Solid State 2 : Course Material - IIT-JEE, AIEEE, Boards Chapters & Discussions

Solid State #2 by Sanjay Sharma

(ii) With the particles in every next row are placed in the depressious between the particles of first row. The particles of third row will show vertical alignment with the first row as shown below -
Such a packing gives hexagonal pattern thus called HEXAGONAL CLOSE PACKING
  • The hexagonal close packing is more efficient than square close packing as here more space is occupied by spheres.
  • The above two packings under category (b) are considered as 2 dimensional packing. The packing which we will discuss now onwords are considered as 3 dimensional packings. As among 2 dimensional packings hexagonal close packings are more efficients, the 3 dimensional packings are based upon this hexagonal close packing patterns. In 2 dimensional packings the voids are also 2 dimensional similarly in 3 dimensional packings the voids are 3 dimensional.
  • The three dimensional packing may be of -
    (a) Hexagonal close packing (hcp)
    (b) Cubic close packing (ccp)
    (c) Body centred cubic packing (bccp)
  • The 2 dimensional hexagonal close packing observation show types of voids in it as shown below -
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    keeping both the types of void in mind in 3 dimensional packing when the IInd row of spheres is placed in such a way that
    its spheres find place in the voids marked X of first row (void marked Y left unoccupied) as shown below-
    In the second layer 2 types of voids seen again-
    (i) Those which lie above the spheres of first layer
    (marked l in above diagram)
    (ii) Those which lie above the voids Y of first layer
    (marked m in above diagram).
    Now when the third layer is placed in such a way that the spheres cover the l voids of second layer, thus a three dimensional cubic packing is obtained where the spheres in every 3rd or alternate layer are vertically aligned i.e.,, 3rd layer is directly above 1st or 4th layer is directly above 2nd.
    The arrangement is also called as ABAB ....... if we designate first layer as A and second layer as B as seen below-
  • Such a ABAB ..... packing is called HEXAGONAL CLOSE PACKING (3 dimensional) e.g., Be, Mg crystals etc.
  • Now, consider the same arrangment described above till 2nd layer, and if a 3rd layer is placed in such a way that spheres cover the void m of second layer, a layer different from A and B described above is seen. Let us call it as layer C. Continuing further, if the forth layer is placed over the 3rd layer in such a way that each sphere of it align vertically with the first layer, the packing is called CUBIC CLOSE PACKING with ABC, ABC ......... pattern of layers as shown below -
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    The packing is also called face centred cubic packing (fccp). The examples of ccp include Iron, Ni,Cu,Ag,Au,and Al etc.

  • Both the above patterns are almost equally efficient and occupy maximum possible space i.e., about 74% of the available volume with variable coordination nos. from 4 to 12 i.e., 4, 6, 8 and 12.
  • The bccp structure has a central atom occuping a lattice site with 8 atoms at the corner of cube. Here the space
  • occupancy is only 68% and coordination no. is almost constant i.e., 8. For example Li, Na, K, Rb and Cs.

  • In closely packed structures the empty space is called interstitial site or void. This void can be a simple triangular space in the case of 2 dimensional packing i.e., when all the spheres lie in same plane such voids present as triangular spaces in 2 dimensional packing patterns are called TRIGONAL VOIDS.
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  • In 3 dimensional close packing patterns the voids can be-
    (a) Tetrahedral (b) Octohedral
  • A tetrahedral void is a simple triangular space surrounded by 4 spheres as shown below -
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  • An octohedral void is a double triangular void surrounded by 6 spheres as shown below -
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  • The sizes of above written voids can be seen in increasing order as -
    Trigonal <>
  • Their sizes can be calculated on the basis of sizes of spheres involved in the formation of solids.
    (i) The size of triagonal void is 0.115 times of the radius
    of bigger sphere involved.
    (ii) The size of tetrahedral void is 0.2247 times of the
    radius of bigger sphere involved.
    (iii) The size of octohedral void is 0.414 times of the
    radius of bigger sphere involved.
  • In a closely packed structure containing x spheres (atoms, or molecules or ions) there are 8 x trigonal voids, 2 x tetrahedral voids and x octohedral voids.
  • The packing fraction i.e., the space occupied in different packings is as follows -
  • (a) In a simple cubic unit cell =21


    (b) In hexagonal close packing and face centre cubic

    structure= 23



  • The density of a unit cell and hence the density of a crystal is given by the formula.
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    Here, Z = No.of particles present per unit cell i.e., 1
    for simple, 2 for Bcc, 4 for Fcc.
    M = Atomic mass of element
    a = Edge of the unit cell
    NA= Avogadro number.
    for ionic crystals the formula used is same and the difference lies in -
    (1) Z = No. of formula units in one unit cell.
    (2) M = formula Mass (Molecular Mass) of the compd.
    (3) a = Edge which is 2 x distance between Na+ and Cl- in NaCl crystal.

  • The ionic radius ratios (of cation and anion) play a very important role in giving a clue to the nature of crystal structure of the ionic substance. This can be clear from the table given below -
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    The ionic crystals may be -

  • Formation of substitutional solid depends upon the electronic structure of impurity while that of latter on the size of impurity.
  • The number (n) of defects per cm3 is given by -
    n = N x ew/2RT
    Where N = no. of sites per cm3
    W = Work of energy requried to produce a
    T = Absolute temperature
    R = Gas constant
    e = base of natural logarithm
  • Solids are classified into 3 groups namely
    (i) Conductors with conductivity range of the
    order of 107 (Wm )-1 e.g Metals
    (ii) Semiconductors with conductivity range
    10-6 - 104(Wm)-1 e.g. Semi metals
    (iii) Insulators with conductivity range 10-10 -10-20
    (Wm)-1 e.g.Non metals
  • In most of the solids conduction is through electron movement under an electric field, however, in some ionic solids the conduction is through ions.
  • In metals conductivity strongly depends upon the no. of valence electrons available per atom. The atomic orbitals form molecular orbitals which are so close to each other as to form a BAND.
  • The conducitivity of solids can be better explained on the basis of energy gap present between the conduction band (HIGHER UNOCCUPIED BAND) and the valence band.
  • (i) In metals the conduction band is almost overlapping
    with the valence band i.e., there is no energy gap present between these two bands or valence band is not completely filled. Then electrons can flow easily under the influence of electric field in both the cases.

    (ii) In the case of semimetals the gap between valence band and conduction band is small and therefore some of the electrons may jump from valence band to conduction band and some conductivity is observed. The conductivity here increases in temperature. The reason for such an increase in lowering of energy gap shown below in the figure.
    The heat is actually responsible for ejection of electrons from their place thus leaving a (+)ve change their (positive hole). The crystal can now conduct electricity because on applying the electric field the electrons, and holes migrate in opposite directions. This type of conductance is called INTRINSIC CONDUCTANCE : For Example Si, Ge etc.
    NOTE :
    (1) For practical pruposes the conductivity of pure Si and Ge is very low at room temperature. In order to increase their conductance the pure substances are carefully doped (introduced with small amount of impurities in the form of elements of the 13 and 15 group of periodic table) .
    (2) The group 15 elements have 1 electron excess to Si or Ge after forming 4 covalent bonds with group 14 member (Si or Ge). this excess free electron is responsible for electrical conductivity in them. Group 14 elements when doped with group 15 elements are called n-type semiconductors. Here n specifies that negative charge flows in them.
    (3) The group 13 elements have 1 electron short to Group 14 elements, thus giving rise to electron deficient band or a hole. Here such holes are responsible for electrical conductivity. Thus group 14 elements when doped with elements of group 13, these are called p-type semiconductors. Here, p specifies that conduction is through positive holes in them.
    (4) Various combiantions of n-type and p-type semiconductors are used to make electronic components. For example, diode is a combination of p and n-type semiconductors and used as rectifier, TRANSISTORS which are pnp or npn sandwich semiconductor are used to detect or amplify radio or audio signals.
    (iii) In the case of non metals (insulators) the energy gap between valence band and conduction band is so large that it can not even covered up by supplying energy in the form of heat.
  • Besides conductivity solids also show magnetic properties and dielectric properties. On the basis of magnetic properties solids can be categorised to -
    (a) Diamagnetic i.e., which are feebly repelled by magnetic field. These have the characteristic absence of unpaired electrons in them i.e., their all electrons are paired. For Example - Non metallic elements (except O2 and S) inert gases and species with paired electrons e.g. T1O2, V2O5, C6H6 NaCl etc.
    (b) Paramagnetic i.e., which are attracted by magnetic field due to the presence of atoms, ions or molecules with impaired electron in them (e.g., O2, Cu++, Fe3+ etc.). In magnetic field these tend to orient themselves parallel to the direction of magnetic field. These are used in electronic appliances.
    (c) Ferromagnetic i.e., which show magnetism even in the absence of magnetic field, Fe, Co and Ni are 3 elements which show ferromagnetism at room temperature. A spontaneous alignment of magnetic moments in the same direction gives rise to ferromagnetism. The ferromagnetism is not seen above a temperature called CURIE TEMPERATURE.
    (d) Antiferromagnetic i.e., those which have net magnetic moment zero due to compensatory alignment of magnetic moments. For example MnO, MnO2, FeO,NiO, Cr2O3 etc.
  • The dielectric properties are seen in insulators, which, when are placed in on electric field, show generation of dipoles in them due to the pulling of electrons & nuclei of atoms or molecules in opposite directions. These dipoles- (a) may align themselves in an ordered manner so that
  • there is a net dipole moment in the crystal.
    (b) may align themselves in such a manner that dipole moments may cancel each other
    (c) It is also possible that there are no dipoles in the crystal but only ions are present.
  • The crystals where situation (a) is found exhibit PEIZOELECTRICITY or PRESSURE ELECTRICITY, which is production of electricity by a polar crystal when mechanical stress is applied to it. Peizoelectric crystals also show development of mechanical stress when electric field is applied to them, thus acting as a mechanical electrical transducer.
    Note :
  • 1. Peizoelectric crystals with permanent dipoles are said
    to have ferroelectricity e.f. Rochelle’s salt, BaTiO3,
    KH2PO4 etc.
    2. Peizoelectric crystals with zerodiople are said to have
    3. Some peizoelectric crystals, when heated produce
    small electric potential or PYROELECTRICITY
  • The phenomenon where electricity passes through a material without resistance is called SUPERCONDUCTIVITY. The material showing this property is said to be SUPERCONDUCTOR. It was KAMMERLINGH ONNES of Netherlands who first reported the case of Hg as superconductor at a very low temperature of 4.2 K. Afterwords some metals alloys and certain organic solids have been reported to be superconducting bUt at very low temperatures. Now a days superconductors are also reported at comperatively higher temperatures for example
    YBa2,Cu3O7 at 90K, Bi2 Ca2Sr2Cu3O10 at 105 K, and Tl2 Ca2 Ba2 Cu3O10 at 125 k etc


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