Spin-Orbit Interaction

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Amos Allen
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1 pin-obit Inteaction Hydogen-like atoms The spin fee hydogen like atom Hamiltonian is Ĥ ez = m πε and we know its eigenfuctions and eigenvalues and we ae inteested in the effect of spinobit inteactions on these. et s fist conside the physical bass of the spin-obit inteaction. We imagine a one-electon atom as an electon moving in the field of a nucleus of atomic numbe Z. Usually we think of the electon as evolving about the nucleus but anothe view is to sit on the electon and watch the nucleus move elative to the electon. The nucleus caies a positive chage and geneates a magnetic field at the location of the electon given by R V F R c ( ) = ( ) We note that the velocity of the nucleus is equal and opposite to the electons so we have Ze ( ) = V whee is diected fom the nucleus to the electon. ince the c πε obital angula momentum of the electon is = mv, ( m is the mass of the electon) Ze we may wite ( ) = whee we measue in multiples of. Using the oh c πε m e magneton, µ = (.975 x - amp M 7 ) and ecognizing that c πε = this m 7 Zµ becomes ( ) =. This magnetic field inteacts with the magnetic dipole moment of the electon which can be witten in tems of the electons spin angula momentum as µ = gµ whee g is the magneto-gyo atio and is appoximately. ote that the spin angula momentum is measued in multiples of. The inteaction 7 gz µ i enegy is E = µ i = = W( ) i. A moe detailed teatment by Thomas. F. Haison Michigan tate Univesity //6

2 7 Z µ H H W i. shows that we must multipl this expession by ½ and thus effectively cancel the magnetogyo atio. o with W( ) = the Hamiltonian becomes ˆ = ˆ + ( ) Commutatos and the pin-obit Hamiltonian We know that ˆ & α whee α = x,y, oz, What about the spin-obit tem? ince W ( ) is a adial function it ˆ ˆ will commute with these opeatos and we need only conside the effect of i. We and obital motion must be a constant of the motion. We wite this vecto opeato as ˆ ˆ ˆ = + and by geneal pinciples we anticipate that the wavefunction fo Ĥ will be an eigenfunction of ˆ ˆ with eigenvalues j( j + ) &m with j being = + Ĥ commutes with the obital angula momentum opeatos know that the atom is isolated so the total angula momentum due to the electons spin, /, Ĵ ( inteest is / /, ) &z and j j j m j in multiples of. ote futhe that ˆ ˆ i so i = ). The eigenvalue poblem of ˆ = ˆ + + ( ˆ ˆ ˆ ( ) ( ˆ ) ˆ W ˆ ˆ H + Ψ = EΨ ecause both the coulomb potential and W ( ) ae adially symmetic we must have Ψ= R ( ) Φ ( θ, φ ) and we can choose ( ) The adial function is then detemined by ( ) ( ) ˆ, ˆ, ˆ Φ θ, φ to be an eigenfunction of & Ŝ. z d d l l+ e Z W ( j ) l( l+ ) j + ( ) R = ER m d d m πε and clealy the enegy will depend on j &lbut not m j so the vaious states will be j + degeneate. Thee ae seveal ways to detemine the explicit fom of Φ, the most ( ). F. Haison Michigan tate Univesity //6

3 geneal using angula momentum coupling ules and the Clebsch-Godon o Wigne coefficients. We simply state the esult fo j = l ± / l± m+ l m+ l+ l+ m m+ Φ jml / =± αyl + βy l ote that j must be positive so when l = we have j = / and the coefficient of W ( ) vanishes and thee is no spin-obit effect fo s states. pin-obit effect fo the n= level of hydogen The n= level of H consists of the s and p obitals and since the s obital is not influenced by the spin-obit petubation we need only conside the p level. Coupling the l =,s = / angula momenta esults in the tems j = / & /. The zeo ode functions ae ( ) ( ) / + m / R Φ = R αy m + βy m m+ /,m,, / l l and ( ) ( ) / m / + R Φ = R αy m + βy m m+ /,m,, / l l whee R ( ) enegies ae is the adial function fo the upetubed system. The fist ode shift in the () E = R ( ) Φ W( )( ˆ ˆ ˆ ) R ( ) Φ = ζ /,m /,m,, / /,m,, / () E = R ( ) Φ W( )( ˆ ˆ ˆ ) R ( ) Φ = ζ /,m /,m,, / /,m,, /. F. Haison Michigan tate Univesity //6

4 = W dand is called a spin obit paamete. If we expess the whee ζ R ( ) ( ) lengths in tems of atomic units we haveζ ( ) ( ) 7 7 Z µ Z. 975 = a 7 Z µ a =. The coefficient =. 68Z au 5.8 cm -. The expectation value fo an abitay hydogen-like obital is Z = l l / l au ( + )( + ) esulting in ζ =. 7Z cm. l =,s= j = ζ j = pin-obit splitting of the n= level of H The obseved splitting fo H is.65 cm-, in excellent ageement with the calculated ζ splitting =. 657cm. The splitting of the = level fo the one electon atoms is pedicted to be E Z = ζ / =. 657Z cm and as the following table shows is ( ) emakably accuate. The expeimental values fo E ae fom Mooe s Tables.. F. Haison Michigan tate Univesity //6

5 Element E( cm ) Z E/Z He i e C O Many Electon Atoms The spin obit petubation opeato fo a many electon atom is appoximated as the sum = i li si of the one electon opeatos, W ξ ( ) ˆ i whee l i & s iae the obital and spin angula momentum opeatos appopiate to electon i and ξ ( i ) has the fom descibed peviously. If one knew the many electon wavefunction fo the state of inteest one could simply detemine the fist ode coection due to Wˆ using degeneate petubation theoy. In this context the enegy shifts to a paticula level ae the eigenvalues of the matix of the petubation fomed within the degeneate subspace. An tem fo an atom would be descibed by the function whee &ae fixed and M & M so we would set up the matix with elements M P W ˆ M ' ' and diagonalize it. o fo example if we wee consideing the level of the cabon atom we would fom the 9x9 matix associated with the 9 degeneate states. A emakable theoem about vecto opeatos (see Tinkham) allows us to do this with impunity. The theoem states that then the expectation value ' ' ˆ ( i) l ' ' ˆ ξ iisi = η( ) i M whee the numbe η () depends on the function ξ ( ) and & but not M and M. ote that this means ˆ i i whee ' ' ' ' ξ( i) li si = ζ ( ). F. Haison Michigan tate Univesity //6 5

6 ζ ( ) = η ( ). The 9x9 matix then has the elements ζ ( ) M M δ ' ' i δ We M MM will use this elationship latte but fo now we can simplify the calculation even futhe. We know fom coupling that the P level can be patitioned into thee moe levels associated with a total angula momentum =, and with the degeneacies, and 5 espectively. These eigenfunctions of ae also eigenfunctions of used as the unpetubed functions. We may fom these functions as Ĥ and as such may be M = M M C M so if we wite M M Wˆ M ' ' ' CM ' ' Ŵ M C M M ' ' s M M M M ' ' And since M Ŵ M = ζ ( ) M i M we have M Wˆ M = ( ) M ( ˆ ˆ = ζ ( ) M ˆ i M = ζ ) M ' ' ' ' ' We see that the petubation matix is diagonal and the enegy shifts fo a given ae E () ˆ ' = M W M = ζ ( ) ( + ) ( + ) ( + ( ) ( ) ) ( )) This is exact fo the simplified fom of the spin-obit petubation that we have used. Fo the time being we will egad ζ ( as a paamete to be detemined by expeiment. ote that fo the P level of cabon = and = so we have E ( ) ( ) so E = ζ,, E = ζ, and fo cabon is ( ) ( ) () ζ (, ) ( ( ) ) = + E = ζ,. The expeimental splitting patten. F. Haison Michigan tate Univesity //6 6

7 P = 7.cm = 6. cm = The theoy developed so fa pedicts that the - tansition is twice the - which is qualitatively what we see. The diffeence is due to vaious elativistic effects that we have not (yet) consideed. ande Inteval Rule The ande inteval ule states that the sepaation of adjacent multiplet levels is popotional to the quantum numbe of the highe of the two levels, E E =ζ. This follows fom the geneal fom of the splitting deived ( ) ( ) ( ) above. As example of its appoximate validity we conside the intevals in the 5 D of C. E ( cm ) Inteval E ( cm ) / This is epesentative of the ageement one sees at this level of theoy.. F. Haison Michigan tate Univesity //6 7

8 Absolute Tem Intevals We now want to elate the paamete ζ ( ) to the electonic stuctue of the atom and associate it with the chaacteistics of the atomic obitals. We have seen that M ξ l s M M M M M ( ) i = ζ ( ) i = ζ ( ) i i i whee is the wavefunction fo a Russell-andes tem, say any one of the 9 components of the P level of cabon. We know that some of these states may be witten as a single deteminant, in paticula =, =,M =,M = = Aˆ sα sβsαsβp αpα so we may evaluate the + expectation value of the petubation using the late-condon ules ˆ ˆ A sα sβ sα sβ p α pα ξ l s Asαsβsαsβp αpα ζ p o we have the identity ζ (, ) = ζ ( ( i) i i ˆ i = ( )( + ) + + p ). This is a vey geneal technique and allows us to detemine the obital contibution to the spin obit paamete. The geneal fom of the ζ x ( l ) elationship is ζ ( ) = fo x < l + and ( ) x ( l ) ζ ζ = fo x > l +. Fo example the lowest tem of the Chomium configuation is 5 so ( d 5 ) ζ ( D) = ζ 5. The expeimental ζ ( ) ( d ) ζ = 7. 7cm s d D = 56. 8cm so the obital paamete is. Using this elationship we constuct the following table. D. F. Haison Michigan tate Univesity //6 8

9 element tem ζ ( tem )cm ζ ( d )cm c D( d ) Ti F( d ) V F( d ) C 5 D( d ) Mn 5 ( d ) A Fe 5 6 D( d ) Co 7 F( d ) i 8 F( d ) Cu 9 D( d ) spin obit paamete fo fist ow tansition elements 8 6 s d ξ(d) ξ(d)cm ξ() numbe of d electons. F. Haison Michigan tate Univesity //6 9

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects