General Paramagnetism

Consider a system consisting of non-interacting atoms in a substance held at absolute temperature , and placed in an external magnetic field, , that points in the -direction. The magnetic energy of a given atom is written

where is the atom's magnetic moment. Now, the magnetic moment of an atom is proportional to its total electronic angular momentum, . In fact,

where

(7.93) |

is a standard unit of magnetic moment known as the

Combining Equations (7.91) and (7.92), we obtain

Now, according to standard quantum mechanics,

(7.95) |

where is an integer that can take on all values between and . Here, , where is a positive number that can either take integer or half-integer values. Thus, there are allowed values of , corresponding to different possible projections of the angular momentum vector along the -direction. It follows from Equation (7.94) that the possible magnetic energies of an atom are

(7.96) |

For example, if (corresponding to a spin-1/2 system) then there are only two possible energies, corresponding to . This was the situation considered in Section 7.3. (To be more exact, a spin-1/2 system corresponds to and .)

The probability, , that an atom is in a state labeled is

(7.97) |

where . According to Equation (7.92), the corresponding -component of the magnetic moment is

(7.98) |

Hence, the mean -component of the magnetic moment is

The numerator in the previous expression is conveniently expressed as a derivative with respect to the external parameter :

(7.100) |

where

is the partition function of a single atom. Thus, Equation (7.99) becomes

In order to calculate , it is convenient to define

which is a dimensionless parameter that measures the ratio of the magnetic energy, , that acts to align the atomic magnetic moments parallel to the external magnetic field, to the thermal energy, , that acts to keep the magnetic moment randomly orientated. Thus, Equation (7.101) becomes

(7.104) |

The previous geometric series can be summed to give

(7.105) |

(See Exercise 1.) Multiplying the numerator and denominator by , we obtain

(7.106) |

or

(7.107) |

where the hyperbolic sine is defined

(7.108) |

Thus,

(7.109) |

According to Equations (7.102) and (7.103),

(7.110) |

Hence,

(7.111) |

or

where

is known as the

The hyperbolic cotangent is defined

(7.114) |

For , we have

(7.115) |

On the other hand, for ,

(7.116) |

Thus, it follows that

when . On the other hand, when , we get

(7.118) |

which reduces to

Figure 7.1 shows how depends on for various values of .

If there are atoms per unit volume then the mean magnetic moment per unit volume (or magnetization) becomes

(7.120) |

where use has been made of Equation (7.112). If then Equation (7.119) implies that . This relation can be written

(7.121) |

where the dimensionless constant of proportionality,

(7.122) |

is known as the magnetic susceptibility. Thus, --a result that is called Curie's law. (See Section 7.3.) If then

(7.123) |

This corresponds to magnetic saturation--a situation in which each atom has the maximum component of the magnetic moment parallel to the magnetic field, , that it can possibly have.

Figure 7.2 compares the experimental and theoretical magnetization versus field-strength curves for three different paramagnetic substances containing atoms of total angular momentum , , and particles, showing excellent agreement between theory and experiment. Note that, in all cases, the magnetization is proportional to the magnetic field-strength at small field-strengths, but saturates at some constant value as the field-strength increases.

The previous analysis completely neglects any interaction between the
spins of neighboring atoms or molecules. It turns out that this is
a fairly good approximation for paramagnetic substances. However, for
*ferromagnetic* substances, in which the spins of neighboring atoms
interact very strongly, this approximation breaks down completely. (See Section 7.17.) Thus, the
previous analysis does not apply to ferromagnetic substances.