# 10.5: Point Groups and Crystal Systems

- Page ID
- 18479

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For the reasons discussed above, symmetry operators can combine in a surprisingly small number of ways. Only 32 combinations are possible; they represent the only combination of symmetry elements that crystals, or arrangements of atoms, can have. This leads to the division of crystals into 32 distinct point groups, also sometimes called the 32 *crystal classes*, each having their own distinct symmetry. They are listed in the table here. We can make drawings of crystal shapes with all 32 possible symmetries, but some of them are not represented by any known minerals. Box 10.4 below discusses in more detail the symbols we use to describe the symmetry of different point groups.

The column on the right in the table lists the names of the general forms for each point group. Sometimes crystallographers use these form names as names for the crystal classes. Although the expression *point group* refers only to symmetry, and *crystal class* refers specifically to the symmetry of a crystal, the semantic difference is subtle and the two phrases are often used interchangeably. Some of the general forms, indicated by blue text, are open forms that must combine with other forms to make a crystal, like the forms we saw in Figure 10.36.

*Although there are 32 possible crystal symmetries, not all are represented by minerals. Chapter 14 in this books describes 180 of the most common minerals. They fall into 24 of the possible 32 crystal classes. About a third of them are monoclinic.*

Each of the 32 crystal classes belongs to one of the seven *crystal systems* (*cubic*, *hexagonal*, *rhombohedral*, *orthorhombic*, *tetragonal*, *monoclinic*, and *triclinic*, listed in the table) based on common symmetry elements. These common symmetries reflect the shapes of the crystal’s unit cells (discussed at the beginning of this chapter and greatly expanded upon in the next chapter).

Cubic unit cells have the most symmetry possible. At the other extreme, triclinic unit cells have shapes equivalent to a squashed box with no edges of equal length and no 90^{o} corners. The most symmetry a triclinic crystal can have is an inversion center (1). In some references, the hexagonal and rhombohedral systems are considered divisions within a larger system instead of being separate. We are not doing that in this book because doing so adds complication and is not useful. We will discuss the systems and their unit cells in more detail in the next chapter.

The 32 Point Groups |
||

system |
point group |
general form |

cubic | ^{4}/_{m}3^{2}/_{m}432 43 m^{2}/_{m}323 |
hexoctahedron gyroid hextetrahedron diploid tetartoid |

hexagonal | ^{6}/_{m}^{2}/_{m}^{2}/_{m}622 6 mm6m2^{6}/_{m}6 6 |
dihexagonal dipyramid hexagonal trapezohedron dihexagonal pyramid ditrigonal dipyramid hexagonal dipyramid hexagonal pyramid trigonal dipyramid |

rhombohedral | 32m32 3 m3 3 |
hexagonal scalenohedron trigonal trapezohedron ditrigonal pyramid rhombohedron trigonal pyramid |

tetragonal | ^{4}/_{m}^{2}/_{m}^{2}/_{m}422 4 mm42 m^{4}/_{m}4 4 |
ditetragonal dipyramid tetragonal trapezohedron ditetragonal pyramid tetragonal scalenohedron tetragonal dipyramid tetragonal pyramid tetragonal disphenoid |

orthorhombic | ^{2}/_{m}^{2}/_{m}^{2}/_{m}222 mm2 |
rhombic dipyramid rhombic disphenoid rhombic pyramid |

monoclinic | ^{2}/_{m}2 m |
monoclinic prism sphenoid dome |

triclinic | 1 1 |
pinacoid pedion |

## Hermann-Mauguin Symbols to Designate Point Groups

The symbols used in this book are based on notations developed by C. H. Hermann and C. V. Mauguin in the early 1900s. They have been used by most crystallographers since about 1930. One, two, or three symbols describe a point group; they combine in different ways for different systems.

crystal system |
possiblefirst symbol |
possiblesecond symbol |
possiblethird symbol |
example of a point group |

cubic hexagonal tetragonal orthorhombic monoclinic triclinic |
4, ^{4}/, 4, 2, _{m}^{2}/_{m}6, ^{6}/, 6, 3, 3_{m}4, ^{4}/, 4_{m}222, mm2,^{2}/_{m}^{2}/_{m}^{2}/_{m}2, ^{2}/, _{m}m1, 1 |
3, 3 2, ^{2}/, _{m}m2, ^{2}/, _{m}m2, ^{2}/, _{m}mblank blank |
2, ^{2}/_{m}, m2, ^{2}/_{m}, m2, ^{2}/_{m}, m2, ^{2}/_{m
blank
blank} |
43m^{6}/_{m}^{2}/_{m}^{2}/_{m}422 mm2^{2}/_{m}1 |

Numbers in the symbols refer to rotation axes of symmetry; a bar over a number indicates a rotoinversion axis. Mirrors, designated by *m*, are perpendicular to an axis if they appear as a denominator (for example, ^{4}/* _{m}*), and parallel to an axis otherwise. When articulating the symbols, they are pronounced just as if they were typographical characters.

^{4}/

_{m}*3*2/

*m,*for example, is read “four over em, bar three, two over em.”

For cubic point groups, the first symbol describes three mutually perpendicular principal symmetry axes, oriented perpendicular to cube faces (if cube faces are present). The second describes four axes oriented at angles of 54°44′ to the principal axes. They correspond to the *body diagonals* of a cube, a diagonal from a corner through the center to the opposite corner. The third symbol, if present, describes six 2-fold axes or mirror planes oriented at angles of 45° to the principal axes. They correspond to *edge diagonals* of a cube, diagonals from the center of edges through the center of the cube to the opposite edge.

For hexagonal point groups, the first symbol describes the single principal axis. The second, if present, describes three secondary rotation axes oriented at 120° to each other and perpendicular to the principal axis, or three mirror planes oriented at 120° to each other and parallel to the principal axis. The third symbol, if present, represents mirror planes or 2-fold axes oriented between the secondary axes.

For tetragonal point groups, the first symbol represents the principal axis. The second, if present, represents two secondary axes perpendicular to each other and to the principal axis, or two mirror planes oriented at 90° to each other and parallel to the principal axis. The third represents axes or mirror planes between the secondary axes.

Only three orthorhombic point groups are possible. Point group 222 has three mutually perpendicular 2-fold axes. Point group *mm*2 has one 2-fold axis with two mutually perpendicular mirror planes parallel to it. Point group ^{2}/_{m}^{2}/_{m}^{2}/_{m }has three perpendicular 2-fold axes with mirror planes perpendicular to each.

For monoclinic point groups, only one symmetry element is included in the Hermann-Mauguin symbols because the only possible symmetries are a 2-fold axis, a mirror, or a 2-fold axis with a mirror perpendicular to it. Similarly, for triclinic crystals, the only possible point groups are 1 and 1.

Figure 10.46 above shows representative drawings for common minerals that belong to each crystal system, and Box 10-5 contains a flow chart that can be used to determine the crystal system and point group of many well-formed crystals. But, sometimes determining the point group and system of a crystal, especially for imperfect crystals, is quite difficult or impossible. Overall, it is much easier to see symmetry in crystals with high symmetry (*e.g.*, cubic or hexagonal crystals) than in crystals with lesser symmetry. Distinguishing monoclinic from triclinic crystals, for example, can be very difficult. Yet, crystals that belong to a given system share characteristics, so we can sometimes identify the crystal system quite quickly (especially for crystals with lots of symmetry). For example, as seen in Figure 10.46:

• cubic crystals tend to be equant (equidimensional)

• multiple perpendicular faces are common in cubic, tetragonal and orthorhombic crystals

• hexagonal and tetragonal crystals may contain prism faces parallel to a long dimension

• hexagonal crystals may contain prism faces that intersect at 120^{o}

• rhombohedral crystals typically show 3-fold symmetry

• monoclinic crystals are commonly tabular (thin in one dimension)

• orthorhombic crystals often have an overall shoe-box shape

For further discussion of the similarities and differences of crystals belonging to different crystal systems, check out the video linked below:

Video 10-4: A closer look at different crystal systems (6 minutes)

Minerals belong to all seven crystal systems. About 10% of the common minerals are cubic, 10% are tetragonal, 10% are triclinic, 20% are hexagonal or rhombohedral; the remainder are 25% orthorhombic and 25% monoclinic. Within each system, different point groups have different amounts of symmetry. Most natural crystals fall into the point group with the highest symmetry in each system. Few belong to the point groups of lowest symmetry.