Linear Algebra/Topic: Crystals/Solutions

SolutionsEdit

Problem 1

How many fundamental regions are there in one face of a speck of salt? (With a ruler, we can estimate that face is a square that is   cm on a side.)

Answer

Each fundamental unit is   cm, so there are about   such units. That gives  , so there are something like   (three hundred million) units.

Problem 2

In the graphite picture, imagine that we are interested in a point   Ångstroms up and   Ångstroms over from the origin.

  1. Express that point in terms of the basis given for graphite.
  2. How many hexagonal shapes away is this point from the origin?
  3. Express that point in terms of a second basis, where the first basis vector is the same, but the second is perpendicular to the first (going up the plane) and of the same length.
Answer
  1. We solve
     
    to get   and  .
  2. Here is the point located in the lattice. In the picture on the left, superimposed on the unit cell are the two basis vectors   and  , and a box showing the offset of  . The picture on the right shows where that appears inside of the crystal lattice, taking as the origin the lower left corner of the hexagon in the lower left.

            

    So this point is in the next column of hexagons over, and either one hexagon up or two hexagons up, depending on how you count them.

  3. This second basis
     
    makes the computation easier
     
    (we get   and  ), but it doesn't seem to have to do much with the physical structure that we are studying.
Problem 3

Give the locations of the atoms in the diamond cube both in terms of the basis, and in Ångstroms.

Answer

In terms of the basis the locations of the corner atoms are  ,  , ...,  . The locations of the face atoms are  ,  ,  ,  ,  , and  . The locations of the atoms a quarter of the way down from the top are   and  . The atoms a quarter of the way up from the bottom are at   and  . Converting to Ångstroms is easy.

Problem 4

This illustrates how the dimensions of a unit cell could be computed from the shape in which a substance crystalizes (see Ebbing 1993, p. 462).

  1. Recall that there are   atoms in a mole (this is Avagadro's number). From that, and the fact that platinum has a mass of   grams per mole, calculate the mass of each atom.
  2. Platinum crystalizes in a face-centered cubic lattice with atoms at each lattice point, that is, it looks like the middle picture given above for the diamond crystal. Find the number of platinums per unit cell (hint: sum the fractions of platinums that are inside of a single cell).
  3. From that, find the mass of a unit cell.
  4. Platinum crystal has a density of   grams per cubic centimeter. From this, and the mass of a unit cell, calculate the volume of a unit cell.
  5. Find the length of each edge.
  6. Describe a natural three-dimensional basis.
Answer
  1.  
  2.  
  3.  
  4.   cubic centimeters
  5.   centimeters.
  6.  

ReferencesEdit

  • Ebbing, Darrell D. (1993), General Chemistry (Fourth ed.), Houghton Mifflin .