This note was split from a 2021 class note on spin, magnetic moment, and gyromagnetic ratio.

Magnetic moment is a vector that describes the strength and direction of an object's magnetic dipole interaction. The object may be a molecule, an atom, a nucleus, or a smaller particle. For the interaction considered here, we use the term magnetic dipole moment (μ); a tiny bar magnet with north and south poles is a useful first picture.

In an external field B0, μ experiences a torque, μ × B0. The torque is the field's action on the magnetic moment, rather than a resistance that μ must overcome. Its potential energy is E = - μ·B0: parallel alignment has the lowest energy, and antiparallel alignment the highest. Both exactly parallel and exactly antiparallel orientations have zero torque, even though their energies differ. OpenStax: force and torque.

Bar magnet dipole next to B0 under torque, then μ parallel to B0 (low energy) and opposed (high energy)
Conceptual picture of a magnetic dipole in B0. Parallel alignment has lower energy than antiparallel alignment; torque changes angular momentum.

For a nuclear magnetic moment associated with angular momentum, torque changes the direction of that angular momentum and produces precession. Torque alone does not make the moment settle into the lowest-energy orientation; relaxation is needed to describe the return to thermal equilibrium. Bloch, 1946.

It also helps to separate μ from magnetization M. μ refers to an individual magnetic moment, while M is the vector sum of many moments per unit volume. The bulk MRI signal depends on this collective magnetization, so the individual contributions can reinforce or cancel one another. Stanford: classical MR, p. 8.

The previous note covered spin. The next is gyromagnetic ratio, the constant relating μ to I.

中文

這篇從 2021 年的課間筆記拆分而來，接續 spin，說明磁矩。

磁矩是一個向量，用來描述物體磁偶極作用的強弱與方向；物體可以是分子、原子、原子核，或更小的粒子。就這裡討論的作用而言，也會使用磁偶極矩（magnetic dipole moment, μ）這個名稱，可以先把它想成一顆有南、北極的小磁鐵。

μ 放進外加磁場 B0 後，會受到 torque（力矩） μ × B0；這個力矩是磁場對磁矩的作用，並不是 μ 必須克服的阻力。位能由 E = - μ·B0 決定，與 B0 同向時最低，反向時最高；值得分開理解的是，完全同向與完全反向時的力矩都為零，但兩者能量並不相同。OpenStax：力與力矩。

對具有角動量的核磁矩來說，力矩會改變角動量的方向，造成進動；單靠力矩，磁矩不會自行停在最低能量的方向，回到熱平衡的過程還需要弛緩來描述。Bloch, 1946。

另外也要把 μ 與磁化向量 M 分開：μ 是個別磁矩，M 則是單位體積內許多磁矩的向量總和。MRI 接收到的是這些磁矩共同形成的訊號，因此個別貢獻可能相加，也可能互相抵消。Stanford：Classical MR，第 8 頁。

上一則談的是 spin，接著要看的旋磁比，就是連結 μ 與 I 的那個常數。