In the International Tables for Crystallography A, table, the trigonal lattices are identified as having space group IT numbers 143 through 167. However, only space groups 148, 155, 160, 161, 166, and 167 have $R$ in their Hermann-Mauguin symbols, the others have $P$ in their HM symbols. All symbols have $3$ or $\overline{3}$ in their symbols, as is required for trigonal symmetry. Section claims that the current 'P' HM symbols were a replacement for an early symbol 'C'-which indicates a "double rectangular C-centered cell", and that the replacement was done "for reasons of consistency", but this doesn't actually illuminate the situation very much.

If I were to attempt a definition of $R$ in the HM symbol, I'd say all groups with a $3$-fold or $\overline{3}$-fold rotational symmetry about a body diagonal, without a $4$-fold axis about a primitive vector. But this appears to contradict the usage in the International tables.

So what does the $R$ indicate?

  • $\begingroup$ $\uparrow$ Direct link to table $\endgroup$ – Qmechanic Sep 1 '17 at 12:26
  • $\begingroup$ That is correct! $\endgroup$ – user14717 Sep 1 '17 at 16:00

All those spacegroups labelled with $R$ have two settings, one with a hexagonal unit cell and one with a rhombohedral unit cell. The complete notation would be e.g. $R3 :\!\!R$ for the latter and $R3 :\!\!H$ for the former. The standard setting is the hexagonal one.

Let's take the example of $R32$. In rhombohedral axes (Hall symbol $P3^*2$), the space group is generated by

  • the threefold rotation about $\renewcommand{\vec}[1]{\mathbf{#1}}\vec{a}+\vec{b}+\vec{c}$, which circularly permutes $\vec{a}$, $\vec{b}$, and $\vec{c}$;
  • a twofold rotation about $\vec{a}-\vec{b}$, which sends $\vec{a}$ to $-\vec{b}$, and $\vec{b}$ to $-\vec{a}$.

In hexagonal axes (Hall symbol $R32''$), it is generated by

  • the threefold rotation about $\vec{c}$, which "generates the hexagon";
  • the twofold rotation about $\vec{a}+\vec{b}$, which permutes $\vec{a}$ and $\vec{b}$;
  • the centring translations $\frac{1}{3}(1,2,2)$ and $\frac{1}{3}(2,1,1)$.

It is not too difficult to see how those transformations are geometrically related.

Now let's take the example of $P3_221$ (Hall $P3_22''$). This is very similar to the above $R32''$, as conveyed by the similarity of the Hall symbols, but with the threefold rotation replaced by a threefold screw about the same axis $\vec{c}$ and an intrinsic translation $\frac{2}{3}\vec{c}$. But contrary to the previous example, there is no way to use rhombohedral axes.

  • $\begingroup$ I'm not sure that this answers the question. The settings merely state which coordinate system the atomic position parameters are measured against. This does not help us distinguish the symmetry operations present in a trigonal lattice which are not present in a rhombohedral lattice. $\endgroup$ – user14717 Aug 31 '17 at 17:17
  • $\begingroup$ I tried to elaborate… $\endgroup$ – user154997 Aug 31 '17 at 20:52

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