Tru Physics

Landé g-Factor

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Introduction

The Landé g-factor, named after Alfred Landé, is a dimensionless quantity that characterizes the magnetic moment and angular momentum of atomic and subatomic particles in quantum physics. This value is a critical part of the Zeeman Effect, where spectral lines split due to an external magnetic field.

The g-Factor in Quantum Mechanics

In quantum mechanics, the g-factor modifies the relationship between a particle’s spin angular momentum (S) and its magnetic moment (\mu). Specifically, the magnetic moment of a particle is given by:

\mu = -g \dfrac{e}{2m} S

where e is the charge of the particle, m is the mass of the particle, and g is the Landé g-factor. The negative sign indicates that the magnetic moment and angular momentum vectors point in opposite directions for particles with a positive charge, such as the proton.

Landé g-Factor in Atoms

For atoms, the Landé g-factor is given by the following formula:

g_J = 1 + \dfrac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}

where J is the total angular momentum, S is the total spin, and L is the total orbital angular momentum.

Landé g-Factor and the Zeeman Effect

The Zeeman Effect refers to the splitting of spectral lines in the presence of an external magnetic field. This is due to the different energy levels that electrons can occupy based on their angular momentum and magnetic moment, which are characterized by the Landé g-factor.

The energy shift due to the Zeeman Effect is given by:

\Delta E = \mu_B g_J m_J B

where \mu_B is the Bohr magneton, B is the magnetic field, m_J is the z-component of the total angular momentum, and g_J is the Landé g-factor.

Conclusion

The Landé g-factor is a critical parameter in understanding how magnetic fields interact with atomic and subatomic particles. By accounting for both spin and orbital contributions to angular momentum, it allows for precise predictions and explanations of phenomena such as the Zeeman Effect. Its understanding is essential for anyone studying quantum mechanics and atomic physics.

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