Each grain is a single crystal with a specific orientation. Imperfections

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1 Crystal Structure / Imperfections Almost all materials crystallize when they solidify; i.e., the atoms are arranged in an ordered, repeating, 3-dimensional pattern. These structures are called crystals and are discernible during microstructural characterization [by optical microscopy, x-ray diffraction, transmission electron microscopy, etc.] of metals. Microstructure is of critical importance to the properties of metals. Aspects of Microstructure : grain size and shape grain orientation phase(s) present size and distribution of second phases nature of grain boundaries and precipitates crystal imperfections Each grain is a single crystal with a specific orientation. Imperfections "FORTUNATELY crystals are seldom, if ever, perfect." These defects or imperfections include : thermal vibrations (0-D) point {vacancies, interstitials, substitutional,... } (-D) line {dislocations} (-D) surface --- {grain/phase boundaries, surfaces,... } (3-D) volume --- {amorphous, liquid state, voids,... } Without the imperfections, materials would have little ductility [malleability] and thus cannot be formed into useful shapes. Increase strength by impurities / second phases Diffusion processes are possible because of vacancies / interstitials Grain boundaries affect mechanical properties [both at low and high temperatures] KL Murty page NE409/509

2 Crystal Structure Lattice - of points is an arrangment of points in 3-D such that each point has identical surroundings Different Arrangements - 7 Systems (symmetry) & 4 Bravais Lattices Unit Cell - Building block describing the 3-D lattice by translation only Type of Arrangement Atomic Bonding Metallic Ionic Covalent Lattice Parameter / Lattice Constant - repeat distance along a, b, c (x, y, z) or (a, a, a 3 ) Relations between directions (a, b, c) and angles (α, β, γ) define the 7 systems Important Parameters SC BCC FCC hcp Number of atoms per unit cell Primitive vs Conventional 4 Coordination Number (CN) : number of equivalent nearest neighbors 6 8 R (atomic radius) vs a (lattice constant) : a = R 3 a=4r a=4r Packing Fraction (Atomic Packing Factor) : (no.atoms/u.c.) (vol.atom) APF (%) = vol. u.c. x00 5% 68% 74.% CPDs <00> <> <0> CPPs {00} {0} {} stacking of CPPs : fcc ABCABCABC vs hcp ABABAB (see figures) KL Murty page NE409/509

3 Diamond structure (Fig. 3.4) C, Si, Ge, etc : FCC with 4 interior atoms (FCC with atoms at each lattice point) CN=4, 8 atoms/uc, cannot be reduced to a primitive unit cell Polymorphism more than one crystal structure as T increases close packed structures transform to open structures In general : hcp fcc bcc as Temp increases (reverse as P increases) Notable exception Fe : bcc at RT (α iron) fcc (γ iron) bcc (δ iron) 90C 400C (see table on p.47 note on the bottom of the table) Zr : hcp (α) bcc NaCl Structure : (Na+ cation and Cl- anion) Fig. 3.0 KCl, MgO, UC a = (RNa + RCl) CN=6 4 ion pairs/uc (like fcc) fcc lattices CsCl Structure : (Cs+ cation and Cl- anion) Fig. 3. AgMg, RbCl, CuZn (β -Brass) a vs R CN=8 ion pair/uc (like sc) sc lattices Fluorite (CaF) Structure : (Ca++ cation and F- anion) Fig. 3. UO 4 Ca (fcc with a) and F (sc with a/) a = ( R + + R ) 4 CaF /uc (like fcc) 3 CN = 8 for Ca and 4 for F advantage over UC for nuclear applications KL Murty page NE409/509

4 Planes and Directions Miller Indices least integers with no common factors Directions : [i j k] <i j k> - family Planes : (h k l) {h k l} family reciprocals of intercepts on the x y z axes Directions are to Planes Examples : cubic group work hcp : hkil so that h+k+i=0 or hk l ---- examples (planes and then directions as normal to the planes) Cubic : Angle between directions : Cosθ = Intersection of planes (direction) : h h h + k + k + l k h + l l cross product of normals to the planes (h k l ) and (h k l ) : [h k l] = [h k l ]x[h k l ] Implies that if [h k l ] falls in a plane (h k l) [h k l ] [h k l]=0 + k + l Bragg s Law : nλ = d hkl sinθ where d hkl is plane separation Cubic : = 4 h + hk + k l + d hkl 3 a c hcp : = h + k + l d hkl a KL Murty page 3 NE409/509

5 KL Murty NE409/509 Kinetics of Vacancy Formation Why vacancies occur or exist? to disrupt the ordered structure or increase randomness increases entropy although work (energy) is required to form or create a vacancy (i.e., to move a host atom from interior to surface). Thus, Energy & Entropy minimize Gibbs Free Energy (G) dg=0 (F = Free Energy, H = Enthalpy) G = F + PV = H - TS = E + PV -TS [E] = ev, J per atom ( ev =.6x0-9 J); Q = Cal per mole & ev=3kcal/mole Equilibrium Vacancy Concentration : G(o) = GFE of perfect lattice and G(n) = GFE of lattice with n vacancies If N = Number of Atoms, the introduction of the vacancies increases the entropy S conf = k lnw configurational entropy with W being the number of different ways of arranging the N atoms and n v vacancies on N+n lattice cites : W = (N+n) C n = (N+n) C N = (N+n)! N! n!, where N! = N(N-)(N-)... Thus G(n) - G(o) = G = n g v - kt lnw, where g v is the GFE to form a vacancy. For thermal equilibrium, G ln W ln W = 0 = gv kt To show = ln N+n n n n n, use Stirling s approximation lnn! = NlnN - N for large N True for n also since in general n is also very large (millions) Thus, ln( n N+n ) = - g v kt n N n N+n = C v = exp(- g v kt ), since N>>n. C v is the vacancy lnc slope = E/k or Q/R concentration and g v = E v + PV v - TS v E v (ore f ) Arrhenius Equation high T /T low T KL Murty page 4 NE409/509

6 Divacancies At very high temperatures, many vacant lattice sites find neighboring site vacant too (Fig. 6. text) divacancies Ε () v = E v - B, B is the binding energy of a divancy (this is the energy required to separate a divacancy into two isolated monovacancies). C () v = β exp (-E v - B kt ), β is the cordination number see text Eq or C() v C v = β exp ( B kt ) Interstitials (self) Similary, n i N = C i = exp(- E i i kt ), n i = interstitial atoms and N i = No. of interstitial sites ( N, no. of atoms). E i >> E v thus the no. of interstitials (self) produced by thermal energy is relatively small. Example Calculation : For Copper (fcc) Q v = 0 kcal/mole B = 7 kcal/mole Q i = 90 kcal/mole T C (K) C v C () v Ci v RT (93).0x0-5.45x x (773).x0-6.80x x (346) (T M = 083) 5.65x0-4 ( vac per 769 atoms).63x0-5.43x0-5 *** ***Self interstitials are produced more easily by radiation KL Murty page 5 NE409/509

7 Ionics (MX and MX types) g f Schottky C vx = C vm = exp(- g C s vm = C im = exp( kt ) Frenkel kt ) C vx = C ix = exp( g f kt ) Divalent Impurity Extrinsic vs Intrinsic e.g. Cation (+ive) vacancies created due to addition of Ca ++ in NaCl charge neutrality. n ppm of Ca++ means nx0-6 Cation vacancies created (these are athermal and not dependent on temperature but equal to the divalent impurity concentration) lncvm C++ exp(-es/kt) /T Can get the relationship (C vm vs T) as before. q D N DM + q M N M = q X N x charge neutrality, where q M = q x (MX type). N DM + N M + N vm = N x + N vx no. sites for M+++M+ = no. X- sites Can do the same calculation for C + v for MX type (such as CaF and/or UO ) - see Text Experimental Determination of C v : Simmons-Balluffi Experiment Quenched-in Resistivity FIM - imaging of atoms (for low C v ) KL Murty page 6 NE409/509