spin-orbit interaction (or “coupling”) h&w chapter 12dc43/ph4021/lectures... · spin-orbit...
TRANSCRIPT
This interaction is weaker than the Coulomb interaction and is
responsible for the fine-structure of atomic lines.
How do we understand the spin-orbit interaction?
• The electron has a spin s, as well as orbital angular momentum l.
• For each angular momentum, there is an associated
magnetic moment, and
• Spin-orbit = interaction between and
Let’s start with the definition of magnetic moment….
lµ
sµ
Spin-orbit interaction (or “coupling”) H&W Chapter 12
lµ
sµ
electron orbit = current loop
the magnetic moment is defined as where I is the current
and A the area of the orbit.
is a vector perpendicular to the plane of the orbit.
Now let’s show that is linked to the orbital angular momentum:
π
ω
ω
π
2
2T
e
T
eI
−=
−==
Magnetic moment lµ
22
2
1
2rer
eAIl ωπ
π
ωµ −=
−=⋅=
AIl ⋅=µ
lµ
lµ
orbit
period
2
00 rmvrmprl ω==×=rrr
orbital
ang. mom.l
m
el
rr
02−=µ
r
v
+
e
lµr
lr
A
o
Bm
e
2
h=µ Bohr magneton
h=l
Note that and point in opposite directions because the electron is
negatively charged.
Let’s consider the case : this is the orbital angular
momentum of the ground state in Bohr’s theory, and is therefore
taken as the unit of angular momentum in atoms.
The modulus of the corresponding magnetic moment is known as
Bohr magneton:
It is unit of magnetic moments in atoms.
lr
lµr
)1()1(2
+=+= llllm
eB
o
l µµ hr
More generally, for a state with quantum numbers l and :
hlZ ml = lml ±±±= ....2,1,0with magnetic quantum number
lBz
o
Zl mlm
eµµ −=−=
2,Component along z
Modulus
lm
)1( += lll hr
We also can apply these definitions to the electron spin….
Electron spin
)2
1(with
±=
=
s
sZ
m
ms h
hhr
2
3)1( =+= sss
• An intrinsic property like charge or mass.
• Its existence proven by Stern-Gerlach experiment (see later), and
justified by Dirac’s relativistic quantum theory.
• It’s an intrinsic angular momentum with quantum number2
1=s
Hence we can apply the same theory that we saw for the orbital
angular momentum (see Hydrogen atom- angular momentum aside).
Component along z
Modulus
Spin magnetic moment sm
eg
o
ss
rr
2−=µ
0023.2=sgWhere is a numerical factor known as the Lande factor
(or “g factor”). Classically it should be 1, but is 2 according to
Dirac’s relativistic quantum theory.
( for orbital angular momentum)1=lg
Next, let’s consider the magnetic interaction of a magnetic
moment with a magnetic field :
Brr
×= µτtorque
(Note that the potential energy is minimised if magnetic moment is
parallel to B, i.e. α=0.)
Magnetic potential energy:
µr
Br
αµr
Br
BB
BdBdBVmag
rr
rr
⋅−=−=
=−==×= ∫∫
µαµ
αµααµαµ απ
α
π
α
π
cos
][cossin)( 2/
2/2/
BVmag
rr⋅−= µ
lssl BVrr
⋅−= µ,
lBr
lBr
s
s
Spin-orbit interaction
r
-rlBr
Orbiting proton
produces a field
at the site of
the electron.lB
The spin, hence , has two possible orientations with respect
to the field: “parallel”, with lower energy, and “anti-parallel” with
higher energy
we expect energy levels in atoms to split in two (fine structure).
sµr
Note that this is not the only magnetic interaction that is taking place in an atom. The nucleus
also has a spin, hence a magnetic moment. The nuclear magnetic moment interacts with the
magnetic field produced by the orbiting electron. This interaction is much weaker than the
spin-orbit interaction considered here, and is responsible for the hyperfine structure of the
energy levels – see later.
Biot-Savart law: ][4
)]([4 3
0
3
0 rvr
Zerv
r
ZeBl
rrrrr×−=−×=
π
µ
π
µ
rvml
vmrlrrr
rrr
×=−
×=
0
0l
mr
ZeBl
rr
0
3
0
4π
µ=
Our goal is to determine an expression for the fine structure starting
from . Let’s start by writing the B-field in terms of the
orbital angular momentum: lssl BVrr
⋅−= µ,
The – sign here is because we are
in the electron’s frame of reference,
see previous slide
lmr
ZeBl
rr
0
3
0
42
1
π
µ×=
from full relativistic
calculation
(“Thomas factor”)
sm
es
m
eg
oo
ss
rrr−≈−=
2µ
)2( ≈sg
)(8
)(32
0
0
2
0
, lsrm
ZeBs
m
eV lsl
rrrr⋅=⋅=
π
µ
sla
V sl
rr
h⋅=
2,32
0
2
0
2
8 rm
Zea
π
µ h=
We define the spin-orbit coupling constant a:
( )
12
13
4
+
+
∝
llln
Za
(With this definition a has the dimension of an energy.)
The constant a is measurable from spectra. We won’t do the
calculation, but it can be shown that
dVrr
ψψ∫=3
*
3
11
n valueexpectatio from
Heavier atoms larger spin-orbit coupling
Larger n smaller spin-orbit coupling
222
222
2,
ˆ ˆ ˆ
)(2
slj
slja
V sl
rrr
h−−=
… let’s now consider the scalar product .
It is possible to express this in terms of the total angular momentum:
),cos( slslslrrrr
=⋅
sljrrr
+= jr
lr
sr
),cos(2222
slslsljrrrrr
++=
(These vectors are operators
in quantum mechanics)
jjjm j −−= ,..,1,
modulus^2: with
projection along z-axis: with
jr
Properties of :
• It is an angular momentum, hence we can apply the results
we saw in the “angular momentum aside”.
new quantum numbers j and such that:jm
Ψ+=Ψ )1(ˆ 22jjj h
Ψ=Ψ jz mj hˆ
... ,2
3 1, ,
2
1
2
integer ==j
[ ] [ ] 0ˆ,ˆ , 0ˆ,ˆ 2222 == sjlj• We have the commutation relations
[ ] [ ] ),, where0ˆ,ˆˆ,ˆ from follow(which 22 zyxisjlj ii ===
hence we can choose a set of common eigenfunctions of
, and calculate on these eigenfunctions….222 ˆ,ˆ,ˆ jsl slV ,
2
1±=±= lslj
[ ])1()1()1(2
)(2
222
2, +−+−+=−−= sslljja
slja
V sl
rrr
h
… and so we can replace the operators with the corresponding
eigenvalues:
[ ])1()1()1(2
, +−+−+= sslljja
V sl
This is our final result for spin-orbit coupling:
shift in energy due
to spin-orbit coupling
But we still haven’t discussed how to calculate j….
From the theory of coupling of angular momenta, it turns out that
e.g. for a p state (l=1)j=1/2
j=3/2
Example:
fine structure of sodium atoms
• state 3s (l=0)
there is no fine structure
(physically, there is no angular momentum,
hence no “current loop” and )
• state 3p
0 and 2/1 , == slVj
0=lB
j=3/2 and
j=1/2 and aa
V sl −=
⋅−−⋅=2
3
2
12
2
3
2
1
2,
22
3
2
12
2
5
2
3
2,
aaV sl =
⋅−−⋅=
fine structure for 3p a/2
aa
2
3j=3/2
j=1/2
3s 3p transition is split into a doublet
(D lines: D1 and D2)
Sodium doublet viewed in a Fabry-
Perot interferometer:
Nomenclature for naming energy terms:
j
sln
12 +
orbital angular momentum
from earlier =s,p,d,f etc...
total angular
momentum
spin term
principal
quantum
number
e.g. ground state of sodium is
(the spin term is sometimes omitted, like in the previous slide, because
it’s the same for all the levels.)
2/1
23 s
Let’s see how the spectral lines split
due to fine-structure:
• series 3s np: pairs of lines
(all p states are double)
• series 3p nd: triple lines
This is because:
d state (l=2)j=5/2
j=3/2
and furthermore we have a new selection rule on j
hence only these three transitions are allowed:0,1±=∆j
2/13p
2/33p
2/5n d
2/3n d
Summary for FINE STRUCTURE
• spin-orbit coupling
• all energy states (except s-states) are split into dublets
• split (larger split in heavy atoms)4
Z∝
sla
BV lssl
rr
h
rr⋅=⋅−=
2, µ
But this is not the end with fine structure….
} 48476 FSE
slrel
nRy
n EEEE
structure fine
,
/
structuregross
2
++=
−
relativistic
mass change
spin-orbit coupling
…Hydrogen is a special case:
Small spin-orbit comparable to relativistic corrections
(remember Bohr-Sommerfeld:
in that model the fine structure was due to relativistic effects)
In heavier atoms, spin-orbit coupling is stronger and relativistic
effects can be neglected.
For the H-atom:
)(8 32
0
0
2
lsrm
e rr⋅
π
µ
−
+−=
njn
EE n
FS4
3
2/1
12α
Result from Dirac’s relativistic theory:
where the dimensionless quantity
is the fine structure constant.
(You can verify that α=v/c with v= velocity of 1st Bohr orbit.)
137
1
2 0
2
==hc
e
εα
Bohr Dirac QED
Example: let’s apply this result to the n=2 level of Hydrogen.
Here we have an s state and a p state:
2/3 ,2/11
2/10
=⇒=
=⇒=
jl
jl Can calculate
for these values of jFSE
(relativistic QM)
l=1, j=3/2
n=2
GHz10≈
l=0,1, j=1/2 MHz1057 2/1
22 p
2/1
22 s
2/3
22 p
This is a further splitting that
can only be explained by
Quantum Electrodynamics (QED)…
Significance of Lamb shift
In QED the electromagnetic field is quantised.
A quantised field has a zero point energy (analogous to the ground
state of the simple harmonic oscillator). This means that even in the
case of number of photons=0 (“vacuum”), there is a fluctuating
electric field.
Due to this fluctuating field the electron performs small oscillations
and its charge is effectively smeared out. It can be shown that the
Coulomb potential acting on the electron is different than that for a
point charge. This difference causes the Lamb shift.
(Note: this fluctuating electric field is also responsible for
spontaneous emission.)
… this is known as the Lamb shift of the s level.
It was first observed in 1947-1952 and was crucial
for the development of QED.