Spin.E2.80.93orbit interaction in atomic energy levels Spin–orbit interaction

1 spin–orbit interaction in atomic energy levels

1.1 energy of magnetic moment
1.2 magnetic field
1.3 magnetic moment of electron
1.4 larmor interaction energy
1.5 thomas interaction energy
1.6 total interaction energy
1.7 evaluating energy shift
1.8 final energy shift

spin–orbit interaction in atomic energy levels

this section presents relatively simple , quantitative description of spin–orbit interaction electron bound atom, first order in perturbation theory, using semiclassical electrodynamics , non-relativistic quantum mechanics. gives results agree reasonably observations. more rigorous derivation of same result start dirac equation, , achieving more precise result involve calculating small corrections quantum electrodynamics.

energy of magnetic moment

the energy of magnetic moment in magnetic field given by

Δ
h
=
−

μ

⋅

b

,


{\displaystyle \delta h=-{\boldsymbol {\mu }}\cdot \mathbf {b} ,}

where μ magnetic moment of particle, , b magnetic field experiences.

magnetic field

we shall deal magnetic field first. although in rest frame of nucleus, there no magnetic field acting on electron, there 1 in rest frame of electron (see classical electromagnetism , special relativity). ignoring frame not inertial, in si units end equation

b

=
−




v

×

e



c

2




,


{\displaystyle \mathbf {b} =-{\frac {\mathbf {v} \times \mathbf {e} }{c^{2}}},}

where v velocity of electron, , e electric field travels through. here, in non-relativistic limit, assume lorentz factor



γ
⋍
1


{\displaystyle \gamma \backsimeq 1}

. know e radial, can rewrite




e

=

|

e

/

r

|


r



{\displaystyle \mathbf {e} =|e/r|\mathbf {r} }

. know momentum of electron




p

=

m

e



v



{\displaystyle \mathbf {p} =m_{\text{e}}\mathbf {v} }

. substituting in , changing order of cross product gives

b

=




r

×

p




m

e



c

2






|


e
r


|

.


{\displaystyle \mathbf {b} ={\frac {\mathbf {r} \times \mathbf {p} }{m_{\text{e}}c^{2}}}\left|{\frac {e}{r}}\right|.}

next, express electric field gradient of electric potential




e

=
−
∇
v


{\displaystyle \mathbf {e} =-\nabla v}

. here make central field approximation, is, electrostatic potential spherically symmetric, function of radius. approximation exact hydrogen , hydrogen-like systems. can that

|

e

|

=



∂
v


∂
r



=


1
e





∂
u
(
r
)


∂
r



,


{\displaystyle |e|={\frac {\partial v}{\partial r}}={\frac {1}{e}}{\frac {\partial u(r)}{\partial r}},}

where



u
=
e
v


{\displaystyle u=ev}

potential energy of electron in central field, , e elementary charge. remember classical mechanics angular momentum of particle




l

=

r

×

p



{\displaystyle \mathbf {l} =\mathbf {r} \times \mathbf {p} }

. putting together, get

b

=


1


m

e


e

c

2







1
r





∂
u
(
r
)


∂
r




l

.


{\displaystyle \mathbf {b} ={\frac {1}{m_{\text{e}}ec^{2}}}{\frac {1}{r}}{\frac {\partial u(r)}{\partial r}}\mathbf {l} .}

it important note @ point b positive number multiplied l, meaning magnetic field parallel orbital angular momentum of particle, perpendicular particle s velocity.

magnetic moment of electron

the magnetic moment of electron is

μ


s


=
−

g

s



μ

b





s

ℏ


,


{\displaystyle {\boldsymbol {\mu }}_{s}=-g_{\text{s}}\mu _{\text{b}}{\frac {\mathbf {s} }{\hbar }},}

where




s



{\displaystyle \mathbf {s} }

spin angular-momentum vector,




μ

b




{\displaystyle \mu _{\text{b}}}

bohr magneton, ,




g

s


≈
2


{\displaystyle g_{\text{s}}\approx 2}

electron-spin g-factor. here




μ



{\displaystyle {\boldsymbol {\mu }}}

negative constant multiplied spin, magnetic moment antiparallel spin angular momentum.

the spin–orbit potential consists of 2 parts. larmor part connected interaction of magnetic moment of electron magnetic field of nucleus in co-moving frame of electron. second contribution related thomas precession.

larmor interaction energy

the larmor interaction energy is

Δ

h

l


=
−

μ

⋅

b

.


{\displaystyle \delta h_{\text{l}}=-{\boldsymbol {\mu }}\cdot \mathbf {b} .}

substituting in equation expressions magnetic moment , magnetic field, 1 gets

Δ

h

l


=



2

μ

b




ℏ

m

e


e

c

2







1
r





∂
u
(
r
)


∂
r




l

⋅

s

.


{\displaystyle \delta h_{\text{l}}={\frac {2\mu _{\text{b}}}{\hbar m_{\text{e}}ec^{2}}}{\frac {1}{r}}{\frac {\partial u(r)}{\partial r}}\mathbf {l} \cdot \mathbf {s} .}

now have take account thomas precession correction electron s curved trajectory.

thomas interaction energy

in 1926 llewellyn thomas relativistically recomputed doublet separation in fine structure of atom. thomas precession rate





Ω


t




{\displaystyle {\boldsymbol {\omega }}_{\text{t}}}

related angular frequency of orbital motion




ω



{\displaystyle {\boldsymbol {\omega }}}

of spinning particle follows:

Ω


t


=

ω

(
γ
−
1
)
,


{\displaystyle {\boldsymbol {\omega }}_{\text{t}}={\boldsymbol {\omega }}(\gamma -1),}

where



γ


{\displaystyle \gamma }

lorentz factor of moving particle. hamiltonian producing spin precession





Ω


t




{\displaystyle {\boldsymbol {\omega }}_{\text{t}}}

given by

Δ

h

t


=


Ω


t


⋅

s

.


{\displaystyle \delta h_{\text{t}}={\boldsymbol {\omega }}_{\text{t}}\cdot \mathbf {s} .}

to first order in



(
v

/

c

)

2




{\displaystyle (v/c)^{2}}

, obtain

Δ

h

t


=
−



μ

b



ℏ

m

e


e

c

2







1
r





∂
u
(
r
)


∂
r




l

⋅

s

.


{\displaystyle \delta h_{\text{t}}=-{\frac {\mu _{\text{b}}}{\hbar m_{\text{e}}ec^{2}}}{\frac {1}{r}}{\frac {\partial u(r)}{\partial r}}\mathbf {l} \cdot \mathbf {s} .}



total interaction energy

the total spin–orbit potential in external electrostatic potential takes form

Δ
h
≡
Δ

h

l


+
Δ

h

t


=



μ

b



ℏ

m

e


e

c

2







1
r





∂
u
(
r
)


∂
r




l

⋅

s

.


{\displaystyle \delta h\equiv \delta h_{\text{l}}+\delta h_{\text{t}}={\frac {\mu _{\text{b}}}{\hbar m_{\text{e}}ec^{2}}}{\frac {1}{r}}{\frac {\partial u(r)}{\partial r}}\mathbf {l} \cdot \mathbf {s} .}

the net effect of thomas precession reduction of larmor interaction energy factor 1/2, came known thomas half.

evaluating energy shift

thanks above approximations, can evaluate detailed energy shift in model. in particular, wish find basis diagonalizes both h0 (the non-perturbed hamiltonian) , Δh. find out basis is, first define total angular momentum operator

j

=

l

+

s

.


{\displaystyle \mathbf {j} =\mathbf {l} +\mathbf {s} .}

taking dot product of itself, get

j


2


=


l


2


+


s


2


+
2


l

⋅

s



{\displaystyle \mathbf {j} ^{2}=\mathbf {l} ^{2}+\mathbf {s} ^{2}+2\,\mathbf {l} \cdot \mathbf {s} }

(since l , s commute), , therefore

l

⋅

s

=


1
2


(


j


2


−


l


2


−


s


2


)
=



ℏ

2


2




(




l

z





l

−







l

+




−

l

z





)


.


{\displaystyle \mathbf {l} \cdot \mathbf {s} ={\frac {1}{2}}(\mathbf {j} ^{2}-\mathbf {l} ^{2}-\mathbf {s} ^{2})={\frac {\hbar ^{2}}{2}}{\begin{pmatrix}l_{z}&l_{-}\\l_{+}&-l_{z}\end{pmatrix}}.}

it can shown 5 operators h0, j, l, s, , jz commute each other , Δh. therefore, basis looking simultaneous eigenbasis of these 5 operators (i.e., basis 5 diagonal). elements of basis have 5 quantum numbers: n (the principal quantum number ), j (the total angular momentum quantum number ), l (the orbital angular momentum quantum number ), s (the spin quantum number ), , jz (the z component of total angular momentum ).

to evaluate energies, note that

⟨


1

r

3




⟩

=


2


a

3



n

3


l
(
l
+
1
)
(
2
l
+
1
)





{\displaystyle \left\langle {\frac {1}{r^{3}}}\right\rangle ={\frac {2}{a^{3}n^{3}l(l+1)(2l+1)}}}

for hydrogenic wavefunctions (here



a
=
ℏ

/

(
z
α

m

e


c
)


{\displaystyle a=\hbar /(z\alpha m_{\text{e}}c)}

bohr radius divided nuclear charge z); and

⟨

l

⋅

s

⟩

=


1
2




(


⟨


j


2


⟩
−
⟨


l


2


⟩
−
⟨


s


2


⟩


)


=



ℏ

2


2




(


j
(
j
+
1
)
−
l
(
l
+
1
)
−
s
(
s
+
1
)


)


.


{\displaystyle \left\langle \mathbf {l} \cdot \mathbf {s} \right\rangle ={\frac {1}{2}}{\big (}\langle \mathbf {j} ^{2}\rangle -\langle \mathbf {l} ^{2}\rangle -\langle \mathbf {s} ^{2}\rangle {\big )}={\frac {\hbar ^{2}}{2}}{\big (}j(j+1)-l(l+1)-s(s+1){\big )}.}



final energy shift

we can that

Δ
e
=


β
2




(


j
(
j
+
1
)
−
l
(
l
+
1
)
−
s
(
s
+
1
)


)


,


{\displaystyle \delta e={\frac {\beta }{2}}{\big (}j(j+1)-l(l+1)-s(s+1){\big )},}

where

β
=
β
(
n
,
l
)
=

z

4





μ

0



4
π




g

s



μ

b


2




1


n

3



a

0


3


l
(
l
+
1

/

2
)
(
l
+
1
)



.


{\displaystyle \beta =\beta (n,l)=z^{4}{\frac {\mu _{0}}{4\pi }}g_{\text{s}}\mu _{\text{b}}^{2}{\frac {1}{n^{3}a_{0}^{3}l(l+1/2)(l+1)}}.}






^ l. h. thomas, motion of spinning electron, nature (london), 117, 514 (1926).
^ l. föppl , p. j. daniell, zur kinematik des born schen starren körpers, nachrichten von der königlichen gesellschaft der wissenschaften zu göttingen, 519 (1913).
^ c. møller, theory of relativity, (oxford @ claredon press, london, 1952).