Chapter Outline How do atoms arrange themselves to form solids? Types of Solids

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1 Chpter Outline How do toms rrnge themselves to form solids? Fundmentl concepts nd lnguge Unit cells Crystl structures Fce-centered cubic Body-centered cubic Hexgonl close-pcked Close pcked crystl structures Density computtions Types of solids Single crystl Polycrystlline Amorphous Types of Solids Crystlline mteril: toms self-orgnize in periodic rry Single crystl: toms re in repeting or periodic rry over the entire extent of the mteril Polycrystlline mteril: crystls or grins comprised of mny smll Amorphous: disordered lck of systemtic tomic rrngement Crystlline Amorphous.8.10 Crystllogrphy Not Covered / Not Tested.15 Anisotropy Not Covered / Not Tested.16 Diffrction Not Covered / Not Tested SiO Crystl structures Why do toms ssemble into ordered structures (crystls)? Energy of intertomic bond 0 Intertomic distnce Let s consider nondirectionl bonding (like in metls) Crystl structure To discuss crystlline structures it is useful to consider toms s being hrd spheres with well-defined rdii. In this hrd-sphere model, the shortest distnce between two like toms is one dimeter of the hrd sphere. 2R - hrd-sphere model We cn lso consider crystlline structure s lttice of points t tom/sphere centers. Energy of the crystl < Energy of the morphous solid 4 1

2 Unit Cell The unit cell is structurl unit or building block tht cn describe the crystl structure. Repetition of the unit cell genertes the entire crystl. Exmple: 2D honeycomb net cn be represented by trnsltion of two djcent toms tht form unit cell for this 2D crystlline structure Exmple of D crystlline structure: Metllic Crystl Structures Metls re usully (poly)crystlline; lthough formtion of morphous metls is possible by rpid cooling As we lerned in Chpter 2, the tomic bonding in metls is non-directionl no restriction on numbers or positions of nerest-neighbor toms lrge number of nerest neighbors nd dense tomic pcking Atomic (hrd sphere) rdius, R, defined by ion core rdius - typiclly nm The most common types of unit cells re fced-centered cubic (FCC) body-centered cubic (BCC) hexgonl close-pcked (HCP). Different choices of unit cells possible, we will consider prllelepiped unit cell with highest level of symmetry 5 6 Fce-Centered Cubic (FCC) Crystl Structure (I) Fce-Centered Cubic Crystl Structure (II) Atoms re locted t ech of the corners nd on the centers of ll the fces of cubic unit cell Cu, Al, Ag, Au hve this crystl structure R The hrd spheres touch one nother cross fce digonl the cube edge length, = 2R 2 The coordintion number, CN = the number of closest neighbors to which n tom is bonded = number of touching toms, CN = 12 Two representtions of the FCC unit cell Number of toms per unit cell, n = 4. (For n tom tht is shred with m djcent unit cells, we only count frction of the tom, 1/m). In FCC unit cell we hve: 6 fce toms shred by two cells: 6 1/2 = 8 corner toms shred by eight cells: 8 1/8 = 1 Atomic pcking fctor, APF = frction of volume occupied by hrd spheres = (Sum of tomic volumes)/(volume of cell) = 0.74 (mximum possible) 7 8 2

3 Fce-Centered Cubic Crystl Structure (III) Let s clculte the tomic pcking fctor for FCC crystl R = 2R 2 Fce-Centered Cubic Crystl Structure (IV) Corner nd fce toms in the unit cell re equivlent FCC crystl hs APF of 0.74, the mximum pcking for system equl-sized spheres FCC is close-pcked structure FCC cn be represented by stck of close-pcked plnes (plnes with highest density of toms) APF = (Sum of tomic volumes)/(volume of unit cell) 4 4 πr Volume of 4 hrd spheres in the unit cell: Volume of the unit cell: = 16 R 2 16 APF = πr 16R 2 = π 2 = 0.74 mximum possible pcking of hrd spheres 9 10 Body-Centered Cubic (BCC) Crystl Structure (I) Body-Centered Cubic Crystl Structure (II) Atom t ech corner nd t center of cubic unit cell Cr, α-fe, Mo hve this crystl structure The hrd spheres touch one nother long cube digonl the cube edge length, = 4R/ The coordintion number, CN = 8 Number of toms per unit cell, n = 2 Center tom (1) shred by no other cells: 1 x 1 = 1 8 corner toms shred by eight cells: 8 x 1/8 = 1 Atomic pcking fctor, APF = 0.68 Corner nd center toms re equivlent 11 12

4 Hexgonl Close-Pcked Crystl Structure (I) HCP is one more common structure of metllic crystls Six toms form regulr hexgon, surrounding one tom in center. Another plne is situted hlfwy up unit cell (c-xis), with dditionl toms situted t interstices of hexgonl (close-pcked) plnes Cd, Mg, Zn, Ti hve this crystl structure Hexgonl Close-Pcked Crystl Structure (II) Unit cell hs two lttice prmeters nd c. Idel rtio c/ = 1.6 The coordintion number, CN = 12 (sme s in FCC) Number of toms per unit cell, n = 6. mid-plne toms shred by no other cells: x 1 = 12 hexgonl corner toms shred by 6 cells: 12 x 1/6 = 2 2 top/bottom plne center toms shred by 2 cells: 2 x 1/2 = 1 Atomic pcking fctor, APF = 0.74 (sme s in FCC) All toms re equivlent c 1 14 Close-pcked Structures (FCC nd HCP) Both FCC nd HCP crystl structures hve tomic pcking fctors of 0.74 (mximum possible vlue) Both FCC nd HCP crystl structures my be generted by the stcking of close-pcked plnes The difference between the two structures is in the stcking sequence FCC: Stcking Sequence ABCABCABC... Third plne is plced bove the holes of the first plne not covered by the second plne HCP: ABABAB... FCC: ABCABCABC

5 HCP: Stcking Sequence ABABAB... Density Computtions Since the entire crystl cn be generted by the repetition of the unit cell, the density of crystlline mteril, ρ = the density of the unit cell = (toms in the unit cell, n ) (mss of n tom, M) / (the volume of the cell, V c ) Atoms in the unit cell, n = 2 (BCC); 4 (FCC); 6 (HCP) Mss of n tom, M = Atomic weight, A, in mu (or g/mol) is given in the periodic tble. To trnslte mss from mu to grms we hve to divide the tomic weight in mu by the Avogdro number N A = toms/mol The volume of the cell, V c = (FCC nd BCC) = 2R 2 (FCC); = 4R/ (BCC) where R is the tomic rdius Third plne is plced directly bove the first plne of toms Thus, the formul for the density is: na ρ = V c N A Atomic weight nd tomic rdius of mny elements you cn find in the tble t the bck of the textbook front cover Polymorphism nd Allotropy Some mterils my exist in more thn one crystl structure, this is clled polymorphism. If the mteril is n elementl solid, it is clled llotropy. An exmple of llotropy is crbon, which cn exist s dimond, grphite, nd morphous crbon. Single Crystls nd Polycrystlline Mterils Single crystl: toms re in repeting or periodic rry over the entire extent of the mteril Polycrystlline mteril: comprised of mny smll crystls or grins. The grins hve different crystllogrphic orienttion. There exist tomic mismtch within the regions where grins meet. These regions re clled grin boundries. Pure, solid crbon occurs in three crystlline forms dimond, grphite; nd lrge, hollow fullerenes. Two kinds of fullerenes re shown here: buckminsterfullerene (buckybll) nd crbon nnotube. Grin Boundry

6 Polycrystlline Mterils Polycrystlline Mterils Atomistic model of nnocrystlline solid by Mo Li, JHU Simultion of nneling of polycrystlline grin structure from (link is ded) Anisotropy Different directions in crystl hve different pcking. For instnce, toms long the edge of FCC unit cell re more seprted thn long the fce digonl. This cuses nisotropy in the properties of crystls, for instnce, the deformtion depends on the direction in which stress is pplied. Non-Crystlline (Amorphous) Solids In morphous solids, there is no long-rnge order. But morphous does not men rndom, in mny cses there is some form of short-rnge order. In some polycrystlline mterils, grin orienttions re rndom, so bulk mteril properties re isotropic Some polycrystlline mterils hve grins with preferred orienttions (texture), so properties re dominted by those relevnt to the texture orienttion nd the mteril exhibits nisotropic properties Schemtic picture of morphous SiO 2 structure Amorphous structure from simultions by E. H. Brndt

7 Summry Mke sure you understnd lnguge nd concepts: Allotropy Amorphous Anisotropy Atomic pcking fctor (APF) Body-centered cubic (BCC) Coordintion number Crystl structure Crystlline Fce-centered cubic (FCC) Grin Grin boundry Hexgonl close-pcked (HCP) Isotropic Lttice prmeter Non-crystlline Polycrystlline Polymorphism Single crystl Unit cell Homework #1: 2.14, 2.15, 2.20,.7, nd.17 Due dte: Mondy, September 6. Reding for next clss: Chpter 4: Imperfections in Solids Point defects (vcncies, interstitils) Disloctions (edge, screw) Grin boundries (tilt, twist) Weight nd tomic composition Optionl reding (Prts tht re not covered / not tested): Microscopy 4.11 Grin size determintion

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