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=µL
J 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= - 2𝑚𝑒
𝑒J[1 +J 2+ S2J 22 – L2]= - g 2𝑚𝑒
𝑒 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= - g1 𝑒 2𝑚𝑒 j1
µj2 = - g2 𝑒 2𝑚𝑒 j1
where g1 and g2 can be obtained from the above equation of g factor.
g1= 1 + j12+ s2𝑗12 – l12
22
g2 = 1 + j22+ s2𝑗22 – l22
22
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+j12 – 2J. j1 = J2 + j12 – 2j1J cos (j1,J) cos (j1,J) = 𝐽2+ j12− j22
2𝑗1𝐽 Similarly,
cos (j2,J) = 𝐽2+ j22− j12
2𝑗2𝐽
Substituting
g = g1
𝐽2+ j12− j22 2𝐽2 + g2
𝐽2+ j22− j12 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 g𝛽eB. 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 (𝜋or 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
∆𝑣̅ = (±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.
