Download Zeeman effect – Studying magnetic fields in star

A little bit of history…
Pieter Zeeman (1865 – 1943)
• An experiment in 1896 showed that two of the lines of
Na were broadened when a Na flame was placed between
strong magnetic poles (Nature 1897, vol. 55, pg. 347)
the 1902 Nobel Prize
Later experiment showed that line broadening was
actually line splitting
Lorentz was able to explain the line splitting with
his theory but the number of split components did
not agree with Lorentz’s prediction
anomalous Zeeman effect
Lorentz’s electron theory
Electron oscillates within the atom
Oscillation is modified by the application of Bfield
No field → spectral line with frequency ν0
Field on → triplet of frequencies: ν0 – δν, ν0,
ν0 + δν, where δν depends on the field strength B
Linear oscillation at ν0 can be resolved into
clockwise and counterclockwise circular motions
(ω=2πν0)
Simple harmonic binding force: F = −kr
1
ν0 =
2π
k
me
B-field cause the additional Lorentz force:
FL = −ev × B
new angular velocityω = 2πν
v = ωr = 2πrν
the centripetal force is
me v
2 2
= 4π ν rme
r
2
Thus we get
4π ν rme = ±2πνreB + kr
2
2
± refers to clockwise and counterclockwise
motion
Using the expression for ν0 to eliminate k we get
the quadratic eqs. for ν:
ν
2
eB
±
ν −ν0 = 0
2π m e
eB
ν = ν0 ±
4πme
So the frequency shift has the form
eB
δν =
4π m e
Note the presence of the factor e/me above !
Semi-classical approach
An added contribution to the potential energy of
the system (atom) due to the B-field is
VM = − • B
µ is a magnetic dipole moment
Let’s consider a circular Bohr orbit for an
electron:
ω = 2πν , A = π r , I = eν
2
µ = IA = πr 2 eν
• Angular momentum is
L = me vr = 2πr meν
2
µ
e
=
= orbital gyromagnetic ratio
L 2me
• Negative charge
e
=−
L
2me
L
v
•
•
r
I
µ
-e
We can also write
= − gµ B
L
g=1 the orbital g-factor and the Bohr magneton is
e
µB =
= 5.788 ⋅ 10 −5 eV T −1
2me
In the presence of an external B-field, a magnetic
dipole experiences a torque
T=
×B
dL
=
dt
where
e
×B = −
L×B =
2me
e
B
=
2me
cf. a
× L, spinning
top
is a constant vector
• So we get the frequency of (Larmor) precession as
ω
eB
=ν =
2π
4πme
Quantum mechanical
approach
L is quantized, µ ∝ L µ is quantized
µ • B the quantization of µ accounts for the
Zeeman splitting in the energy levels of the atom
If there is no B-field, we have
−
2
2µ
∇ + V (r ) ψ nlml = Enψ nlml ,
2
where V denotes the Coulomb potential energy,
ψnlm is the stationary-state eigenfunction and En is
the energy eigenvalue
Let’s put the B-field on, and see what happens
(B=Bez):
VM = − • B = − µ z B
In QM, we have
L ≡ −i r × ∇
∂
Lz = −i
∂φ
So we get another differential operator
− µzB =
gµ B B
∂
Lz = −igµ B B
∂φ
Now we need a new stationary state
2
Ψ = ψe
∂
ψ = Eψ
−
∇ + V (r ) − igµ B B
2µ
∂φ
2
Since Lz = ml , where ml=-l,…,l is the
magnetic quantum number, we note that
∂
− i ψ nlml = mlψ nlml
∂φ
− iEt /
2
∂
−
∇ + V (r ) − igµ B B
ψ nlml = Enψ nlml + gµ B Bmlψ nlml
2µ
∂φ
2
• So the new energy eigenvalue is
E = En + gµ B Bml
• States with adjacent values of ml are spaced in
energy by an amount
δ E M = gµ B B
Zeeman splitting of the energy levels causes a
frequency shift in the radiation from the atom
Selection rule for ml is
∆ml = 0 or ± 1
three distinct emitted-photon energies
∆E − δEM , ∆E , ∆E + δEM
A shift in frequency is (once again)
δν =
δE M
h
gµ B B
eB
=
=
2π
4πme
• The previous cases described the so-called
normal Zeeman effect:
special case in which the total electron spin of
an atom = 0 and the equally-spaced Lorentz triplet
is observed
The electron spin
Spin is needed in QM to explain the fine structure
of spectral lines and the anomalous Zeeman effect
A new quantum mechanical variable for the
electron is a spin vector S:
1
S z = ms , where ms = ± is a spin quantum number
2
The Stern-Gerlach experiment implies that both L
and S have their own unique magnetic moments
and g-factors
The new spin magnetic moment of the electron is
analogous to L
S
= −gS µB
S
,
where gS≈2 is the spin g-factor
The total angular momentum of the atom is
J =L+S
The Zeeman effect
Both µL and µS contribute to µ:
=
L
+
S
=−
µB
(L + 2S)
The expectation value of the magnetic-moment
interaction is
VM = − µ z B =
µB
Lz + 2 S z B = µ B B (ml + 2ms )
The component of µ along J is
µJ =
=−
µB
J
•J
µB
=−
(L + 2S) • (L + S)
J
J
( L2 + 2 S 2 + 3S • L)
J 2 = (L + S) • (L + S) = L2 + 2L • S + S 2
3 2
µJ = −
L + 2 S + ( J − L2 − S 2 )
2
J
µB
=−
µB
J
2
2
(3 J 2 + S 2 − L2 )
B
Jz
µ⊥B
µ⊥J
µ
θ
θ
µJ
J
µz =
→
J
J
+
⊥J
+
cos θ
J z = J cos θ
⊥B
cos θ
Now we can write the following expectation value
µz J
2
= J z Jµ J = −
µB
2
J z (3 J + S − L )
2
2
Next we use the following eigenvalue properties
=
2
j ( j + 1)
=
2
s ( s + 1)
L =
2
J
S
2
2
2
l (l + 1)
2
The resulting equality can be written as
µz = −
= − gµ B
µ B 3 j ( j + 1) + s ( s + 1) − l (l + 1)
2
j ( j + 1)
Jz
The coefficient
j ( j + 1) + s ( s + 1) − l (l + 1)
g = 1+
2 j ( j + 1)
is called the Landé g-factor
Jz
• The Zeeman energy shift becomes now
VM = − µ z B = gµ B B
Jz
= gµ B Bm j ,
m j = − j ,− j + 1,..., j − 1, j
There is as many possible transitions as allowed
by the selection rule
∆m j = 0, ± 1
Thus the anomalous pattern of Zeeman spectral
lines is observed, instead of the normal Lorentz
triplet
• The more general case is
known as the anomalous
Zeeman effect:
the total orbital and spin
angular momenta of an
atom are both ≠ 0 and the
line splitting is more
complex (> 3 components)
∆m j = 1
∆m j = −1
∆m j = 0
lines are right - circularly polarized (σ R )
lines are left - circularly polarized (σ L )
lines are linearly polarized (π )
• Observing along the B-field, the π component
vanish and only the σ components, which are
polarized in opposite senses, are observed
these are the components that are used in
practise to determine the B-fields
When dealing with a many-electron atom, the
magnetic moment is constructed from the orbital
and spin contribution of each electron:
=
Z
i =1
i
=−
µB
(L i + S i )
i
For some molecules, µ can also come from
localized charges or from the nuclear spins (I),
and its value is generally close to the nuclear
magneton
e
µB
µN =
=
2m p 1836
B-field probes
B-fields in the atomic envelopes of molecular
clouds are observed through the Zeeman effect in
HI (2S1/2 , F=1-0 @ 1.420 GHz, Z=2.80 Hz µG –1 )
Molecular line observations are necessary to probe
the denser regions
To measure B-field accurately, a molecular probe
should have a relatively large µ
For molecules with grand total angular momentum
F (=J+I), the perturbation of the state energy from
its value at B=0 is
∆EB = − g F µ B M F B
M F = F,
g F = g Jα J + g I β I
J ( J + 1) + S ( S + 1) − N ( N + 1)
gJ =
J ( J + 1)
F ( F + 1) + J ( J + 1) − I ( I + 1)
αJ =
2 F ( F + 1)
g I (H ) = 2.792847
F ( F + 1) + I ( I + 1) − J ( J + 1)
βI =
2 F ( F + 1)
,− F
Dividing the previous energy eq. by h, we get the
splitting in terms of frequency:
Z
∆ν B =
B,
2
Z
gF µBM F
=−
2
h
Z is known as the Zeeman factor
(Zeeman splitting coefficient, Zeeman sensitivity)
Especially important are molecules with one unpaired
electron (Λ≠0 for diatomic molecules), classified chemically
as free radicals
The most widely used species of this kind is OH
(2Π3/2, J=3/2, F=1+-1- @ 1.6654018 GHz, Z=3.27 Hz µG –1 )
(2Π3/2, J=3/2, F=2+-2- @ 1.6673590 GHz, Z=1.96 Hz µG –1 )
However,
OH is highly reactive
[OH/H2] decreases significantly at higher densities
OH serves as a tracer only of rather diffuse material (n(H2)
≈ 103-4 cm-3) (small A-coefficient small ncrit)
• Other candidates for Zeeman detection are, e.g.,
CH (2Π3/2, J=3/2, F=2-2 @ 0.701677 GHz,
Z=1.96 Hz µG -1)
C4H (2Σ, N=1-0, J=3/2-1/2, F=1-2 @ 9.497616 GHz,
Z=1.40 Hz µG -1)
C2S (3Σ-, JN=10-01 @ 11.19446 GHz, Z=0.84 Hz µG -1)
SO (3Σ-, JN=22-11 @ 86.094 GHz, Z=0.47 Hz µG -1)
CN (2Σ, N=1-0, J=3/2-1/2, F=5/2-1/2 @ 113.49115 GHz,
Z=0.56 Hz µG -1)
C2H (2Σ, N=1-0, J=3/2-1/2, F=2-1 @ 87.31723 GHz,
Z=1.40 Hz µG -1)
• Z-factors are small ∆νB is very small for typical
cloud field strengths! (in masers ∆νB > ∆ν)
• Especially, CN offers the best opportunity to
measure B-fields in dense cores (n(H2) ≈ 105-6 cm-3)
(Crutcher et al. 1996, ApJ, 456, 217)
• CN (N=1→0) transition at ν≈113.5 GHz has a total
of 9 hf-components (of which 7 are strong and of
these 4 have a strong Zeeman effect)
• The very different Z values of the hf. lines helps to
distinguish the real Zeeman effect and instrumental
effects
Measurement of B-fields
Molecular lines are brodened at least thermally
∆ν therm
ν0
vs
≈
c
vs ∝ Tkin is the sound speed in the cloud
∆ν B
ZB c
B
−3
≈
≈ 10
G
∆ν therm ν 0 vs
Tkin
10 K
−1 / 2
Since B-fields in space are small, the well- separeted
lines are not seen
Thus, a more indirect approach is needed
Molecular line can be observed through two
circularly polarizing filters of opposite helicity
The resulting line profiles,
I R (ν − ν 0 )
I L (ν − ν 0 )
are identical in shape, but shifted in frequency by ∆νB
The intensity difference of these two components
is known as the Stokes V-parameter
The V-spectrum of the source is
I R (ν − ν 0 ) − I L (ν − ν 0 )
Both IR and IL are reduced in magnitude from the
total intensity by cos θ, where θ is the angle
between the B-field and the line of sight
θ
B
Now we can write the difference as
I R (ν −ν0 ) − IL (ν −ν0 ) = cosθ [I (ν −ν0 − ∆ν B ) − I (ν −ν0 + ∆ν B )]
dI
≈ −2 cosθ∆ν B (ν −ν0 )
dν
dI
= −ZBcosθ (ν −ν0 )
dν
dI
= −ZB|| (ν −ν0 )
dν
1)
2)
3)
The final expression represents a practical means
of determing the Blos :
first measure directly the V-spectrum on the LHS
the derivative on the RHS is best to evaluate using
the average of IR and IL, i.e., ½ times the Ispectrum (the Stokes I-parameter is the sum of the
two circularly polarized components: IR + IL)
determine Blos through a fitting procedure (find the
best-fit value for Blos)
Note:
a positive value of Blos
a negative value of Blos
∆νB > ∆ν
B-field points away the observer
B-field points toward the observer
the splitting is directly measurable and provides B itself
It is important to know what the polarization
properties of the telescope are !
Important instrumental effect are
beam squint = small offset of the beam in L and R
circular polarization in the presence of vgradients in the cloud, this produce a pseudoZeeman splitting
the polarization leakage between L and R
circularly polarized receiver channels
See more from Heiles and Crutcher 2005, Cosmic
magnetic fields, Lecture notes in Physics, 664, 137
(astro-ph/0501550)
Statistically, for a random orientation of B with
respect to the los, the most probable values for B
and B2 are B=2Blos and B2=3B2los
π /2
B cos θ sin θdθ
Blos =
0
=
π /2
sin θdθ
1
B
2
0
π /2
( B cos θ ) 2 sin θdθ
2
los
B
=
0
π /2
sin θdθ
0
1 2
= B
3
Some results: HMSFRs
W3(OH)
OH, 2Π3/2 J=7/2 F=3+-3Λ-doublet absorption line
Blos=3100±
±400 µG
(Güsten et al. 1994, A&A,
286, L51)
NGC 6344A
OH 1665 MHz
Blos=148±
±20 µG
OH 1667 MHz
Blos=162±
±33 µG
HI 1420 MHz
Blos=47±
±15 µG
(Sarma et al.
2000, ApJ, 533,
271)
DR21(OH)
CN hf. components
Blos=-450±
±150 µG
(Crutcher et al. 1999,
ApJ, 514, L121)
Some results: LMSFRs
L1544
OH 18 cm
Blos=10.8±
±1.7 µG
(Crutcher & Troland 2000, ApJ, 537, L139)
TMC-1 CP
(Turner & Heiles 2006, ApJS, 162, 388)
C4H F=0-1 & F=1-2
Blos=14.5±
±14.0 µG
Cf.
The general B-field of the Galaxy ∼ 2-5 µG
Average IMF @ 1 AU ∼ 60 µG
Earth’s B-field @ Helsinki ∼ 0.6 G
B-field in a typical US home ∼ 1 mG – 1 G