EE-145L Properties of Materials Laboratory

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1 Univesity of Califonia at Santa Cuz Jack Baskin School of Engineeing EE-145L Popeties of Mateials Laboatoy Sping 2003 Holge Schmidt Developed by Ali Shakouti, based on the notes by Pof. Emily Allen, San Jose State Univesity and Pof. David Rutledge, Califonia Institute of Technology. 1

2 Univesity of Califonia at Santa Cuz Jack Baskin School of Engineeing Electical Engineeing Depatment EE-145L: Popeties of Mateials Laboatoy Lab 1: Stuctue of Cystalline Solids Sping Leaning Objectives Holge Schmidt Afte successfully completing this laboatoy wokshop, including the assigned eading, the lab bluesheets, the lab quizzes, and any equied epots, the student will be able to: 1. Geneate a potential enegy cuve fom knowledge of the enegy function of a bonded system. 2. Detemine the bond enegy and equilibium bond length in a solid o molecule fom a potential enegy cuve. 3. Distinguish simple, face-centeed, and body-centeed elemental cubic cystal stuctues. 4. Distinguish between cystal stuctues with an elemental basis and those with moe complex bases (alloys o compounds). 5. Identify assigned planes and diections in a cubic solid using the Mille index notation. 2.0 Refeences S.O Kasap. Electical Engineeing Mateials and Devices, Chapte 1, Theoy of Atomic Aangements 3.1 Bonding Thee ae thee pimay types of bonds in cystalline solids: ionic, covalent, and metallic. The mechanical and electonic popeties of solids vay significantly depending on which type of bonding the solid has. Ceamic mateials have ionic bonds, which ae the stongest type of bonds, poducing vey had mateials. Semiconductos have covalent and sometimes ionic bonds which ae diectional and thus also vey had; metals have metallic bonds which ae spheically symmetical and thus allow easy movement of atoms, o defomation, to occu. 3.2 Metallic Bonding In metals, the bonds ae isotopic o spheical. Metallic bonding can only occu among a lage aggegate of atoms, such as in cystal. On the othe hand a covalent bond can occu between only 2

3 two atoms, in an isolated molecule. Fo example in face-centeed cubic and hexagonal closepacked metals, each atom has 12 neaest neighbos and thus is bonded in all diections. In bodycenteed cubic metals thee ae 8 neaest neighbos. The valence electons fom each atom ae shaed thoughout the cystal. The valence electons ae vey loosely attacted to the nucleus of the atom, and they ae spead out so fa fom the nucleus that the may be close to anothe nucleus in the solid. Thus all the electons ae hence fee to tavel thoughout the cystal, esulting in the lage electical conductivity of metals. The atoms in metals can slide easily by each othe, because the bonds ae not esticted to one diection o a stict angle, making it easy to defom most metals. This is why we can make so many stuctual pats fom metal. Most metal have a facecenteed o body centeed cubic stuctue, which povides the most dense packing of atoms, thus the highest density solids. 3.3 Covalent Bonding Covalent solids ae mainly fomed fom non-metallic elements. In covalent mateials, the bonded atoms shae electons between them. Most semiconductos ae covalent o mixed covalent and ionic. The atom must have a half-filled p-obital. Fo example, silicon, with 14 electons, is covalently bonded. Each silicon atom is bonded to 4 othes in a tetahedal bond, which leads to the diamond cubic cystal stuctue. The electonic stuctue of Si is 1s 2 2s 2 2p 6 3s 2 3p 2. When the fou Si atoms ceate tetahedal covalent bonds, the 3s and 3p electons fom a new set of hybid obitals called 3sp. Thus the electonic configuation becomes 1s 2 2s 2 2p 6 (3sp) 4. Gemanium (Ge) is anothe covalent semiconducto, with the stuctue 1s 2 2s 2 2sp 6 3s 2 3p 6 3d 10 (4sp) 4. Tetahedal bonds ae highly diectional and thee is little pobability of an electon being outside the vicinity of this bond. High tempeatue o othe souce o enegy is needed to emove an electon fom the stong covalent bond. This is why semiconductos have elatively low electical conductivities unless they have special impuities added. Because of the diectionality of the bond, atoms in a covalent solid cannot be easily displaced fom thei equilibium positions, making covalent solids vey bittle. 3

4 3.4 Ionic Bonding Solids with moe than one type of atom often possess ionic bonds. This includes ceamic mateials, such as oxides and silicates, as well as salts. In an ionic bond an electon is given by the cation to the anion; this then ceates an electostatic attaction between them, ceating a vey stong ionic bond. Electonegative atoms ae those that have a few empty p-obitals; they tend to acquie electons and become negative anions. Electopostive atoms have only a few electons in an oute shell, and tend to give up electons, becoming cations. Thus none of the atoms in an ionic solid ae neutal; all atoms in the cystal ae ions with eithe a plus chage (cation) o a minus chage (anion). The electon swapping lowes the enegy of the cystal by poviding each ion with an electon configuation close to a filled oute shell. Fo example in NaCl when the Na gives up one electon (and become Na + ), it has a filled 3s shell and becomes moe stable. When the Cl accepts the electon fom the Na (becoming Cl - ), it now has a filled 3p shell and is moe stable. Not all combinations of elements can fom ionic bonds: only pais which complement each othe can combine. It is difficult to defom ionic solids because of the stong electostatic foce between the ions. Thus ceamic mateials ae vey bittle and cannot defom easily in the solid sate. The electical conductivity in geneal is vey low because thee ae no fee electons to conduct cuent. Howeve, some ionic solids have ionic conductivity, in which small mobile ions can conduct cuent. Once the cation and anion have fomed, thee is an electostatic attaction between them. This attactive foce inceases as the ions come close to each othe. Howeve when the ions gets too close to each othe, thei electonic clouds stat to ovelap and a epulsive foce aises. At any given distance apat, thee is a net foce between the ions which is simply the sum of the attactive and epulsive foces. The net foce between the ions is plotted as a function of, the inteionic distance, in Figue 1. When the attactive and epulsive foce ae equal, the net foce is zeo, and the ions ae said to be at thei equilibium inteionic distance. This can be consideed to be the bond length in the solid 0, shown in Figue 1. 4

5 It is convenient to think about the potential enegy between the two ions instead of the foces. Since potential enegy (V) is the integal of foce (F) ove distance: V net = Fnetd = Fattactived + Fepulsived (1) V net = V A + V R The potential enegy of the pai deceases as they ae bought close togethe. The attactive enegy is consideed negative, since deceasing ( the inteionic distance makes the absolute value of the potential enegy lage. Thus the attactive potential can be expessed as: V atttactive = A (2) The epulsive enegy is consideed positive, since deceasing makes the epulsive enegy lage. It can be expessed as: V epulsive = B m (3) whee m has a value aound 8 o 9. The net total potential enegy can be witten in the fom: V total = A + B m (4) Figue 2 shows the plot of potential enegy vesus inteionic distance; this type of enegy function is known as potential well. The minimum in the cuve occus at 0, the bond length, and the value of the potential enegy at 0 is the bond enegy, V0. The deepe the well, the stonge the bond between the two ions. The lage the value of 0, the longe the bond length between the ions. Note by compaing Figues 1 and 2, that when the foce between the ions is zeo, the potential enegy is a minimum (not zeo). Even though not all solids ae ionically bonded, we can use this idea of a potential well to descibe loosely the potential enegy distibution between atoms as well as the equilibium inteatomic distance in all types of solids. 5

6 F Inteionic Foce Attactive Foce 0 Inteionic Distance Repulsive Foce Figue 1 V Inteionic Potential Enegy Inteionic Distance Figue Cystalline Solids When atoms come togethe to fom solids they may be aanged in many diffeent ways. In a cystalline solid the atoms ae aanged in a peiodic fashion and have long ange ode. By tanslating an atom o goup of atoms in thee dimensions a cystal stuctue is fomed. The cystal stuctue of a mateial is based on the cystal lattice which, is an aay of imaginay points in space. This aay of points is not abitay but follows a set of otational and tanslational ules. Each lattice point may have one o moe atoms, ions o molecules associated with it called a basis o motif. 6

7 The smallest goup of lattice points that displays the full symmety of the cystal stuctue is called the unit cell (see Fig. 1.37, p. 46 text). The unit cell has all the popeties found in the bulk cystal. The geomety and the aangement of lattice points define the unit cell. By tanslating the unit cell in thee dimensions the entie cystal stuctue is fomed. The geomety of a unit cell can be epesented by a paallelepiped with lattice paametes a, b, and c and angles α,β, and γ. By vaying the lattice paametes and angles, seven distinct cystal systems can be fomed. The seven cystal systems ae cubic, tetagonal, othohombic, hexagonal, hombohedal, monoclinic, and ticlinic. Thee ae 14 ways to place the lattice points in these systems to ceate Bavais lattices. Most of the metals, ionic salts, and semiconductos studied in this couse ae membes of the cubic cystal system. The cubic cystal system has lattice paametes a= b = c and angles α= β = γ = Theefoe, the lattice paamete is efeed to as a and the angles ae ignoed. The thee Bavais lattices associated the cubic system ae simple cubic (SC-sometimes called pimitive cubic), body centeed cubic (BCC), and face centeed cubic (FCC) (see Figs. 1.28, 1.29, pp text).. The distinction between the Bavais lattices is in the numbe and position of the lattice points. SC has a lattice point at each of the cube cones. BCC also has lattice points at its cones and one in the cente of the cube. FCC has lattice points at the cones and one point on each of the cube faces. The diffeent cystal stuctues that can be fomed fom these lattices depends on the basis o motif. The basis is the smallest numbe of atoms that can be placed at the lattice points to build the cystal stuctue. Evey lattice point has the exact same basis. Many of the metallic elements fom solids that ae BCC and FCC. The basis in the metal lattice is typically one atom centeed at each lattice point. Some stuctues have moe than one atom o ion associated with a lattice point. A quick calculation can help detemine the basis. Numbe of atoms in the basis = numbe of atoms in the unit cell. numbe of lattice points in the unit cell 7

8 This can be a tial and eo pocess if you do not know the cystal lattice. Howeve thee ae only 14 Bavais lattices and x-ay diffaction data can limit some of the choices. The numbe of atoms bonded to one paticula atom is called the coodination numbe. These ae the neaest neighbo atoms and ae assumed to be touching each othe. This is a good assumption fo building models of metals and ionic compounds but it is not the case fo covalently Bonded mateials. By using x-ay diffaction data the bond lengths can be detemined and the unit cell paametes calculated. The coodination numbe gives infomation about the envionment aound a paticula atom (i.e. electon enegy states and physical popeties). One popety that can be calculated fom knowing the aangements of atoms in the cystal stuctue and the adius of the atoms is the atomic packing facto (APF). The APF is the numbe of atoms in the unit cell multiplied by the volume of the atom and divided by the volume of the unit cell. Atomic Packing Facto = (#of atoms) x (atom volume)/unit cell volume) This is the amount of space that is occupied by atoms in the unit cell. Knowing the atomic weight of the element and the cystal stuctue, one can calculate the density of a mateial. An example of how the cystal stuctue can affect density is by compaing Ca and Rb. The element Ca has a FCC cystal stuctue and an atomic weight of The element Rb has a BCC cystal stuctue and an atomic weight of The density of Ca is 1.4 g/cm 3. The unit cell volumes fo Ca and Rb ae 1.72 x cm 3 and 1.85 x cm 3 espectively. The diffeence is that thee ae only 2 Rb atoms pe unit cell, while thee ae 4 atoms pe unit cell in Ca. 8

9 3.6 Identifying Planes and Diections in Cystals To undestand the popeties of cystalline mateials, we need a common way of discussing the symmety popeties of the cystal. Since the atoms o molecules ae aanged the same way thoughout the cystal, we can use cetain planes of atoms, which ae two-dimensional slices though the cystal, to descibe the cystal. Sometimes we also need to discuss cetain diections Though the cystal, because popeties may be anistopic, o diffeent in diffeent diections. 3.6a Identifying Cystalline Planes Mille indices ae commonly accepted method of identifying specific planes within a cystal. To find Mille indices, fist visualize o sketch the cystal stuctue of inteest. If the basis is a single atom, then dawing only the lattice points aanged on a coodinate axis will be sufficient. The placement of the oigin in a coodinate system is abitay, as long as we use the ight-hand ule. To detemine the indices of a specific plane, follow these steps: 1. Sketch the cystal lattice and mak the plane of inteest. 2. Assign an oigin and mak x, y, and z axes. 3. If the place eithe intesects all thee axes, o is paallel to one o moe of the axes, go on to step If the plane is not paallel to an axis, but does not intesect it, move the oigin until step 2 is fulfilled 5. Recod the value of each coodinate intecept, in factional fom. A plane which is paallel to an axis has an intecept of infinity. 6. Take the ecipocal of the intecepts and place them in paentheses. Negative intecepts have a ba ove the numeal. 7. Clea factions by multiplying by the least common denominato. 9

10 8. A plane is thus descibed by the indices h, k and l, as (hkl). These ae called the Mille indices of the plane. 9. In a cubic cystal, a family of planes is a set with the same thee indices, in any ode, and egadless of sign. Thus the goup o family of planes with the indices (hkl) may be genealized and witten {hkl}. Such a family will have the same measuable popeties on evey plane of that family. 3.6b Identifying Cystalline Dietions To identify a cystallogaphic diection, follow these steps: 1. Sketch the cystal lattice and mak the diection of inteest; it should be consideed a vecto with a specific diection. 2. Assign an oigin and mak the x, y, and z axes. 3. Move the vecto so that its tail is at the oigin; o move the oigin. 4. Recod the value of the pojection of the vecto onto each coodinate axis. If the vecto is nomal to an axis, its pojection is zeo. 5. Multiply though by the least common denominato and educe to integes. 6. Place the educed numeals in squae backets. Negative intecepts have a ba ove the numeal. 7. A diection is thus descibed by the indices [uvw]. 8. In a cubic cystal, a family of diection is a set with the same thee indices, egadless of sign, and in any ode. Thus the family of diections with the indices [uvw] may be genealized and witten <uvw>. Such a family will have the same meauable popeties in evey diection of that family. 10

11 4.0 Pelab Execises 4.1 Identifying Planes Ty identifying the planes shown below, then check you answes with the bottom of the page. Fo plane (a), notice whee the plane intesects x, y, and z-axis. In case (a) it is necessay to move the oigin to the font left cone. Then the intecepts ae 1,, and 1. We take the ecipocal of each intecept, esulting in the plane named: (101). 11

12 4.2 Identifying Cystalline Diections Ty the execise (a)-(e). Look at the diection epesented by (a). The x-, y-, and z-axis pojections ae ½, ½, 1. We multiply by the lowest common denominato 2, then suound squae backets, esulting in the diection named [112]. Ty the othe diections youself then compae to the answes below. In the next execises (f)-(h), some of the diections ae negative and some do not begin at the oigin of ou coodinate system. Fo example, look at the diection epesented by (f). Fist we need to move ou oigin to the cone whee the tail of the vecto is. Then the x-, y-, and z-axis pojections ae 1,0,-1. This esults in the diection named [101]. Ty the othe diection youself, then compae to the answes below. [111] 12

13 Lab Section 1 Coodination Numbe Using the Solid State Model Kits: Helpful Hints: 1. The two plastic bases have maks on them, one is yellow semicicle and the othe is a geen cicle. These symbols match the symbols on the letteed templates. 2. If the holes on the template do not match up, tun the template If you tied hint 2 and they still don t line up ty the othe base. 4. Do not foce the ods into the bases holes. They should slide in easily. 5. Do not foce the balls down the ods. 6. The colo of the balls used fo each model is displayed at the bottom of each page. 7. The numbes fo each laye of the model coespond to the balls at the bottom of each page. 8. At the top of the page thee ae instuctions fo building each model and the template you should use. Good Luck! Coodination Numbe (CN): Build the model fo CN 8,6, and 4 on page 93 of the Model Kit Manual. Build the model fo CN 8 pg.100. Build the model fo CN 4 on pg

14 Answe these questions about coodination numbes: 1. Which set of stuctues that you just built epesent compounds and why? 2. What is the maximum numbe of neaest neighbos you can have fo a stuctue with a single element? 3. How many neaest neighbos do an octahedal and tetahedal atom have? 14

15 Lab Section 2a Cystal Diections Execise I Daw the following diections in the cubic unit cells shown below: (A) [100] [010] [001] (all in the same unit cell) (B) [111] [111] [111] (all in the same unit cell) (C) [121] [112] [211] (all in the same unit cell) 15

16 Lab Section 2b Cystal Planes Execise 2 Daw the following planes in the cubic unit cells shown below: (A) (100) (010) (001) (B) (110) (101) (011) (C) (121) (211) (321) 16

17 Lab Section 3 Cystal Systems and Bavais Lattices: Build the models fo simple cubic (SC) pg. 9, Body Centeed Cubic pg., 18, and Face-Centeed Cubic pg.27 and answe questions in the table below. Simple Cubic Body Centeed Cubic Face Centeed Cubic # of atoms in the unit cell? # of lattice points in the unit cell? # of atoms pe basis? Coodination Numbe? Lattice Paamete a? Atomic Packing Facto? # of atoms in the [111] diection # of atoms on the (110) plane? Which plane has the highest atom density? 17

The geometric construction of Ewald sphere and Bragg condition:

The geometric construction of Ewald sphere and Bragg condition: The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied. This can be done as follows: i) Daw a wave vecto k in