous and indued proesses. When we speak of lifetimes, we mean in general the lifetime of an exited state, whih is ended by the spontaneous emission of a photon. For the partiles that leave state, we have an equation analogous to equ.(.): d A dt. (.6) By integrating equ.(.6) and onsideration of the initial ondition (t ) we obtain: exp ( A t). (.6) The time average of the funtion (t) is the mean lifetime τ of the partiles in the exited state t () t t dt τ () t dt ( ) t exp At dt ( ) exp A t dt A. (.6) From that we see that the time /A, after whih (t) has been redued to /e of its initial value, is equal to the average lifetime τ of the partiles. Out of the measurement of the lifetime of exited states it is possible to diretly speify the emission probabilities and alulate the Einstein oeffiient B using equ.(.7). The standard deviation Δt refers to the root-mean-square deviation from the average lifetime τ. The standard deviation of the lifetime Δt in this ase is also τ beause (Δt) ( τ) ( ) t t dt () t dt ( t τ) t exp dt τ t exp τ dt τ (.6) A priniple of quantum mehanis formulated by Werner Heisenberg in 97 states that the produt of the unertainty of any two mutually anonially onjugated quantities suh as loation and momentum or energy and time an never be smaller than 4π divided by Plank s onstant h: Spetrosopy D. Freude Chapter "Radiation", version June 6

14 Chapter, page 4 h ΔE Δt 4π h. (.64) With ΔE h Δν and the result from equ.(.6) we get Δν 4π Δt 4πτ (.65) as the smallest limit for the unertainty in the frequeny. From a mathematial viewpoint, this is a standard deviation. To derive a lassial relationship with a similar result to that of equation (.65), we make use of Jean Baptiste Joseph Fourier s transformation. It is the basis of Fourier spetrosopy whih we will enounter time and again in the following setions. Fourier s original form from 8 was oneived to desribe the spatial distribution of temperature. In spetrosopy, it is mainly used to transform signals from the time domain into the frequeny domain and vie-versa. The symmetri form of the Fourier transformation is written and + π exp i dω (.66) g(t) f ( ω) ( ωt) f(ω) () ( ω ) π + gt exp i t d t. (.67) Let us now onsider the funtion g(t) exp( t/t d ) os ω t, (.68) whih, with t > and < /T d «ω, desribes an osillation of frequeny ω and exponential damping with time onstant T d. For t <, g(t). In our further onsiderations, the realvalued funtion g(t) an be replaed by the omplex funtion [note: exp(iω t) osω t + i sinω t] g(t) exp( t/τ d + i ω t). (.69) The Fourier transform of this funtion using equ.(.67) an be found in referene books: ( ω ω) ( ω ω) i Td T T d d f (ω) + i i π + ( ω ) ω T π d + π ω Td ω + Td The omplex funtion f(ω) has been separated into real and omplex parts on the right hand side of equation (.7). The frequeny dependent real part is. (.7) π f ( ω) T d + ω ω ( ) T d f Lorentz. (.7) Spetrosopy D. Freude Chapter "Radiation", version June 6

15 Chapter, page 5 This is the Lorentz urve, named after Hendrik Antoon Lorentz, in whih /T d Δω ½ is the simple half width, and /T d δω ½ is the full width at half maximum (FWHM). In spetrosopy, the latter quantity is often simply referred to as the half-width. f Lorentz / Δω / δω / /T d ω Fig..5 Lorentz urve and it s halfwidths Δω / and δω /. ω Our further onsiderations follow a similar development to those of hapter.. The Lorentz urve f ' (ω) in equ.(.7) has the same frequeny dependeny as the imaginary part of the index of refration in equ.(.), and the imaginary part f " (ω) is similar to the real part of the index of refration in equ.(.). In the derivation of the equations for the index of refration, we started with the differential equation (.) of a damped osillator under the influene of an external eletri field. The free osillator is desribed by the orresponding homogenous differential equation, in other words, the external field has zero amplitude: m d dt + mγ d y dt + mω y (.7) y With the initial onditions y and dy/dt at t, the real valued solution of the differential equation (.7) is y(t) exp( tγ/) [osωt + (γ/ω)sinωt] where ω ω If the damping is weak, < γ «ω, and it holds that ω ω, we get γ. (.7) 4 y(t) exp( tγ/) osω t (.74) for the damped osillation of frequeny ω with amplitude exp( tγ/). The approahes (.68) and (.69) also represent real and omplex solutions, respetively, when T d /γ. If we multiply equation (.7) by dy/dt, we get d m dy m dy y + m dt dt + ω γ. (.75) dt The two terms in square brakets orrespond respetively to the kineti and potential energies, and therefore to the total energy W of the osillation. From that it follows from equations (.74) and (.75) for the radiant intensity (radiant power) that dw mγω exp( γt) sin ω t. (.76) dt Spetrosopy D. Freude Chapter "Radiation", version June 6

16 Chapter, page 6 The time average of the sin funtion over a omplete period of rotation is ½. With that we arrive at an average power dw dt ½mγω exp( γt) (.77) that is proportional to the square of the amplitude funtion exp( γt/). The power falls to /e of its initial value after a time t /γ. The time onstant /γ an now be onsidered to be the average lifetime of a large number of undamped but time limited vibrating osillators. In analogy to the lifetime of a state τ, we set /γ τ. By omparing the time funtion (.68) to the orresponding frequeny funtion (.7) and onsidering T d /γ, we an see that an osillator of average lifetime τ produes a Lorentz urve of half-width δω ½ /τ. Expressed in frequenies, it holds δν ½ πτ. (.78) This lassially derived equation is similar to the quantum mehanial unertainty relation (.65). These relations annot, however, be transformed into eah other. For example, the omparison of equ.(.6) with (.7) shows that the standard deviation of a Lorentz urve diverges. It is nevertheless generally true that the natural profile of a spetral line is a Lorentz urve whose half width is speified by the finite lifetime τ through equation (.78). Up to now we have always assumed that the partiles only have finite lifetimes in state. If state is not the ground state, the partiles have finite lifetime in both states and we have to replae the value τ in equ. (.78) by +. (.79) τ τ τ The lifetimes of exited optial states range from the pioseonds to seonds for forbidden transitions. The line width in equ.(.78) hanges aordingly. Quotients of frequenies and line widths or of lifetimes and osillation periods always produe large values. For example, for the prodution of the Fraunhofer line D, the ground state of sodium s S / and the exited state p P / with a lifetime τ 6 ns are involved. The wavelength of the line is λ /ν 589, nm. With these numbers we get a frequeny of around 5 4 Hz, and from equ. (.78) it follows that δν ½ 7 Hz; the quotient of frequeny and line width is therefore 5 million. On the other hand, sine τ/t τν 8 6, we see that the amplitude of the emitted radiation is only notieably redued after a few million osillations..7 Doppler Effet Broadening, Homogenous/Inhomogenous Broadening, Saturation The natural line broadening onsidered in hapter.6 is the lower limit of the line width. Observed line profiles an be broadened by the measurement apparatus or by saturation from strong inoming radiation. Additionally, broadening ours as a onsequene of atomi and moleular motion within the substane under study. Elasti and inelasti ollisions between the partiles result in the so alled ollision broadening or pressure broadening. If ollision indued transitions our, the lifetime of a state is shortened, and the line broadening an be Spetrosopy D. Freude Chapter "Radiation", version June 6

17 Chapter, page 7 alulated from equ.(.78). Broadening effets our in various forms. Here we will examine Doppler Broadening, whih dominates in low pressure gases. The priniple referred to in 84 by Christian Doppler and proven a few years later in both aoustis and optis says that a frequeny hange takes plae if the tone (or radiation) soure and observer (or reeiver) are in relative motion to eah other with the speed v. If k is the wave vetor, and we neglet relativisti effets, the differene between the observed frequeny ω and the emitted frequeny ω is given to us by the relationship ω ω kv. Let s onsider a wave moving in the x diretion, k (k x,,). Beause k ω / it follows ω ω ( + v x / ). Rearranging v x gives v x ω ω. (.8) ω The Maxwell-Boltzmann veloity distribution gives for partiles of mass m and most probable speed v p v p v p kt m at temperature T. The number of partiles n(v) dv with speed lying between v and v + dv is n(v)/ onst. (v/v p ) exp( [v/v p ] ). If we now onsider the x omponent v x alone instead of the absolute value of the speed v we get the one-dimensional equation n( v ) x v onst. exp v x p (.8). (.8) The intensity I(ω) of the absorbed or emitted radiation depends on the number of partiles that absorb or emit a ertain frequeny and therefore, beause of the Doppler effet, also depends on v x. Putting equation (.8) into (.8) we get: ( v ) n With that, x onst. exp ω ω ωv p. (.8) I(ω) I(ω ) ω ω exp. (.84) ωv p The Intensity distribution orresponds to the Gauss bell urve φ(z) exp ( z /) / π, whih was introdued by Karl Friedrih Gauß as the probability density of the normal distribution. The half-width δω ½ is determined from equation (.8) with v p to be δω Doppler ½ ω 8kT ln. (.85) m By using Avogadro s number A, the molar mass M A m, the gas onstant R A k and the speed of light in a vauum it follows that Spetrosopy D. Freude Chapter "Radiation", version June 6

18 Chapter, page 8 δω ω Doppler / T/ Kelvin M/ Gram 7,6 7. (.86) For example, for the a D -Line at 589, nm at 5 K mentioned above, we get δω ½ Doppler /π,7 9 Hz and therefore broadening by a fator of 7 as ompared to the natural line width. Homogenous and inhomogeneous line broadenings our, by definition, when all partiles under onsideration moving between states E i E k have, respetively, the same or differing transition probabilities. A typial example of homogenous broadening is the natural line width, and a typial example for inhomogeneous broadening is aused by the Doppler effet. If we have a Doppler broadened line, we ould input one frequeny v x that only auses transitions in a ertain interval of relative speed v x, while leaving the other parts of the line unaffeted so that no absorption takes plae. Suh proesses an be seen in the saturation behavior of the line. Saturation ours in absorption spetra when the differene in the oupation numbers of the two levels under onsideration is signifiantly hanged by the input energy, in the extreme ase the oupation numbers beome equal. If the initial values of the oupation numbers are maintained by suffiient spontaneous or indued emission, we speak of linear absorption. In this ase, the absorbed power is proportional to the input power. a b Fig.6 (a) Saturation behavior of a homogenously broadened line. The solid line has the Lorentz profile of an unsaturated line. The dotted absorption line has been inreased on the outsides by doubling the input power, but is redued in the middle. (b) saturation behavior of an inhomogenously broadened line. The unsaturated absorption line here has a Gauss profile. The dotted line is aused by saturation and has a Lorentz profile with the natural line width. Saturation hanges the line form. A homogenously broadened line is more strongly saturated in the middle sine the energy transfer is greatest there. This auses line broadening. In Fig.6 a, the solid urve portrays the measured absorption profile without saturation. To reate the dotted urve, the input power was doubled and the saturation effets were taken into aount. The half-width of the line is visibly inreased. A saturation effet that ours in inhomogenously broadened lines is used as a fundamental property for the definition of inhomogeneous broadening: if we san the absorption profile with a seond variable frequeny of small amplitude and simultaneous strong (saturating) input at ω s, a hole is burnt by the saturation in the line profile. In Fig..6 b, both the homogenous and inhomogeneous line widths an be seen in the saturated absorption line..8 Lines and Band Intensities in Optial Spetra Pierre Bouguer determined in 76 that the attenuation of a beam of light in an absorbing medium is proportional to the intensity of the beam and the length of the medium. Johann Heinrih Lambert desribed in 76 this fat with an equation and August Beer found in 85 through absorption measurements on rarefied solutions that the transmittane of a material Spetrosopy D. Freude Chapter "Radiation", version June 6

19 Chapter, page 9 with a onstant ross setion only depends on the amount of material through whih the light shines. These realizations led to the fundamental law for quantitative optial spetrosopy, whih is named after the last, the last two, or all three disoverers. If the light intensity transmitted by an absorbing layer is I D I, then it follows for the logarithm of the transmittivity and the transmittivity log I I ε ν M d and D exp ( ε ν M d ln ). (.87) The ε ν in the Beer law is the frequeny dependent molar extintion oeffiient and d is the thikness of the layer of the substane (normally in m). M is the onentration of the absorbing substane, normally given in Mol per Liter and often alled the molarity. The dimension of ε ν is volume mole thikness of layer, where the usual units are mole/liter and m for the thikness of the layer, thus m mol. The SI unit of m per mole (m mol ), whih is times larger, is rarely used. Unfortunately, the molar extintion oeffiient is often given with no units at all, even though it has dimensions, in ontrast to the extintion ε ν M d. There is also the possibility for onfusion by the usage of the natural extintion oeffiient ε ν n, where in equ. (.87) in plae of the deade logarithm the natural logarithm is used. It is valid that ε ν n ε ν ln ε ν,. The empirially introdued extintion oeffiient depends on the imaginary part of the index of refration, whih desribes the damping of the eletri field as it passes through a medium, equ. (.4). Sine the radiant energy is proportional to the square of the amplitudes of the field strength, see hapter., from the omparison equations (.4) and (.87), this relation follows: n"ω/ ε ν n M m ν, (.88) in whih ε ν n M is the natural extintion module m ν (extintion per unit length). Equation (.) inserted into (.88) shows that the extintion near the resonane is represented by a Lorentz urve. In pratial appliations it is advantageous to measure the integral extintion of a line or band. For the derivation of the orresponding relation we will limit ourselves to linear absorption. In that ase, for a beam inident perpendiular to the surfae F, that the inident spetral radiant intensity I w ν F. The absorbing spetral intensity follows from the relation I I exp( m ν x), see equations (.87) and (.88), di w ν F m ν dx. (.89) With F as unit area we get the spetral absorbed intensity I abs per unit volume after integration of the x-oordinate over the unit length: I abs w ν m ν, (.9) whih is a funtion of the frequeny. In the frequeny range of a line (or band), the spetral energy density w ν of the inident eletromagneti wave an be onsidered onstant. Through that we get after integration over the frequeny range of a line the integral absorbed intensity, whih is a power density: Spetrosopy D. Freude Chapter "Radiation", version June 6

20 Chapter, page dw abs dt line end line end wν mν dν wν mν dν wν s. (.9) line begin line begin The integral absorption oeffiient, introdued here and the integral extintion module s an be rewritten as integral extintion oeffiient after division by the onentration of the substane. This is more useful for pratial appliations than the frequeny dependent value. We must of ourse be areful not to integrate over the whole spetrum but rather over the line or band under onsideration. The net rate of the transitions i k, see equ. (.7), is B ik w ν i in a transition between non degenerate energy levels with E k > E i and k «i. For every transition, the energy hν ik is absorbed. i is the number of states per unit volume. We then have for the absorbed power per unit volume: d dt w abs hν ik B ik w ν i. (.9) With equations (.9), (.9) and (.59) it follows that s h B ν ik ik i π ν ik i M ε h ik. (.9) These equations ombine the integral absorption oeffiient with the dipole moment of the transition, i.e. a parameter determined by an experimentally measured spetrum with the quantum mehanial expetation value. The later is not easy to alulate and depends on the symmetry of the moleule or solid body building blok and perturbations of this symmetry. For this reason, quantitative statements from optial spetra are problemati. With the above equations, it is possible to derive the onnetion between the Einstein oeffiients and the osillator strengths: By ombining equations (.88), (.9) and (.9) we get: s h B ν ik ik i Line end Line begin Line end 4πν ik mν dν n dν. (.94) Line begin ow let us return to equation (.6) and onsider only frequenies near resonane in the ombination ik. With that we get a relation similar to equation (.). For pratial reasons we replae the angular frequenies with frequenies and end up with: n" e i fik γ ik 6πεmνik ( νik ν) + ( γ ik / ). (.95) (The half-widths γ used here differ from those used in equations. to.6 by the fator π). Spetrosopy D. Freude Chapter "Radiation", version June 6

21 Chapter, page Finally, we ombine equ. (.95) with equ. (.94) and alulate the integral. We get B ik e 4ε ν f ik. (.96) mh ik With that, the relationship between the measurable integral absorption oeffiients, the osillator strengths from the lassial standpoint, the Einstein oeffiients, and the dipole moment of the transition is made. In omparing these relationships with similar equations found in publiations, take note that the use of a different basis leads to different forms of the equations. This is true when radiation densities instead of spetral energy densities are used, frequenies are replaed by wavelengths or wave numbers, and even by the use of angular frequenies in plae of frequenies. Literatur H. Haken und H.C. Wolf: Atom- und Quantenphysik, 8. Aufl. Springer 4, ISD H. Haken, H.C. Wolf: Molekülphysik und Quantenhemie, 4. Aufl., ISD W. Demtröder: Experimentalphysik, Atome Moleküle und Festkörper,. Aufl., Springer, ISB P.W. Atkins: Physial Chemistry, 6 th edition inluding a CD version, Oxford 999 P.W. Atkins: Physikalishe Chemie,. Aufl., Wiley-VCH, ISB Meshede, D. (Ed.) Gerthsen Physik,. Aufl., Springer, Landau/Lifshitz: Theoretishe Physik, Band II Elektrodynamik Spetrosopy D. Freude Chapter "Radiation", version June 6