show that the mirror planes are only compatible with the reactangular unit cell
Answers
Explanation:
Bravais Lattice refers to the 14 different 3-dimensional configurations into which atoms can be arranged in crystals. The smallest group of symmetrically aligned atoms which can be repeated in an array to make up the entire crystal is called a unit cell. There are several ways to describe a lattice.
Answer:
Specifically, in finite objects, there are a number of operations (elements of symmetry) describing repetitions. In the wall-drawing (shown above) we find translational operations (the motif is repeated by translation). The repetition of the petals in the flowers show us rotational operations (the motif is repeated by rotation) around a symmetry axis (or rotation axis). And, although not exactly, the symmetry shown in the phrase or in the music fragment (shown above), lead us to consider other symmetry operations known as symmetry planes (reflection planes, or mirror planes); the same operation that occurs when you look into a mirror. Similarly, for example, if we look at the relationship between the three-dimensional objects in some of the pictures shown below, we will discover a new element of symmetry called center of symmetry (or inversion center), which is an imaginary point between objects (or inside the object) as shown in some drawings below.
Generally speaking, and taking into account that pure translational operations are not strictly considered as symmetry operations, we can say that finite objects can contain themselves, or may be repeated (excluding translation) by the following symmetry elements:
The identity operation is the simplest symmetry element of all -- it does nothing! But it is important because all objects at the very least have the identity element, and there are many objects that have no other symmetry elements.
The reflection is the symmetry operation that occurs when we put an object in front of a mirror. The image is found perpendicular to the reflection plane and equidistant from that plane, on the opposite side of the plane. The resulting object can be distinguishable or indistinguishable from the original, normally distinguishable, as they cannot be superimposed. If the resulting object is indistinguishable from the original, is because the reflection plane is passing through the object.
The inversion operation occurs through a single point called the inversion center. Each part of the object is moved along a straight line through the inversion center to a point at an equal distance from the inversion center. The resulting object can be distinguishable or indistinguishable from the original, normally distinguishable, as they cannot be superimposed. If the resulting object is indistinguishable from the original, is because the inversion center is inside the object.
The rotation operations (both proper and improper) occur with respect to a line called rotation axis. a) A proper rotation is performed by rotating the object 360°/n, where n is the order of the axis. The resulting rotated object is always indistinguishable from the original. b) An improper rotation is performed by rotating the object 360°/n followed by a reflection through a plane perpendicular to the rotation axis. The resulting object can be distinguishable or indistinguishable from the original, normally distinguishable, as they cannot be superimposed. If the resulting object is indistinguishable from the original, is because the improper rotation axis is passing through the object.
In addition to the name of the symmetry elements, we use graphical and numerical symbols to represent them. For example, a rotation axis of order 2 (a binary axis) is represented by the number 2, and a reflection plane is represented by the letter m.
Explanation:
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