tetragonal crystal system

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In crystallography, the tetragonal crystal system is one of the 7 crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square base (a by a) and height (c, which is different from a).

Bravais lattices

There are two tetragonal Bravais lattices: the primitive tetragonal and the body-centered tetragonal.

class=wikitable ! Bravais lattice ! Primitive tetragonal ! Body-centered tetragonal
align=center ! Pearson symbol \| tP \| tI
Unit cell \| File:Tetragonal.svg \| File:Body-centered tetragonal.svg

class=wikitable

! Bravais lattice

! Primitive
tetragonal

! Body-centered
tetragonal

align=center

! Pearson symbol

| tP

| tI

Unit cell

| File:Tetragonal.svg

| File:Body-centered tetragonal.svg

The body-centered tetragonal lattice is equivalent to the primitive tetragonal lattice with a smaller unit cell, while the face-centered tetragonal lattice is equivalent to the body-centered tetragonal lattice with a smaller unit cell.[http://www.aem.umn.edu/people/faculty/shield/hane/tet.html Cubic-to-Tetragonal Transition]

Crystal classes

The point groups that fall under this crystal system are listed below, followed by their representations in international notation, Schoenflies notation, orbifold notation, Coxeter notation and mineral examples.[http://webmineral.com/crystal/Tetragonal.shtml Webmineral data]Hurlbut, Cornelius S.; Klein, Cornelis, 1985, Manual of Mineralogy, 20th ed., pp. 73–78, {{isbn|0-471-80580-7}}

class="wikitable" ! rowspan=2 width=60\|# ! colspan=5\| Point group ! rowspan=2\| Type ! rowspan=2\| Example ! colspan=2\| Space groups
Name{{cite web \|title=The 32 crystal classes \|url=https://www.tulane.edu/~sanelson/eens211/32crystalclass.htm \|access-date=2018-06-19}} ! Intl ! Schoen. ! Orb. ! Cox. ! Primitive ! Body-centered
align=center ! 75–80 \| Tetragonal pyramidal \| 4 \| C₄ \| 44 \| [4]⁺ \| enantiomorphic polar \| pinnoite, piypite \| align=left\| P4, P4₁, P4₂, P4₃ \| I4, I4₁
align=center ! 81–82 \| Tetragonal disphenoidal \| {{overline\|4}} \| S₄ \| 2× \| [2⁺,4⁺] \| \| cahnite, tugtupite \| align=left\| P{{overline\|4}} \| I{{overline\|4}}
align=center ! 83–88 \| Tetragonal dipyramidal \| 4/m \| C_4h \| 4* \| [2,4⁺] \| centrosymmetric \| scheelite, wulfenite, leucite \| align=left\| P4/m, P4₂/m, P4/n, P4₂/n \| I4/m, I4₁/a
align=center ! 89–98 \| Tetragonal trapezohedral \| 422 \| D₄ \| 224 \| [2,4]⁺ \| enantiomorphic \| cristobalite, wardite \| align=left\| P422, P42₁2, P4₁22, P4₁2₁2, P4₂22, P4₂2₁2, P4₃22, P4₃2₁2 \| I422, I4₁22
align=center ! 99–110 \| Ditetragonal pyramidal \| 4mm \| C_4v \| *44 \| [4] \| polar \| diaboleite \| align=left\| P4mm, P4bm, P4₂cm, P4₂nm, P4cc, P4nc, P4₂mc, P4₂bc \| I4mm, I4cm, I4₁md, I4₁cd
align=center ! 111–122 \| Tetragonal scalenohedral \| {{overline\|4}}2m \| D_2d (V_d) \| 2*2 \| [2⁺,4] \| \| chalcopyrite, stannite \| align=left\| P{{overline\|4}}2m, P{{overline\|4}}2c, P{{overline\|4}}2₁m, P{{overline\|4}}2₁c, P{{overline\|4}}m2, P{{overline\|4}}c2, P{{overline\|4}}b2, P{{overline\|4}}n2 \| I{{overline\|4}}m2, I{{overline\|4}}c2, I{{overline\|4}}2m, I{{overline\|4}}2d
align=center ! 123–142 \| Ditetragonal dipyramidal \| 4/mmm \| D_4h \| *224 \| [2,4] \| centrosymmetric \| rutile, pyrolusite, zircon \| align=left\| P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4₂/mmc, P4₂/mcm, P4₂/nbc, P4₂/nnm, P4₂/mbc, P4₂/mnm, P4₂/nmc, P4₂/ncm \| I4/mmm, I4/mcm, I4₁/amd, I4₁/acd

class="wikitable"

! rowspan=2 width=60|#

! colspan=5| Point group

! rowspan=2| Type

! rowspan=2| Example

! colspan=2| Space groups

Name{{cite web |title=The 32 crystal classes |url=https://www.tulane.edu/~sanelson/eens211/32crystalclass.htm |access-date=2018-06-19}}

! Intl

! Schoen.

! Orb.

! Cox.

! Primitive

! Body-centered

align=center

! 75–80

| Tetragonal pyramidal

| 4

| C₄

| 44

| [4]⁺

| enantiomorphic polar

| pinnoite,
piypite

| align=left| P4, P4₁, P4₂, P4₃

| I4, I4₁

align=center

! 81–82

| Tetragonal disphenoidal

| {{overline|4}}

| S₄

| 2×

| [2⁺,4⁺]

| cahnite, tugtupite

| align=left| P{{overline|4}}

| I{{overline|4}}

align=center

! 83–88

| Tetragonal dipyramidal

| 4/m

| C_4h

| 4*

| [2,4⁺]

| centrosymmetric

| scheelite, wulfenite, leucite

| align=left| P4/m, P4₂/m, P4/n, P4₂/n

| I4/m, I4₁/a

align=center

! 89–98

| Tetragonal trapezohedral

| 422

| D₄

| 224

| [2,4]⁺

| enantiomorphic

| cristobalite, wardite

| align=left| P422, P42₁2, P4₁22, P4₁2₁2, P4₂22, P4₂2₁2, P4₃22, P4₃2₁2

| I422, I4₁22

align=center

! 99–110

| Ditetragonal pyramidal

| 4mm

| C_4v

| *44

| [4]

| polar

| diaboleite

| align=left| P4mm, P4bm, P4₂cm, P4₂nm, P4cc, P4nc, P4₂mc, P4₂bc

| I4mm, I4cm, I4₁md, I4₁cd

align=center

! 111–122

| Tetragonal scalenohedral

| {{overline|4}}2m

| D_2d (V_d)

| 2*2

| [2⁺,4]

| chalcopyrite, stannite

| align=left| P{{overline|4}}2m, P{{overline|4}}2c, P{{overline|4}}2₁m, P{{overline|4}}2₁c, P{{overline|4}}m2, P{{overline|4}}c2, P{{overline|4}}b2, P{{overline|4}}n2

| I{{overline|4}}m2, I{{overline|4}}c2, I{{overline|4}}2m, I{{overline|4}}2d

align=center

! 123–142

| Ditetragonal dipyramidal

| 4/mmm

| D_4h

| *224

| [2,4]

| centrosymmetric

| rutile, pyrolusite, zircon

| align=left| P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4₂/mmc, P4₂/mcm, P4₂/nbc, P4₂/nnm, P4₂/mbc, P4₂/mnm, P4₂/nmc, P4₂/ncm

| I4/mmm, I4/mcm, I4₁/amd, I4₁/acd