# Hexagonal ferrite

Hexagonal ferrites or hexaferrites are a family of ferrites with hexagonal crystal structure. The most common member is BaFe$_{12}$O$_{19}$, also called barium ferrite, BaM, etc. BaM is a strong room-temperature ferrimagnetic material with high anisotropy along the c axis. All the hexaferrite members are constructed by stacking a few building blocks in a certain order.

## Basic building blocks
### S block

The S block is very common in hexaferrites, which has a chemical formula of Me$^S$$_6$O$_8$$^{2+}$. Me$^S$ are smaller metal cations, for example, Fe and other transition metals or noble metals. The S block is essentially a slab cut along the $(111)$ plane of an AB$_2$O$_4$ spinel. Each S block has one A layer and one B layer. The A layer features Me$^S$-centered tetrahedron and Me$^S$-centered octahedron, while the B layer is made up of edge-sharing Me$^S$-centered octahedron. Both A and B layers have the same chemical formula of Me$^S$$_3$O$_4$$^{2+}$.

### R block

The R block has a chemical formula of Me$^L$Me$^S$$_6$O$_{11}$$^{2-}$. Me$^L$ are larger metal cations, for example, alkaline earth metals (Ba, Sr,) rare earth metals, Pb, etc. The point group symmetry of the R block is ${\bar {6}}m2$. The large metal cations are located in the middle layer of the three hexagonally packed layers. This block is also composed of face-sharing Me$^S$-centered octahedra and Me$^S$-centered trigonal bipyramids.

### T block

The T block has a chemical formula of Me$^L$$_2$Me$^S$$_8$O$_{14}$$^{2-}$. The point group symmetry of the T block is $3m$. One T block consists of 4 oxygen layers with the two Me$^L$ atoms substituting two oxygen atoms in the middle two layers. In one T block, there are both Me$^S$-centered octahedra and Me$^S$-centered tetrahedra.

## Family nembers
### M-type ferrite

M-type ferrite is made up of alternating S and R blocks in the sequence of SRS*R*. (* denotes rotating that layer around the c axis by 180°.) The chemical formula of M-type ferrite is Me$^L$Me$^S$$_{12}$O$_{19}$. Common examples are BaFe$_{12}$O$_{19}$, SrFe$_{12}$O$_{19}$. It exhibits $P6_{3}/mmc$ space group symmetry. For BaFe$_{12}$O$_{19}$, a = 5.89 Å and c = 23.18 Å. M-ferrite is a very robust ferrimagnetic material, thus widely used as fridge magnets, card strips, magnets in speakers, magnetic material in linear tape-open.

### W-type ferrite

W-type ferrite, like the M-type, consists of S and R blocks, but the stacking order and the number of blocks are different. The stacking sequence in a W-ferrite is SSRS*S*R* and its chemical formula is Me$^L$Me$^S$$_{18}$O$_{27}$. It exhibits $P6_{3}/mmc$ space group symmetry. One example of W-type ferrite is BaFe$_{18}$O$_{27}$, with a = 5.88 Å and c = 32.85 Å.

### R-type ferrite

R-type ferrite has a chemical formula of Me$^L$Me$^S$$_6$O$_{11}$ with a space group of $P6_{3}/mmc$. Unlike other hexaferrites, R-type ferrite doesn't have an S block. Instead, it only has single B layers extracted from the S block. The stacking sequence is BRB*R*.

### Y-type ferrite

Y-type ferrite has a chemical formula of Me$^L$$_2$Me$^S$$_{14}$O$_{22}$ with a space group of $R{\bar {3}}m$. One example is Ba$_2$Co$_2$Fe$_{12}$O$_{22}$ with a = 5.86 Å and c = 43.5 Å. Y-type ferrite is built up with S and T blocks with an order of 3(ST) in one unit cell. There is no horizontal mirror plane in a Y-type ferrite.

### Z-type ferrite

Z-type ferrite has a chemical formula of Me$^L$$_3$Me$^S$$_{26}$O$_{41}$ with a space group of $P6_{3}/mmc$. It has a complicated stacking of SRSTS*R*S*T* in one unit cell. Some Z-type members may have sophisticated magnetic properties along different directions. One example is Ba$_3$Co$_2$Fe$_{24}$O$_{41}$ with a = 5.88 Å and c = 52.3 Å.

### X-type ferrite

X-type ferrite has a chemical formula of Me$^L$$_2$Me$^S$$_{30}$O$_{46}$ with a space group of $R{\bar {3}}m$. The stacking order is 3(SRS*S*R*) in one unit cell. One example is Sr$_2$Co$_2$Fe$_{28}$O$_{46}$ with c = 83.74 Å.