SIF3002 Lecture 15: Exchange Energy and Spin States

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Questions and Answers

What is the total energy expression for the helium atom in terms of exchange energy?

$E = -\frac{4E_R}{n_1^2} + \frac{1}{n_2^2} - D_{\alpha \beta} + J_{\alpha \beta}$

$E = -\frac{4E_R}{n_1^2} + \frac{1}{n_2^2} + D_{\alpha \beta} - J_{\alpha \beta}$ (correct)

$E = -\frac{4E_R}{n_1^2} + \frac{1}{n_2^2} - D_{\alpha \beta} - J_{\alpha \beta}$

$E = -\frac{4E_R}{n_1^2} + \frac{1}{n_2^2} + D_{\alpha \beta} + J_{\alpha \beta}$

What is the significance of the + and - signs in the total energy expression for the helium atom?

The + sign applies to triplet states and the - sign applies to singlet state. (correct)

The + sign applies to singlet state and the - sign applies to triplet states.

The + sign indicates higher spin alignment and the - sign indicates lower spin alignment.

The + sign indicates higher energy level and the - sign indicates lower energy level.

What is the relationship between the energies of singlet and triplet states in terms of exchange symmetry?

The energies of singlet and triplet states differ by 3𝐽𝛼𝛽 due to exchange symmetry.

The energies of singlet and triplet states are equal due to exchange symmetry.

The energies of singlet and triplet states differ by 𝐸𝑅 due to exchange symmetry.

The energies of singlet and triplet states differ by 2𝐽𝛼𝛽 due to exchange symmetry. (correct)

Why is exchange splitting considered a significant effect in the structure of the helium atom?

Exchange splitting directly impacts the alignment of the spins, leading to changes in energy levels. Signup and view all the answers

How is exchange energy sometimes expressed to show its relationship with spin alignment?

$\Delta E_{exchange} = -2J_{\alpha \beta} S_1ҧ. S_2ҧ$ Signup and view all the answers

In the presence of a magnetic field, the Zeeman effect causes the splitting of spectral lines into several components. What quantum number is associated with this splitting?

Orbital angular momentum quantum number, L Signup and view all the answers

Which regime of interaction between atoms and a magnetic field depends on quantum mechanical effects?

Weak magnetic fields Signup and view all the answers

What is the formula to predict the frequencies for the split component lines in the Zeeman effect?

$𝑣 = 𝑣0 ± 4 𝜋 𝑚 𝑐$ Signup and view all the answers

What is the net spin of the active electron of an atom in the Normal Zeeman effect?

Equal to zero Signup and view all the answers

What is the interaction energy between the magnetic moment and a uniform magnetic field in the Normal Zeeman effect?

$∆ 𝐸 = −𝜇.𝐵$ Signup and view all the answers

Study Notes

Total Energy Expression for Helium Atom

Total energy incorporates kinetic energy, potential energy, and exchange energy due to electron interactions.
Exchange energy quantifies the effect of electron spin configuration on total system energy.

Significance of Signs in Total Energy Expression

Positive signs indicate repulsion between like-spin electrons, leading to increased energy.
Negative signs indicate attraction between opposite-spin electrons, lowering energy.

Relationship between Energies of Singlet and Triplet States

Singlet states exhibit lower energy due to paired spins and reduced electron repulsion.
Triplet states have higher energy owing to parallel spins that increase repulsive interaction.

Importance of Exchange Splitting

Exchange splitting is crucial for explaining the stability and configuration of electrons in helium.
It influences spectral properties and the interaction of atoms with external magnetic fields.

Exchange Energy and Spin Alignment

Exchange energy is often represented by terms that account for antiparallel versus parallel spin configurations.
Positive exchange energy favors parallel spins (triplet), while negative favors antiparallel spins (singlet).

Zeeman Effect and Spectral Line Splitting

Zeeman effect manifests as spectral line splitting when atoms are exposed to a magnetic field.
Quantum number associated with this splitting is the magnetic quantum number (m).

Quantum Mechanical Regime of Interaction

Quantum mechanical effects dominate in the weak field regime during atom-field interactions.
Strong fields can lead to classical behavior where quantum effects become less significant.

Formula for Frequencies in Zeeman Effect

Frequencies of split components are calculated using the relation: ( \Delta E = g \mu_B B )
Here, ( g ) is the Landé g-factor, ( \mu_B ) is the Bohr magneton, and ( B ) is the magnetic field strength.

Net Spin in Normal Zeeman Effect

In the Normal Zeeman effect, the net spin of the active electron is ( \frac{1}{2} ).
This results from the intrinsic spin of single electrons in the atom.

Interaction Energy in Normal Zeeman Effect

Interaction energy in the Normal Zeeman effect is given by: ( H = -\vec{\mu} \cdot \vec{B} )
( \vec{\mu} ) represents the magnetic moment related to the electron's spin and angular momentum.

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Description

Test your understanding of exchange energy and spin states in the context of SIF3002 Session 2023/24. Explore the calculation of total energy and the splitting of spin states, and their significance in the exchange symmetry.