Unit cell, defining vectors

The vector OP points along the [120] direction in the orthorhombic unit cell defined by abc. 

Corresponding to any crystal lattice, we can construct a reciprocal lattice, so called because many of its properties are reciprocal to those of the crystal lattice. Let the crystal lattice have a unit cell defined by the vectors ai, a2, and 83. Then the... 

Let (/) (r) be a periodic function in three dimensions, so that (r) = (r + R) with R = Mjai+/i2a2+ 3a3 witha/O = 1,2,3) being three vectors that characterize the three-dimensional periodicity and nj any integers (see Section 4.1). The function is therefore characterized by its values in one unit cell defined by the three a vectors. Then the integral (1.37) vanishes if the volume of integration is exactly one unit cell. 

Let us fix the parameter 2A = 0.34 nm which is equal to the graphite inter-layer separation. The chirality and the radius of single-wall carbon nanotube are uniquely specified by the chiral vector C/, = iai +112 2 - ( i. 2)> where ni,ri2 are integers and ai, a2 are the unit cell basis vectors of graphite [1]. The chiral vector C is a circumferential lattice vector defined on nanotube surface, and C is perpendicular to the tube axis. For armchair nanotubes n =ri2 = n, and the tube radius r is defined by r = C/, /2ti = a f3n/2n, where a = 0.249 nm is the lattice constant for graphite. These values of r are used in our calculations. 

Unit Cell The vectors a, b, c define a cell. There is, in principle, an infinite number of ways to define a unif cell in any crysfal laffice. Buf, as in many areas of crysfallo-graphy, there is a convention ... 

Figure 3.8 A cubic unit cell defined by three vectors...

Consider a solid with unit cell defined by lattice vectors , b and c, indicated in Fig. 1. Each point within the solid, whose is formed from the replication of the unit cell in all direction, as shown in Fig. 1, can be written as eq. (2), in which R corresponds to an arbitrary position within the solid. 

The unit cell can be defined in tenns of tluee lattice vectors (a, b, c). In a periodic system, the point x is equivalent to any point x, provided the two points are related as follows ... 

In this section, we concentrate on the relationship between diffraction pattern and surface lattice [5], In direct analogy with the tln-ee-dimensional bulk case, the surface lattice is defined by two vectors a and b parallel to the surface (defined already above), subtended by an angle y a and b together specify one unit cell, as illustrated in figure B1.21.4. Withm that unit cell atoms are arranged according to a basis, which is the list of atomic coordinates within drat unit cell we need not know these positions for the purposes of this discussion. Note that this unit cell can be viewed as being infinitely deep in the third dimension (perpendicular to the surface), so as to include all atoms below the surface to arbitrary depth. 

To verify effectiveness of NDCPA we carried out the calculations of absorption spectra for a system of excitons locally and linearly coupled to Einstein phonons at zero temperature in cubic crystal with one molecule per unit cell (probably the simplest model of exciton-phonon system of organic crystals). Absorption spectrum is defined as an imaginary part of one-exciton Green s function taken at zero value of exciton momentum vector... 

Note that the denominator in each case is equal to the volume of the unit cell. The fact that a, b and c have the units of 1/length gives rise to the terms reciprocal space and reciprocal latlice. It turns out to be convenient for our computations to work with an expanded reciprocal space that is defined by three closely related vectors a , b and c, which are multiples by 2tt. of the X-ray crystallographic reciprocal lattice vectors ... 

Fig. 1. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The chiral vector OA or Cf, = nOf + tnoi defined on the honeycomb lattice by unit vectors a, and 02 and the chiral angle 6 is defined with respect to the zigzag axis. Along the zigzag axis 6 = 0°. Also shown are the lattice vector OB = T of the ID tubule unit cell, and the rotation angle 4/ and the translation r which constitute the basic symmetry operation R = (i/ r). The diagram is constructed for n,m) = (4,2).

It is, however, more revealing in the context of monodromy to allow/(s, ) to pass from one Riemann sheet to the next, at the branch cut, a procedure that leads to the construction in Fig. 4, due to Sadovskii and Zhilinskii [2], by which a unit cell of the quantum lattice, with sides defined here by unit changes in k and v, is transported from one cell to the next on a path around the critical point at the center of the lattice. Note, in particular, that the lattice is locally regular in any region of the [k, s) plane that excludes the critical point and that any vector in the unit cell such as the base vector, marked by arrows, rotates as the cell is transported around the cycle. Consequently, the transported dashed cell differs from that of the original quantized lattice. 

We also need to define how large the unit-cell is in terms of both the length of its sides and its volume. We do so by defining the unit-cell directions in terms of its "lattice unit-vectors". That is, we define it in terms of the x, y, z directions of the unit cell with specific vectors having directions corresponding to ... 

In general, we use only the lattice constants to define the solid structure (unless we are attempting to determine its S5nnmetry). We can then define a structure factor known as the translation vector. It is a element related to the unit cell and defines the basic unit of the structure. We will call it T. It is defined according to the following equation ... 

Vector notation is being used here because this is the easiest way to define the unit-cell. The reason for using both unit lattice vectors and translation vectors lies in the fact that we can now specify unit-cell parameters in terms of a, b, and c (which are the intercepts of the translation vectors on the lattice). These cell parameters are very useful since they specify the actual length eind size of the unit cell, usually in A., as we shall see. 

As vectors a, b and c we choose the three basis vectors that also serve to define the unit cell (Section 2.2). Any translation vector t in the crystal can be expressed as the vectorial sum of three basis vectors, t = ua + vb + wc, where u, v and w are positive or negative integers. 

An infinite three-dimensional crystal lattice is described by a primitive unit cell which generates the lattice by simple translations. The primitive cell can be represented by three basic lattice vectors such as and h defined above. They may or may not be mutually perpendicular, depending on the crystal... 

In the above relation, quantum states of phonons are characterized by the surface-parallel wave vector kg, whereas the rest of quantum numbers are indicated by a the latter account for the polarization of a quasi-particle and its motion in the surface-normal direction, and also implicitly reflect the arrangement of atoms in the crystal unit cell. A convenient representation like this allows us to immediately take advantage of the translational symmetry of the system in the surface-parallel direction so as to define an arbitrary Cartesian projection (onto the a axis) for the... 

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