Spin-1/2 Paramagnetism

Our atom can be in one of two possible states. Namely, the state in which its spin points up (i.e., parallel to ), or the state in which its spin points down (i.e., antiparallel to ). In the state, the atomic magnetic moment is parallel to the magnetic field, so that . The magnetic energy of the atom is . In the state, the atomic magnetic moment is antiparallel to the magnetic field, so that . The magnetic energy of the atom is .

According to the canonical distribution, the probability of finding the atom in the state is

(7.13) |

where is a constant, and . Likewise, the probability of finding the atom in the state is

(7.14) |

Clearly, the most probable state is the state with the lower energy [i.e., the state]. Thus, the mean magnetic moment points in the direction of the magnetic field (i.e., the atomic spin is more likely to point parallel to the field than antiparallel).

It is apparent that the critical parameter in a paramagnetic system is

(7.15) |

This parameter measures the ratio of the typical magnetic energy of the atom, , to its typical thermal energy, . If the thermal energy greatly exceeds the magnetic energy then , and the probability that the atomic moment points parallel to the magnetic field is about the same as the probability that it points antiparallel. In this situation, we expect the mean atomic moment to be small, so that . On the other hand, if the magnetic energy greatly exceeds the thermal energy then , and the atomic moment is far more likely to be directed parallel to the magnetic field than antiparallel. In this situation, we expect .

Let us calculate the mean atomic moment, . The usual definition of a mean value gives

(7.16) |

This can also be written

(7.17) |

where the hyperbolic tangent is defined

(7.18) |

For small arguments, ,

(7.19) |

whereas for large arguments, ,

(7.20) |

It follows that at comparatively high temperatures, ,

(7.21) |

whereas at comparatively low temperatures, ,

(7.22) |

Suppose that the substance contains
atoms (or molecules) per unit volume.
The *magnetization* is defined as the mean magnetic moment per unit
volume, and is given by

(7.23) |

At high temperatures, , the mean magnetic moment, and, hence, the magnetization, is proportional to the applied magnetic field, so we can write

(7.24) |

where is a dimensionless constant of proportionality known as the

(7.25) |

The fact that is known as

(7.26) |

so the magnetization becomes independent of the applied field. This corresponds to the maximum possible magnetization, in which all atomic moments are aligned parallel to the field. The breakdown of the law at low temperatures (or high magnetic fields) is known as