1 / 16

Lecture 3: More on Structures, Miller Indeces, Stereographic Projection

Lecture 3: More on Structures, Miller Indeces, Stereographic Projection. PHYS 430/603 material Laszlo Takacs UMBC Department of Physics. Crystal structure data from http://cst-www.nrl.navy.mil/lattice/. The three most essential crystal structures FCC, Cu , A1, cF4 HCP, Mg, A3, hP2

unity
Download Presentation

Lecture 3: More on Structures, Miller Indeces, Stereographic Projection

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 3: More on Structures, Miller Indeces, Stereographic Projection PHYS 430/603 material Laszlo Takacs UMBC Department of Physics

  2. Crystal structure data fromhttp://cst-www.nrl.navy.mil/lattice/ The three most essential crystal structures • FCC, Cu , A1, cF4 • HCP, Mg, A3, hP2 • BCC, W, A2, cI2 What other information is there? • Space group/ set of symmetries • Number of space group (arbitrary) • Other substances with the same structure • Primitive translational vector of the lattice (of mathematical points) • Basis vectors describing the position of atoms in the basis • Different ways to view the structure

  3. Some compound structures that derive from fcc with insterstitial sites filled:TiN (NaCl structure) N at octahedral sites, as many as atomsTiH2 (CaF2 structure) H at tetrahedral sites, twice as many as atomsTiH (ZnS, zinc blende) H only in every other tetrahedral site; related to diamondTi alone is hcp.

  4. The size of atoms varies amazingly little, given the large increase in the number of electrons. Especially true for the meals used in typical alloys.Notice the small sizes of H, B, C, N. Defining the size of atoms is not trivial, they are not rigid balls. The atom-atom distance depends on the character of the bond. Ionic radii are much different, small for positive, Large for negative.

  5. The metal lattice is rearranged to create a larger interstitial site. The Al2Cu (C16) structure of Fe2B

  6. The NRL site on Al2Cu • Al2Cu prototype. • tI12: Body centered tetragonal with 12 atoms in the unit cell. • C16: The 16th A2B structure. • I4/mcm: Space group symbol. • 140: Arbitrary number assigned to the space group. • X, Y, Z: Unit vectors in the directions of the unit cell axes; a, c lattice parameters. • Atom positions within the unit cell are given for only one of the equivalents. Notice that x has to be determined from measurement.

  7. The elementary vectors of translation, i.e. the edges of the elementary cell, define the unit vectors of our coordinate system. Directions and planes are defined in this coordinate system. It is identical to the ordinary rectangular coordinate system with identical scales only for cubic structures. c b a

  8. Defining directions and planes Directions defined by a vector: r = ua + vb + wc Usually we are interested in directions where u, v, w are small integers. Standard notation for a direction: [u v w] Notation for all equivalent directions: < u v w > There is no comma between numbers, “-” sign put over the numbers. Set of parallel planes described by Miller indeces • Note the intersections with the axes in units of a, b, c; typically small integers. Put down infinity, if the plane is parallel to the axis. • Take the reciprocal of the three numbers, 0 for infinity. • Multiply with the smallest number that gives a set of integers. Notation for a single plane: (m n q) Notation for a set of equivalent planes: {m n q} There is no comma between numbers, “-” sign put over the numbers. In the cubic system (and only there) [m n q] is the normal of (m n q).

  9. Some planes of the [0 0 1] zone(A zone is the set of all planes that are parallel to the given direction.Weiss zone law: hu + kv + lw =0)

  10. Complications with the hcp structureIn the ideal case, lattice parameters a and c are related:a = edge of hexagon = diameter of atoms c = twice the distance between layersThe red tetrahedron is regular, thusc/a = 1.62 for Mg, Co; 1.86 for Zn; 1.58 for Ti In order to reflect symmetry in the basal plane, rather than (h k l) Miller indeces defined with two primitive vectors in the basal plane, the four-index Miller-Bravais notation is used: (h k -(h+k) l) Similarly, a four-index notation is used for directions: [m n p q] with m+n+p = 0 and r = ma1 + na2 + pa3 +qc

  11. In order to represent a direction (or the normal of a plane) in a stereographic projection, intersect the reference sphere with the direction (P) then project P from the “South pole” of the sphere onto the equatorial plane.

  12. [0 0 1] direction points up. The intersection of directions with the sphere are projected to the equatorial plane from the [0 0 -1] point; the upper hemisphere is imaged.

  13. The more detailed standard [0 0 1] projection of a cubic lattice.Every type of direction appears in the shaded triangle, the rest relates by symmetry operations.

  14. [0 1 1] z y Projection from the [0 1 1] direction x

  15. Pole figure of rolled aluminum. The sample is looked at from the normal direction, the rolling direction (RD) and the transverse direction (TD) are the vertical and horizontal axis. Shade shows he probability of the indicated direction pointing in the given direction. http://www.labosoft.com.pl/texture_standards.htm

  16. Texture of iron processed by ECAE, Equal Channel Angular Extrusion M.A. Gibbs, K.T. Hartwig, R.E. Goforth, Department of Mechanical Engineering, Texas A&M University E.A. Payzant, HTML, Oak Ridge National Laboratory T F

More Related