Cubic crystal system

In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.

There are three main varieties of these crystals:

  • Primitive cubic (abbreviated cP[1] and alternatively called simple cubic)
  • Body-centered cubic (abbreviated cI[1] or bcc),
  • Face-centered cubic (abbreviated cF[1] or fcc, and alternatively called cubic close-packed or ccp)

Each is subdivided into other variants listed below. Note that although the unit cell in these crystals is conventionally taken to be a cube, the primitive unit cell often is not.

Pyrite Cubes
A rock containing three crystals of pyrite (FeS2). The crystal structure of pyrite is primitive cubic, and this is reflected in the cubic symmetry of its natural crystal facets.
Kubisches Kristallsystem
A network model of a primitive cubic system
FCC primative-cubic cells
The primitive and cubic close-packed (also known as face-centered cubic) unit cells

Bravais lattices

The three Bravais lattices in the cubic crystal system are:

Bravais lattice Primitive
Pearson symbol cP cI cF
Unit cell Cubic Cubic-body-centered Cubic-face-centered

The primitive cubic system (cP) consists of one lattice point on each corner of the cube. Each atom at a lattice point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom (​18 × 8).
The body-centered cubic system (cI) has one lattice point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell (​18 × 8 + 1).
The face-centered cubic system (cF) has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell (​18 × 8 from the corners plus ​12 × 6 from the faces). Each sphere in a cF lattice has coordination number 12. Coordination number is the number of nearest neighbours of a central atom in the structure.

The face-centered cubic system is closely related to the hexagonal close packed (hcp) system, where two systems differ only in the relative placements of their hexagonal layers. The [111] plane of a face-centered cubic system is a hexagonal grid.

Attempting to create a C-centered cubic crystal system (i.e., putting an extra lattice point in the center of each horizontal face) would result in a simple tetragonal Bravais lattice.

Crystal classes

The isometric crystal system class names, point groups (in Schönflies notation, Hermann–Mauguin notation, orbifold, and Coxeter notation), type, examples, International Tables for Crystallography space group number,[2] and space groups are listed in the table below. There are a total 36 cubic space groups.

# Point group Type Example Space groups
Name[3] Schön. Intl Orb. Cox. Primitive Face-centered Body-centered
195–197 Tetartoidal T 23 332 [3,3]+ enantiomorphic Ullmannite P23 F23 I23
198–199 P213 I213
200–204 Diploidal Th 2/m3
3*2 [3+,4] centrosymmetric Pyrite Pm3, Pn3 Fm3, Fd3 I3
205–206 Pa3 Ia3
207–211 Gyroidal O 432 432 [3,4]+ enantiomorphic Petzite P432, P4232 F432, F4132 I432
212–214 P4332, P4132 I4132
215–217 Hextetrahedral Td 43m *332 [3,3] Sphalerite P43m F43m I43m
218–220 P43n F43c I43d
221–230 Hexoctahedral Oh 4/m32/m
*432 [3,4] centrosymmetric Galena Pm3m, Pn3n, Pm3n, Pn3m Fm3m, Fm3c, Fd3m, Fd3c Im3m, Ia3d

Other terms for hexoctahedral are: normal class, holohedral, ditesseral central class, galena type.

Voids in the unit cell

Visualisation diamond cubic
Visualisation of a diamond cubic unit cell: 1. Components of a unit cell, 2. One unit cell, 3. A lattice of 3 x 3 x 3 unit cells

A simple cubic unit cell has a single cubic void in the center.

A body-centered cubic unit cell has six octahedral voids located at the center of each face of the unit cell, and twelve further ones located at the midpoint of each edge of the same cell, for a total of six net octahedral voids. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void, for a total of twelve net tetrahedral voids. These tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells.

A face-centered cubic unit cell has eight tetrahedral voids located midway between each corner and the center of the unit cell, for a total of eight net tetrahedral voids. Additionally, there are twelve octahedral voids located at the midpoints of the edges of the unit cell as well as one octahedral hole in the very center of the cell, for a total of four net octahedral voids.

One important characteristic of a crystalline structure is its atomic packing factor. This is calculated by assuming that all the atoms are identical spheres, with a radius large enough that each sphere abuts on the next. The atomic packing factor is the proportion of space filled by these spheres.

Assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be ​a2 and the atomic packing factor turns out to be about 0.524 (which is quite low). Similarly, in a bcc lattice, the atomic packing factor is 0.680, and in fcc it is 0.740. The fcc value is the highest theoretically possible value for any lattice, although there are other lattices which also achieve the same value, such as hexagonal close packed (hcp) and one version of tetrahedral bcc.

As a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common. (Loosely packed arrangements do occur, though, for example if the orbital hybridization demands certain bond angles.) Accordingly, the primitive cubic structure, with especially low atomic packing factor, is rare in nature, but is found in polonium.[4][5] The bcc and fcc, with their higher densities, are both quite common in nature. Examples of bcc include iron, chromium, tungsten, and niobium. Examples of fcc include aluminium, copper, gold and silver.

Multi-element compounds

Compounds that consist of more than one element (e.g. binary compounds) often have crystal structures based on a cubic crystal system. Some of the more common ones are listed here.

Caesium chloride structure

CsCl crystal
A caesium chloride unit cell. The two colors of spheres represent the two types of atoms.

The space group of the caesium chloride (CsCl) structure is called Pm3m (in Hermann–Mauguin notation), or "221" (in the International Tables for Crystallography). The Strukturbericht designation is "B2".[6]

One structure is the "interpenetrating primitive cubic" structure, also called the "caesium chloride" structure. Each of the two atom types forms a separate primitive cubic lattice, with an atom of one type at the center of each cube of the other type. Altogether, the arrangement of atoms is the same as body-centered cubic, but with alternating types of atoms at the different lattice sites. (See picture here.) Alternately, one could view this lattice as a simple cubic structure with a secondary atom in its cubic void.

In addition to caesium chloride itself, the structure also appears in certain other alkali halides when prepared at low temperatures or high pressures.[7] Generally, this structure is more likely to be formed from two elements whose ions are of roughly the same size (for example, ionic radius of Cs+ = 167 pm, and Cl = 181 pm).

The coordination number of each atom in the structure is 8: the central cation is coordinated to 8 anions on the corners of a cube as shown, and similarly, the central anion is coordinated to 8 cations on the corners of a cube.

Other compounds showing caesium chloride like structure are CsBr, CsI, high-temp RbCl, AlCo, AgZn, BeCu, MgCe, RuAl and SrTl.

Rock-salt structure

NaCl polyhedra
The rock-salt crystal structure. Each atom has six nearest neighbors, with octahedral geometry.

The space group of the rock-salt (NaCl) structure is called Fm3m (in Hermann–Mauguin notation), or "225" (in the International Tables for Crystallography). The Strukturbericht designation is "B1".[8]

In the rock-salt or sodium chloride (halite) structure, each of the two atom types forms a separate face-centered cubic lattice, with the two lattices interpenetrating so as to form a 3D checkerboard pattern. Alternately, one could view this structure as a face-centered cubic structure with secondary atoms in its octahedral holes.

Examples of compounds with this structure include sodium chloride itself, along with almost all other alkali halides, and "many divalent metal oxides, sulfides, selenides, and tellurides".[7] More generally, this structure is more likely to be formed if the cation is somewhat smaller than the anion (a cation/anion radius ratio of 0.414 to 0.732).

The coordination number of each atom in this structure is 6: each cation is coordinated to 6 anions at the vertices of an octahedron, and similarly, each anion is coordinated to 6 cations at the vertices of an octahedron.

The interatomic distance (distance between cation and anion, or half the unit cell length a) in some rock-salt-structure crystals are: 2.3 Å (2.3 × 10−10 m) for NaF,[9] 2.8 Å for NaCl,[10] and 3.2 Å for SnTe.[11]

Other compounds showing rock salt like structure are LiF[12], LiCl, LiBr, LiI, NaF[12], NaBr, NaI, KF[12], KCl, KBr, KI, RbF, RbCl, RbBr, RbI, CsF, MgO, PbS, AgF, AgCl, AgBr and ScN.

Zincblende structure

A zincblende unit cell

The space group of the Zincblende structure is called F43m (in Hermann–Mauguin notation), or 216.[13][14] The Strukturbericht designation is "B3".[15]

The Zincblende structure (also written "zinc blende") is named after the mineral zincblende (sphalerite), one form of zinc sulfide (β-ZnS). As in the rock-salt structure, the two atom types form two interpenetrating face-centered cubic lattices. However, it differs from rock-salt structure in how the two lattices are positioned relative to one another. The zincblende structure has tetrahedral coordination: Each atom's nearest neighbors consist of four atoms of the opposite type, positioned like the four vertices of a regular tetrahedron. Altogether, the arrangement of atoms in zincblende structure is the same as diamond cubic structure, but with alternating types of atoms at the different lattice sites.

Examples of compounds with this structure include zincblende itself, lead(II) nitrate, many compound semiconductors (such as gallium arsenide and cadmium telluride), and a wide array of other binary compounds.

Other compounds showing zinc blende-like structure are α-AgI, β-BN, diamond, CuBr, β-CdS, BP and BAs.

Weaire–Phelan structure

12-14-hedral honeycomb
Weaire–Phelan structure

The Weaire–Phelan structure has Pm3n (223) symmetry.

It has 3 orientations of stacked tetradecahedrons with pyritohedral cells in the gaps. It is found as a crystal structure in chemistry where it is usually known as the "Type I clathrate structure". Gas hydrates formed by methane, propane, and carbon dioxide at low temperatures have a structure in which water molecules lie at the nodes of the Weaire–Phelan structure and are hydrogen bonded together, and the larger gas molecules are trapped in the polyhedral cages.

See also


  1. ^ a b c P. M. de Wolff, N. V. Belov, E. F. Bertaut, M. J. Buerger, J. D. H. Donnay, W. Fischer, Th. Hahn, V. A. Koptsik, A. L. Mackay, H. Wondratschek, A. J. C. Wilson and S. C. Abrahams (1985). "Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Symmetry". 41. Acta Crystallographica Section A: 278. doi:10.1107/S0108767385000587.CS1 maint: Multiple names: authors list (link)
  2. ^ Prince, E., ed. (2006). International Tables for Crystallography. International Union of Crystallography. doi:10.1107/97809553602060000001. ISBN 978-1-4020-4969-9.
  3. ^ Crystallography and Minerals Arranged by Crystal Form, Webmineral
  4. ^ Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. ISBN 978-0-08-037941-8.
  5. ^ The original discovery was in J. Chem. Phys. 14, 569 (1946).
  6. ^ The CsCl (B2) Structure Archived 2008-09-15 at the Wayback Machine
  7. ^ a b Seitz, Modern Theory of Solids (1940), p.49
  8. ^ The NaCl (B1) Structure Archived 2008-10-19 at the Wayback Machine
  9. ^ Sundquist, J. J.; Lin, C. C. (1981). "Electronic structure of the F centre in a sodium fluoride crystal". Journal of Physics C: Solid State Physics. 14 (32): 4797–4805. Bibcode:1981JPhC...14.4797S. doi:10.1088/0022-3719/14/32/016.
  10. ^ Abrahams, S. C.; Bernstein, J. L. (1965). "Accuracy of an automatic diffractometer. Measurement of the sodium chloride structure factors". Acta Crystallogr. 18 (5): 926–932. doi:10.1107/S0365110X65002244.
  11. ^ Kao, W.; Peretti, E. (1970). "The ternary subsystem Sn4As3-SnAs-SnTe". Journal of the Less Common Metals. 22: 39–50. doi:10.1016/0022-5088(70)90174-8.
  12. ^ a b c J. Aigueperse, P. Mollard, D. Devilliers, M. Chemla, R. Faron, R. Romano, J. P. Cuer, "Fluorine Compounds, Inorganic" (section 4) in Ullmann’s Encyclopedia of Industrial Chemistry, Wiley-VCH, Weinheim, 2005. doi:10.1002/14356007.a11_307.
  13. ^ L. Kantorovich (2004). Quantum Theory of the Solid State. Springer. p. 32. ISBN 1-4020-2153-4.
  14. ^ Birkbeck College, University of London
  15. ^ The Zincblende (B3) Structure Archived October 19, 2008, at the Wayback Machine

Further reading

  • Hurlbut, Cornelius S.; Klein, Cornelis, 1985, Manual of Mineralogy, 20th ed., Wiley, ISBN 0-471-80580-7

External links


Alabandite or alabandine is a rarely occurring manganese sulfide mineral. It crystallizes in the cubic crystal system with the chemical composition Mn2+S and develops commonly massive to granular aggregates, but rarely also cubic or octahedral crystals to 1 cm.

Architectonic and catoptric tessellation

In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations (or honeycombs) of Euclidean 3-space and their duals, as three-dimensional analogue of the Platonic, Archimedean, and Catalan tiling of the plane. The singular vertex figure of an architectonic tessellation is the dual of the cell of catoptric tessellation. The cubille is the only Platonic (regular) tessellation of 3-space, and is self-dual. There are other uniform honeycombs constructed as prismatic stacks (and their duals) which are excluded from these categories.

The pairs of architectonic and catoptric tessellations are listed below with their symmetry group. These tessellations only represent four symmetry space groups, and also all within the cubic crystal system. Many of these tessellations can be defined in multiple symmetry groups, so in each case the highest symmetry is expressed.


Auricupride is a natural alloy that combines copper and gold. Its chemical formula is Cu3Au. The alloy crystallizes in the Cubic crystal system and occurs as malleable grains or platey masses. It is an opaque yellow with a reddish tint. It has a hardness of 3.5 and a specific gravity of 11.5.A variant called tetra-auricupride (CuAu) exists. Silver may be present resulting in the variety argentocuproauride (Cu3(Au,Ag)).It was first described in 1950 for an occurrence in the Ural Mountains Russia. It occurs as low temperature unmixing product in serpentinites and as reduction "halos" in redbed deposits. It is most often found in Chile, Argentina, Tasmania, Russia, Cyprus, Switzerland and South Africa.


Bunsenite is the naturally occurring form of nickel(II) oxide, NiO. It occurs as rare dark green crystal coatings. It crystallizes in the cubic crystal system and occurs as well formed cubic, octahedral and dodecahedral crystals. It is a member of the periclase group.

It was first described in 1868 for a sample from a hydrothermal nickel-uranium vein from Johanngeorgenstadt, Erzgebirge, Saxony, Germany and named for German chemist Robert William Eberhard Bunsen (1811–1899). Other occurrences include west of the Scotia talc mine near Bon Accord, Barberton district, Transvaal, South Africa and from Kambalda south of Kalgoorlie, Western Australia. The South African occurrence has evidence of thermal metamorphism of a nickel-rich meteorite. It occurs associated with native bismuth, annabergite, aerugite, xanthiosite in Germany; and with liebenbergite, trevorite, nickeloan serpentine, nickeloan ludwigite, violarite, millerite, gaspeite, nimite and bonaccordite in the South African occurrence.


Fletcherite is a rare thiospinel sulfide mineral with formula Cu(Ni,Co)2S4. It is an opaque metallic steel gray mineral which crystallizes in the cubic crystal system. It is a member of the linnaeite group.

It was first described in 1977 for an occurrence in the Fletcher Mine, Viburnum Trend (New Lead Belt), near Centerville, Reynolds County, Missouri.It occurs as a dissemination within copper sulfide minerals in mineralization replacing dolostone at the type locality in the Fletcher

mine where it is associated with vaesite, pyrite, covellite, chalcopyrite, bornite and digenite. In an occurrence in Kalgoorlie, Australia it is found in black slate associated with pyrrhotite.


Fukuchilite, Cu3FeS8, is a copper iron sulfide named after the Japanese mineralogist Nobuyo Fukuchi (1877–1934), that occurs in ore bodies of gypsum-anhydrite at the intersection points of small masses of barite, covellite, gypsum and pyrite, and is mostly found in the Hanawa mine in the Akita prefecture of Honshū, Japan where it was first discovered in 1969. It occurs in masses within the third geologic unit of the Kuroko type deposits within the mine.

As a copper, iron sulfide, it is placed in the same group as bornite and chalcopyrite, and most fukuchilite locations are found in relatively close proximity to these minerals. Fukuchilite was found to have a reflection color very similar to bornite and bright pinkish brown in air, while being a purplish brown in oil. Also, it was found to have a reactivity lower than pyrite, but distinctly higher than bornite. It has a Mohs hardness of 4-6, a specific gravity of 4.9, and a sub metallic luster, composed of 11.1% iron, 37.9% copper, and 51.00% of sulfur.It is in the isotropic cubic crystal system with symmetry: (2/m3), space group P a3. Much relating to the structure of the mineral is still under debate, and some believe that fukuchilite might actually be a form of villamaninite (Cu,Ni,Co,Fe)S2, but fukuchilite currently still holds its mineral status as there is currently not enough evidence to discredit an already accepted and titled mineral.


Galena, also called lead glance, is the natural mineral form of lead(II) sulfide. It is the most important ore of lead and an important source of silver.Galena is one of the most abundant and widely distributed sulfide minerals. It crystallizes in the cubic crystal system often showing octahedral forms. It is often associated with the minerals sphalerite, calcite and fluorite.


Haxonite is an iron nickel carbide mineral found in iron meteorites and carbonaceous chondrites. It has a chemical formula of (Fe,Ni)23C6, crystallises in the cubic crystal system and has a Mohs hardness of ​5 1⁄2 - 6.It was first described in 1971, and named after Howard J. Axon (1924–1992), metallurgist at the University of Manchester, Manchester, England. Co-type localities are the Toluca meteorite, Xiquipilco, Mexico and the Canyon Diablo meteorite, Meteor Crater, Coconino County, Arizona, US.It occurs associated with kamacite, taenite, schreibersite, cohenite, pentlandite and magnetite.

Hermann–Mauguin notation

In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann (who introduced it in 1928) and the French mineralogist Charles-Victor Mauguin (who modified it in 1931). This notation is sometimes called international notation, because it was adopted as standard by the International Tables For Crystallography since their first edition in 1935.

The Hermann–Mauguin notation, compared with the Schoenflies notation, is preferred in crystallography because it can easily be used to include translational symmetry elements, and it specifies the directions of the symmetry axes.


The term isometric comes from the Greek for "having equal measurement".

isometric may mean:

Isometric projection (or "isometric perspective"), a method for the visual representation of three-dimensional objects in two dimensions; a form of orthographic projection, or more specifically, an axonometric projection.

Isometric graphics in video games and pixel art, a near-isometric parallel projection used in computer art.

Isometry and isometric embeddings in mathematics, a distance-preserving representation of one metric space as a subset of another. Like congruence in geometry.

Isometric exercise, a form of resistance exercise in which one's muscles are used in opposition with other muscle groups, to increase strength, for bodybuilding, physical fitness, or strength training.

Isometre, a rhythmic technique in music.

Isometric joystick, a type of pointing stick, a computer input option

Isometric process, a thermodynamic process at constant volume (also isovolumetric)

Cubic crystal system, also called isometric crystal system

Isometric platform game, a video game sub-genre.

Isometric scaling, the opposite of allometry, which occurs when changes in size (during growth or over evolutionary time) do not lead to changes in proportion.

"Isometric (Intro)", a song by Madeon from the album, Adventure

Isotropic solid

An isotropic solid is a solid material in which physical properties do not depend on its orientation. It is an example of isotropy which can also apply to fluids, or other non-material concepts. The properties could be felt or seen such as the index of refraction or mechanical. For example an isotropic solid will expand equally in all directions when heated. Heat will conduct equally well in any direction, and sound will pass at the same speed. The physics of these solids are much easier to describe. Some examples are glass with random arrangements that average out to uniform. The opposite is anisotropic solids. All crystal structures, including the cubic crystal system, are actually anisotropic.

Isotropy should not be confused with homogeneity, which is a property of independence of position. Isotropy of materials also depends on scale. For example, at micro level involving a few crystals, steel is not isotropic. But at macro level, for example a steel rod using in the building construction, steel is isotropic.


Kangite is an exceedingly rare scandium mineral, a natural form of impure scandium oxide (Sc2O3), with the formula (Sc,Ti,Al,Zr,Mg,Ca,□)2O3. It crystalizes in the cubic crystal system diploidal class. In terms of chemistry it scandium-analogue of tistarite. Both kangite and tistarite were discovered in the Allende meteorite.


Krut'aite (also spelled Krutaite) is a rare mineral with the formula CuSe2. It crystallises in the cubic crystal system. It is part of the pyrite group, being composed of Cu2+ ions and Se22− ions. The mineral is most often found as a dark grey aggregate consisting of tiny crystals no more than a millimeter in size. The crystals are opaque in any size. It has no industrial use, but it is a prized collector's item.


Oldhamite is a calcium magnesium sulfide mineral with formula (Ca, Mg)S. Ferrous iron may also be present in the mineral resulting in the formula: (Ca,Mg,Fe)S. It is a pale to dark brown accessory mineral in meteorites. It crystallizes in the cubic crystal system, but typically occurs as anhedral grains between other minerals.

Pyrite group

The pyrite group of minerals is a set of cubic crystal system minerals with diploidal structure. Each metallic element is bonded to six "dumbbell" pairs of non-metallic elements and each "dumbbell" pair is bonded to six metal atoms.The group is named for its most common member, pyrite (fool's gold), which is sometimes explicitly distinguished from the group's other members as iron pyrite.

Pyrrhotite (magnetic pyrite) is magnetic, and is composed of iron and sulfur, but it has a different structure and is not in the pyrite group.

Skull crucible

The skull crucible process was developed at the Lebedev Physical Institute in Moscow to manufacture cubic zirconia. It was invented to solve the problem of cubic zirconia's melting-point being too high for even platinum crucibles.

In essence, by heating only the center of a volume of cubic zirconia, the material forms its own "crucible" from its cooler outer layers. The term "skull" refers to these outer layers forming a shell enclosing the molten volume. Zirconium oxide powder is heated then gradually allowed to cool. Heating is accomplished by radio frequency induction using a coil wrapped around the apparatus. The outside of the device is water-cooled in order to keep the RF coil from melting and also to cool the outside of the zirconium oxide and thus maintain the shape of the zirconium powder.

Since zirconium oxide in its solid state does not conduct electricity, a piece of zirconium metal is placed inside the gob of zirconium oxide. As the zirconium melts it oxidizes and blends with the now molten zirconium oxide, a conductor, and is heated by RF induction.

When the zirconium oxide is melted on the inside (but not completely, since the outside needs to remain solid) the amplitude of the RF induction coil is gradually reduced and crystals form as the material cools. Normally this would form a monoclinic crystal system of zirconium oxide.

In order to maintain a cubic crystal system a stabilizer is added, magnesium oxide, calcium oxide or yttrium oxide as well as any material to color the crystal. After the mixture cools the outer shell is broken off and the interior of the gob is then used to manufacture gemstones.


Spinel ( ) is the magnesium aluminium member of the larger spinel group of minerals. It has the formula MgAl2O4 in the cubic crystal system. Its name comes from Latin "spina" (arrow).Though spinels are often referred to as rubies, as in the Black Prince Ruby, the ruby is not a spinel. Balas ruby is an old name for a rose-tinted variety of spinel.

Tin(II) sulfide

Tin(II) sulfide is a chemical compound of tin and sulfur. The chemical formula is SnS. Its natural occurrence concerns herzenbergite (α-SnS), a rare mineral. At elevated temperatures above 905K, SnS undergoes a second order phase transition to β-SnS (space group: cmcm, No. 63). in recent years, it has become evident that a new polymorph of SnS exist based upon the cubic crystal system, π-SnS (space group: P213, No. 198).

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