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Week 3: Jan 30 and Feb. 1

CRYSTALLOGRAPHY (cont'd)
  • Indicating directions in a crystal - something similar to the Miller Index can also be used to indicate a direction in a crystal. Directions are denoted by [uvw]. (So, a face is denoted by (hkl), forms by {hkl} and directions by [uvw].) The integers describing the direction are the node coordinates through which the line passes, appropriately simplified as in the Miller Index. (See figure 2.27 of your textbook.) In this way, the a-axis can be denoted as the [100] direction, the b-axis as the [010] direction and the c-axis as the [001] direction.
  • Zone - a group of faces with parallel lines of intersection. Zones are named according to the direction of the lines of intersection.
MINERAL HABIT - minerals do not always form well-formed crystals with distinct crystal faces. Habit refers to the particular way a mineral grain or group of grains grow,
  • Terms referring to the degree of development of crystal faces:
    • euhedral - consisting of well-formed faces
    • anhedral - lacking crystal faces
    • subhedral - have some faces developed
  • Terms describing the relative dimensions of individual grains (see fig. 2.47 of your book):
    • equant
    • columnar
    • acicular
    • fibrous
    • stubby
    • platy
    • scaly
    • tabular
    • bladed
  • Terms for aggregates of crystals: parallel, radiating, felted, foliated, plumose, granular, dendritic, reticulated, drusy, colloform (botryoidal, mammillary, globular, reniform), concentric,banded, pisolitic, oolitic, massive, amygdaloidal, concretionary
CRYSTAL CHEMISTRY AND CRYSTAL STRUCTURE
  • Review of the atomic model
    • atom = the smallest subdivision of matter that retains the properties of the chemical element
      • nucleus - composed of
        • protons (+ charge)
        • neutrons (neutral charge)
      • electron "cloud" (negative charge)
    • size of atoms - determined by radii of electron orbits (affected by coordination number as we shall see later)
      • on the order of 2 x 10E-10 m
      • nucleii are around 10,000 times smaller
      • nucleus was "discovered" by Rutherford in alpha-particle--gold film experiment
    • mass of atoms
      • atomic mass (A) is sum of atomic number (Z, number of protons) and N (number of neutrons); electrons are essentially massless
      • amu - atomic mass unit; where 12 amu = weight of 12-Carbon
    • The Bohr model of the atom
      • electrons have discrete shells that they occupy around the nucleus corresponding to some energy level
      • "jumping" of an electron from one orbital to another requires the absorption (inner to outer orbital) or release (outer to innner orbital) of energy
    • Schrodinger's model
      • A more "fuzzy" model of electron structure: Probabilities are given for finding an electron at a given distance from the center of the atom
      • Electrons occupy orbitals instead of shells
    • (We will re-visit the electron structure of atoms when we cover the origins of color, chemical bonding, and analytical x-ray techniques.)
  • Ions - atoms that have lost or gained an electron(s)
    • Valence electrons - those electrons that are lost or gained by atoms
    • Cations - positively charged ions; have lost electrons
    • Anions - negatively charged ions; have gained electrons
  • Ionic radii - if the electron clouds around ions are "fuzzy", what is the size of an ion?
    • Oppositely charged ions will tend to attract each other. The force of this attraction is directly proportional to the charges of the ions and inversely proportional to the square of the distance between the ions
    • As the oppositely charged ions approach each other, however, the repusion of the electron clouds between the two ions comes into play.
    • The two ions settle at an "equilibrium" distance where these two forces balance
    • The size of an ion therefore depends on what type and how many other ions it is bonded to (the coordination number, which we look at next)
    • (Despite the details about (limited) variation in ionic size, the rigid "billiard ball" model of ions is useful as a first step in understanding the way ions organize themselves to from crystalline structures.) So, we now have...
  • Coordination of ions - cations will tend to attract to themselves (bond with) the maximum number of anions that size permits
    • The number of anions that a cation can attract to itself is limited by the repulsion of the anions against themselves
    • To take extreme cases, on the one hand an extremely small cation may have difficulty holding three anions to itself because the anions may be closer to each other than they are to the central cation. On the other hand a very large cation will be able to keep more anions about itself because the anions can all "touch" the central cation.
    • The stable number of anions that can surround a cation depends therefore on the relative sizes of the ions (the radius ratio = radius of cation/radius of anion).
    • Coordination Number = the number of anions around the cation
    • The assemblage of anions around a cation forms a simple shape that resembles a line, triangle or polyhedron (with the anions occupying the corners):
      Radius Ratio
      Coordination Number
      Coordination Type
      <0.155
      2
      Linear
      0.155 to 0.225
      3
      Triangular
      0.225 to 0.414
      4
      Tetrahedral
      0.414 to 0.732
      6
      Octahedral
      0.732 to 1
      8
      Cubic
      1
      12
      Cubic or Hexagonal Close Packing
  • Common elements in the Earth's crust - We now look at the specific cations and anions that we find in the Earth's crust, and which therefore combine to make the common minerals.

    •  


    Element
    Weight %
    Molecular %
    Volume %
    Common oxidation state
    Common Coordination Number with Oxygen
    O
    47
    63
    94 !
    - 2
    ---
    Si
    28
    21
    		  
    + 4
    4
    Al
    8
    6
    		  
    + 3
    4 or 6
    Fe
    5
    2
    		  
    + 2 or + 3
    6
    Ca
    4
    2
    		  
    + 2
    6 or 8
    Na
    3
    3
    		  
    + 1
    8
    K
    3
    1
    		  
    + 1
    8 or 12
    Mg
    1
    2
    		  
    + 2
    6

  • Pauling's Rules - A set of rules that describe the way ions combine in a crystal structure. 
    • Rule 1: The Coordination Principle - Cations will surround themselves with anions as size permits, as discussed above.
    • Rule 2: The electrostatic valence principle - In a stable crystal structure, the total strength of the bonds that reach an anion from all neighboring cations (or vice versa) is equal to the charge of the anion
      • electrostatic valency ("e.v.") = (charge of ion / coordination number)
      • small highly charged cations coordinating with larger less strongly charged anions will form firmly bonded groups (e.g., CO3)
    • Rule 3: Sharing of edges and faces by polyhedra is not favored (sharing of corners is more favored)


    (to be continued ...)
     

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