Appendix 4: Spectral Data for Mercury
MERCURY
The prominent mercury lines are at 435.835 nm (blue), 546.074 nm (green), and a pair at
576.959 nm and 579.065 nm (yellow-orange). There are two other blue lines at 404.656 nm and
407.781 nm and a weak line at 491.604 nm.
http://physics.nist.gov/PhysRefData/Handbook/Tables/mercurytable2.htm
1000 P,c3983.931Hg IISR01
400 P
4046.563Hg I BAL50
60
4339.223Hg I BAL50
100
4347.494Hg I BAL50
1000 P
4358.328Hg I BAL50
12 c
5128.442Hg IISR01
15
5204.768Hg IISR01
80 P
5425.253Hg IISR01
500 P
5460.735Hg I BAL50
200 P
5677.105Hg IISR01
50
5769.598Hg I BAL50
60
5790.663Hg I BAL50
HgSpec.1
Appendix 4: Spectral Data for Mercury
http://physics.nist.gov/PhysRefData/Handbook/Tables/mercurytable3.htm
HgSpec.2
Appendix 4: Spectral Data for Mercury
400 P 4046.563Hg IBAL50
60
4339.223Hg IBAL50
100
4347.494Hg IBAL50
1000 P4358.328Hg IBAL50
500 P 5460.735Hg IBAL50
50
5769.598Hg IBAL50
60
5790.663Hg IBAL50
Polarization and the Zeeman Effect: Nothing has been presented about the spin
angular momentum of the electron so only comments about the polarization of the
light emitted during transitions is to be presented, not the details of the energy
levels. First, the z axis direction is set whenever a measurement is made or a
direction is made distinct. For the Zeeman problem, a magnetic field is applied to
the atom defining the field direction as the z direction with the result that the
energy of the states shift by m effective. In a 210 to 100 transition, the electron
density oscillates along the z direction and the emitted light is polarized along the
projection of the z direction onto a plane perpendicular to the plane of propagation.

3D to 2P Transitions with the levels split by an applied B field.
The three lines are the m = 0, 1 transitions.
Splittings are of order 60 (eV) or 14 GHz in a 1 Tesla field.
HgSpec.3
Appendix 4: Spectral Data for Mercury
For a 211 to 100 transition, the electron density circles around the z direction at the
difference frequency 2 - 1. Viewing in along the z direction the light is circularly
polarized. View from a direction in the x-y plane, only the electron density
projection is observed – motion back and forth along a line – so the light is linearly
polarized. Again, consider the projection of the circular motion onto a plane
perpendicular to the propagation direction of the light. For light propagating at
angles  relative to the z direction, the projection on the motion onto the plane
yields an elliptical path, and the emitted light is elliptically polarized.
Begin by imagining the field line pattern due to a static point charge at the origin.
Next imagine the charge executing uniform circular motion about the origin in
the x-y plane. The information about the charges motion propagates out along the
field lines at speed c. The transverse components are greatly exaggerated. Observer
A views inward along the x axis and sees horizontal electric field components.
Observer B views inward along the z axis and observes a transverse electric field
with a direction that rotates in the plane parallel to the x-y plane. If the intensity
observed along the z axis is normalized to 2, then the net intensity radiated along a
direction at angle  relative to the z axis is 1 + cos2. The light is circular
polarized for lines of sight at  = 0 or ; it is linearly polarized for  = ½; and it is
elliptically polarized for intermediate angles.
B is in the z direction.
In the lab we view in along the x axis, the
y axis is vertical, and the z axis is
horizontal and perpendicular to our line
of sight.
HgSpec.4
Appendix 4: Spectral Data for Mercury

y
z
x
x

Appendix II: THE SOURCE OF E-M RADIATION: ACCELERATED CHARGE
HgSpec.5
Appendix 4: Spectral Data for Mercury
The static (Coulomb) electric field due to a charge at rest is ECoul  4qr
 o r3
where r is
the displacement from the position S of the source charge q, to the field point P where
the field is to be specified. In addition, a slowly moving charge has a Biot-Savart
r
magnetic field given by BBS  4qv
. Finally, an accelerated charge has a radiation
 r3
contribution to the electric field that is approximated by the relation ERad   4qac r 


 Ret

2
(for v << c). The subscript Ret directs that the source charge acceleration a, the
component of the acceleration perpendicular to the line of sight at the observation
point, should be evaluated at the retarded time t' = t - r/c where r is the distance
between the observer now and the source location at the time the radiation currently
being detected was emitted by the source. That is you want to evaluate a at the time
that the light left the source S in order to reach P at time t. Note that Erad isto the
line of sight and that it is directed oppositely to the retarded value of a and that the
radiation field falls off as r -1 at large distance while ECoul and BBS fall off as r -2. [There
are corrections to ERad of order v/c and r -2, etc.] The energy flow is described by the
Poynting Vector S  1 E  B which is proportional to the product of ERad and BRad . At
large r the net energy flow out is proportional to S 4πr 2 = (1/µo) E B 4πr 2. As radiation
must be capable of carrying energy to infinity, the radiation contributions to the net E
and B fields must fall off no faster than r -1 with distance. We assume v << c which
means that only non-relativistic sources are to be considered.
HgSpec.6
Appendix 4: Spectral Data for Mercury
HgSpec.7
Appendix 4: Spectral Data for Mercury
The form of the radiation field can be motivated using Gauss's Law. Electric field lines
begin and end only on charges. For a charge in uniform motion, the lines are directed
radially away from the instantaneous position of the charge. A short burst of
acceleration therefore puts kinks on the field lines that propagate outward at c. In the
kinks, the field has a component perpendicular to the line of sight in addition the
Coulomb field along the line of sight. If the charge accelerates to the right from time 0
HgSpec.8
Appendix 4: Spectral Data for Mercury
to t, the field lines must adjust such that those inside a distance ct are directed away
from the instantaneous position of the source charge while the lines outside a distance
c(t+t) are directed away from the instantaneous position that the charge would have
had if it had not been accelerated. Between ct and c(t+t), the lines must be continuous
(no starting or stopping in the absence of charge). They join with straight line segments
if the acceleration is constant over the intervalt. Note that this behavior adds a
transverse component to the electric field that is directed oppositely to the projection of
the acceleration perpendicular to the line of sight at the retarded time t'=t-r/c. Please
remember that the sketch is exaggerated as v = a t << c. The two spheres (circles)
should appear nearly concentric! The center separation of centers is v t = t (a t). For
the lines to be continuous, the ratio of the transverse (Radiation) and longitudinal
(Coulomb) field components must be the distance that the source has traveled since
acceleration divided by the distance that light traveled during the time elapsed during
the acceleration. ERad = ECoul v t/c t where t = r/c and v = a t.
Radiation Exercises:
1.) The radiation contribution to the magnetic field can be represented as
 qa  r
BRad   3 2
4 c r
Find an analogous expression for ERad . It requires a double cross product! Show that
rˆ  ERad  c BRad where r̂ is the direction of propagation.
2.) Consider a charge q with acceleration a = - 2 d cos(t) in the z direction. Find an
expression for ERad (r ) in terms of spherical components and unit vectors with the
coordinates centered on the charges location.
HgSpec.9
Appendix 4: Spectral Data for Mercury
3.) Compute Poynting's vector ( S  01 [ E  B] ) for the charge discussed in #2.
Describe the dependence of the radiation on the angle measured relative to the
direction of the acceleration. Compute the average power radiated during one cycle.
4.) Devise an argument that shows that the transverse component ERad (r ) falls off as
r -1 at large distance if the radial component ECoul (r ) falls off as r -2 using the picture
above these exercises. Motivate equation 3.
Vector Triple Product:  A , B , C   A   B  C   B  C  A  C   A B 








5.) Consider a charge q with acceleration a = - 2 d cos(t) in the z direction. The
time dependent dipole moment is
p  qd cos t  kˆ  p0 cos t  kˆ .
average power radiated by the charge q by computing
Compute the
 S  nˆ dA over a large sphere
concentric with the charge and averaging over one cycle. Express the result in
terms of po and  plus other factors.
p
2
0
e
2
f r i
2
.)
Ans: Paverage =
(For the E-M transition problem,
p02  
    c 
546 nm Green
HgSpec.10
Appendix 4: Spectral Data for Mercury
The Zeeman pattern is nine lines with line
to line spacing of ½ h-1B. The pi lines
are the center three, and they are polarized
horizontally as viewed along a horizontal
line at right angles to the field direction.
The sigma lines are polarized vertically
when viewed along the same line. When
viewed from along the direction of the
field, the sigma lines have right and left
circular polarizations. The relative
intensities are shown for the green lines.
The levels are inverted in this drawing! 3S1is the upper.
HgSpec.11
Appendix 4: Spectral Data for Mercury
HgSpec.12
Appendix 4: Spectral Data for Mercury
HgSpec.13
Appendix 4: Spectral Data for Mercury
HgSpec.14
Appendix 4: Spectral Data for Mercury
S state is upper level for 546.1 nm
3
E0 ( S1 )  2  B Bext
3
E0 ( S1 )
3
E0 ( S1 )  2  B Bext
3
S1
g=2
mJ = -1, 0, 1
E  E0 (3 S1 )  E0 (3 P2 )    B Bext
  2 to2 by ½
3
E0 (3 P2 )  3 B Bext
E0 (3 P2 )  3 2 B Bext
P2
E0 (3 P2 )
E0 ( P2 )  3 2 B Bext
E0 (3 P2 )  3 B Bext
g = 3/2;
3



mJ = -2, …. , 2
HgSpec.15
Appendix 4: Spectral Data for Mercury
vo - h-1BB
 Lines
m = 0
vo + h-1BB
Mercury 577 nm Zeeman Structure
ERROR
vo
h-1BB
h-1BB
In each set of three lines
The line to line spacing is:
1
/6 h-1BB
Lines
m = 1,-1
21
HgSpec.16
Appendix 4: Spectral Data for Mercury
vo - h-1BB
 Lines
m = 0
vo + h-1BB
Mercury 577 nm Zeeman Structure
No intensity information.
vo
h-1BB
h-1BB
In each set of three lines
The line to line spacing is:
1
/3 h-1BB
Lines
m = 1,-1
HgSpec.17
Appendix 4: Spectral Data for Mercury
ERROR Mercury 577 nm Zeeman Pattern
|2 2ñ
ED + 7/3 BB
|2 1ñ
ED + 7/6 BB
3
D2
|2 0ñ
ED
|2 -1ñ
1
P1
ED - 7/6 BB
|2 -2ñ
ED - 7/3 BB
|1 1ñ
EP + BB
EP
EP - BB
|1 0ñ
|1 -1ñ

7
g = /6 ;
mJ = - 2, …. , 2
g=1
mJ = - 1, 0, 1
REVISED Mercury 577 nm Zeeman Pattern
|2 2ñ
ED + 8/3 BB
|2 1ñ
ED +4/3 BB
3
D2
|2 0ñ
ED
|2 -1ñ
ED – 3/4 BB
|2 -2ñ
1
P1
ED -8/3 BB
|1 1ñ
|1 0ñ
|1 -1ñ
4
g = /3 ;
mJ = - 2, …. , 2

EP + BB
EP
EP - BB
g=1
mJ = - 1, 0, 1
HgSpec.18
Appendix 4: Spectral Data for Mercury
Mercury 579 nm Zeeman Pattern
3
D1
|1 1ñ
ED + 1/2 BB
|1 0ñ
ED
|1 -1ñ
ED - 1/2 BB
|1 1ñ
1
P1
EP + BB
g=1
|1 0ñ
|1 -1ñ
g =1 /2 ;
mJ = - 1,0,1
EP
EP - BB

Mercury Violet 436 nm
Upper: 3P1
gu = 3/2
mJ = - 1, 0, 1
Lower: 3S1 g = 2
HgSpec.19
Appendix 4: Spectral Data for Mercury
g  1
J ( J  1)  L( L  1)  S ( S  1)
2 J ( J  1)
Mercury 436 nm Zeeman Pattern
|1 1ñ
3
P1
EP + 3/2 BB
|1 0ñ
EP
|1 -1ñ
EP - 3/2 BB
|1 1ñ
3
S1
ES + 2BB
g=2
|1 0ñ
|1 -1ñ
g =3 /2 ;
mJ = - 1,0,1
ES
mJ = - 1, 0, 1

ES - 2BB




 




 

} +
} 
} 

Mercury deep violet 405 nm
Upper: 3P0
gu = *
Lower: 3S1 g = 2
HgSpec.20
Appendix 4: Spectral Data for Mercury
g  1
J ( J  1)  L( L  1)  S ( S  1)
;
2 J ( J  1)
* the g factor for a J = 0 state can be set to 3/2 or any other finite value.
Mercury 405 nm Zeeman Pattern
3
P0
|1 0ñ
EP
|1 1ñ
3
S1
ES + 2BB
g=2
|1 0ñ
|1 -1ñ
g =3 /2 ;
mJ = 0
ES
mJ = - 1, 0, 1









ES - 2BB
+



HgSpec.21
Appendix 4: Spectral Data for Mercury
Sodium
unicorn.ps.uci.edu/.../Sodium/sodium.html
HgSpec.22
Appendix 4: Spectral Data for Mercury
http://internal.physics.uwa.edu.au/~stamps/2006Y3Lab/SteveAndBlake/theoretical.html
HgSpec.23
Appendix 4: Spectral Data for Mercury
Sodium Energy Levels
HgSpec.24
Appendix 4: Spectral Data for Mercury
26.) Assume that the 577 nm line in the Hg spectrum has a frequency of about
520,000 GHz. Identify the frequency shift of each line in an applied field of
strength one Tesla. Prepare a spectral drawing that displays the  lines one way
and the  lines in a distinct way. For the experimental setup used in our course,
prepare a spectrum drawing that would be observed if the polarizer were passing
the vertical polarization.
…….
The horizontal polarization.
Mercury 577 nm Zeeman Pattern
|2 2ñ
ED + 7/3 BB
|2 1ñ
ED + 7/6 BB
3
D2
|2 0ñ
ED
|2 -1ñ
1
P1
ED - 7/6 BB
|2 -2ñ
ED - 7/3 BB
|1 1ñ
EP + BB
EP
EP - BB
|1 0ñ
|1 -1ñ

7
g = /6 ;
mJ = - 2, …. , 2
g=1
mJ = - 1, 0, 1
27.) Assume that the 579 nm line in the Hg spectrum has a frequency of about
518000 GHz. Identify the frequency shift of each line in an applied field of
strength one Tesla. Prepare a spectral drawing that displays the  lines one way
and the  lines in a distinct way. For the experimental setup used in our course,
prepare a spectrum drawing that would be observed if the polarizer were passing
the vertical polarization.
HgSpec.25
Appendix 4: Spectral Data for Mercury
…….
The horizontal polarization.
Mercury 579 nm Zeeman Pattern
3
D1
|1 1ñ
ED + 1/2 BB
|1 0ñ
ED
|1 -1ñ
ED - 1/2 BB
|1 1ñ
1
P1
EP + BB
g=1
|1 0ñ
|1 -1ñ
g =1 /2 ;
mJ = - 1,0,1
EP

EP - BB
mJ = - 1, 0, 1

29.) The distinctive cadmium red arises from the transition from a 1D2 (5s5d) state
to a 1P1 (5s5p) state.
a.) Prepare a term diagram similar to the ones in the previous two problems, but
appropriate for the Cd red transition.
b.) .) Assume that the 579 nm line in the Hg spectrum has a frequency of about
466000 GHz. Identify the frequency shift of each line in an applied field of
strength one Tesla. c.) Prepare a spectral drawing that displays the  lines one way
and the  lines in a distinct way. For the experimental setup used in our course,
prepare a spectrum drawing that would be observed if the polarizer were passing
the vertical polarization.
…….
The horizontal polarization.
d.) The Cd red line displays a normal Zeeman effect. What characteristics
determine whether the effect is normal or anomalous?
HgSpec.26
Appendix 4: Spectral Data for Mercury
30.) The sodium yellow doublet arises from transitions from either a 3P3/2 or a 1P1/2
state to the common 3S1/2 state.
a.) Prepare term diagrams similar to the ones in the previous two problems, but
appropriate for the Na yellow transitions.
b.) .) Each line in the Na yellows has a frequency of about 466000 GHz. Identify
the frequency shift of each line in an applied field of strength one Tesla. c.) Prepare
a spectral drawing that displays the  lines one way and the  lines in a distinct
way. For the experimental setup used in our course, prepare a spectrum drawing
that would be observed if the polarizer were passing the vertical polarization.
…….
The horizontal polarization.
31.) The Na yellow lines have an average wavelength of about 589.3 nm. The lines
share a common lower state while the corresponding upper states have energies
that differ by 0.0021 eV. Correct to 0.1 nm, what are the wavelengths of the two
lines in the Na yellow doublet?
Modeled Line Shapes for Hg (546 nm) Green:
Model as a sum of exponential shapes centered at a and with width parameter w.
 ( xa )
2
line[ x, a, w]  e
w2
. For the sigma components of the green line, we expect lines
on the plus and minus sides at separations of BB, 3/2 BB and 2 BB with relative
intensities of 6:3:1. The variable x is in units of ½ BB as is the width parameter
w. The lines shape is plotted for several values of w and the diameter of the ring
measured for the positive peak to the negative side peak is identified. (Choose the
value of w that yields the best approximation of the line profile that you observe.
HgSpec.27
Appendix 4: Spectral Data for Mercury
Report the values that you use.) Be aware that there will be background light
added to the observed signal light.
line[x_, a_, w_] = Exp[-(x - a)^2/w^2]
w = .5; Plot[6 line[x, 2, w] + 3 line[x, 3, w] + line[x, 4, w] +
6 line[-x, 2, w] + 3 line[-x, 3, w] + line[-x, 4, w], {x, -5, 5}]
w = 0.5
Plot[6 line[x, 2, w] + 3 line[x, 3, w] + line[x, 4, w] +
6 line[-x, 2, w] + 3 line[-x, 3, w] + line[-x, 4, w], {x, 1.85,
2.35}]
HgSpec.28
Appendix 4: Spectral Data for Mercury
For w = 0.5, The  to  ring separation is modeled to be 4.04 in units of ½ BB.

For w = 0.6, The  to  ring separation is modeled to be 4.08 in units of ½ BB.
HgSpec.29
Appendix 4: Spectral Data for Mercury
For w = 0.7, the ring diameter is modeled to be 4.18 in units of ½ BB.
6
5
4
3
2
1
4
2
2
4
For w = 0.8, The  to  ring separation is modeled to be 4.30 in units of ½ BB.
HgSpec.30
Appendix 4: Spectral Data for Mercury
7
6
5
4
3
2
1
4
2
2
4
For w = 0.9, the  to  ring separation is modeled to be 4.4 in units of ½ BB.
7
6
5
4
3
2
1
4
2
2
4
For w = 1.0, the  to  ring separation is modeled to be 4.48 in units of ½ BB.
HgSpec.31
Appendix 4: Spectral Data for Mercury
6
5
4
3
2
1
4
2
2
4
For w = 0.2, the  to  ring separation is modeled to be 4.00 in units of ½ BB.

6
5
4
3
2
1
4
2
2
4

For w = 0.3, The  to  ring separation is modeled to be 4.00 in units of ½ BB.
HgSpec.32
Appendix 4: Spectral Data for Mercury
6
5
4
3
2
1
4
2
2
4


For w = 0.4, the  to  ring separation is modeled to be 4.00 in units of ½ BB.
Modeled Line Shapes for Hg (436 nm) Blue-Violet Line:




 


 


 

} +
} 
} 
For the outer sigma components; model the peaks as having an intensity ratio
of 2:1 and w =0.6
HgSpec.33
Appendix 4: Spectral Data for Mercury
2.0
1.5
1.0
0.5
4
2
2
4
For w = 0.6, the  to  ring separation is modeled to be 6.1 in units of ½ BB.
2.0
1.5
1.0
0.5
4
2
2
4
For w = 0.8, the  to  ring separation is modeled to be 6.3 in units of ½ BB.
HgSpec.34
Appendix 4: Spectral Data for Mercury
2.5
2.0
1.5
1.0
0.5
4
2
2
4
For w = 1.0, the  to  ring separation is modeled to be 6.5 in units of ½ BB.
Relative Intensities
Multi-electron Atom Selection Rules
L  1
S  0
J  0,1
(0 to 0 forbidden)
M J  0,1 (0 to 0 forbidden for J  0)
Relative Intensities for Zeeman components (Electric Dipole Radiation):
Following Shore and Menzel in Principles of Atomic Spectra
For transitions from and initial state i to a final state f the intensity of the radiation
is
N ( J M ) ck2
4
J Q
(1)
 J 
2
2
( J M ,1m|JM )2 ½(1  cos  ) m  1

2
2 J 1
m0
 sin 
HgSpec.35
Appendix 4: Spectral Data for Mercury
The initial state i has quantum numbers J Mwhere  is all the quantum
numbers
except for the angular momentum specific values J M
The final state f has quantum numbers JMwhere  is all the quantum numbers
except for the angular momentum specific values J M
J M) is the number of atoms in the initial state
k is the wave-number
 J Q (1)  J  is the reduced matrix element, a value independent of M and M.
(J Mm|JM) is the Clebsch-Gordan coefficient for adding JMand m to get JM.
 is angle between the magnetic field and the line of sight. For the case of our
Zeeman
experiment,  = 90o.
Relative Intensities in the Zeeman Effect:
This section demonstrates a computation method that reproduces the relative
intensities claimed in the manual for the Pacific Scientific experiment. A
revised version will be attempted in which the perturbation only operates on
the orbital portions of the wavefunction. The predictions of the two methods
are to be compared with experiment.
Consider the 5460.7 Å 3S  3P2 line in the spectrum of mercury. If the mercury is
1
placed in a strong uniform magnetic field, the 3S and 3P2 levels are each split into
1
several levels.
Into how many levels is the 3S level split? splits on mJ2J + 1 
1
Into how many levels is the 3P2 level? splits on mJ2J + 1= 5.
HgSpec.36
Appendix 4: Spectral Data for Mercury
Compute the g factors for the 3S {J=1; S=1; L=0; g = 2}. 3P2 {J=2; S=1; L=1; g =
1
3
/2}
The selection rule for allowed level to level transitions is: m = 0, 1
The magnetically split levels as corresponding to energies:
E0 (3 S1 )   k B Bext  E0 (3 S1 )  g B Bext mJ  E0 (3 S1 )  2 B Bext mJ
E0 (3 P2 )   j B Bext  E0 (3 P2 )  g B Bext mJ  E0 (3 P2 )  3 2 B Bext mJ
3
|1 1ñ
S1
3
|1 0ñ
|1 -1ñ

|1 -1ñ
P2
g=2
mJ = -1, 0, 1
E  E0 (3 S1 )  E0 (3 P2 )    B Bext
|1 0ñ
|1 +1ñ
3
E0 ( S1 )  2  B Bext
3
E0 ( S1 )
3
E0 ( S1 )  2  B Bext
|2 2ñ
|2 1ñ
|2 0ñ
|2 -1ñ
|2 -2ñ
  2 to2 by ½
E0 (3 P2 )  3 B Bext
E0 (3 P2 )  3 2 B Bext
E0 (3 P2 )

E0 (3 P2 )  3 2 B Bext
E0 (3 P2 )  3 B Bext

g = 3 /2 ;
mJ = -2, …. , 2
The relative intensities of the lines depend on the direction from which they are
viewed. For that reason, the form of the transition rate formulae prior to averaging
over directions must be used.
ba3
I ab 
 oc3
 f JfMf
p  eˆ  i Ji M i
2
3
e2 ba

 oc3
 f J f M f r  eˆ  i J i M i
2
HgSpec.37
Appendix 4: Spectral Data for Mercury
The overall angular momentum quantum numbers for the initial and final states are
displayed explicitly. The symbols i and f represent all the other quantum numbers
that remain fixed as the angular momentum options are investigated. For our
discussion, the only requirement is that i and f represent states of opposite parity.
The assumption is that the matrix element factors. This conjecture is supported by
the results for the n = 2 to n = 1 transition rates for atomic hydrogen that are
computed in the appendix.
I ab
3
e2 ba

 oc3
 f r i
2
J f M f rˆ  eˆ J i M i
2
Note that ê is the polarization direction of the electric field.
The relative intensities follow as they are just the absolute squares of the ratios of
the angular momentum matrix elements. Let’s recall the angular momentum nature
of the position vector.
r  xiˆ  y ˆj  z kˆ  r (sin  cos  iˆ  sin  sin  ˆj  cos kˆ
r r

2
rˆ 
2
3 {Y1
3 {Y1
1
1
( , )  Y11 ( , )}iˆ  i
( , )  Y11 ( , )}iˆ  i
2
2
1
3 {Y1
1
3 {Y1
( , )  Y11 ( , )} ˆj 
( , )  Y11 ( , )} ˆj 
4
4
0
3 Y1
0
3 Y1
( , ) kˆ

( , ) kˆ
HgSpec.38
Appendix 4: Spectral Data for Mercury
y
z
x
HgSpec.39
Appendix 4: Spectral Data for Mercury
Consider the  lines which, when viewed from the transverse direction (in along
the x axis) are polarized parallel to the magnetic field. That is: they are polarized in
the z direction.
J f M f rˆ  kˆ J i M i
2
 JfMf
4
0
3 Y1 ( ,  ) J i M i
2
The perturbation z has a Y10 or |1 0ñ angular momentum character. The radial parts
of the wavefunctions are the same for the various Ym as long as  remains fixed.
For the chase above, Y10 or |1 0ñ is to be combined with |Ji Miñ to make |Jf Mfñ.
The three  lines have the following angular momentum characters:
| 1 1ñ  | 1 0ñ | 2 1ñ| 1 0ñ  | 1 0ñ | 2 0ñ and | 1 -1ñ  | 1 0ñ | 2 -1ñ.
HgSpec.40
Appendix 4: Spectral Data for Mercury
Using the 1  1 Clebsch-Gordan table:
| 1 1ñ  | 1 0ñ | 2 1ñ| 1 0ñ  | 1 0ñ | 2 0ñ and | 1 -1ñ  | 1 0ñ | 2 -1ñ.
11  10 
10  10 
2
3
1  1  10 
21 
1
2
1
2
11
20  0 10 
1
2
2 1 
1
2
1
3
00
11
Squaring the amplitudes, the intensities are predicted to be in the ratios:
1
/2 : 2/3 : 1/2 or 3:4:3.
For the + lines we only see the y projection of the polarization when viewed
transversely. Use i
2
1
3 {Y1
( , )  Y11 ( , )} . Note that the coefficients for the Y’s
is less by a factor of the square root of two (
2
3
vs.
4
3
) which is equivalent to
a factor of 2 in the predicted relative intensity.
| 1 1ñ  | 1 1ñ | 2 2ñ| 1 0ñ  | 1 1ñ | 2 1ñ and | 1 -1ñ  | 1 1ñ | 2 0ñ.
11  11  22
This 100% times 1/2 means the same as the 1/2 lines in the  line set.
HgSpec.41
Appendix 4: Spectral Data for Mercury
10  11 
1  1  11 
1
6
1
2
21 
20 
1
2
1
2
11
10 
1
3
00
Using the 1  1 Clebsch-Gordan table, the matrix elements are 1, 2-½, 6-½
corresponding to ratios of 1: 1/2 : 1/6 or, dividing by 2 to scale to the  lines,
1
/2: 1/4 : 1/12. For the collection of all the lines, we have:
1
/12: 1/4 : 1/2:1/2 : 2/3 : 1/2:1/2: 1/4 : 1/12
Scaling by 3/2 these become: 1/8: 3/8 : 3/4:3/4: 1 : 3/4:3/4: 3/8 : 1/8

! PERFECT AGREEMENT WITH THE CHINESE MANUAL!
Relative Intensities of the lines in a Zeeman pattern
First, consider an atom with a single active electron. Following the lab setup, the
active perturbation is z for the  lines and y for the  lines. 
Consider a mythical atom with a fine structure that leads to a 2S1/2 and 2P1/2 states
with the same energy with no applied magnetic field. Lying at a higher level is a
HgSpec.42
Appendix 4: Spectral Data for Mercury
2
P3/2 state. The level diagram is not drawn to scale. The energy separation between
the 2P3/2 state and the 2S1/2 and 2P1/2 states is E2 - E1 = 0.5 eV.
Mythical Atom Fine Structure/Zeeman Level Diagram
mj= 3/2
mj= 1/2
2
E2
P3/2
g B B
mj= -1/2
mj= -3/2
n0
2
2
E1 2
S1/2
P1/2
g' B B
g'' B B
S1/2 mj= 1/2
2
P1/2 mj= 1/2
2
P1/2 mj= -1/2
2
S1/2 mj= -1/2
Exercise: a.) Compute the g factors for the levels. b.) What is the frequency of a
0.5 eV photon? c.) A magnetic field of strength 1 Tesla is applied. Label each
Zeeman level by its frequency shift per Tesla relative to the zero field level. For
example, the 2P3/2 , mj = - 3/2 state might be at – 28 GHz/T. d.) Suppose that the atom
HgSpec.43
Appendix 4: Spectral Data for Mercury
is initially in the 2P3/2; mj = 3/2 level. What are the allowed transitions that the
atom/electron can make? e.) Repeat for the other three mj levels.
Electric Dipole Selection Rules state that P to S transitions are allowed while P to
P transitions are forbidden. Remember that spins must not flip. Compute the
angular part of the matrix elements for the allowed transitions. The angular parts
are associated with the orbital angular momentum, not the total angular
momentum. Find the relative intensities of the allowed transitions.
The Battle Plan:
The intensities are computed for viewing along the x axis with the magnetic field
in the z direction. The  lines are horizontally polarized (z direction) while the 
lines appear vertically polarized (y direction).
1.) Decompose each level into its orbital angular momentum ket multiplying its
spinor representation. For example, 2P3/2; mj = ½ 
1
3
11 ½  ½ 
2
3
10 ½½ .
In each pair, the first ket is the orbital ket related to the spatial coordinates  and ,
and the second is the spin ket related to the internal coordinates of the electron.
2.) The operators z and y are to be represented in terms of the spherical harmonics
revealing their ,  character. y 
Equivalently, y 
2
3 r{ 11
2
3 r{Y1
 1 1 }; z 
1
( , )  Y11 ( , )}; z 
4
3r
4
0
3 rY1
( , ) .
10 . These perturbations act in
the coordinate range of the orbital angular momentum and act as the identity
operation, multiplication by one, in spinor space. The Clebsch-Gordan tables
can be used to find the orbital kets equivalent to the operators acting on the orbital
kets. For example, z |11ñ
4
3r
10 11 
4
3r

1
2
21 
1
2

11 .
HgSpec.44
Appendix 4: Spectral Data for Mercury
3.) Compute all the initial to final matrix elements. The ket above is multiplied by
its spinor ket as are the final state orbital bras. The inner products are evaluated are
the product of the orbital bra-kets and the spinor bra-kets. When the operator y is
used, either the plus or minus term alone is considered depending on whether the
intensity of the + or the - line is in question.
Extended Battle Plan: The Zeeman components in the 546 nm transitions of
mercury. Mercury has two valence electrons in the spin symmetric S =1
configuration for this line. Match this with an anti-symmetric two particle spatial
functions. The dipole operator must include both electrons, p  er1  er2 . Proceed
using the components of the previous battle plan with the two particle
complications.
??? What is the two particle coordinate perturbation
HgSpec.45
Appendix 4: Spectral Data for Mercury
HgSpec.46