Paper No. : Laser, Atomic and Molecular Spectroscopy

Module: Atoms in External fields

Development Team

Production of Courseware

- Contents For Post Graduate Courses

Principal Investigator: Dr. VinayGupta , Professor

Department Of Physics and Astrophysics, University Of Delhi, New

Delhi-110007

Paper coordinator: Dr. Devendra Mohan, Professor

Department of Applied Physics

Guru Jambheshwar University of Science And Technology, Hisar-125001

Content Writer: Dr. Devendra Mohan, Professor

Department of Applied Physics

Guru Jambheshwar University of Science And Technology, Hisar-

125001

Content Reviewer: Ms. KirtiKapoor

Department of Applied Physics

Guru Jambheshwar University of Science And Technology, Hisar-

125001

Description of Module Subject Name Physics Paper Name Atomic, Molecular and Laser,Spectroscopy Module Name/Title Paschen Back Effect

Module Id

Contents:

1. Zeeman Effect

2. Paschen Back Effect

3. The Stark Effect

The students will be able to learn about

Zeeman Effect, The Stark Effect, Paschen- Back effect and its comparison with

Anomalous Zeeman effect

1. Zeeman Effect

Zeeman effect is named after the great scientist P.Zeeman who in1876

observed that when a light source is brought into a magnetic field, each spectral

line is splitted into number of components. This implies that the energy levels of

the atom, those are involved in the transition, in the presence of a magnetic field,

must split into several components. Thus the interaction of the electronic magnetic

moment of the atom with the magnetic field results in splitting.

It is known that the ratio of magnetic and mechanical moment of an electron

in an orbit is

µ1

𝐼 = -

𝑒

2𝑚𝑒 = -

2𝜋 𝑔1 𝛽𝑒

ℎ

Also, the electron also has orbital angular momentum l and a spin angular

momentum s. The ratio of magnetic and mechanical moment for the spinning

electron is

µ𝑠

𝑆 = - 2

𝑒

2𝑚𝑒 = -

2𝜋 𝑔𝑠 𝛽𝑒

ℎ

L-S coupling

It is assumed that the interaction between the spin angular momentum of the

electrons on one hand, the interaction between their orbital motions on the other

hand is large compared with the interaction between spin and orbital angular

momenta of each electron and thus LS coupling holds (in case the atom has more

than one valence electron).

The magnetic moment µL is

µL = - 𝑒

2𝑚𝑒L

L is the total orbital angular momentum of all electrons. The magnetic moment µs

then

µS= - 2 𝑒

2𝑚𝑒S

S is the total spin angular momentum.

The vector L and S precess together around their resultant J in the absence of a

magnetic field.

When a magnetic field B is applied, L and S couple with it and in the absence of

coupling between L and S, the latter precess independently around B.

However, energy corresponding to coupling of L and S with B is smaller spin-orbit

interaction energy, in weak magnetic field. Thus, B does not perturb the coupling

between L and S under such condition. The L and S precess about their resultant J.

Because of torque, J precesses around B at a rate small compared with the

precession rate of L and S about J. The figure depicts the Precession of J about the

field direction z in a weak magnetic field B

The total electronic magnetic moment µ = µL + µsof the atom is not oriented in the

same direction as the total angular momentum J = L + S. This is because of

different dependences of µL and µs on L and S, respectively. The figure illustrate

vectors L,S and J and the associated magnetic moment vectors.

μ μJ μL

S

μs

J L

φ

θ

M

Z B

L

J S

As L and S precess rapidly about J, µL and µs precess rapidly as well, resulting µ

to zero and the component parallel to –J remains a constant in magnitude µJ. The

component of µ along –J axis from the above figure is

µJ = µLcos𝜃 + µS cos𝜑

wherer J =L + S or S = J –L

S2= (J – L). (J – L) = J2 + L2 – 2J.L

2JLcos𝜃= J2 + L2 – S2

Cos𝜃 = J 2+ L2 – S2

2𝐽𝐿

Similarly,

Cos𝜑 = J 2+ L2 – S2

2𝐽𝑆

Therefore,

µJ=µLJ 2+ L2 – S2

2𝐽𝐿 + µS

J 2+ S2 – L2

2𝐽𝑆

substituting the values of µLand µS

µJ= - 𝑒

2𝑚𝑒[J 2+ L2 – S2

2𝐽 + 2

J 2+ S2 – L2

2𝐽]

µJ= - 𝑒

2𝑚𝑒J[1 +

J 2+ S2 – L2

2J 2]= - g

𝑒

2𝑚𝑒 J

where

g = 1 + J 2+ S2 – L2

2J 2 = 1 +

𝐽(𝐽+1)+𝑆( +1)−𝐿(𝐿+1)

2𝐽(𝐽+1)

This is called Lande’s splitting factor

jj coupling

In case where interaction between spin and orbital motion of the electron is large

compared with the interaction between the spin angular momenta of the electrons

on one hand and the interaction between their orbital motions on the other hand, jj

coupling holds and hence the s and l of each electron are coupled together to form

their own resultant j.

The figure presents the vector model for jj-coupling in a weak magnetic field. j1

and j2 couples together to give the resultant J

J

j2

j1

B

The magnetic momenta for the two electrons from µJ= - 𝑒

2𝑚𝑒J[1 +

J 2+ S2 – L2

2J 2]= - g

𝑒

2𝑚𝑒

J are

µj1= - g1𝑒

2𝑚𝑒 j1

µj2 = - g2 𝑒

2𝑚𝑒 j1

where g1 and g2 can be obtained from the above equation of g factor.

g1= 1 + j12+ s1

2 – l12

2𝑗22

g2 = 1 + j22+ s2

2 – l22

2𝑗22

µJis obtained from the projection of j1 and j2 on J.

µJ= µj1 cos (j1, J) + µj2 cos (j2, J) = -[g1j1cos (j1, J) +g2j2cos (j2, J)]𝑒

2𝑚𝑒

But

µJ= -g 𝑒

2𝑚𝑒 J

On comparing the two equations

g J = g1j1 cos (j1, J) + g2j2cos (j2, J)

The angles between j1, j2 and J are constant and J = j1 + j2 or J – j1 = j2. Taking

dot product of j2 with j2

j22 = J2+j1

2 – 2J. j1 = J2 + j12 – 2j1J cos (j1,J)

cos (j1,J) = 𝐽2+ j1

2− j22

2𝑗1𝐽

Similarly,

cos (j2,J) = 𝐽2+ j2

2− j12

2𝑗2𝐽

Substituting

g = g1𝐽2+ j1

2− j22

2𝐽2 + g2

𝐽2+ j22− j1

2

2𝐽2

The magnetic interaction energy resulting from the interaction between the

electronic magnetic moment of the atom and an external magnetic field B (directed

along the z axis) is

∆E = -µJ.B

Here g is chosen depending whether the coupling is LS or jj. J cos (JB) is the

projection of J on B that is equal to Jz. The allowed values of Jz are Mħ (M being

the magnetic quantum number takes values from + J to – J, a total of 2J +1 value).

So,

∆E = g 𝑒ℎ

4𝜋 𝑚𝑒 MB = g𝛽eMB

This clearly mentions that the magnetic field lifts the degeneracy giving rise to 2J

+ 1 equidistant sublevels those are called Zeeman sublevels. The distance between

two consecutive sublevels is g𝛽 eB. Let E0 is the energy of the atom without

magnetic field, and then the energy in the magnetic field is

E =E0 + g𝛽eMB

It is known that there is a relationship between term value T and energy E i.e. T =

-E/hc. The interaction energy in wave-numbers becomes

∆𝐸

ℎ𝑐 = -∆T = g

𝛽𝑒

ℎ𝑐 MB (6.23)

-∆T = gML (6.24)

Where L = 𝛽e B/hc is called Lorentz unit and its value is 0.467B cm-1 when B is

measured in Webers per square metre (tesla). The separation between two

neighbouring sublevels is gL and is determined by magnetic field B and g-factor

belonging to the energy level.

Examples

Example 1. Consider the Transition between 2D5/2 and 2P3/2. Using g = 1 +

J 2+ S2 – L2

2J 2 = 1 +

𝐽(𝐽+1)+𝑆( +1)−𝐿(𝐿+1)

2𝐽(𝐽+1) the g values of 2D5/2and2P3/2are 4/5 and 4/3,

respectively. The splitting of the levels and the allowed transition are shown in the

Figure that is Weak field Zeeman splitting. The allowed selection rules for

transition between magnetic sublevels are

A

2P3/2

2D5/2

1/2

-3/2

3/2

-1/2

1/2

-3/2

3/2

-1/2

-5/2

5/2

∆M = 0 (p component or π component)

∆M = ±1 (s component or 𝜎 component)

∆J = 0, with M =0 → M = 0 is not allowed

The 2D5/2 →2P3/2transition spits up into twelve components ( four p components and

eight s components) .

Example 2. Consider the transition between 1F3 – 1D2. As 1F3 – 1D2are singlet,

therefore their g values are equal to 1. Fig. shows the splitting of the levels in weak

magnetic field and allowed transition between them.

A

1D2

1F3 a

-3

3

a

2

-2

The allowed transitions from 1F3 – 1D2in weak magnetic field are shown in the

figure.

M of initial state→

M of final state↓ 3 2 1 0 -1 -2 -3

2 A+a A A-a X X X X

1 X A+a A A-a X X X

0 X X A+a A A-a X X

-1 X X X A+a A A-a X

-2 x X x x A+a A A-a

This indicates that there are only three distinct energies at A+a, A and A-a. Thus

the spectral line corresponding to 1F3 – 1D2transition splits into three components in

the weak magnetic field, one at the same position and other two are displaced on

either side of undisplaced line.

Now it is clear that when levels involved in a transition are singlet, only three lines

are observed corresponding to normal Zeeman effect.

If the levels in a transition involved are different from singlet, the anomalous

Zeeman effect is observed.

Conclusevly, when the contribution of spin towards magnetic moment is zero, the

normal Zeeman effect is observed and the spin contribution is taken in account,

anomalous Zeeman effect is observed.

Representation of transitions between Zeeman sublevels on a normal energy level

diagram can be understood with following arrangement.

Write in a row all the values of M, for 3P3/2 ---------2S1/2 transition. Below each

value of M put the displacement Mg of the magnetic level of the initial spectral

term. Similarly write down the displacement of the levels belonging to the final

term in the next row. The vertical arrows represent the transitions ∆M = 0 (π or p)

Forbidden Transition is represented by arrows of greater inclination. The line

positions are expressed as multiples of 1/r, where r is called Runge denominator

and is the least common multiple of the denominator in the values of Mg. The

usual notation for a Zeeman pattern consists of a long line with Runge denominator

beneath it with integer above it to show at what multiple of 1/r the components lie.

The table demonstrates the procedure for obtaining the Zeeman pattern for

2P3/2→2S1/2 transition.

The g values for 2P3/2and 2S1/2are 4/3 and 2, respectively. The vertical differences

( 𝜋 or p component) are 3/6 and -3/6 while the diagonal difference ( 𝜎 or s

components) are 6/6, 5/6, -5/6 and -6/6. In Runge notation it is written as

∆�̅� = (±3),±5,±6

6 Lcm-1

With two p components being set in parentheses, followed by the four s

components.

Intensity rules

(i) The intensities of the components of one line are symmetrical relative to

the position of the original line.

(ii) The sum of the intensities of combinations of a level characterized by

the magnetic quantum number M with level M-1, M and M+1 is

independent of M.

(iii) Sum of the intensities of all p components is equal to the sum of the

intensities of all s components

Table intensity rules

M→M + 1, I = A(J + M + 1) (J - M)

J→J M→M - 1, I = A(J - M + 1) (J + M)

M→M, I = 4AM2

M→M + 1, I = B(J – M) (J – M - 1)

J→J - 1 M→M - 1, I = B(J + M) (J + M - 1)

M→M, I = 4B (J2 – M2)

M→M + 1, I = B(J + M + 1) (J + M + 2)

J→J + 1 M→M - 1, I = B(J - M + 1) (J - M + 2)

M→M, I = 4B(J + M + 1) (J - M + 1)

When Zeeman Effect is observed in a direction perpendicular to the

magnetic field, only half of the intensity of the s component is observed. The

other half is observed in a direction parallel to the magnetic field. Therefore, in

studying intensities s components need to be multiplied by 2. A and B are

constants in the above table that need not be determined for relative intensities

within each Zeeman pattern.

The table holds for any coupling scheme. To determine the strongest

component the following approximate rule is useful. In case where J1≠ J2, the

vertical differences in the middle of the scheme and diagonal differences at the

ends give, respectively, the strongest p and s components. If J1 = J2,the vertical

differences at the end of the scheme and the diagonal differences at the centre

give, respectively the strongest p and s components with the restriction that M =

0 to M = 0 is forbidden.

There are Polarisation rules

Viewed ⊥ to the field----- ∆ M =±1; plane polarised ⊥ to B; s or 𝜎

component

∆M = 0; plane polarised||to B; p or π component

Viewed ||to the field……. ∆M =±1; circularly polarised; s or 𝜎 component

∆M = 0; forbidden; p or π component

2. Paschen Back Effect

In describing the anomalous Zeeman Effect, it is assumed that external magnetic

field is weak compared to the internal fields; consequently the interaction

between J and H is weak compared to the interaction of orbital and spin magnetic

fields. The interaction between l and s starts loosening or gets broken at

sufficiently large value if the strength of the external magnetic field is increased to

the extent that it starts competing with the internal field.

Under this condition J starts losing its significance and l & s interact independently

with the external magnetic field resulting in the independent precessions of l & s

about J; their precession is much faster than the precession of residual interaction

of l & s about j. J processes slowly around H in comparison to l and s about J. This

is known as Paschen-Back Effect. The separation between the spin components or

between the Zeeman components is a measure of the corresponding processional

frequency.

In the anomalous Zeeman Effect, the precession of l & s about j is faster and

therefore the average values of their components normal to j are assumed as zero in

order to avoid the perturbation of other precessions.

However, when the external magnetic field is of the same order as that of the

internal fields, the case is different and the relation -∆T= gLmj no longer holds

good. Further, l & s will precess independently about H as the field is increased

and hence will become quantized independently in the direction of field H. The

figure depicts the precession of l & s about external magnetic field H and their

space quantization.

The projection of l on H takes integral values ml = +l,l-1,….0, -1,……-l due to

which there are (2l+1) values of for a given value of l . The projection of

s on H takes two values; ms =±1/2. .

As each of the electrons takes each of the two values of ms there are

2(2l+1) different quantum states; each corresponding to different combination of

quantum numbers. The figure depicts space quantization for l=1 and s=1/2 . When

the ls-interaction is significantly loosened or broken, major part of the total energy

of the atom consists of the energies due to the precession of l around H plus energy

due to precession of s about H. Thus, the major energy shift is

∆Emag.int. =∆El,H+∆Es,H=H.μ1+H.μs = H.μl cos(l,H)+Hμs(s,H) ------------[1]

Using equations for μl and μs respectively and substituting the values of cos(l,H)

and cos(s,H) , the equation (1) reduces to

∆Emag.int.=h(ml+2ms) 𝑒 𝐻

2𝑚𝑐 = HμB(ml+2ms),μB isBohr magneton -------------[2]

In terms of wave numbers, -∆T=(ml+2ms)Lcm-1 ---------------[3]

Though l & s precess independently, but each produces magnetic field on the

electron causing some perturbation to other’s motion.

Though, the ls- interaction is small compared to the effect of the external magnetic

field but need to be taken into consideration. The separation due to this residual

interaction is of the same order of magnitude as the field free fine structure doublet

separations. Therefore, using equation 3 and that the j is vector sum of l and s, its

contribution can be written as

-∆Tl, s =α ls cos (l,s) where α = 𝑅𝛼2𝑧4

𝑛3(𝑙+1

2)(𝑙+1)

---------------[4]

In field free l,s interaction, l and s rotate together as a rigid system about j thereby

the angle between l and s is constant rendering easy evaluation of cos(l,s). In the

present case angle between l and s is varying continuously because of the

anomalous behavior of spin (s precess faster than l). This necessitate the use of

average value of cos(l,s) that can be evaluated using trigonometry theorem, viz

cos(l,s)=cos(l,H)×cos(s,H). Using this theorem along with the values

of cos(l,H) and cos(s,H) from the figure (1), one finds

---------------[5]

Adding this to equation , total energy shift becomes

---------------[6]

A general relation for the term values may, thus, be written as

---------------[7]

T0 is the term value of the hypothetical center of gravity of the fine structure

doublet. As J is no longer well defined quantum number; therefore, Paschen Back

effect is described in terms of quantum number (ml+2ms).

Paschen Back effect of Sodium D1 & D2 lines: quantum numbers for the states

involved in the transitions exhibiting Paschen Back effect are as:

Two highlighted states have same value of Paschen Back quantum

number and will, therefore, constitute a single component of the

state . States corresponding to these quantum numbers are shown in the figure 2.

Selection rules allow only six out of possible ten transitions. If

resolution of the detecting system is small enough to neglect the interaction,

the Paschen Back will constitute three lines; each is a coinciding pair (a coincides

with with and with ) since the values of for each of the

components of a pair of lines are , 0 and respectively (fig 2). Dotted lines

represent the forbidden transitions. Three pair of lines is obtained under the

assumption that ls- interaction is zero.

Fig. 2. Transitions showing Paschen Back effect in the absence of ls-interaction.

The effect of the external magnetic field (weak to strong with respect to internal

magnetic field) on the spectrum can be summarized as: as long as the external

magnetic field is unable to perturb the inner precession, one gets anomalous

Zeeman spectrum. On the other side, if the strength of the external field yields the

magnetic resolution

Fig. 3. Transition from anomalous Zeeman effect to Paschen Back effect.

more than the spin –orbit fine structure, that is normal Zeeman triplet, shown in

the fig. 3 (transition of interaction for WEAK to STRONG field). In a way, these

are the two extreme situations; what about the intermediate fields, i.e. during the

transition from relatively weak to strong external field? Usually this transitional

zone is referred to as the Paschen Back effect.

It is noteworthy that the Paschen Back spectrum is not as simple as discussed; it is

somewhat complicated and separation between the various magnetic field

components is a function of external magnetic field.

The situation can be addressed by using the classical laws of the conservation of

angular momentum. The conservation law demands that , projection of on ,

that describes the internal magnetic field & is well defined under weak field, must

be equal to the total angular momentum when the field is relatively strong.

That is, conservation law says that . In addition, in keeping with the

quantum mechanics, levels with same must not cross in the correlation between

the two extreme conditions. Correlation between the magnetic levels of the

transitions are shown in the fig 4.

Fig. 4 Correlation between the magnetic levels of the transitions

In a relatively strong field, as discussed above, the extreme limit (spin is not

coupled to orbital motion) of the Zeeman effect leads to normal Zeeman triplet.

Quantum numbers and , since are independently & strongly coupled

to B, are well defined and, therefore, keep their physical significance. Conservation

of angular momentum accounts for the selection

rules: for the electric dipole radiation. Since the

orientation of the spin does not alter during any radiation process (emission or

absorption), the selection rules and for dipole radiation hold well.

Together with these, selection rules and allow us to ignore spin. Or

one may argue that on substituting in the relation (2), that is magnetic

energy shift , the shift turns out to be proportional to ; that is

spin is completely dropped out. Consequently, only three lines corresponding

to are observed. Implication of the selection rule is that the

polarization remains unaltered throughout all the magnetic field strengths. It may

be remarked that should , there will be residual spin-orbit coupling yielding

group of several transitions around each of the three components.

NOTE:

It is noteworthy that the weak or Strong external magneticfField is not in absolute

term but is relative to the net internal

A week external field for one atom could be strong for some other atom. Magnetic

field of 30k Gauss produces Zeeman separation of 2 cm -1 in sodium; while the ls

interaction yields J1/2 and J3/2 with a separation of 17.8 cm-1 (a measure of internal

field) indicating that internal field is much stronger than the external field. On the

other hand, same magnetic field applied to H-atom, Zeeman separation (measure of

external field) is much larger than the ls interaction separation (measure of internal

field) that is less than half a wavenumber. Thus the field which is weak in case of

sodium is strong for H-atom.

3. The Stark Effect

When a source of light is placed in an electric field, the spectral lines split up into

number of components. The Stark effect is due to the interaction between the

electric moment of the atom and the external electric field. The term to the

interaction energy is,

W = -p. E,

Here p is the electric dipole moment and E is the external field.

Electric dipole moment in atoms arises as a consequence of the way that charge is

distributed within atoms. There exist two aspects of the Stark effect:

the linear effect and

the quadratic effect.

The linear effect is due to a dipole moment that arises from a naturally occurring

non-symmetric distribution of electron charge, while the quadratic effect is due to a

dipole moment that is induced by the external field. For simplicity the effects of

fine and hyperfine structure are ignored.

Linear Stark Effect:

In the Figure, the transition from the ground level (n= 1) to the first excited level

(n=2) of atomic hydrogen is the simplest case of the linear effect wherein the

excited level splits into three equally spaced sublevels under the influence of the

external field.

The spacing between sublevels increases linearly with the applied field. The

central sublevels of n = 2 and the ground level do not shift in response to the field.

NOTE: As a general case, an n level splits into 2n – 1 sublevels that separate in

the field at a rate that scales with n. Thus the higher n states are more sensitive to

the external field.

As the linear effect is the result of the interaction between the electric dipole

moment due to the distribution of charge with- in the atom and the external field.

The dipole moment can be understood by considering an eccentric elliptical orbit.

m

ћω12 Energy

22S, 22P

12S

|E|

3ea0

0

+1, 0, -1

0

3ea0

Increasing electric field

0

|E|=0

|E|

|E|≠0

|E|=0

ω12-∆ ω12+∆ ω12

ω

ω

Here the nucleus is at one focus and the electron follows a Kepler orbit that sweeps

out equal areas in equal times.

The electron moves fastly when it is near the nucleus and slowly when far from the

nucleus and as a result the electron’s position averaged over an orbit is not

centered on the nucleus. The separation between positive and negative charge

centers here referred to a dipole moment. The size of the naturally occurring dipole

moment corresponding to a highly eccentric orbit can be estimated, given that the

mean radius of an atom in a state of principal quantum number n is on the order of

n2a0, where a0 is the Bohr radius. Thus the dipole moment, which is the product of

the charge and the charge separation, is en2a0. This estimate is comparable with the

observed moment of the stark Level, 3en(n-1) ao/2

So, W=3/2 enkao and k is called electric quantum number that ranges from +(n-|m|-

1) to –(n-|m|-1) for all possible values of m (i.e. from +(n-1) to -(n-1).

K=+/- 1that have the smallest dipole moments corresponding to circular orbits.

Fast

Effective center of

electron charge

Nucleus

slow

Summary:

The Zeeman and Stark Effects refer, respectively, are due to effects of

external magnetic and external electric fields on the structure of atoms or

molecules. The effects are observed through the modification of spectral

features such as strength, polarization, width and position of emission or

absorption lines.

The structure of the anomalous effect is more complicated than that of the

normal effect. The field dependence is also more complicated.

In weak fields, the levels separate linearly with the strength of the applied

field as they did for the normal effect; however, the rates of separation of

levels are different, than those observed for the normal effect. In moderate

fields, the levels shift in a complicated way that is not easily described by

any simple power law. In strong fields, the levels again shift in proportion to

the field strength but this time with rate of separation that are the same as

those of the normal effect.

The anomalous effect was a mystery until 1925 when S. Goudsmit and G.

Uhlenbeck introduced the concept to electron spin. Electron spin was

conceived to explain why the fine structure separation of the 2P levels of

alkali metal atoms were so much larger than the corresponding levels of

hydrogen. Goudsmit and Uhlenbeck suggested that electrons have both an

intrinsic angular momentum and an intrinsic magnetic moment. Following

this assumption, the fine structure splitting was shown to be the result of the

magnetic interaction between the intrinsic magnetic moment of the electron

and an internal magnetic field produced by the electron’s orbital motion.