Crystal twinning

Twin lattice (2D). The long horizontal red line is the composition plane where the two crystal segments join. The upper crystal lattice is the reflection of the lower crystal lattice. The red points are the shared crystal lattice points.

Twinning is a form of symmetrical intergrowth between two or more adjacent crystals of the same mineral. It differs from the ordinary random intergrowth of mineral grains in a mineral deposit, because the relative orientations of the two crystal segments show a fixed relationship that is characteristic of the mineral structure. The relationship is defined by a symmetry operation called a twin operation.[2][3]

The twin operation is not one of the normal symmetry operations of the untwinned crystal structure. For example, the twin operation may be reflection across a plane that is not a symmetry plane of the single crystal.[2][3]

On the microscopic level, the twin boundary is characterized by a set of atomic positions in the crystal lattice that are shared between the two orientations. [2][3] These shared lattice points give the junction between the crystal segments much greater strength than that that between randomly oriented grains, so that the twinned crystals do not easily break apart.[4]

Twin laws are symmetry operations that define the orientation between twin crystal segments. These are as characteristic of the mineral as are its crystal face angles. For example, crystals of staurolite show twinning at angles of almost precisely 90 degrees or 30 degrees.[4] A twin law is not a symmetry operation of the full set of basis points.[3]

Twin laws include reflection operations, rotation operations, and the inversion operation. Reflection twinning is described by the Miller indices of the twin plane (i.e. {hkl}) while rotational twinning is described by the direction of the twin axis (i.e. <hkl>). Inversion twinning is typically equivalent to a reflection or rotation symmetry.[2]

Rotational twin laws are almost always 2-fold rotations, though any other permitted rotation symmetry (3-fold, 4-fold, or 6-fold) is possible. The twin axis will be perpendicular to a lattice plane.[5] It is possible for a rotational twin law to share the same axis as a rotational symmetry of the individual crystal if the twin law is a 2-fold rotation and the symmetry operation is a 3-fold rotation. This is the case for spinel law twinning on <111>: The spinel structure has a 3-fold rotational symmetry on <111> and spinel is commonly twinned by 2-fold rotation on <111>.[2]

The boundary between crystal segments is called a composition surface or, if it is planar, a composition plane. The composition plane is often, though not always, parallel to the twin law plane of a reflection law. If this is the case, the twin plane is always parallel to a possible crystal face.[3]

Common twin laws

Spinel law contact twinning. A single crystal is shown at left with the composition plane in red. At right, the crystal has effectively been cut on the composition plane and the front half rotated by 180° to produce a contact twin. This creates reentrants at the top and lower left of the composition plane.[4]

30° twin of staurolite

90° twins of staurolite

Pyrite iron cross twin

In the isometric system, the most common types of twins are the Spinel Law (twin plane, parallel to an octahedron) <111>, where the twin axis is perpendicular to an octahedral face, and the Iron Cross <001>, which is the interpenetration of two pyritohedrons, a subtype of dodecahedron.[6]

In the hexagonal system, calcite shows the contact twin laws {0001} and {0112}. Quartz shows the Brazil Law {1120}, and Dauphiné Law <0001>, which are penetration twins caused by transformation, and Japan Law {1122}, which is often caused by accidents during growth.[6]

In the tetragonal system, cyclical contact twins are the most commonly observed type of twin, such as in rutile titanium dioxide and cassiterite tin oxide.[6]

In the orthorhombic system, crystals usually twin on planes parallel to the prism face, where the most common is a {110} twin, which produces cyclical twins, such as in aragonite, chrysoberyl, and cerussite.[6]

In the monoclinic system, twins occur most often on the planes {100} and {001} by the Manebach Law {001}, Carlsbad Law [001], Braveno Law {021} in orthoclase, and the Swallow Tail Twins (Manebach law) {001} in gypsum.[6]

In the triclinic system, the most commonly twinned crystals are the feldspar minerals plagioclase and microcline. These minerals show the Albite and Pericline Laws.[5][6]

The most common twin operations by crystal system are tabulated below. This list is not exhaustive, particularly for the crystal systems of lowest symmetry, such as the triclinic system.[7][2][6]

System	Law	Operation	Examples
Triclinic	Albite law Pericline law Carlsbad law Baveno law Manebach law	{010} <010> <001> {021} {001}	Plagioclase
Monoclinic	Carlsbad law Baveno law Manebach law	<001> {021} {001} {100} <031> <231>	Orthoclase Gypsum Staurolite
Orthorhombic		{110} {101} {011}	Aragonite, cerrusite; often cyclic
Tetragonal		{110} {101} {011}	Cassiterite, rutile
Hexagonal	Brazil law Dauphine law Japan law	{0112} {0001} {1011} {1120} <0001> {1122}	Calcite Quartz
Isometric	Spinel law Iron cross law	<111> {111} {001} <001>	Spinel Pyrite

Chrysoberyl showing cyclic twinning

Simple twinned crystals may be contact twins or penetration twins. Contact twins meet on a single composition plane, often appearing as mirror images across the boundary. Plagioclase, quartz, gypsum, and spinel often exhibit contact twinning.[4] Merohedral twinning occurs when the lattices of the contact twins superimpose in three dimensions, such as by relative rotation of one twin from the other.[8] An example is metazeunerite.[9] Contact twinning characteristically creates reentrant faces where faces of the crystal segments meet on the contact plane at an angle greater than 180°.[4]

In penetration twins the individual crystals have the appearance of passing through each other in a symmetrical manner.[4] Orthoclase, staurolite, pyrite, and fluorite often show penetration twinning. The composition surface in penetration twins is usually irregular and extends to the center of the crystal.[3]

Contact twinning can arise from either reflection or rotation, whereas penetration twinning is usually produced by rotation.[3]

If several twin crystal parts are aligned by the same twin law they are referred to as multiple or repeated twins. If these multiple twins are aligned in parallel they are called polysynthetic twins. When the multiple twins are not parallel they are cyclic twins. Albite, calcite, and pyrite often show polysynthetic twinning. Closely spaced polysynthetic twinning is often observed as striations or fine parallel lines on the crystal face. Rutile, aragonite, cerussite, and chrysoberyl often exhibit cyclic twinning, typically in a radiating pattern.[4][3]

For rotational twinning the relationship between the twin axis and twin plane falls into one of three types:[10]

1. parallel twinning, when the twin axis and compositional plane lie parallel to each other,
2. normal twinning, when the twin plane and compositional plane lie normally, and
3. complex twinning, a combination of parallel twinning and normal twinning on one compositional plane.

Parallel growth of pyromorphite

There are three modes of formation of twinned crystals. Growth twins are the result of an interruption or change in the lattice during formation or growth due to a possible deformation from a larger substituting ion. Annealing or transformation twins are the result of a change in crystal system during cooling as one form becomes unstable and the crystal structure must re-organize or transform into another more stable form. Deformation or gliding twins are the result of stress on the crystal after the crystal has formed. Because growth twins are formed during the initial growth of the crystal, they are described as primary, whereas transformation or deformation twins are formed in an existing crystal and are described as secondary.[11]

Parallel growth describes a form of crystal growth that produces the appearance of a cluster of aligned crystals. Close examination reveals that the cluster is actually a single crystal. This is not twinning, since the crystal lattice is continuous throughout the cluster. Parallel growth likely takes place because it reduces system energy.[12]

Transformation twinning

Illustration of transformation twinning in 2-D. The orthorhombic crystal at left transforms to a monoclinic crystal at right, with two parallel twinning planes (polysynthetic twinning)

Transformation twinning takes place when a cooling crystal experiences a displacive polymorphic transition. For example, leucite has an isometric crystal structure above about 665 °C (1,229 °F), but becomes tetragonal below this temperature. Any one of the three original axes of a crystal can become the long axis when this phase change takes place. Twinning results when different parts of the crystal break their isometric symmetry along a different choice of axis. This is typically polysynthetic twinning, which enables the crystal to maintain its isometric shape by averaging out the displacement in each direction. This produces a pseudomorphic crystal that appears to have isometric symmetry. Potassium feldspar likewise experiences polysynthetic twinning as it transforms from a monoclinic structure (orthoclase) to a to triclinic structure (microcline) on slow cooling.[2]

Deformation twinning

Animation of deformation twinning of crystal

Deformation twinning is a response to shear stress. The crystal structure is displaced along successive planes of the crystal, a process also called glide. The twinning is always reflection twinning and the glide plane is also the mirror plane. Deformation twinning can be observed in a calcite cleavage fragment by applying gentle pressure with a knife blade near an edge. This particular glide twinning, {102}, is found almost universally in deformed rock beds containing calcite.[2]

Twinning and slip are competitive mechanisms for crystal deformation. Each mechanism is dominant in certain crystal systems and under certain conditions.[13] In fcc metals, slip is almost always dominant because the stress required is far less than twinning stress.[14]

Twinning can occur by cooperative displacement of atoms along the face of the twin boundary. This displacement of a large quantity of atoms simultaneously requires significant energy to perform. Therefore, the theoretical stress required to form a twin is quite high. It is believed that twinning is associated with dislocation motion on a coordinated scale, in contrast to slip, which is caused by independent glide at several locations in the crystal.

Compared to slip, twinning produces a deformation pattern that is more heterogeneous in nature. This deformation produces a local gradient across the material and near intersections between twins and grain boundaries. The deformation gradient can lead to fracture along the boundaries, particularly in bcc transition metals at low temperatures.

Fivefold twinning in a gold nanoparticle (electron micrograph).

Of the three common crystalline structures bcc, fcc, and hcp, the hcp structure is the most likely to form deformation twins when strained, because they rarely have a sufficient number of slip systems for an arbitrary shape change. High strain rates, low stacking-fault energy and low temperatures facilitate deformation twinning.[1]

If a metal with face-centered cubic (fcc) structure, like Al, Cu, Ag, Au, etc., is subjected to stress, it will experience twinning. The formation and migration of twin boundaries is partly responsible for ductility and malleability of fcc metals.[15]

Twin boundaries are partly responsible for shock hardening and for many of the changes that occur in cold work of metals with limited slip systems or at very low temperatures. They also occur due to martensitic transformations: the motion of twin boundaries is responsible for the pseudoelastic and shape-memory behavior of nitinol, and their presence is partly responsible for the hardness due to quenching of steel. In certain types of high strength steels, very fine deformation twins act as primary obstacles against dislocation motion. These steels are referred to as 'TWIP' steels, where TWIP stands for twinning-induced plasticity.[16]

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