32-5 Magnetism and Electrons#

Prompts

What are spin angular momentum \(\vec{S}\) and spin magnetic dipole moment \(\vec{\mu}_s\)? Why are they called intrinsic properties of electrons?
Explain why we can only measure the \(z\)-components \(S_z\) and \(\mu_{s,z}\), not the full vectors. What values can \(m_s\) take for an electron?
Calculate the \(z\)-component of the spin magnetic dipole moment of an electron in terms of the Bohr magneton \(\mu_B\). What is the orientation energy \(U\) when the spin is in an external magnetic field \(\vec{B}\)?
An electron has orbital angular momentum \(\vec{L}_{\text{orb}}\) and orbital magnetic dipole moment \(\vec{\mu}_{\text{orb}}\). How are \(L_{\text{orb},z}\) and \(\mu_{\text{orb},z}\) related to the orbital magnetic quantum number \(m_\ell\)? What values can \(m_\ell\) take?
An atom is in an external magnetic field. Walk me through finding the orientation energy \(U\) of the orbital magnetic dipole moment.

Lecture Notes#

Overview#

Magnetic materials get their magnetism from electrons
Two sources of magnetic dipole moment per electron: spin (intrinsic) and orbital (from motion in atom)
Both are quantized: only certain values of \(z\)-components can be measured

Spin magnetic dipole moment#

Spin \(\vec{S}\): intrinsic angular momentum of electron (not rotation—quantum property)
Spin magnetic dipole moment \(\vec{\mu}_s\): \(\vec{\mu}_s = -\frac{e}{m}\vec{S}\) (opposite directions)

Quantization: Only \(S_z\) can be measured:

(214)#\[ S_z = m_s \frac{h}{2\pi}, \quad m_s = \pm \frac{1}{2} \]

Spin up (\(m_s = +1/2\)): \(S_z\) parallel to \(z\); spin down (\(m_s = -1/2\)): antiparallel

Bohr magneton:

(215)#\[ \mu_B = \frac{eh}{4\pi m} \approx 9.27 \times 10^{-24}\,\text{J/T} \]

\(z\)-component of spin dipole moment:

(216)#\[ \mu_{s,z} = \pm \mu_B \]

Orientation energy in external field \(\vec{B}_{\text{ext}}\):

(217)#\[ U = -\vec{\mu}_s \cdot \vec{B}_{\text{ext}} = -\mu_{s,z} B_{\text{ext}} \]

Physical intuition

Lower energy when \(\vec{\mu}_s\) aligns with \(\vec{B}\). Spin-up (\(\mu_{s,z} = +\mu_B\)) has lower \(U\) when \(B > 0\); spin-down has higher \(U\).

Orbital magnetic dipole moment#

Orbital angular momentum \(\vec{L}_{\text{orb}}\): from electron’s motion in atom
Orbital magnetic dipole moment \(\vec{\mu}_{\text{orb}} = -\frac{e}{2m}\vec{L}_{\text{orb}}\)

Quantization:

(218)#\[ L_{\text{orb},z} = m_\ell \frac{h}{2\pi}, \quad m_\ell = 0, \pm 1, \pm 2, \ldots \]

(219)#\[ \mu_{\text{orb},z} = -m_\ell \mu_B \]

Orientation energy:

(220)#\[ U = -\vec{\mu}_{\text{orb}} \cdot \vec{B}_{\text{ext}} = -\mu_{\text{orb},z} B_{\text{ext}} \]

Three types of magnetism#

Type	Source of moment	Response to \(\vec{B}_{\text{ext}}\)
Diamagnetism	Induced (all materials)	Weak; opposite to field
Paramagnetism	Permanent atomic dipoles, random	Partial alignment with field
Ferromagnetism	Permanent, exchange-coupled	Strong alignment; can persist

Summary#

Spin: \(\vec{\mu}_s = -\frac{e}{m}\vec{S}\); \(S_z = m_s \hbar\), \(m_s = \pm 1/2\); \(\mu_{s,z} = \pm \mu_B\)
Orbital: \(\vec{\mu}_{\text{orb}} = -\frac{e}{2m}\vec{L}_{\text{orb}}\); \(L_{\text{orb},z} = m_\ell \hbar\); \(\mu_{\text{orb},z} = -m_\ell \mu_B\)
Orientation energy: \(U = -\vec{\mu} \cdot \vec{B}_{\text{ext}}\)
Three types: diamagnetic, paramagnetic, ferromagnetic