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?

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

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

    <p>$\Delta E_{exchange} = -2J_{\alpha \beta} S_1ҧ. S_2ҧ$</p> 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?

    <p>Orbital angular momentum quantum number, L</p> Signup and view all the answers

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

    <p>Weak magnetic fields</p> Signup and view all the answers

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

    <p>$𝑣 = 𝑣0 ± 4 𝜋 𝑚 𝑐$</p> Signup and view all the answers

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

    <p>Equal to zero</p> 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?

    <p>$∆ 𝐸 = −𝜇.𝐵$</p> 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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    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.

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