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Subject Chemistry
Paper No and Title 8/ Physical Spectroscopy
Module No and Title 10/ Zeeman effect
Module Tag CHE_P8_M10
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 10 (ZEEMAN EFFECT)
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TABLE OF CONTENTS
1. Learning Outcomes 2. Introduction
3. Spin-orbit coupling
4. Zeeman effect
5. Selection Rules
6. Summary
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 10 (ZEEMAN EFFECT)
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1. Learning Outcomes
In this module, we shall learn about the quantitative aspects of spin-orbit coupling and how to recognize Russell-Saunders coupling in an atom. This will be followed by a discussion of the Zeeman effect, the additional splitting of lines in a magnetic field.
2. Introduction
The splitting of lines in the presence of a magnetic field is called the Zeeman effect. This important effect helps in the assignment of transitions. Historically, Zeeman observed that each transition is split into three. Later observations showed splitting into many more lines and this was termed the anomalous Zeeman effect, and the splitting into three lines as the normal Zeeman effect. However, the so-called anomalous Zeeman effect is actually the rule and the ‘normal’ Zeeman effect a special case for singlet states.
We first begin with a discussion of the quantitative aspects of Russell-Saunders coupling.
3. Spin-Orbit Coupling
2S+1 The term symbol LJ emphasizes the fact that in Russell-Saunders coupling, the energy depends on the three quantum numbers, S, L and J.
The spin-orbit coupling Hamiltonian is given by
ˆ ˆ ˆ H s−o = AL.S
where A is a constant, known as the spin-orbit coupling constant. It adds an additional energy term to the total energy of an atom. " ! ! Since J = L + S , we may write
! " " 2 " " ! ! J 2 = (L + S ) = L2 + S 2 + 2L.S ! ! 1 ! ! ! ⇒ L.S = J 2 − L2 − .S 2 2 [ ]
which implies that
A Hˆ ψ = ALˆ.Sˆψ = Jˆ 2 − Lˆ2 − Sˆ 2 ψ s−o 2 [ ]
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 10 (ZEEMAN EFFECT)
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which implies that the spin-orbit interaction energy is
A 2 Eint = [J (J +1) − L(L +1) − S(S +1)]! 2
Thus, the energy of a term depends upon the values of L, S and J.
The energy separation between energy levels of a multiplet also depends on J:
A 2 ΔEJ J 1 = ! [(J +1)(J + 2) − L(L +1) − S(S +1) − (J(J +1) − L(L +1) − S(S +1))] → + 2 = A!2(J +1)
This is called the Landé interval rule, which states that the separation between two adjacent levels within a term is proportional to the larger of the two J-values involved. This rule can be used to determine the value of J and also to determine to what extent Russell-Saunder’s coupling scheme is valid.
Consider the carbon 2p2 configuration, giving rise to a 3P ground state. For this state, L = 1 and S = 1. Substitution in equation (1) gives
! 2 Eint = A [J(J +1) − 2 − 2] 2
with the following energies:
J 2 ΔEint / ! 2 A 1 -A 0 -2A
The energy gap between adjacent J levels is the order of 50 – 800 cm-1, much less than the spacing between the main spectroscopic terms, which is the order of 10,000 cm-1. For carbon, the 3 3 -1 3 3 -1 separation between the P0 and P1 terms is 16.4 cm , and that between P0 and P2 is 43.5 cm CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 10 (ZEEMAN EFFECT)
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3 3 3 (Fig. 3.8). According to the Landé’s interval rule, the separation between P2 and P1, and P1 and 3 P0 should be in the ratio 2:1, which is only approximately obeyed (1.65:1).
The spin-orbit coupling constant is different for different terms and can be of either sign. It follows that A > 0 for less than half-filled subshells, while A < 0 for more than half-filled subshells, according to Hund’s third rule. For example, for oxygen, which has a 2p4 electron 3 3 3 configuration, and hence the same P ground state as carbon, the spacing between the P2 and P1 -1 3 3 -1 3 terms is 158 cm , and that between P2 and P0 is 227 cm . The ratio of the spacings between P2 3 3 3 and P1, and P1 and P0 is then 158/69 (2.29), which is close to the expected value of 2.0. Note that the amount of splitting increases as the number of electrons (and hence the electron 3 repulsion) increases, and that the energy order is reversed, i.e. P2 is lowest in energy since the spin-orbit coupling constant is negative for more than half-filled states.
Figure 1 shows schematically the interactions by which the 15 states of the ground-state configuration of carbon are split. The strong electrostatic interactions (including exchange) lead to a splitting into three terms 3P, 1D and 1S. The weaker spin-orbit interaction splits the ground 3P 3 3 3 term into three components, P0, P1 and P2.
Each term component can be further split into the (2J + 1) MJ levels by an external magnetic field, an effect known as the Zeeman effect (Fig. 2), which we shall discuss in the next section.
Figure 1 The splitting of lines in ground state carbon CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 10 (ZEEMAN EFFECT)
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Each J value is associated with 2J+1 values of MJ, which are degenerate in the absence of an external field. Let us see what happens when an external magnetic field is applied. The resultant splitting is known as the Zeeman effect.
4. Zeeman Effect
Each atom is like a tiny magnet, whose magnetic moment is proportional to the magnitude of the angular momentum.
e! µ J = − g J J (J + 1) = −βg J M J 2me
e! where β = − is a combination of fundamental constants and is called the Bohr magneton. Its 2me magnitude is 9.27400968×10−24 J T-1. Obviously, if J = 0, the atom is diamagnetic, and paramagnetic otherwise.
The Landé g factor is given by
J (J +1) + S)S + 1) − L)L + 1) g J =1 + 2J (J + 1)
There are 2J + 1 values of MJ associated with each J. Before a magnetic field is applied, all directions are equal and there is no preferred direction, but the moment a magnetic field is applied, its direction becomes unique, and the magnetic moment of the atom may either be aligned with the field or against it. The component of the magnetic moment in the direction of the ! field B is given by
e! µ B = − g J M J 2mec
The perturbation due to the magnetic field is thus given by
ˆ ! ! H'= −µ B.B
and the interaction energy by
Eint = βg J M J B
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 10 (ZEEMAN EFFECT)
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The interaction energy is therefore proportional to the strength of the magnetic field B and the value of MJ. More negative values of MJ are therefore lower in energy.
.5. Selection rules
We already know the selection rules for J, L and S, i.e. ΔJ = 0, ±1, ΔL = ±1, ΔS = 0. In the presence of a magnetic field, an additional selection rule, ΔMJ = 0, ±1 also operates, but ΔMJ cannot be zero if ΔJ is also zero. This rule is easily understood in terms of the conservation of angular momentum. A photon must cause a change of one unit in either the total angular momentum or its component because it has an intrinsic angular momentum of one unit.
This additional selection rule causes a splitting of lines in a magnetic field and this is called Zeeman effect. In an electric field, the term used is the Stark effect.
1 1 As an example, let us consider the P1 ← S0 transition in the absence and presence of a magnetic field. This is an allowed transition, having ΔJ = 1, ΔL = 1, ΔS = 0. In the absence of a magnetic field, only one line will be observed, but this is split into three lines when an external magnetic field is applied due to the Zeeman effect. The splitting of lines into three was first observed by Zeeman and is called the normal Zeeman effect.
However, later findings showed that this is a special case for singlet states. In other cases, more than three lines are observed and this is the general rule, but for historical reasons, the splitting into more than three lines is called the anomalous Zeeman effect.
1 1 Figure 2 The P1 ← S0 transition in the absence and presence of a magnetic field
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 10 (ZEEMAN EFFECT)
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All three transitions are allowed because ΔJ ≠ 0.
As another example, consider the 3p ← 3s transition of sodium. The 3s1 configuration gives rise 2 1 2 2 to the term 3 S1/2, whereas the excited 3p configuration gives two terms 3 P1/2 and 3 P3/2, of which the former is lower in energy since the configuration is less than half-filled.
In the absence of a magnetic field, only two lines are observed, and these are the famous D-lines. In the presence of a magnetic field, however, these are split into 10. Of the possible 12 lines, only two: MJ = -3/2 → +1/2 and MJ = +3/2 → -1/2 are forbidden because they violate the ΔMJ = ±1 selection rule.
Figure 3 The splitting of the sodium D lines in the presence of a magnetic field
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 10 (ZEEMAN EFFECT)
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6. Summary
• The electrostatic interactions split an electron configuration into states of different energy.
• The spin-orbit coupling further splits these terms into different J values.
• Each J value has 2J + 1 MJ values associated with it, which are degenerate in the absence of an external field.
• In the presence of a magnetic field, each J level is split into 2J + 1 levels, leading to splitting of lines, known as the Zeeman effect.
CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 10 (ZEEMAN EFFECT)