Electromagnetism II (spring term 2020)
Lecture 7
Microscopic theory of magnetism
E Goudzovski [email protected] http://epweb2.ph.bham.ac.uk/user/goudzovski/Y2-EM2
Previous lecture
The electric displacement field is defined as
Gauss’s law for the electric displacement field:
or equivalently
For linear, isotropic and homogeneous (e.g. most) dielectrics,
Boundary conditions: and , leading to
Energy density of the electric field: 1 This lecture
Lecture 7: Microscopic theory of magnetism
Magnetization and magnetic susceptibility Magnetic properties of atoms; the Bohr magneton Theory of diamagnetism Theory of paramagnetism
2 Types of magnetism
In external magnetic field, materials acquire non-zero magnetization
, i.e. induced magnetic dipole moment per unit volume [A/m].
For isotropic homogeneous materials, the magnetic susceptibility is defined as
Magnetism is linked to the charged constituents in materials:
1) orbital angular momenta of electrons; 2) spin angular momenta of electrons; 3) spin angular momenta of nuclei (a much smaller contribution).
Three main types of magnetic materials
Diamagnetic: M is opposite to external B; small and negative. B Paramagnetic: M is parallel to external B; small and positive. B Ferromagnetic: very large and non-linear magnetization M. Permanent magnets are magnetized in absence of external B. 3 Magnetic properties of atoms (1)
A fundamentally quantum effect: classical physics predicts absence of magnetization phenomena. Ze Bohr’s semiclassical model
Angular momentum of the electron: e
Magnetic dipole moment of a current loop (lecture 1)
(T is the rotation period; S is the area; e>0 is the elementary charge)
Gyromagnetic ratio of electron orbital motion: 4 Magnetic properties of atoms (2)
Quantization of angular momentum:
Quantization of the magnetic moment of electron orbital motion:
Bohr magneton:
Electron spin (intrinsic angular momentum):
The associated magnetic moment is B, therefore the gyromagnetic ratio is twice larger than for orbital motion:
e/m. spin 5 Diamagnetism
For each electron, B Diamagnetic materials: electrons of atoms/molecules have Ze zero total angular momentum, i.e. F no permanent magnetic dipole moment e In the absence of external field, Newton’s second law:
;
In external magnetic field, Lorentz force Newton’s second law:
; 6 Induced magnetic moment
Let’s define the Larmor angular frequency as
(in practice, ) Then The positive solution:
If is in the direction of , it increases by . If is in the opposite direction, it decreases by . In both cases, the electron acquires an additional angular momentum 2 L = mLr in the direction of , and an induced magnetic moment:
7 Diamagnetism as induction effect Magnetic forces do not change angular speed: Larmor rotation arises due to electromagnetic induction when the field is switched on.
Faraday’s law (lecture 2):
Induced E-field in the electron orbit along the direction of motion:
therefore
Equation of motion:
Torque Moment of inertia
;
Universal diamagnetism (B<0) is expected from the Lenz rule. 8 Magnetization of diamagnetics
Induced magnetic moment for a single electron:
For an atom,
(Z: atomic number; r0: root-mean-square radius of electron orbit;
1/6 comes from , due to the spherical symmetry). Magnetization: induced magnetic dipole moment per unit volume
(n: density of atoms [1/m3])
Magnetic susceptibility for diamagnetics:
Depends only on density (not temperature or pressure).
Measurements of B can be used to determine atomic sizes. 9 9 E.g. Neon at STP: B=3.76×10 leads to r0=0.49×10 m. 9 Paramagnetism Paramagnetic materials: non-zero permanent magnetic dipole moment (odd number of electrons; ions and ionic molecules). Permanent magnetic moments tend to align with the field (lecture 1)
Typical atomic magnetic dipole moments are of the order For a very large field (B=10 T), work required to revert the direction of magnetic moment : (lecture 1)
Much smaller than thermal motion energy at room temperature:
Polarization of a polar dielectric (lecture 5):
The computation for paramagnetics is analogous: 10 Magnetization of paramagnetics
Magnetic susceptibility for paramagnetics:
Paramagnetic term Diamagnetic term due to permanent due to induced magnetic moments magnetic moments
For most paramagnetics, the paramagnetic term dominates over the diamagnetic term.
This leads to the Curie’s law (B~1/T), which is valid for gaseous paramagnetics for .
We have assumed Bexternal=Blocal (this will be justified in lecture 8). 11 Summary External magnetic fields induce magnetization of materials, which is characterized by the magnetic susceptibility:
Diamagnetism (B<0) is a universal phenomenon: electrons acquire an induced angular momentum and therefore an induced magnetic dipole moment.
Magnitude of the effect: the Larmor frequency
Paramagnetism (B>0) in materials with non-zero permanent atomic magnetic moments is due to the alignment of magnetic moments with the external magnetic field.
Molecular theory leads to
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