Title: Radiative transitions and spectral broadening
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1 Lecture 6 Ttle: Radatve transtons and spectral broadenng Objectves The spectral lnes emtted by atomc vapors at moderate temperature and pressure show the wavelength spread around the central frequency. Ths s known as wdth of the spectral lnes. In ths lecture at frst we wll understand the orgn of radatve transtons. We wll establsh the relatons between Ensten coeffcents B 12, B 21 and A 21 We wll dscuss the dfferent mechansm for the broadenng of the spectral lnes emtted by the atomc source. We wll fnd out the reasons for these broadenng.
2 Page-1 Radatve transtons and spectral broadenng: We now know that, Bohr s theory descrbed that the emsson of radaton from atoms was from the hgher energy state to lower energy state of atom. Ths s known as quantum jump Bohr tred to descrbe t through the correspondence prncple Accordng to Bohr s theory, me E n ( n 1) h Equaton 6.1 And E n 2 me 2 2 ( n) h 2 4 Equaton 6.2 Thus me 1 1 n 1 En E [ ] h ( n) ( n 1) me for n >> 1 h n E E h The emtted frequency 1 1, n n n n Equaton 6.3 However, Bohr s theory fals to descrbe the mechansm of transtons between the two statonary states. It also does not provde ratonal explanaton for dervng the ntensty and polarzaton of the emtted spectral lnes.
3 Page-2 Ensten transton probabltes: The frst major step to understand the transtons between two statonary states and the correspondng radaton was taken by Ensten. As shown n Fgure 6.1, let us assume that E 1 and E 2 are the statonary energy levels of an atomc system. Let us consder that ths system s exposed to a radaton hν. E 2 E2E1 h Absorpton Stmulated emsson Spontaneous emsson E 1 Fgure 6.1 Process 1: If an atom absorbs a photon of energy E2E1 h, t s excted from the lower energy level E 1 to the hgher energy level E 2. The process s called nduced absorpton. The dp12 probablty per second that the atom wll absorb the photon, dt Ths s proportonal to the number of photons of energy hν per unt volume = the spectral densty of the radaton feld ( ) dp12 So, dt B12 ( ), B 12 = Ensten coeffcent of nduced absorpton. Equaton 6.4
4 Page-3 Process 2: The radaton feld can nduce the atoms to make a transton from excted states E 2 to lower energy state E 1. Snce the system decreases the energy, the extra energy wll be released as the emsson of a photon of energy hν. Ths process s known as nduced emsson or stmulated emsson. The nduced photon of hν wll have the same nature as the photon caused ths dp12 emsson. The probablty that one atom emts one nduced photon per second s dt dp dt 21 B21 ( ) Equaton 6.5 B 21 = Ensten coeffcent of nduced emsson. Process 3: An excted atom n the excted state E 2 can spontaneously jump nto the lower energy states E 1 by emttng a photon of energy E2E1 h Note: Spontaneous radaton can be emtted n any arbtrary drecton. The probablty per second depends on the nature of the statonary energy state and ndependent of the external feld. dp21 So, dt spont A 21 Equaton 6.6 A 21 = Ensten coeffcent of spontaneous emsson and also known as spontaneous transton probablty.
5 Page-4 Relaton between B 12, B 21 and A 21 Let us assume that the system s havng N number of atoms and dstrbuted n dfferent energy levels E such that N N At thermal equlbrum of temperature (T), accordng to Boltzmann dstrbuton, N N g e E kt Equaton 6.7 Where g s the degeneracy of level E. In a statonary feld, The number of photons absorbed per unt volume per second = the emsson state B N ( ) [ B ( ) A ] N Usng N g g e e N1 g1 g1 ( E2E1) h 2 2 kt 2 kt We get, A 21 B 21 ( ) g h 1 B12 e kt g2 B21 1 Equaton 6.8 The radaton densty gven by plank s law, ( ) 3 8 h 1 c 3 h e kt 1 Equaton 6.9
6 Equatng we get, B g B g1 Equaton 6.1 It mples that wth the equal weghts g 2 g 1, the probablty of nduced emsson s equal to that of nduced absorpton. We also get A 8 h c 3 B Equaton h Ths provdes the number of modes per unt volume and unt frequency nterval n( ) 3 c A21 thus, B21h n( ) 3 and Ths means that the rato of the nduced to the spontaneous emsson rate n an arbtrary mode n equal to the number of photons n ths mode.
7 Page-5 Relaton between lfetme and spontaneous transton probablty An excted atom n the excted state E 2 can spontaneously jump nto the lower energy states E 1 by emttng a photon of energy E2E1 h dp21 So, dt spont A 21 E A 2 When there are several pathways then A 1 A A k k A E 2 The populaton of the excted state wll decrease. E 1 So dn A N dt E We get N ( t) N () e Where N () s the populaton densty at t = At N N () The populaton N () wll be N () / eafter tme 1 as shown n Fgure 6.2. Ths A N ()/e represents the mean spontaneous lfetme of the level E. t Fgure 6.2
8 Page-6 Semclasscal approach of radatve transtons: In ths approach, we consder the radaton as classcal electromagnetc wave E E Cos( t kz) Equaton 6.12 On the other hand, we treat the atom quantum mechancally. That means, the atoms absorb or emt radaton when they jump between the two quantzed states. Snce the dmenson of the atom s much smaller (~.5 nm) than the wavelength of lght (~5 nm), we take t t E E Cos t e e ( ) E 2 Equaton 6.13 In the dpole approxmaton, the nteracton energy V p. E pecos( t) where the dpole p er The general soluton ( rt, ) of the tme dependent Schrodnger Equaton H( r, t) ( rt, ) t Equaton 6.14 Can be expressed as m ( r, t) C ( t) ( r) e m m m E t / Equaton 6.15 Where () r s the egen functon of the tme ndependent Schrodnger equaton m H ( r) E ( r) m m m Equaton 6.16
9 Page-7 For two level system as shown n Fgure 6.3, ( r, t) a( t) ( r) e b( t) ( r) e a Eat / Ebt / b Equaton 6.17 Where a(t) and b(t) are the tme dependent probablty ampltude of the states a and b and 2 2 a( t) b( t) 1 Substtutng n the Schrodnger equaton, we get da( t) db( t) e e a( t) V e b( t) V e dt dt Eat / Ebt / Eat / Ebt / a b a b Integratng over the space we get Fgure 6.3 b da() t dt a b a( t) Vaa b( t) e ( E E ) t / Equaton 6.18 a a And db() t dt a b b( t) Vbb a( t) e ( E E ) t / Equaton 6.18 b a * * Where V V d ee r d ab b b a The dpole matrx element * Dab e rbd a Snce r has odd party, we get D D and ab ba * Dab Dba e r a bd Ths depends on the statonary state wavefunctons () r and () r and need to calculate for understandng the transtons between the two states. a b Intensty of the transton I D ab 2
10 Page-8 Wdths and Profles of Spectral lnes As shown n Fgure 6.4, the spectral lnes observed from atomc vapors are never strctly monochromatc. Even wth very hgh senstve nstrument, the observed lnes show a wavelength spread around the lne center. Ths s known as wdth of the spectral lnes. Wdth of the spectral lnes Central frequency ( E E ) / correspondng to a transton a The lne profle I( ) around s known as lne profle 1 and 2 where the ntensty s the half of the maxmum ntensty ( I /2) 2 1 s known as the Full Wdth at Half Maxmum (FWHM) b I /2 I Fgure 6.4 /2 Equaton 6.19 c c Snce then Equaton 6.2 The relatve half wdth Equaton 6.21
11 Page-9 Manly there are two reasons for the broadenng of the spectral lnes (a) Wdth caused by the atomc source (b) Wdth arses due to the nstrumental lmtatons (a) There are basc three reasons for the broadenng caused by the atomc source 1. Natural lne broadenng 2. Doppler lne broadenng 3. Pressure / Collsonal lne broadenng Dependng on the effect on the ndvdual atoms, there are two types of broadenng. If all the atoms are experencng the same type of effect, then the broadenng caused by ths s known as Homogeneous broadenng. On the other hand, f ndvdual atoms are experencng dfferent effect then ths s known as nhomogeneous broadenng.
12 Page-1 1. Natural lne broadenng An excted atom can emt the radaton spontaneously. Ths phenomenon can be treated classcally as damped oscllator dscussed n the prevous lecture. The ntensty profle I( ) A( ) A ( ) I * 2 2 ( ) ( / 2) 1 Equaton 6.22 Where I s the maxmum ntensty and 2. Ths lne profle s known as Lorentzan profle and can be wrtten as L( ) ( ) ( / 2) In ths, full wdth half maxmum (FWHM) = or 2 Ths also can be understood from the Hesenberg Uncertanty Prncple Equaton 6.23 Equaton E h where s the lfetme and E s the uncertanty n energy. So, when s small, 2 E s large. Ths broadenng s known as natural lnewdth of the spectral lne for the excted state whose lfetme s fnte. The broadenng of the spectral lne can not be smaller than ths E even a hgh resoluton nstrument s used to measure t. Ths type of bradenng s Homogeneous broadenng.
13 Page-11 1 Natural Wdth Lfe Tme of Excted States Equaton 6.25 At hgh pressure, due to collsons lfe of the excted state s decreased and the broadenng occurs. n 1 n 1 lfetme of the excted state t t tme for atom n the excted state Equaton 6.26 If collson occurs t looses some exctaton energy and for that f t decreases then the natural wdth ncreased. For example: If the lfetme of a partcular excted state s 1-9 sec then the energy broadenng from Hesenberg uncertanty prncple Et E J 11 J t 9 1
14 Page Doppler lne broadenng Atoms emttng electromagnetc radaton are not statonary (gas). For an observer n the laboratory frame of reference, the emsson must be consdered as comng from a source n moton, so necessary to take Dopper effect. Speed V of atom s small compared to c, so t s possble to use classcal expresson for the Doppler effect. Let be the angle between observaton drecton & velocty vector, the change of frequency between the exact frequency and the frequency seen by the observer, Fgure 6.5 V Detector V cos Vx Equaton 6.27 c c V x s the component n the drecton of the observer. If we assume that the temperature of the gaseous source of lght s unform, the dstrbuton of speeds of the atoms s a Maxwell dstrbuton. So the number of atoms whose velocty V x s between V x and Vx dvx x. dn N f V d V N Total no of Atoms f V s the probablty densty for the component v x x x M M x 2 RT 2RT 2 exp V f V M molecular weght, R perfect gas constant x Equaton 6.28 Now, d Vx c dv c So, x Equaton 6.29
15 Page-13 Let Pd s the power emtted n the band frequences between and d s proportonal to the number of atoms between velocty V x and Vx dvx, f proportonalty constant s K, P d K N f V d V x d K N f c c K ' f c d x 2 2 M c So, P K ' f c K "exp 2 2RT Equaton 6.3 Ths profle as gven n Equaton 6.3 & as shown n Fgure 6.6 s the Gaussan lne profle. P, the wdth at half heght of the curve. D 2 1 K /2 2 2 M c 1 exp 2RT 2 2 c 2RT M ln Fgure 6.6 So the wdth 2 2RT D ln 2 c M Equaton 6.31 So, Doppler Wdth 1 M D T Equaton 6.32
16 nm.57 cm Na gas at 5 K 6 1 c 1 1 Natural Lne Wdth 1 So the Doppler wdth s much greater than the Natural lne wdth. Doppler broadenng s nhomogeneous broadenng.
17 Page Pressure / Collsonal lne broadenng For a gas at a gven pressure radatng atoms nteract wth the neghborng atoms va collson and ths affect the emsson lne wdth strongly. For example, an atom A wth energy levels E 1 and E 2 approaches to another atom B. Because of the nteracton, the energy levels get perturbed and the shft of these energy levels occur. Due to ths the lne profle gets broadened. Ths broadenng depends on the collson dameter. FWHM = 1 where s the mean flght tme between two successve collson.
18 Page-15 (b) Wdth arses due to the nstrumental lmtatons The basc set up for the emsson experment s Monochromator Detector Fgure 6.7 The basc confguraton of a monochromator S 1 () S 2 () Source f 1 slt aperture Dspersng element f 2 Screen dx dn Lnear Dsperson = f(, f2) d d Fgure 6.8 Slt wdth = x 1 Equaton 6.33 Equaton 6.34 I Image wdth = x 1 (f 2 /f 1 ) Equaton 6.35 x Dffracton due to aperture
19 Page-16 Spectral Resolvng Power I( ) I( )[ Sn(( ) / 2)( ) / 2] I( ) I ( ) I ( ) Equaton x 2 Fgure 6.9 Raylegh crteron.8 of I max I x 2 = f 2 (/a) Where a s the aperture Equaton 6.37 x And the resolvng power = Equaton 6.38
20 Page-17 The Gratng monochromator. Fgure 6.1 () Lght gatherng power / numercal aperture / speed of monochromator Acceptance Angle = d/f 1 () The spectral transmsson of the optcal systems T() or R() () Spectral resolvng power = / mnmum separaton of two spectral lnes that can be resolved (v) Free Spectral range : wavelength range n whch the wavelength can be unambguously determned from x() = If N s the no. of grooves per nch and m s the order then mn
21 Page-18 Recap In ths lecture we understood the classcal and quantum descrpton of the radatve transtons. We establshed the relatons between Ensten coeffcents B 12, B 21 and A 21 We came to know that the mnmum spectral lne wdth s governed by the lfetme of the excted state. Now we know that the orgn of homogeneous broadenng and nhomogeneous broadenng. We had brefly gone through the monochromator whch s an essental nstrument for spectroscopy experment. In ths module, we prepare ourselves the basc understandng of quantum mechancs through the development of concept of atom. In the next module, we wll start applyng these concepts to understand varous observatons of atomc spectroscopy.
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