Atomic Structures of Minerals
                                  And
                                                Ionic Substitution
               Stable atomic structures of minerals can be explained to a large extent by the
                5 rules of Linus Pauling--Pauling's rules and the principles of ionic substitution
               (Goldschmidt) are important concepts used to classify minerals
   Pauling's rule
        - in the atomic structure of minerals, anions and cations are bonded together--
            hence anions surround cations and cations surround anions  
      1. Rule 1--coordination number principle
          -in the atomic structure of a mineral, a set number of anions surround each cation--that
           number is dependent on the ionic radius of the cation and anion--if the atoms are mostly
           spherical in shape when combining, spherical geometry can be used to derive the number
           of anions surrounding a cation and the shape of this configuration--the number of atoms
           surrounding a centrally coordinated atom is referred to as the Coordination Number (CN)
          -the CN depends on the radius ratio (ionic radius of cation/ionic radius of anion)
          -also in the atomic structure of a mineral, cations surround anions--this is treated in
            Rule # 2
           --the following table relates the radius ratio to the CN and the geometric shape taken by the
           anions around the cation in a single bond in a mineral--
         
range of radius ratio (rr) geometric shape of anions around the cation number of anions around the  cation(CN)
0.155- 0.225 equilateral triangle 3
0.225-0.414 tetrahedron 4
0.414-0.732 octahedron 6
0.732-1.00 cube 8
1.00 closest packing 12


         -use the atomic chart to find the ionic radii of the appropriate valence state of the elements-
         -in the mineral halite, NaCl (Na-Cl bond) how many Cl-1 surround each Na+1?--
          rr = (0.95/1.81)= 0.53--see the table above
          -this falls between the radius ratio representing a CN of 6--hence there should be 6 Cl
          atoms surrounding each Na atom in octahedral coordination in the array of atoms in halite
         -the calculated and actual CN of a bond may differ--two important factors can cause this--
          (1) pressure and more importantly temperature of formation of the mineral-- that is, if
          the radius ratio is borderline, a higher temperature of formation will favor the smaller 
          CN and vice versa; (2) if the percent ionic character of the cation-anion bond is too 
          low--unfortunately, there are no specific absolute values for either 1 or 2 which will allow
          complete confidence in the calculated CN--use of the molecular hybridization or molecular
          orbital concepts may give better results if the bond lacks adequate ionic character. Temperature
          of formation affects only anisodesmic bond types as will be discussed soon.
         --examples of cations which bond to oxygen in minerals which can have more than 1 CN
            with O based on the 2 factors above are: Al-O(4 or 6), K-O( 8,12),
            Na-O(6, 8) Ca-O(6, 8), Ba-O(8, 12) and Rb-O(8, 12) since all r.r. numbers are 
            borderline and theses are of the isodesmic bond type. 
         -in the calculation above concerning halite, the rr of 0.53 is not borderline and the
          calculated CN of 6 should be the same as the actual which it is--likewise, the percent ionic
          character of the bond is high enough for the calculated CN to be valid--from the Periodic
          Table of the Elements, the difference between the electronegativities of Na(0.9) and that
          for Cl (3.0) is 2.1 which equates to a 67% ionic bond character--
        -Rule 1 concerns the number of anions surrounding each cation--but how many cations 
         surround each anion? Rule 2 helps explain this-

     
         
      2.  Rule 2--electrostatic valency principle

          -the sum of electrostatic valence numbers of bonds reaching a centrally  
           coordinated anion are equal to the valence number (absolute value of that  
           centrally coordinated anion)--the electrostatic valence number of a bond (e.v.) =  
           (absolute valence number of the cation (z)/(CN)
          -an example is a centrally coordinated atom of F-1, in CaF2 in which the sum of the e.v.
           from all bonds reaching the F atom must equal  exactly 1
          --the ev principle also applies to any centrally coordinated cation as in CaF2 above
          in which the sum of e.v. from all anions reaching the Ca atom must equal exactly 2
         
       -now let us look at an example of the application of Rules 1 and 2  for a mineral--
        the number of anions surrounding each cation and the number of cations around each 
        anion
          -determine the atomic structure hematite(Fe2O3)--the rr between Fe+3(ionic radius=0.65
            and O-2(ionic radius=1.40) is 0.46 or a CN of 6 (from Rule 1)--hence there must be 6 oxygens
            around and bonded to each Fe in octahedral coordination from Rule1
           -from Rule 2, the e.v. of the Fe-O bond = 3/6 or 1/2 
           -there must be 4 Fe (tetrahedral coordination) surrounding each O to equal the valence 
            number of 2 (4x1/2)--this shows the octahedron and tetrahedron (polyhedrons) are
            connected by oxygen in the atomic structure of hematite

          -there are 3 bond types related to e.v.---the last two have special significance:
                      a.   isodesmic bond
                          -a bond in which the e.v. number is less than 1/2 the valence number of the
                           anion--an example is the Na-Cl bond in halite: from above, the CN = 6 and the
                           e.v. would be 1/6, or less than 1/2 of 1(1= valence number of Cl) as mentioned above, 
                          -temperature of mineral formation can result in a different CN than calculated 
                            for this type of bond if the r.r. of the bond is borderline--also, the CN can be  
                            affected by the low ionic character of the bond--examples of cations which
                            bond to oxygen in minerals which can have more than 1 CN with O based 
                            on temperature of mineral formation are: Al-O(4 or 6), K-O( 8,12), Na-O(6, 8) 
                            Ca-O(6, 8), Ba-O(8, 12) and Rb-O(8, 12) since all r.r. numbers are 
                            borderline .   
                      b. mesodesmic bond
                           -a bond in which the e.v. number is equal to1/2 the valence number of the
                            anion--an example is the Si-O bond in silicate minerals: the CN of the Si-O
                            bond (Si+4 and O-2) is 4 (rr = 0.29) and the e.v. number is 4/4 or 1 , which is 
                            exactly equal to1/2 of 2 (valence number of O)
                           -the presence of this bond type in minerals is paramount  
                             large polymer-like units form--an example of this is the large units comprising the
                            different silicate subclasses (see pages 436 and 437 in text)--the silicate
                            (Si-O) and borate (B-O) classes form polymerization structures-
                            -effects of temperature or pressure on the CN of this type of bond cannot
                              happen since the r. r. is not borderline in both cases--the same is true
                              for low % ionic character since both bonds have a very high % ionic character 
                             -Si-O bond and B-O bond always have a 4 and 3 CN, respectively  
                      c.   anisodesmic bond
                           -a bond in which the e.v. number is greater than 1/2 the valence number of the
                            anion--an example is the C-O bond in carbonate minerals: the CN of the
                            C-O bond(C+4 and O-2) is 3 and the e.v. number is 4/3, which is more than
                            1/2 of 2 (valence number of O)
                           -the presence of this bond type in minerals is paramount since complex anions
                            (oxyacid anions) when present in minerals define many mineral classes--
                            the correct CN of the bond or the number 
                            of anions (O) surrounding each cation in the anionic group is the same as the
                            subscript number associated with the O--what is important is this allows the
                            determination of the CN without (1) a radius ratio calculation and (2) con-
                            sideration of the % ionic character of the bond or if the rr is borderline--
                            The # of O bonded to each cation in the anionic group is verified on pages 400-406
                            in the text (23rd edition).
                            Do not perform Rules 1 or 2 above on this type of bond --you may get an incorrect 
                            CN
                              --the table below shows many of the mineral classes with this type of bond. Know
                                 the contents of the table--it can save you some computation in problem 
                                 solving and also it can save you from reporting an incorrect structure
                                                         
     
bond mineral class anionic group CN(cation-anion)
S-O sulfate (SO4)-2 4
N-O nitrate (NO3)-1 3
C-O carbonate (CO3)-2 3
P-O phosphates (PO4)-3 4
V-O vanadates (VO4)-3 4
Mo-O molybdates (MoO4)-2 4
As-O arsenates (AsO4)-3 4
W-O tungstates (WO4)-2 4

              -the e.v. number also reflects the relative strength of the bond--cleavage through mesodesmic
               and anisodesmic bonds is unlikely--breakage of isodesmic bonds which are aligned can
               result in cleavage along that alignment--minerals with a mixture of mesodesmic and isodesmic
               bonds or anisodesmic and isodesmic bonds break along the weaker isodesmic bond

       

       3. Rule 3--sharing of polyhedra-1
            -this explains how geometric units (polyhedra) fit together in an atomic array of a mineral
            -polyhedral are most stable if they are connected at points--the sharing of edges and part-
             icularly faces decreases stability--this does not exclude the presence of polyhedral units
             sharing edges in some minerals--the explanation of this preferred occurrence is based on
             the proximity of cations and + sign replusion in adjacent polyhedra--simply stated,
             cations are situated at a greater distance if polyhedra share points, (less repulsion) and closer 
             if they share faces (more repulsion)

      4. Rule 4--sharing of polyhedra-2
           -in minerals containing different cations, those of high valence numbers and small CN tend
            not to share polyhedra with each other--if they do, shared edges shrink and cations are
            displaced away from shared edges

      5.  Rule 5--the principle of parsimony
           -the number of different cations, anions or anionic groups tends to be small--although
            elements can substitute for each other during mineral formation, this tends to be limited

 

     To determine the atomic structure of a mineral:
             1. define each individual bond in the mineral--C1-A,....C2-A....  where C1 and C2....is (are)  
                 the cation bond(s) with the anion, A.
             2. use Pauling's rule #1 and r.r. to determine the CN--the # of anions around each cation
                 which defines the shape of each polyhedron--do this r.r calculation only if the bond type is
                 isodesmic----if mesodesmic or anisodesmic, refer to the notes under these bond types to
                 explain how to obtain or determine the number of anions around each cation giving the
                 corresponding polyhedron shape or configuration  
             3. use Pauling's rule #2 and establish the e.v. for each C-A bond--surround the anions with
                 the corresponding cation(s) so as the sum of all the e.v. numbers equals the valence number
                 of the anion--the configuration of the # of cations around each anion also defines the  
                 polyhedron in the table under Pauling's rule #1
             4. join polyhedra via a common anion. (See rule #3--join at points by default)

             As a summary,
             1.  all isodesmic bonds reaching an O have ev values less than 1---all mesodesmic
                  bonds reaching an O, equal to 1---all anisodesmic bonds reaching an O, greater than 1.
             2.  in a mineral structure there can be only 1 anisodesmic bond to an O, all other bonds to that O
                  must be comprised of 1 or more isodesmic bond(s).
             3.  in a mineral structure there can be 1 or 2 mesodesmic bonds reaching an O---if only 1,
                  there is in addition, 1 or more isodesmic bonds reaching the O----if 2, then no additional
                  isodesmic bonds can reach that O.

     IONIC SUBSTITUTION)

           Ionic Substitution
             -is the ability of ions of different elements to take or occupy sites (proxy) in the atomic structure 
               of a mineral during the formation of the mineral--cation proxy on a large scale is most important
               since this is the basis for solid solution (discussed later)-                        
             -ionic substitution can exist with an abundant ion substituting (proxying) for the preferred ion in a
              large number of atomic sites.  The substituting ion is shown in the mineral formula.
             -if trace concentrations of cations proxy in the atomic structure, the cations cannot be shown in the
               mineral formula--however, the study of trace concentrations of cations in minerals which proxy  
               for a major cation can be paramount in certain geologic studies

   
                   factors influencing ionic substitution:

                       1.   C.N.
                            -cations which can proxy must form the same (predetermined) C.N. with the 
                             anion--as discussed in Pauling's rules the cations must have the appropriate
                             ionic radius (cations of different elements must have similar ionic radii) and
                             the temperature and pressure of mineral formation must be appropriate if rr is
                             borderline

                                  consider the Al-O and Si-O bond
                                  Al+3 can proxy for Si+4 in silicates only at high temperatures as in a
                                  magma but cannot do so during the formation of clay minerals--explain
                                  in detail why this is fact

                      2.   Ionic Charge and electroneutrality
                             -any cation which may proxy for a preferred cation must have the same or
                               similar valence number--if the number is too different, the energy for sub-
                               stitution is too great and will not occur--if there is a proxy and a difference
                               in valence number, to adhere to the electroneutrality rule, another proxy
                               for a second element in the formula must occur--an example is the ionic
                               substitution of Ca+2 and Na+1 in plagioclase feldspar--for every Ca+2
                               that substitutes for a Na+1 in the mineral, NaAlSi3O8 there must be a
                               simultaneous substitution of an Al+3 for a Si+4 to keep electroneutrality
                               and the mineral, CaAl2Si2O8 will form--this is very important to know

                          3.   Electronegativity
                              -if the electronegativity (numbers) between the preferred element and the
                               substituting element is too great and would result in a much less ionic
                               character with the anion, the substitution will not occur. An example is
                               Cu+1 substituting for Na+1 in a Na mineral as NaAlSi3O8--even though both ions 
                               have the same ionic size and therefore should form the same CN with the
                               anion, O (condition # 1 above) and both have the same valence,
                               +1(condition # 2 above), the electronegativities are quite different (Cu = 1.9   
                                and Na = 0.9) resulting in a much weaker ionic bond with the anion, O if
                                Cu were to substitute--a slight difference in electronegativities would allow
                                substitution to occur as a .1 or .2 difference    
                              --hence, the substitution of Cu+1 for Na+1 will not occur.  
                              

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   -the next section deals with important concepts in Mineralogy, some of which are more fully
   understood with a knowledge of Pauling's rules and ionic substitution--some of these concepts are 
   isomorphism, solid solution, and polymorphism, all of which are important in classifying minerals