Engineering Physics-I Crystal Physics- Atomic rad., Coord. No.,APF for SC and BCC

Transcription

1 Intoduction Most of the mateials in solid state ae cystalline. Among these many ae in the polycystalline state. To obtain single cystal one has to employ a suitable cystal gowth method. This may vay fom one mateial to anothe. To undestand the diffeent mateial popeties, the stuctue of the mateial is vey essential. The unit cell which is the building block of a cystalline mateial can be made of the basic units namely the basis o the patten unit o the motif. In this section we study the numbe of atoms pe unit cell, how they ae aanged in simple stuctues to ende the atomic adius, the numbe of fist neaest neighbous and how the basic units ae packed to fom diffeent cystal stuctues. Leaning Objectives On completion of this chapte you will be able to: 1. calculate the numbe of atoms pe unit cells of simple stuctues 2. define atomic adius and coodination numbe 3. deive the packing factos fo SC and BCC Calculation of numbe of atoms pe unit cell The numbe of atoms pe unit cell depends on the cystal stuctue and the type of basic elements (basis o patten unit o motif) that fom the cystal. Fo example: The numbe of atoms in a simple cubic (SC) cystal is one in the case of polonium. The numbe of atoms in a body cented cubic (BCC) cystal is two in the case of sodium. In SC unit cell, the atoms occupy only the cones. In BCC unit cell, the atoms occupy the cones as well as the body cente of the unit cell. In FCC unit cell, the atoms occupy the cones as well as the face cente of the unit cell. In the case of hexagonal closed packed (HCP) stuctue, atoms occupy the 12 cones, the cente of the bottom and top faces and the body cente of the altenate one sixth of the unit cell. The atoms available at the cones and in the face centes ae shaed by moe than one unit cells. Only the atoms pesent well within the unit cell solely belong to the unit cell. Thus the numbe of atoms pe unit cell is equal to the poduct of 'the numbe of atoms pe basis' and 'the numbe of basis pe unit cell'. In geneal, if the basis has n numbe of atoms and the numbe of basis pe unit cell is m, then the numbe of atoms in the unit cell will be mn. The numbe of atoms in the FCC, HCP and diamond unit cells ae espectively 4, 6, and 8. Atomic adius In solid state physics, the atomic adius is half the distance between two neaest neighbou atoms. In the case of SC all the adjacent cone atoms in a cubic unit cell ae touching each othe along the edges (Fig.1). Let 'a' and '' be the side of the cubic unit cell and adius of the atoms in the unit cell espectively. The atomic adius in the case of FCC, and HCP ae 2a/ 4 and a/2 espectively. Mateial pepaed by: < Physics faculty > Session No: < 3 > Page 1 of 5

2 a Figue 1 In SC, 'a' and '' ae elated by, = a/2 Figue 1 In BCC the atoms along the body diagonal ae touching each othe as shown below (Fig.2). Figue 2 Mateial pepaed by: < Physics faculty > Session No: < 3 > Page 2 of 5

3 Co-odination numbe: In BCC, 'a' and '' ae elated by, = 3a/4 It is the numbe of neaest neighbou atoms with espect to any atom in the lattice. It vaies with one stuctue to anothe. As this numbe inceases the density of packing of the atoms/ions/molecules in the available space to fom a cystal stuctue inceases. The coodination numbes fo SC, BCC, FCC, HCP and the diamond stuctue ae 6, 8, 12, 12, and 4 in the same ode. Packing facto: volume occupied by the atoms of the unit cell v Definition: packing facto= volume of the unit cell V Packing facto fo SC: v = Numbe of atoms in the unit cell x volume of one atom = 1 x Fo SC, = a/2 This is obtained by consideing the atoms along a cube edge (Fig.3). 2 = a Figue 3 V =a 3 Substituting the expessions fo v and V in the packing facto expession and simplifying, one gets packing facto= 6 =0.52 This indicates that 52 % of the available space is occupied by the atoms foming the SC lattice. Mateial pepaed by: < Physics faculty > Session No: < 3 > Page 3 of 5

4 Packing facto fo BCC: v = Numbe of atoms in the unit cell x volume of one atom = 2 x a Figue 4 Fo BCC, = 3a/4 This is obtained by consideing the atoms along a body diagonal (Fig.4). V =a 3 Substituting the expessions fo v and V in the packing facto expession and simplifying, one gets packing facto= 3 8 =0.68 This indicates that 68 % of the available space is occupied by the atoms foming the BCC lattice. Thus BCC is moe dense than SC. Mateial pepaed by: < Physics faculty > Session No: < 3 > Page 4 of 5

5 Check you undestanding 1. State tue o false: The coodination numbe and packing facto ae connected someway. 2. State tue o false: Does the coodination numbe decide the mechanical stength of the mateial? 3. State tue o false: Do the atomic adius of a mateial dependent on stuctue? 4. What decides the density of a mateial? 5. Can one have diffeent cystal stuctues with the same stating mateial? If not, why? Summay On completion of this chapte you have leaned : 1. how to calculate the numbe of atoms in SC, BCC, FCC and HCP 2. to define atomic adius, coodination numbe and packing facto 3. how to calculate the packing factos fo SC and BCC cystals. Suggested Reading 1. Engineeing Physics- P.K.Palanisamy (SCITECH PUBLICATIONS (INDIA) PVT. LTD., Chennai) 2. Foundations of mateials science and engineeing William F. Smith (McGaw Hill) 3. Elements of mateials science and engineeing- Lawence Van Vlack (Peason Education India). Mateial pepaed by: < Physics faculty > Session No: < 3 > Page 5 of 5