# When do we have a coherent magnetization rotation? (single-domain particles)

So I am studying micromagnetics + spintronics and I am little bit confused with certain concepts regarding the physics of magnetic domains (especially single domain particles):

What I already knew: For sufficiently small particles (at least in the nm regime), the magnetization is most probably single domain. This means the magnetization is uniform throughout the sample and the constiuting spins rotate in unison. The simplest model to describe the static properties of these particles is the Stoner-Wohlfarth model developed in 1948.

What is confusing me: There are magnetic characteristic lengths, below which interesting phenomenon can be observed. For example:

• The exchange length $l_{ex}$: The length below which exchange interaction dominates the dipolar interaction. Thus for $L<l_{ex}$, exchange dominates, where as for $L>l_{ex}$, dipolar dominates.

• The critical single domain diameter $d_{sd}$: The length below which a material is single domain. Thus for $L<d_{sd}$, the sample is single domain, whereas for $L>d_{sd}$, the sampel is multi-domain.

Now, consider these scenarios:

1. $L<l_{ex}<d_{sd}$: The sample is single domain, since $L<d_{sd}$, and thus the magnetization is uniform. Moreover, exchange dominates dipolar interaction, since $L<l_{ex}$, which means exchange interaction wins and penalizes any non-uniformities in the sample and will cause coherent rotation of magnetization.
2. $l_{ex}<L<d_{sd}$: The sample is still single domain, since $L<d_{sd}$, and thus the magnetization is uniform. However, dipolar dominates exchange interaction, since $L>l_{ex}$, which means dipolar interaction wins and there will be non-uniformities (+ multi-domains?!) and will cause non-coherent rotation of magnetization (e.g., curling, buckling, fanning, etc.)

Is not case 2. contradicting? If single domain means a uniform magnetization, and exchange interaction (which penalizes non-uniformities) is weaker than dipolar interaction (which causes splitting of domains + non-uniformities), how can one have a uniformly magnetized specimen? Will the magnetization coherently rotate under the application of a magnetic field?

• The spins don't "rotate in unison", unless you are at very low temperatures and in very high fields. At $1T$ field the Larmor frequency of electrons in a magnetic field is on the order of $27GHz$, far below $kT$ and the local fields of solids is on the order of $10T$, or so, if I remember correctly, so you always have to do the spin statistics as a function of temperature. – CuriousOne Apr 2 '16 at 13:48