Understanding Crystal Symmetry and Rotational Axes in Solid State Physics, Slides of Solid State Physics

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MULTIPLE SYMMETRIES OF CRYSTAL STRUCTURE

Solid materials are formed from densely-packed atoms, with intense interaction forces between them. These interactions are responsible for the mechanical (e.g. hardness and elasticity), thermal, electrical, magnetic and optical properties of solids. Depending on the material involved and the conditions in which it was formed, the atoms may be arranged in a regular, geometric pattern (crystalline solids, which include metals and ordinary water ice) or irregularly (an amorphous solid such as common window glass).

CRYSTAL

STRUCTURE

Crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry. Patterns are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters.

 SYMMETRY defines the order resulting from how atoms

are arranged and oriented in a crystal.

(OR)

A symmetry element is a line, a plane or a point in or

through an object, about which a rotation or reflection

leaves the object in an orientation indistinguishable from

the original

 OPERATIONS is movement of any number of points

of lattice

 SYMMETRY OPERATIONS are those operations that

take the structure into itself.(without any change)

 SYMMETRY GROUP ON SPACE GROUP is the set of all

operations which leaves bravais lattice unchanged.

 POINT OPERATION are those symmetry operations

during which at least one lattice point does not move.

 POINT GROUP is the subset of the full symmetry group of

a bravais lattice that include all the point operations of that

bravais lattice.

REFECTION

SYMMETRY

(MIRROR SYMMETRY)

A mirror symmetry operation is an imaginary operation that can be performed to reproduce an object. The operation is done by imagining that you cut the object in half, then place a mirror next to one of the halves of the object along the cut. If the reflection in the mirror reproduces the other half of the object, then the object is said to have mirror symmetry. The plane of the mirror is an element of symmetry referred to as a mirror plane , and is symbolized with the letter m

The rectangles shown

below have two planes

of mirror symmetry.

The rectangle on the

left has a mirror plane

that runs vertically on

the page and is

perpendicular to the

page. The rectangle on

the right has a mirror

plane that runs

horizontally and is

perpendicular to the

page. The dashed parts

of the rectangles below

show the part the

rectangles that would

be seen as a reflection

in the mirror.

Symmetry is when one

shape becomes exactly like

another if you flip, slide or

turn it. The simplest type of

Symmetry is "Reflection" (or

"Mirror") Symmetry, as

shown in this picture of my

dog Flame.

 If an object can be rotated about an axis and repeats

itself every 90

o of rotation then it is said to have an axis

of 4-fold rotational symmetry. The axis along which

the rotation is performed is an element of symmetry

referred to as a rotation axis. The following types of

rotational symmetry axes are possible in crystals.

3-Fold Rotation Axis - Objects that repeat

themselves upon rotation of 120o^ are said to

have a 3-fold axis of rotational symmetry

(360/120 =3), and they will repeat 3 times in

a 360o^ rotation. A filled triangle is used to

symbolize the location of 3-fold rotation axis

4-Fold Rotation Axis - If an object repeats

itself after 90o^ of rotation, it will repeat 4 times

in a 360o^ rotation, as illustrated previously. A

filled square is used to symbolize the location

of 4-fold axis of rotational symmetry.

6-Fold Rotation Axis - If

rotation of 60o^ about an axis

causes the object to repeat itself,

then it has 6-fold axis of

rotational symmetry

(360/60=6). A filled hexagon is

used as the symbol for a 6-fold

rotation axis.