Applying Atomic Physics To Obtain The Lande g-Factor And Zeeman Effect

Recall the  Zeeman effect is  a broadening, i.e. of a spectral line, e.g.  from the Sun,  due to strong magnetic fields such as in sunspots. An example is depicted below:

      

The left image shows the   Zeeman splitting of a spectral line associated with a sunspot. The right image shows two electron spins  (m _s    = + 1/2)  associated with differing energy levels in an atom. The assumption the electron has a magnetic moment of 1  Bohr magneton( u _B ) in spite of the fact the spin is only 1/2 ħ   was first  advanced by Goudsmit and Uhlenbeck simultaneously with the hypothesis of electron spin and leads to a complete explanation of splitting in all other cases, including for the Zeeman and anomalous Zeeman effects.. 

In the case of certain spectral lines, the Lande g-factor figures prominently as a means to calculate the relative splitting of different energy levels, i.e. in weak magnetic fields.  In general we refer to 'g' as the g-factor (generic) which depends on the values of the quantum numbers L, S and J, see e.g. 

http://brane-space.blogspot.com/2014/08/an-introduction-to-quantum-mechanics-2.html

Note that both the orbital and spin angular momentum contribute to the magnetic moment of an atomic electron so that:

    m   =    - (e/ 2m)  L          and        m _s   =   -  g   S (e/ 2m)    

So the interaction energy of an atomic electron can then be written:

D(E) =   (e/ 2m)  [L +  2S ]  · B

Basically,  the  splitting of energy levels in an atom increases the number of spectral lines.  If each energy level is split into 2j + 1 components - i.e. one for each of the values of    m _J  (see preceding link)- then the magnitude of the splitting will be different for levels with different Lande g factors.

Note that for the cases where spectral lines are split into three components (normal Zeeman effect as shown in top left image)  the transitions Δ m _J  = 0, + 1   lead to only 3 spectral lines because there are only 3 possible energy differences for these transitions. (Because these cases correspond to transitions between states for which s = 0)

Thus, in the image shown the spectral line appears in classic "triplet" form.  That is,  there exists a normal (unaffected)  line of wavelength  l_o  on either side of which are lesser and greater wavelength lines, hence "splits".  The triplet wavelengths are as follows:

l_o   +   D  l _H 

l_o

l_o  -   D  l _H  

George Ellery Hale was the first to apply the Zeeman splitting of a solar spectral line to the problem of quantifying the strength of the magnetic field associated with a sunspot.  He thereby arrived at the following cgs version of the equation:

D  l _H  =     (l_o)²  e H / 4 π  m_e c²   

Here:  H is the intensity of the sunspot magnetic field (to be found),  e is the electron charge in electrostatic  units (e.s.u.),  m_e    is the electron mass in grams and c is the speed of light in cm/sec.

 The formula for the Lande g factor is:

g  = 1 +  [J (J + 1) + S (S + 1) - L (L + 1)/ 2 J(J+1)]

And is present in the Zeeman splitting  formula above except "disguised" since g = 1.  This is for the particular case s = 0  applied to the spin orbital angular momentum, whence: 

S = [s(s + 1)] ^1/2  ħ    = [0 (0 + 1)] ^1/2  ħ   =  ħ  

Then the electron magnetic dipole moment would be:

m _s = - g(s) m _B     S  / ħ     =   - m _B  

  The more general case in which this doesn't apply is the anomalous Zeeman effect.

To obtain a derivation for the Lande g  we consider again the magnetic moment (m ) but this time in terms of two components – for the spin and angular momentum such that:

m  =    m _s +  m _L = - e (2S)/ 2m + (-e (L)/ 2m)

=  -e (J + S)/ 2m

Since:  L + S = J

The vector projection of m on J is:

m · J / | J | = (-e/ 2m) [J·J + J·S ] / | J 

|

L + S = J

So:  L  =  J – S

Then:  

L·L = (J – S) (J – S)      =   J·J + S·S   - 2 J·S

Re-arranging:

  J·S   = ½ ( J·J + S·S   - L·L)

Then we can write:

m · J / | J |  = (-e/2m) [J·J + ½ ( J·J + S·S   - L·L)/ | J |]

After substituting specific values for the J·J, etc. one finds:

m · J / | J | =  g  (-e ħ /2m) Ö J (J + 1)

 Where:

g  = 1 +  [J (J + 1) + S (S + 1) - L (L + 1)/ 2 J(J+1)]

Example Problem (1):

Find the Lande g-factor for an atom in the state 1D _3/2.

Solution:

The 1D _3/2 state implies: L = 2, S = 1 with J = L + S

= 2 + 1 = 3

then:

g = [3(3 + 1) + 1(1 + 1) - 2( 2+ 1)/ 2(3)(3 + 1)] + 1 = 1.3

Is the Lande g-factor

Anomalous Zeeman Effect:

In the normal or semi-classical case, the Zeeman effect appears as triplet line splitting as we saw at the top. This is associated with the precession of the magnetic moment m about some external magnetic field B. The stronger the field the faster the precession and the greater the separation between the three spectral lines. 

When the L·S  coupling is strong compared to the interaction of either vector with B, then S and L precess rapidly about J producing a rapid precession of m about J. (Previous section).  This system then precesses slowly about B.

In this way the anomalous Zeeman effect arises, given it depends upon the component of  m  along J

Example Problem (2):

Determine the value of the energy splitting of an atom in a magnetic field B if it is assumed that the splitting depends only on the component of m  along J .

Solution:

The component of m  along J  is:

m _J =  g  (-e ħ /2m) Ö J (J + 1)

Also expressed in vector notation: 

m _J =   m _J    J / | J |  =  g  (-e ħ /2m) Ö J (J + 1) [  J/ Ö J (J + 1) ħ =   

-e ħ /2m  (g J)

The energy splitting is then given by:

D E =  -  m _J  B = e ħ /2m  g J·B

=  e ħ /2m  g B J _z  =  e ħ /2m  g B M _J

And we know already from quantum mechanics that:

M _J   = J, J – 1,……, -J + 1, - J

So that for a given field intensity B each energy level will split into 2J + 1 sublevels with the amount of splitting determined by the  g –factor.

Problems for energetic readers:

1)      Find the Lande g-factor for an atom in each of the following states:  ³ F ₃ ,  ³ F ₂      and   ³  F ₄  

2)     a) Assuming the L·S    interaction to be much stronger than the interaction with an external magnetic field, calculate the anomalous Zeeman splitting of the lowest energy states:

² S _1/2 ,  ² P _1/2  and   ² P _3/2  

In the hydrogen atom for a field of 0.05T

Present a table with the results of the calculations showing the energy states in the extreme left side column under ‘State’, with the headers of the other columns, viz: 
L, S, J,  g  ,  M _J ,  D E (in eV x 10 ^-5  )

2(b) Given that: m _s  = - e (2S)/ 2m and:

m _{L  =}    (-e (L)/ 2m)

Show in a vector diagram that m  and J  are not parallel.

Selected Questions- Answers From All Experts Astronomy Forum (The Zeeman Effect And Sunspot Magnetic Fields)

Question: I've just been reading a book on the Sun ('Our Sun' by Donald Menzel )  and found references to sunspots having magnetic fields of up to 4000 gauss. How can astronomers or solar physicists obtain such values? How do they know this? 

Answer:

The Zeeman effect is  a broadening, i.e. of a spectral line from the Sun,  due to strong magnetic fields such as in sunspots. An example is depicted below:

      

The left image shows the photo of the line-centered sunspot, i.e. the sunspot for which a spectral line has been obtained at line center, and in classic "triplet" form.  That is there exists a normal (unaffected)  line of wavelength  l_o  on either side of which are lesser and greater wavelength lines, hence "splits".  Thus the triplet is presented in terms of the normal wavelength as follows:

l_o   +   D  l _H 

l_o

l_o  -   D  l _H  

George Ellery Hale was the first to apply the Zeeman splitting of a solar spectral line to the problem of quantifying the strength of the magnetic field associated with a sunspot.  He thereby arrived at the following cgs version of the equation:

D  l _H  =     (l_o)²  e H / 4 π  m_e c²   

Here:  H is the intensity of the sunspot magnetic field to be found,  e is the electron charge in electrostatic  units (e.s.u.),  m_e    is the electron mass in grams and c is the speed of light in cm/sec.  To obtain the intensity in Gauss then, we first need to use basic algebra to solve for H:

H =   4 π  m_e c ²   D  l _H  /    (l_o)²  e

We then must pay attention to the units, so that we have:

e =    4.8 x  10  -10  esu 

  m_e     =   9.1   x 10^-28    gram

c  =  3   x 10  ¹⁰ m/s 

To  illustrate the application we will let the  undisturbed solar line  ( l_o ) be the H-alpha line which  has wavelength:  6.62  x 10 ^{- 5}  cm.   We then let the line displacement (shift owing to H) on either side be:  D  l _H    = 0.05 A = 5.0 x 10 ^-10 cm

  The equation with units substituted in for computation, then becomes:

H  = 

4 π (9.1   x 10^-28  g) (3  x 10  ¹⁰ cm/s )²  (5.0 x 10 ^-10 cm) / (6.62  x 10 ^{- 5}  cm)2 (4.8 x  10  -10  esu) 

The calculated sunspot magnetic field intensity is:  H = 2440 G  approximately.

An interesting further exercise is to compute the field strength in Tesla (T) instead of Gauss. Tesla is the S.I. unit of measurement for the magnetic field intensity.  To do this basically requires changing all the units used above to consistent S.I. units.   Thus, cm now becomes meters (m), and the e.s.u. becomes coulombs (C). The electron mass is now in kg instead of grams and so on.

Looking at Stellar Emission and Absorption Processes (2)

In approaching stellar line formation we will be looking at a number of related equations, including:

1)      The Boltzmann equation

2)     The Saha equation

3)     Combined Boltzmann and Saha equations

These will enable us to form a picture of spectral line formation which can then be generalized for different atoms and energy transitions.  We start then with the Boltzmann equation, which we already introduced in the previous chapter:

N₂ / N₁   =     [g₂ / g₁ ]   exp (- E2 – E1) / kT

     That is, for the atoms of a given element in a specified state of ionization, the ratio of the number of atoms N₂ with energy E2, to the number of atoms N₁   with energy E1, in different states of ionization is given by the above formula. The same form of the equation can also be used to find the ratio of probabilities, i.e. that the system will be found in any of the  g₂   degenerate states with energy E2 to the probability that the system is in any of the g₁   degenerate states E1, viz.

P(E2) / P(E1)   =     [g₂ / g₁ ]   exp (- E2 – E1) / kT

Thus, the Boltzmann equation can be posed in two forms.  In statistical mechanics we could have also seen the partition function:

Z  =   å _j   exp ( - e _j )/ t

Which is just the summation over the Boltzmann factor (exp ( - e _j )/ t ) for all states j for which the number of particles (N) is constant. We will find it useful to rewrite it:

Z = g₁ +  å^¥ _{j = 2}   g _j  exp (- E _j – E1) / kT

Of interest now are the relative numbers of atoms in ionization stage i, which is written:

N _e N _{i + 1} / N _i  =  2 Z _{i + 1} / Z _i (2 p m _e kT/ h ²) ^1.5  e ^{- c i/ kT}

    

This is the Saha equation, named after the Indian astrophysicist who first derived it.  Here,  N _e  is the number of free electrons per unit volume and c _i  is the ionization potential of the ith ionization stage. Thus, the equation relates the number of atoms in two successive  ionization stages to the quantities that are relevant. As per our introduction to quantum mechanics, the factor ‘2’ in the equation refers to the two possible spins of the free hydrogen election with spin quantum number:

m _s =  +½.

 Recall that for thermodynamic equilibrium, the rate of ionization cannot exceed the rate of recombination[1].  In other words, the rate at which atoms in the ith stage are ionized (i.e. to the i +1st stage) must equal the rate at wich ions in that i +1st stage are recombining with free electrons to form ions in the ith stage. The latter depends on N _e N _{i + 1}   and the former on N _i . Hence, Saha’s equation simply expresses the fact these two processes must occur at the same rate.

One can also rewrite the equation in a more manageable logarithmic form if one substitutes the numerical constants:

log(N _e N _{i + 1} /N _i)  = 

15.38 + log (2 Z _{i + 1} / Z _i ) + 1.5 log T – 5040 c _i/ T

The units here are important to note, and are consistent with the ionization potential being measured in electron volts (eV). Therefore N _e  must be in particles per cubic centimeter.

    Yet another way to express the Saha equation is to introduce the electron pressure, P _e . This acknowledges that each separate species of particle makes its own contribution to the total gas pressure. The free electrons in a gas therefore produce a pressure given by: P _e = N _e kT.

Then we may write another log form of the Saha equation:

log(P _e N _{i + 1} /N _i)  = 

-0.48  + log (2 Z _{i + 1} / Z _i ) + 2.5 log T – 5040 c _i/ T

There are also two distinct processes by which lines can be formed:

1)      Bound-bound transitions

2)     Bound-free transitions

In the first case, the photon goes from one bound atom to another. Also, in case (1) the photon has a good chance of being scattered, i.e. emitted in the same downward transition.  This may be at the same frequency as the absorption, in which case we say the scattering is coherent, or not, in which case it is non-coherent.

Details of bound-bound transitions differ in significant ways from bound-free transitions. In the first case, the transitions are also affected by a broadening function which is not so important for continuous emission. If we write out the equation for absorption in more detail we get:

a _u  =  [1 - e ^{-  h u o / kT}] (p e²/ mc) f f _u

which yields units in  cm² / atoms at lower level. Two other absorption derivative values are possible from the preceding:

i)                   The absorption coefficient per unit length (cm^-1)

ii)                 The mass absorption coefficient  k _u .

      The value for (i) is just a _u  multiplied by the number of absorbing atoms per unit volume. The value for  k _u  is just  multiplied by the number of absorbing atoms.  

The value for a _o   is just:  a _o =  a _u  /  f _u.  

Sometimes referred to as a “fudge factor”, f is known as the oscillator strength or f-value of the line. It is basically the transition probability for the line and is to be computed by quantum mechanics or measured in the laboratory.

       The broadening function  f _u  is:

f _u  =   1/ Öp   [exp (u  -u _o)  /D u _D ]²  D u_D

Which can also be rewritten as:   f _u  du  =

1/ Öp   [exp (u  - u _o)  /D u_D ]²  du / D u_D

This would be the probability that the absorbed photon lies between u  and  u +  du, assuming equal intensities for all frequencies. Thus the integral:

ò f _u  du  =  1

Where du is over all frequencies.  The value of f _u  is larger near  u _o  the frequency of the line center, as may be deduced from the line profile diagram below:

Absorption line profile showing the core and "wings"

Note here that  D u_D  is the Doppler half-width of the line. As can be seen on inspection, f _u is very large at line center and falls off in the “wings”, i.e. at larger and smaller frequencies.

 The three important types of line broadening are doppler effect, natural and pressure broadening. We will confine our attention to the first type which is given by the broadening probability equation, provided the velocity is Maxwellian and that the frequency at line center  u _o is also observed for some u . The Maxwellian will display the distribution of velocities as shown below where the central line defines the most probable.

Maxwellian showing distribution of velocity with proportion of particles.

For the Sun, solar physics, it is also necessary to consider the Zeeman effect, a broadening due to strong magnetic fields such as in sunspots. An example of this applied to a sunspot is depicted below:

The left image shows the line-centered sunspot for which the Zeeman effect in classic "triplet" form (right image) is detected and measured. The greater the spectral line splitting the greater the magnitude of the associated magnetic field.  George Ellery Hale, who discovered the effect, posed the quantitative relationship in terms of the original wavelength   l_o  (undisturbed line) and the spread of wavelengths, D  l:

 D  l =     (l_o)²  e H/ 4 π  m_e c²  

Where H is the intensity of the sunspot magnetic field in gauss (to be found), e is the electron charge in e.s.u.,  m_e  is the mass of the electron in grams, and c the velocity of light in cm/ sec.

Problems:

1) For a  hydrogen plasma find:

(P _e N _{i + 1} /N _i)  =  P _e N _HII /N _HI   

at a temperature of 5040 K, given the hydrogen partition functions are:

Z _{i + 1}  =  Z ₂    =   1   and:

Z _i =    Z ₂    =   2      with  c _i =   13.6 eV

2) For the temperature and conditions of problem (1) of the previous set, find the ratio of the probability that the system will be found in any of the eight degenerate states of energy level E2, to the probability that the system will be in any of the two degenerate states of energy level E1.

3) An H-alpha line undergoes triplet splitting in the vicinity of a sunspot. The undisturbed line is measured at:   l_o  =    6.62 x 10  ^-5 cm. The line shift on either side is: + 0.0025 Å.   Use this information to find the strength of the sunspot magnetic field: a) in gauss, b) in Tesla.

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[1] This must hold if the excitation and ionization equations are assumed valid, hence the numbers of atoms in a given level must not change with time.