2 Absorption and Emission of Radiation uni leipzig de

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Chapter 2 page 2, The energy density w energy per unit volume is ED for an electric field or BH for a. magnetic field The time average of cos2 t is From that we obtain for the energy density of. a linearly polarised electromagnetic wave,w r 0 Ey2 r 0Hz2 2 05. The Poynting vector S which refers to the energy flux density through a unit area points. along the x direction and is the product of the energy density and the speed of light. Sx wc 2 06, From that we come to the important conclusion that the energy flux of the radiation in the. direction of propagation is proportional to the squares of the amplitudes of the field strengths. Fig 2 2 Energy flux of an electromagnetic,wave propagating in the x direction The wave. vector k and the Poynting vector S also point in,the x direction In the time t the energy in the.
A cube flows through the surface A If the time t,is chosen to be one second and the area A is the. unit area energy density and power density,have the same numerical value. 2 2 Dipole Moments and other Quantities from Electrodynamics. To clarify a few terms such as dipole moment and y. polarizability we use the so called multipole expansion obervation. which describes the electric potential V r of a charge r r. distribution The charges qn are located at r n and the r. origin of the coordinate system is located within or not x. far from the charge distribution The left figure of the. water molecule shows the charge of the oxygen nucleus. located at r and the observation point at r Electron charges are drawn in orange grey. V r 4 0 r r,n 1 xi 2 xi x j r,0 1 2 q n 3 r 5 ij xi x j 2 07. r n r r i j, The factor 4 0 is introduced on the left so that the potential has the SI unit of volts The. potential in equ 3 03 is for large distances between the point of observation and the charge. i e r r r n It is expanded by powers of 1 r by taking the derivatives with respect to r the. point under consideration The series development shows that the potential of any charge. distribution can be represented by a sum of multiples. Spectroscopy D Freude Chapter Radiation version June 2006. Chapter 2 page 3, Let us now consider an electrically neutral molecule in which the positive nuclear charge and.
the negative electron charge compensate each other In this case the first term of the. expansion 0 is zero 1 is the dipole moment and 2 the quadrupole moment Here we stop. the expansion 1 can be written as r r3 or er r2 where er is the unit vector in the. r direction,qn rn 2 08, is defined as the dipole moment of a charge distribution It does not depend on the location. of the origin in the case of neutrality of the charge cloud of atoms or molecules The unit of. the dipole moment is Asm The old cgs unit named after Peter Debye in which 1 D 3 33564. 10 30 Asm is still in use because the dipole moments of small molecules are in the range of. 1 D H2O 1 85 D HCl 1 08 D Be careful not to confuse the Debye with atomic unit. ea0 8 478 10 30 Asm which refers to the elementary charge e 1 602 10 19 As and the. Bohr radius a0 5 292 10 11 m 2 in equ 3 describes the potential of a quadrupole The. quadrupole moment is,qn 3 xn i xn j rn2 ij 2 09, at the origin ij is the Kronecker symbol From equ 2 07 follows ij ji and from equ. 2 09 we can see that the quadrupole tensor has no trace. Even though single magnetic charges do not exist we can write a relationship for magnetic. potential analogous to equ 2 07 The magnetic moment which is also represented by. plays together with the electric dipole moment an important role in the electromagnetic. dipole radiation, Now we consider dielectric material consisting of particles without permanent dipole. moments e g CO2 or CH4 A moment ind can be induced through the electric polarizability. units Asm2 V 1 under the influence of an external electric field E The corresponding. polarizability tensor is defined by,ind E 2 10, This linear effect is sufficient for the consideration of weak fields The basis of the. consideration of the non linear optics NLO is the extended equation ind E E2 E3. Non linear effects play an important role in the laser spectroscopy For weak fields we. have the electric polarization P as the induced dipolar moment per unit volume cf equ. P e 0 E 2 11, where the electric susceptibility e is a scalar dimension less unit for isotropic material The.
electric field E is static if it is caused by a direct current DC source applied to a capacitor. If we replace DC by AC alternating current with the frequency we get the corresponding. magnetization of the field only in the case if the charges can change their orientation quickly. enough The electronic polarizability produced by shifting the positively charged nucleus with. respect to the negative electron shell takes place in less than 10 14 s The polarization by the. shifting or vibration of the ions in a molecule or lattice ion polarization distortion. polarization happens a thousand times more slowly on the order of 10 11 s Both types of. polarization are united under the term of displacement polarization The orientation. polarization is much slower and therefore plays no role for the index of refraction optical. Spectroscopy D Freude Chapter Radiation version June 2006. Chapter 2 page 4, range It is caused by the lining up of permanent molecular dipoles which are present even in. the absence of an external field The dielectric relaxation of the orientation polarization can be. experimentally examined with high frequencies and helps determine the dynamics of the. system DC spectroscopy is the frequency dependent measurement of the relative dielectric. 2 3 Absorption and Dispersion, The phase speed c which is defined as the product of wavelength and frequency is. reduced in comparison to speed of light in a vacuum c0 when the electromagnetic wave. travels through a medium with an index of refraction n 1 The reduced value is c c0 n We. will show that the frequency dependency of n leads to a dispersion which can be described. using a classical model It will be also shown that the imaginary part of a complex index of. refraction describes the damping of an electromagnetic wave. For this presentation we consider an electric field with the amplitude vector AE 0 E0 0. which has a complex time dependency exp i t instead of the cos t of equ 2 01 The. differential equation of a damped oscillation forced by an external field is. m 2 m m 02 y q E0 exp i t 2 12, where the mass of the oscillator is m the charge q and the characteristic frequency 0 is the. damping constant With an exponential trial solution of y y0 exp i t we obtain. as the complex amplitude of the oscillation An induced electric dipole moment ind appears. in the y direction,y q y exp i t 2 14,With N oscillators per unit volume we obtain. Pind e 0 E N 2 15, as the induced electric polarization and with that a complex susceptibility.
0 m 02 2 i, The real and imaginary parts of e are not independent as can be shown by multiplying. numerator and denominator of equ 2 04 by the conjugated complex of the parenthesis In a. vacuum we have r r 1 From the definition n c0 c and equ 2 04 it follows that. n r r 2 17, Spectroscopy D Freude Chapter Radiation version June 2006. Chapter 2 page 5, Since we are not considering ferromagnetic materials we can set r 1 with sufficient. accuracy and we obtain the Maxwell relation,n r 1 e 2 18. We should note that these quantities are frequency dependent For example the orientation. polarization mentioned earlier has no effect on the susceptibility in the optical frequency. From equations 2 16 and 2 18 it follows that the index of refraction represents the complex. 0 m 02 2 i, To separate this into real and imaginary components different conventions are in use.
n n i n 2 20, When n 1 which is the case in gaseous media we can make the approximation. n2 1 n 1 n 1 2 n 1 Near the resonant frequency 0 0 or. 0 2 0 2 With that we have,4 0 m 0 0 2,8 0 m 0 0 2 2. The frequency dependent quotient in equ 2 22 is described by a function in the form of. y 1 1 x2 which is commonly called Lorentz curve The parameter is the full width at. half maximum of the curve The Lorentz curve will be described in more detail in chapter 2 6. The meaning of the real and imaginary components can be clarified by the following. considerations Analogous to equ 2 01 we have for a wave propagating in the x direction. Ey AyE exp i t kxx 2 23, The wave vector k can be replaced by nk0 where k0 with k0 c0 is the wave vector in the. vacuum From equation 2 20 and 2 23 it follows that. Ey AyE exp i t k0x n in x AyE exp n x c0 exp ik0x c0 t n x 2 24. The first exponent on the right side of equ 2 24 describes a damping of the wave Later we. will show how the absorption described by the imaginary part of the index of refraction n is. related to experimentally measurable extinction coefficient The second exponent describes. the dispersion In connection with equ 2 21 we get from that the dependency of the phase. speed on the frequency, Spectroscopy D Freude Chapter Radiation version June 2006. Chapter 2 page 6, If we set the charge of the oscillator q to be the elementary charge e equation 2 22.
describes the total absorption of atoms with a single valence electron The electrons Ni in state. i can through absorption move into new states k including non discrete states in the. continuum For this reason only a portion fik of the total absorption has to be considered for. the transition from the state i to the state k For these so called oscillator strengths it holds. fik 1 2 25, With the oscillator strengths fik the discrete transitions can be introduced into the classically. derived equation The imaginary part of the index of refraction then becomes. Ni e2 f ik ik, Here the half width of the absorption line for the transition from i k is ik and it has to be. summed over all possible excited levels k Since the frequencies ik stretch over a wide range it. is not possible to input a single frequency that fulfils the condition ik ik for all values. of k For this reason we did not make use of the approximation that ik ik and. 0 2 0 2 in the derivation of equ 2 26 in contrast to the procedure followed in the. derivation of equations 2 21 and 2 22 We cannot therefore directly compare equ 2 26. with equ 2 24 We will return to an explanation of extinction coefficients in chapter 2 8. 2 4 Spontaneous and Induced Transitions Radiation Laws. A spontaneous event needs no external influence to occur The light of a thermal radiator. which we can visually see occurs when a substance at high temperature spontaneously emits. quanta of light An induced or stimulated event only occurs with external influence. Accordingly absorption is always induced stimulated But emission can be induced if a. frequency equal to that of the light to be emitted is externally input. Let us now consider two energy levels of an isolated particle see below Since the following. considerations are applicable to any states we will label them with i and j Here and in the. next two sections we will set i 1 and j 2 Let E2 E1 and E2 E1 h where h 6 626. 10 34 Js denotes the Planck constant The occupation numbers of the states are N2 and N1. The number of particles which go,from state 1 to state 2 is. h dN1 B12 w N1 dt 2 27,E1 N1 where B12 w is the absorption. Absorption Induced emission Spontaneous emission probability with the spectral energy. Fig 2 3 Absorption induced and spontaneous emission. Spectroscopy D Freude Chapter Radiation version June 2006. Chapter 2 page 7, The energy absorbed by the particles for the transition is given by.
dWabs h dN1 2 28, The energy emitted in the form of radiation by the transition from 2 to 1 is. dWem h dN2 2 29, For the balance of particles that go from 2 to 1 we need to consider a spontaneous transition. probability A21 in addition to the transition probability B21 w. dN2 B21w A21 N2 dt 2 30, The probability A21 does not depend on external fields The probability of an induced. transition however does depend on the external field It is the product of the B coefficients. with the spectral energy density w of the external fields in the frequency range from to. d The spectral energy density w has the units of energy per volume and frequency Instead. of this quantity the spectral beam density L is often used L is the power in the frequency. range to d that is emitted per unit area in a cone of solid angle 1 A solid angle. 1 would mean that 1 m2 is cut out of the total surface area of 4 m2 of a sphere with a radius. of 1 m The aperture angle of the cone is about 66 In a vacuum where the speed of light is. c0 it holds that,L w c0 4 2 31, B12 and B21 are the Einstein coefficients for absorption and induced emission With the help of. these coefficients Albert Einstein could find a simple and secure proof of the radiation law in. 1917 The radiation law was discovered at the end of 1900 by Max Planck through an. interpolation of the behavior of the second derivative of the entropy with respect to the. energy between Wien s radiation law and Rayleigh Jeans radiation law. Einstein s derivation starts with a closed cavity in a heat bath at the temperature T Because of. equilibrium we have for two arbitrary states between which transitions occur that the number. of absorbed and emitted quanta of energy must be equal w is in this case the spectral energy. 2 Absorption and Emission of Radiation 2 1 Electromagnetic Radiation In the year 1886 Heinrich Hertz experimentally demonstrated the existence of electromagnetic waves and their equivalence to light waves After Hertz s achievement the electromagnetic theory of James Clerk Maxwell developed from 1861 to 1864 became the basis of examining optical absorption and dispersion phenomena As

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