Doublet structure of Alkali spectra:

Transcription

1 Doublet stuctue of : Caeful examination of the specta of alkali metals shows that each membe of some of the seies ae closed doublets. Fo example, sodium yellow line, coesponding to 3p 3s tansition, is a close doublet with sepaation of 6A 0 while potassium (K) has a doublet sepaation of 34A 0 and so on. Futhe investigations show that only the S-tems ae singlet, while all the othe tems P, D, F etc. ae doublets. Such doublet stuctue in enegy is obseved fo all the atoms possessing a single valence electon i.e., in the oute most shell. Usually the doublet spacing is small compaed to the tem diffeence (fo Na the main D line is canteed at 5893A 0 ; D = 5890A 0 and D = 5896A 0 ) and hence it is called fine stuctue. To explain this featue Uhlenback and Goudsmith fist poposed the hypothesis of electon spin, which was late on obtained as a natual consequence of Diac s elativistic theoy. Spin is essentially a quantum phenomenon. The spin of the electon is found to be h and S s ( s ) = + h whee s =, the quantum numbe fo spin. (Li, Na, K, Rb, Cs, F) Explanation of doublet stuctue of alkali atom: The Hamiltonian fo the lone electon of an alkali atom elative to the atomic coe is given by, Whee, p H0 = + V( )...() µ u P = momentum of the lone electon and V( ) = Potential.. = Distance of the lone electon fom the cente of the atomic coe, i.e., nucleus. Whee Now consideing the spin obit inteaction the total Hamiltonian is given by, p H = + V( ) + H µ S-O H = spin obit inteaction tem S-O...()

2 An electon with obital angula momentum l and spin s will behave a total angula momentum j = l + s Which gives the quantum numbes fo j as j = ( l+ s),...,( l s). Since fo a single electon s = o j = l + and l, fo l = 0; j = only. The coupling of spin with obital angula momentum gives ise to the fine stuctue splitting of spectal lines and it gives a full account of the doublet stuctue of alkali specta. This inteaction is called spin obit inteaction. The existence of electon spin and the doublet stuctue comes as a natual consequence of the elativistic theoy but classical electodynamics gives a non-ealistic desciption. Conside an electon of chage e moving with velocity v at a distance fom the nucleus of chage Ze (o an entie atomic configuation of effective chage Ze). Viewed fom electons est fame the entie atomic configuation (consisting of the nucleus and the est of the electonic chage cloud be taken as igid coe) is moving with a velocity of V and thei effective field also moves with the same velocity. Associated with the v moving electic field aising fom elativistic tansfomation equation. To fist ode in. c the magnetic field is Β= E v u u c and, E = φ; φ = scala potential u Β= Φ v c u =+ V v ec whee V = - e φ = potential. Now fo a spheical symmetic potential u dv V =. d dv Β =. v ec d dv u =. p p = mv mec d dv u =. l l = p = a ngula momentum. mec d

3 If the coodinate system is fixed with the atom the effective magnetic field is to be multiplied by a facto. This is known as Thomas coection obtained fom a spinning top model of the electon. dv Β =. l mec d The inteaction enegy of the electonic magnetic moment with the magnetic field due to the motion of chage paticle is S O uu H = µ s. Β e dv e =.. ls. µ s = s mc mec d mc This equation holds in the est fame of both electon and the nucleus as to fist ode in v, the enegy is same. Thee is no tem popotional to l as in the est fame of the electon; c thee is no obital magnetic moment. Now we can wite using quantum mechanical opeato fo l and s ( as h l and h s ). S O h dv H =. l. s mc d = ξ ( ) l. s...() Whee l and s ae dimensionless vecto opeatos, h dv ξ ( ) =. mc d The total Hamiltonian of the alkali atom consideing spin obit inteaction is given by, p H = + V( ) + ξ ( )( l. s)...(3) µ Fo a pue Coulombic potential, Ze V ( ) = dv Ze = 3 d Ze ξ ( ) = h....(4) 3 mc Now the spin obit tem can be consideed as petubation tem. The contibution of the spin obit tem can be calculated by consideing its expectation value. (hee we conside only the fist ode coection)

4 E = ψ H S O ψ...(5) Whee ψ is the gound state wave function of the effective one paticle system. In an atom the total angula momentum j is always conseved even if individual S O l and s may not be conseved. Hence we can wok in a epesentation o system whee H is diagonal. Also we choose nl jm j epesentation, whee individual l and s ae combined to fom the conseved quantity j Thus, We have, E = ψ nljm j S O H ψ nljm j = ψ nl ( ) ξ ( ) ψ nl ( ) ψ lsjm l. s ϕ j lsjm = ζ nl ls....(5). Whee, ( ) ( ) ( ) ζ ψ ξ ψ nl = nl nl = Rnl ( ) ξ ( ) d 0 Ze h = ( ) Rnl d 3 m c 0 Ze h = 3 m c j

5 To calculate the angula tem ls. Hee fo a given l, 3 Z h = ; a 3 0 = 3 3 nl( l+ ) me l+ a 0 4 Z We can wite, ζnl = Ryα 3 nl( l+ ) l+ 4 me Whee, R y = = Rydbeg constant and h e α = = fine stuctue constant. hc we poceed as follows: ls. uu u uu = j l s ψlsjm ls. ψ. j lsjm = ls j = j l s = ( ) ( ) ( ) j j+ l l+ s s+ 3 = j( j+ ) l( l+ ) s = 4 Q j = l+ andl. Thus each level split into two sublevels with j = l+ andl.

6 Fo $ j = l ls. = ( ) l+ l+ l l+ 4 3l 3 3 = l + + l+ l l 4 4 l = Fo $ j = l 3 3 ls. $ = l l l( l ) = l l l 4 4 = ( l + ) Thus the enegy levels ae given by, l E $ j = l+ = E0 + ζ nl and ( l ) E $ + j = l = E0 ζ nl The sepaation is given by, E = E E = ζ nl l+ Thus the splitting of enegy levels i.e., sepaation between the two levels fo each value of l is ( l + ) ζ nl. This is the so called fine stuctue of spectal lines fo spin obit inteaction of enegy levels. Now we may daw the enegy level diagam fo the non elativistic theoy then the coection to the enegy levels due to elativistic vaiation of mass and at last coection to the enegy level due to the pesence of spin obit inteaction.

7 Now fo l = 0, j =± but j can neve be ve. Hence j = +. Theefoe s level doesn t split up fo ( S-O) inteaction. 3 Fo l =, j =, 5 3 l =, j =, 7 5 l = 3, j =, and so on Thus except l = 0 (which has no S O inteaction) the othe enegy levels split up into two levels (doubles) as shown below due to S O inteaction. The mechanism esponsible to the doublet splitting is S O inteaction. Each enegy level has got a spin multiplicity s+ =. + = + =

8 In spectoscopic notation, each enegy level is epesented as, s + n L j n = Pincipal quantum numbe. L = obital angula momentum quantum numbe. J = Total angula momentum quantum numbe = l+ s s + = spin multiplicity. l = 0,,.n m l = -l. +l j = l + s j = (l + s), (l +s ), (l s).