ZEEMAN EFFECT: Zeeman Effect is a magneto-otical henomenon discovered by Zeeman in 1896. He observed that when an atom (light soce) is laced in an external magnetic field, the sectral lines it emits are slit into several comonents. For field less than several tenths of 1 tesla, the slitting is roortional to the strength of the field. This effect of magnetic field on the atomic sectral lines is called Zeeman Effect. A singlet sectral line viewed normal to the field is slit into three lane olarized comonents; a central unslifted line with the electric vector vibrating arallel to the field (called π -comonent) and two other lines equally dislaced one on either side with electric vector erendicular to the field (called σ-comonent). This is called normal Zeeman Effect. The fine structe comonents of a multilet sectral line, however, show a comlex Zeeman attern. For examle, the D 1 and D comonents of sodium yellow doublet give fo and six lines resectively in the Zeeman attern (fig b). This is called anomalous Zeeman effect. Hamiltonian for one electron atom in the resence of an external magnetic field: Let us consider an atom with the valence electron i.e., one electron in the outermost shell and all other inner shells are totally filled u. H-like atom and alkali atom fall into this category. Now, in the absence of any internal magnetic field the Hamiltonian for that electron in the atom is given by: H0 = T + V = + V( r)...(1). m Eigenfunction for this Hamiltonian is secified by n,l,s,m l, ms where m l = l to +l and m s =, Now if we take into account the sin orbit interaction term then the Hamiltonian will be; i
H0 = + V ( r) + ξ ( r)( L. S)...(). m dv ( r) Ze h Where ξ ( r ) =. = 3 mc r dr mcr Then the eigen states are nls,,, jm, where m = -J to +J, (j + 1) nos. When we have the sin orbit term in the Hamiltonian, the Eigen states have a degeneracy in the rojection of j. There are (j + 1) fold degeneracy. All this (j+ 1) states have the same energy eigen function and the total angular momentum will be conserved. Now let this atom with a single electron at the outside core, is laced in a constant uniform magnetic field, the energy level under the influence of external magnetic field are slit u into comonents giving characteristics Zeeman attern. Now the interaction energy which roduces these dislacements consists of the two arts, that arising from the orbital motion of the electron and that arising from the sin of the electron. The effect of the external magnetic field on the orbital motion of the electron is obtained by using the vector otential A where, B = A The Hamiltonian or K.E for a article of charge e in a magnetic field of otential A is e obtained by writing + A in lace of in the resective exressions c 1 e K.E. T = + A Where A = vector otential m c 1 e e = + A + A m c c e e = + ( A+ A) + A...(3). m mc mc Now does not commute with A in general and r ( A) ϕ = ih(. A) ϕ i = h r = ih( ϕ.. A+ A. ϕ) Now if we choose the coulomb gauge. A = 0 then we have ( PA. ) ϕ = ih( A.. ϕ ) = A( ih ) ϕ u = AP. ϕ u PA. = AP. e u T = + ( A. ) + e A m mc mc ii
Using this we have from equation () the Hamiltonian as, e e r r H = + ( A. ) + A +V ( r ) +ξ ( r) ( LS. )...(5). m mc mc Now the magnetic moment of an electron due to its sin is given by, u r e 1 µ s = gsβ. s, where β = and gs = ( s+ 1) =. + 1 = mc Until u to this oint we have not taken into account the intrinsic magnetic moment of the electron, which also interacts with the external field B. Thus the interaction of the intrinsic sin magnetic moment with the magnetic field is given by, u H = µ s. B r = gsβ sb. r = β B.s Thus we have the comlete Hamiltonian as, e u e r r u r H == + ( A. ) + A +V ( r ) + ξ( r) ( LS. ) + βb. s...(6) m mc mc Again, for a uniform magnetic field we can write, 1 r A= ( B r) e u e 1 r e r e ( A. ) = ( B r). = B( r ) = BL. mc mc mc mc = e BL. = β BL. mc Using the above we have from (6); rr e rr u r H = + V( r ) + βbl. + A + ξ ( r) ( LS. ) + βb.s m mc r e = H0 + β B( L+ s) + A mc r r e H = + + ξ( r)( Ls) + βb( L+ s) + A m mc V r....(7) Equation (7) gives the total Hamiltonian of a single electron in an atom in the resence of external magnetic field. e Now the term A. in the exression (equation 7) is neglected as the external field is mc small enough comared to the field roduced by the electron and the nucleus. iii
r H = H0 + β B L+ s...(8). r H0 V r r Ls. = unertbed Hamiltonian in the absence of external m magnetic field. And r H = β B L+ s = Pertbed Hamiltonian in the resence of external magnetic field. Where, = + +ξ Now let us consider the external alied magnetic field be weak enough. So that β.b is small comared to the contribution to the Hamiltonian due to the sin orbit couling term. u r ie., β. B L+ s ξ r Ls. Under these circumstances the ertbed Hamiltonian is u H = β. B( L+ s) (9) Now if we aly the magnetic field in the Z-direction i.e., B x = B y = 0 and B z = B Then, H = β B L + s = β B J + s z z z z z z <, L-S couling occ and out of S one S remains. The entry shift u to the first order is E = ϕβb L+ S ϕ s 0 In weak magnetic field ( H H ) ( ) = ϕβb( J + S) ϕ = βb ϕ J + S ϕ u = βb ϕ Jz + Sz ϕ as B in Z direction = β B Jz + Sz...(10). From the symmetry the average value of S is a vector arallel to J as this is the only vector which is conserved. S = constant J = CJ J. S = CJ iv
Now, J S J S J. S. C = = = J J J JS.. S = J...(11). J J = L+ S J L= S J + S JS = L J + S L JS. = J + S L ( 11 ) S =. J J From equation (10) we have, E = β B Jz + S z SLJM J u J + S L = βbmh + βb Jz. J { j( j+ 1) + s( s+ 1) l( l+ 1) } h = βbmh+ βbmh j( j+ 1) h { j( j+ 1) + s( s+ 1) l( l+ 1) } h = β Bmh 1+ j( j+ 1) h = β Bmg h E =β Bm h g { j( j+ 1) + s( s+ 1) l( l+ 1) } h Where g = 1 + j( j+ 1) h Since the diagonal elements of the ertbation oerator are the only non vanishing one, the energy of the atom in the 1 st ertbation aroximation is given by, Enljm = Enj mhβ Bg Where m = 0, ± J The (J + 1) fold degeneracy is thus lifted in the resence of the magnetic field. The shift of the level is symmetric with resect to the unertbed energy level E nj. The distance between the neighboring substances, is E = mh β Bg, which is roortional to the magnetic field strength and to the Lande s factor which deends on the quantum number j, l and s. The slitting of the energy levels determined by the above equation is called the anomalous Zeeman Effect. v
For a sineless article s = 0 and the Lande s factor g = 1. In that case, the distance between neighboring sublevels doesn t deends on all the character of the state and equal to E = mh β Bg. Such a slitting of energy level is called the normal Zeeman Effect. Normal Zeeman Effect is secial case of anomalous Zeeman Effect for which s = 0. Thus for weak field, each energy level is slitted symmetrically into (j + 1) equally saced states the slitting being roortional to the magnetic field and is indeendent of the total quantum number n of the atom. Examle: slitting of Na-D line under the influence of magnetic field: Term P1 3 1 l = 1, j =, S = P 1 l = 1, S =, J = S 1 l = 0, S =, J = No of Zeeman Levels (j + 1) 4 g 4 5 3 Mj +J -J J, J-1, -J 3 3,,,,, Zeeman shift 6 6,,, 5 5 5 5, 3 1, -1 The selection rules for transitions are mj =± 1 σ -comonent. = 0 π comonent. and L = ± 1 vi
j m =± 1 σ Comonent = 0 π omonent D 1 line slits into 4 lines and D line slits into 6 lines. vii