438
8 Electric and Magnetic Phenomena
in Sect. 8.4.3 via Eq. (8.4.19a) to aid in the description of a magnetizable system
brought quasistatically into an externally-maintained constant magnetic induction
field. For present purposes, we shall prefer to refer to this internal energy as U spin .
If we now consider N isolated (i.e., distinguishable by virtue of being at specified
locations in a solid lattice) I =
1
2 atoms, the macroscopic magnetization M will be
given by
M = Nμ,
(8.5.8a)
in which μ represents the macroscopic average of the magnetic dipole moments
of the N spin1
2 atoms. As both μ x and μ y precess about the z-axis (with a typical
NMR frequency), the equilibrium values of M x and M y average to zero on a typical
NMR time scale, and only the z-component of M, given via Eq. (8.5.6) as
M z = Nμ z =
1
2 Ng I μ N tanh(
1
2 βg I μ N B 0 ),
(8.5.8b)
survives. We shall hereinafter denote M z as M.
The characteristic function [8] for a canonical ensemble 4 of N distinguishable
nuclear spin1
2 atoms is the Helmholtz energy, A spin , given by
A spin (T , B 0 ) = −k B T ln Z spin (N, T , B 0 ),
(8.5.9a)
in terms of the canonical partition function Z spin (T , B 0 ) for N atoms, or as
A spin (T , B 0 ) = −Nk B T ln z spin (T , B 0 ),
(8.5.9b)
in terms of the single-nucleus partition function z spin (T , B 0 ) given by Eq. (8.5.4).
In this manner, the Helmholtz energy for a spin1
2 particle in a magnetic induction
field B 0 is obtained as an explicit function of T and B 0 , namely
A spin (T , B 0 ) = −Nk B T ln[2 cosh(
1
2 βμ N g I B 0 )].
(8.5.10)
It is perhaps worth noting that the extensivity of the Helmholtz energy is here
represented specifically by the factor N.
The thermodynamic expression for the Helmholtz energy is, according to the
development in Sect. 8.4.3,
A spin (T , V , B 0 ) ≡ U spin (S, V , B 0 ) − T S
= U(S, V , M) − T S − B 0 M.
(8.5.11a)
4 The importance of making an appropriate identification of the characteristic function for the
canonical ensemble and an unambiguous connection with macroscopic thermodynamics has been
emphasized by Castellano [8], who points out, in particular, that this connection is not a minor
issue, as the identification must also hold for systems of interacting spins.
8 Electric and Magnetic Phenomena
in Sect. 8.4.3 via Eq. (8.4.19a) to aid in the description of a magnetizable system
brought quasistatically into an externally-maintained constant magnetic induction
field. For present purposes, we shall prefer to refer to this internal energy as U spin .
If we now consider N isolated (i.e., distinguishable by virtue of being at specified
locations in a solid lattice) I =
1
2 atoms, the macroscopic magnetization M will be
given by
M = Nμ,
(8.5.8a)
in which μ represents the macroscopic average of the magnetic dipole moments
of the N spin1
2 atoms. As both μ x and μ y precess about the z-axis (with a typical
NMR frequency), the equilibrium values of M x and M y average to zero on a typical
NMR time scale, and only the z-component of M, given via Eq. (8.5.6) as
M z = Nμ z =
1
2 Ng I μ N tanh(
1
2 βg I μ N B 0 ),
(8.5.8b)
survives. We shall hereinafter denote M z as M.
The characteristic function [8] for a canonical ensemble 4 of N distinguishable
nuclear spin1
2 atoms is the Helmholtz energy, A spin , given by
A spin (T , B 0 ) = −k B T ln Z spin (N, T , B 0 ),
(8.5.9a)
in terms of the canonical partition function Z spin (T , B 0 ) for N atoms, or as
A spin (T , B 0 ) = −Nk B T ln z spin (T , B 0 ),
(8.5.9b)
in terms of the single-nucleus partition function z spin (T , B 0 ) given by Eq. (8.5.4).
In this manner, the Helmholtz energy for a spin1
2 particle in a magnetic induction
field B 0 is obtained as an explicit function of T and B 0 , namely
A spin (T , B 0 ) = −Nk B T ln[2 cosh(
1
2 βμ N g I B 0 )].
(8.5.10)
It is perhaps worth noting that the extensivity of the Helmholtz energy is here
represented specifically by the factor N.
The thermodynamic expression for the Helmholtz energy is, according to the
development in Sect. 8.4.3,
A spin (T , V , B 0 ) ≡ U spin (S, V , B 0 ) − T S
= U(S, V , M) − T S − B 0 M.
(8.5.11a)
4 The importance of making an appropriate identification of the characteristic function for the
canonical ensemble and an unambiguous connection with macroscopic thermodynamics has been
emphasized by Castellano [8], who points out, in particular, that this connection is not a minor
issue, as the identification must also hold for systems of interacting spins.