Course Title : Metallic condensedCourse Title : Metallic condensed matters

Lecture 2: Crystalline structure of metals

Lecture Plan:Lecture Plan:1. PC, BCC, FCC, HCP lattices : unit cell, number of atoms per a

cell, coordination number. 2. Crystal system. Types of crystal lattices. y y yp y3. Conventions : Crystallographic points (basis), directions and

planes. 4. Reciprocal lattice, Evalds representation

Diff ti i lid d diff ti t h i5. Diffraction in solids and diffraction techniques6. Linear and planar densities7. Some properties of real pure metals connected with their

structure

1

structure

Important Terms and Concepts

Adjacent , Allotropy - Amorphous A i t

Diffraction grating face-centered cubic (FCC) -

Anisotropy - atomic packing factor (APF) body centered cubic (BCC)

grain - grain boundary hexagonal close packed (HCP)body-centered cubic (BCC)

-Braggs law collinear

hexagonal close-packed (HCP) Incident - ()i t i collinear

coordination number

isotropic - lattice - lattice parameters

crystalline - crystal structure

long-range order Miller indices noncrystalline -

crystal system Deceleration

noncrystalline - Polycrystalline -Polymorphism -

2

diffraction

Polymorphism single crystal

Unit cell is a smallest group of atoms possessing the symmetry of crystal which when repeated in all directions will develop the crystal

1. Primitive, BCC, FCC, HCP lattices : Definitions

crystal which when repeated in all directions will develop the crystal lattice. Unit cell is selected having the highest level of geometrical symmetry.

Number of atoms in a unit cell : Here are shown 8 unit cells of Primitive cubic type (PC).The atom A doesnt belong uniquely to one unit cell but is a part of all eight unit cellssurrounding it. Therefore it can be said that only one-eight part of the corner atom A belongst it ll 1 ll i t f 8 t th t i 1 t llto any one unit cell. 1 cell consists of 8 corner atoms, that is 1 atom per a cell.Close-packed directions and planes - on which atoms are as closely spaced as possible.In PC lattice 3 closed-packed directions.Coordination Number equals the number of nearest neighbors that an atom possesses inCoordination Number equals the number of nearest neighbors that an atom possesses inthe lattice. It means that the nearest atoms are located at surface of imaginary sphere withthe center in the atom location. The number of 1-st coordination sphere in PC is 6.Cell edge length (lattice parameter or lattice constant).F PC l tti i t f t i di R 2Ra =For PC lattice in terms of atomic radius R : 2Ra =Atomic packing factor (APF) ( ) is a sum of sphere volumes of all atoms within a unit cell (assuming the atomic hard-sphere model) divided by the unit cell volume

3

the unit cell volume.)1(

volumecellunit totalcellunit ain atomsofvolume

=APF

BCC lattice - body-centered cubic

Basic data:

BCC structure. a) representation of the model of atomichard-spheres, b-c) unit cell of BCClattice.

Basic data:

2 atoms per a cellClose-packed directions: 4 cube diagonalsClose-packed directions: 4 cube diagonals.Coordination number - 8APF equals 0.68.Cell edge length (lattice parameter or lattice constant) -

4Ra =4

Cell edge length (lattice parameter or lattice constant)3

a

FCC lattice face-centered cubic

(d)

FCC structure. a) representation of the model ofatomic hard-spheres, b-c) unit cell of FCClattice?d closed-packed directions

Basic data:In FCC lattice number of corner atoms in a cell is 8, numberof atoms at centers of grains is 6 and these grain centeredatoms are shared between 2 neighbor cellsatoms are shared between 2 neighbor cells.4 atoms per a cellClose-packed planes: 8, among these 4 crystallographically different

(c)

y g p yClose-packed directions 3 in each plane.Coordination number - 12APF equals 0.74

5

Cell edge length (lattice parameter) - 22Ra =

HCP lattice hexagonal close packed lattice

Hexagonal close-packed crystal structure. (a) a reduced-sphere unit cell (a and c represent the short and long edge lengths, respectively) (b) an respectively), (b) an aggregate of many atoms.

Basic data:In HCP lattice has the unit cell signified by letters ABCDEFHG, which contains 8 corneratoms and one internal atom.atoms and one internal atom.2 atoms per a cellClose-packed planes: horizontal face planesClose-packed directions 3 in each plane.p pCoordination number - 12APF equals 0.74Cell edge length (lattice parameter) - , the ratio a/c should be 2Ra =

6

1.633; for some HCP metals this ratio deviates from the ideal value.

Atomic Radii and crystal structures for 16 metals

7

Crystal system. Types of crystal lattices.

A unit cell with x, y, and z coordinate axes, showing axial lengths (a, b and c) and interaxial g g ( , )angles (, , ).

The unit cell geometry is completely defined in terms of six parameters: the three edge lengths a b and c of six parameters: the three edge lengths a, b, and c, being the lattice periods on each axis and the three interaxial angles (, , ). These are termed as lattice parameters of a crystal structureparameters of a crystal structure.

Unit cell volume [ ]cbaV =2

1222 )coscoscos2coscoscos1( += abcV

8caVHCP

286,0=3aVCubic =

Braves lattices

There are if we consider all realized ways of a unit cell filling there areThere are if we consider all realized ways of a unit cell filling there areknown 14 types of Brave lattices.

The Brave lattice is a group of translations characterizing the arrangement of material particles (atoms, group of atoms) in space.Selection of the Brave unit cell must satisfy three criteria.1.Symmetry of a unit cell corresponds to the symmetry of the crystal;unit cell edges are to be translations of the crystalline lattice.2.Unit cell must contain the maximal possible number of direct anglesor equal angles and edges of equal lengths.3. Unit cell must have the minimal volume.These conditions must be satisfied sequentially, i.e. the first is more i t t th th d d important than the second and so on.

There are seven different possible combinations of lattice parameters, each of which represents a distinct crystal system (or group of each of which represents a distinct crystal system (or group of translations) : cubic, tetragonal, hexagonal, orthorhombic, rhombohedral, monoclinic and triclinic.

9

Four types of sites arrangement in Bravais unit cell: primitive, base-centered, body-centered and face-centered.

14 Bravais lattices

10

Crystallographic points, directions, planes Conventions Conventions

Bases vectors in bcc, fcc and hcp lattices.

A massive of identical sites belonging to one unit cell where each site

C di ti f b it i C t i di t t

A massive of identical sites belonging to one unit cell, where each siterepresents its own primitive lattice, forms a basis of crystalline lattice. Axistranslation of base sites reconstruct the whole crystal.

Coordination of base sites in Cartesian coordinate system :in bcc [[ 000, ]] in fcc [[ 000, 0 ]] ,[[ 0 , 0]]in hcp [[000 ]]in hcp [[000, ]]

A site [[m n p]] is determined by radius-vector R = ma +nb +

11

A crystallographic direction is defined as a line between two points, or a vector.

Crystallographic direction

Directional indices are computed in terms of the vector projection on each of the coordinate axes.The following steps are used to determine the three directional indices: 1 k t l ti f th d i d t t th i i f th di t t1. make a translation of the a desired vector to the origin of the coordinate system; 2. measure the length of the vector projection on each of the three axes in terms of the unit cell dimensions a, b, and c;3 if these three numbers are multiplied or divided by a common factor to reduce them3. if these three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values;4. three indices are enclosed in square brackets, thus: [uvw].

T i if ll i l t di ti l b k t To signify all equivalent directions use angle brackets .For example to signify a family of all four directions ([] [],[],[]) write. Axial directions [100], [00], [010], [00], [001], and [00] are grouped as[ ], [ ], [ ], [ ], [ ], [ ] g p.

The [100], [110], and [111] directions within a unit cell. The Miller indices of x axis [100] , y [010], z [001]. Here The Miller indices of x axis [100] , y [010], z [001]. Here only the smallest integers that can give a direction parallel to the desired direction.

12

Planes are identified by sets of integers obtained from intercepts that the plane has with three axis Usually crystallographic planes are specified by

Crystallographic plane

plane has with three axis. Usually crystallographic planes are specified by three Miller indices as (hkl) : h :k :l = 1/m:1/n:1/k . Any two planes parallel to each other are equivalent and have identical indices.

Examples of a series of crystal-lographic planes

To signify all equivalent (i e having the

lographic planes (a) (001), (b) (110), and (c) (111).

To signify all equivalent (i.e. having the same atomic packing) planes use braces{hkl}. For example, in cubic crystals the

() ( ) () () () ()() ( ), (), (),(), (), planes all belong to the {111} family.

In cubic lattice indices of a plane coincide with indices of a normal

13to the plane.

Coordinate axis system for a Coordinate axis system for a hexagonal unit cell (MillerBravais scheme).

(a)the [0001], [ 1-100], and [11-20] directions;

(b)the (0001), (10-11), ( )

( ) ( ), ( ),and (-1010)planes. ( )= uu 231

( )u= 231( )

tuu =( )+= ut

ww =

++

khkh

u

22

1~'

'

tuu =t=

Conversion from the three-index (Miller) system to the four-index (Miller-Bravais) system ( )

+

+

khl

ca

khw

223

2'

2

2ww =

The reciprocal

14

(Miller Bravais) system, [u w ] [ u t w]

( )conversion the normal to a plane

Reciprocal latticeReciprocal lattice is a mathematical image used for establishing of relation p g gbetween different parameters of a crystalline lattice which define physical properties of the crystal.

cba rrr ,, - primitive vectors of translation in ordinary lattice, so basic vectors of the reciprocal lattice are

rr brr

bb rrrr

)(*

cbaacb rrrrr

= )(*

cbabac rr

r

=

h th d i t i l f it ll i th di l tti

Vcb

cbacba rr

r =

=)(

*

where the denominator is a volume of unit cell in the ordinary lattice

1*** === ccbbaa rrrrrr

*c1ccbbaa0****** ====== bcaccbabcaba

rrrrrrrrrrrr

c

*c

a b15

a

The rule : Vector with coordinates [hkl]* in reciprocal lattice is normal to the atomic plane with the same indices (hkl) in ordinary lattice. The p ( ) yplane (UVW)* in reciprocal lattice has a normal with coordinates [UVW] in atomic lattice.

A scalar product of any vector lying in the plane (hkl) and a vector of reciprocal A scalar product of any vector lying in the plane (hkl) and a vector of reciprocal lattice with indices [hkl] gives zero.

( ) 0*** = ++= accWbVaUACGUVWrr

rrrr ( ) 0

++hl

cWbVaUACGUVW

( ) 0*** =

++=

ha

kbcWbVaUABGUVW

rrrrrr ( )

hk

UVW

0=

=

lW

hUACGUVW

r

lh0=

=

hU

kVABGUVW

r

Orientation of the reciprocal space n

lW

kV

hU

===nhU =nkV =

n=1,

16

Orientation of the reciprocal space vector GUVW and corresponded plane (hkl) in direct space

lkh nkVnlW =

Thus, the reciprocal lattice is a massive of sites, every of which is related to the family of parallel atomic planes and has the same indices .

The rule: Any family of atomic planes (hkl) with interplanar spacing dThe rule: Any family of atomic planes (hkl) with interplanar spacing din reciprocal lattice relates to the vector normal to these planes andhaving coordinates hkl and length 1/d.

hkl dGRrr( ) ndnR =0r

r

hklhkl

hkl ndG

=r( )hklGhkl r

( )hkl

hkl

ndGh

clbkahna=

++r

rrrr ***

Orientation of a vector G of hklhkl G

d r1=

17

Orientation of a vector G of reciprocal lattice and perpendicular planes of direct lattice

hklG

Reciprocal lattice of BCC, FCC and HCP-lattices

( ) ( )*|| hklhklIn cubic lattice [ ] [ ]*|| hklhkl

18

DiffractionThe diffraction patters can be formed as a result of The diffraction patters can be formed as a result of interaction with crystalline lattice of beams with wave -lengths comparable with interatomic spacing (10-10m).

( ) ( ) EkeVEhc /4.12==& wavelength of 1A corresponds to the

energy of ~ 12000 eV.

X-rays are generated both by deceleration of electrons in metal targets and by inelastic excitation if the core electrons in the atoms of the target The first process gives a broad continuous spectrum the second target. The first process gives a broad continuous spectrum, the second gives sharp lines. The radiation from a copper target bombarded by electrons shows a strong line (K1 line) at 1.541A.

( ) ( )eVEmEh

mh /12

2==

& for electrons wavelength of 1 A

corresponds to energy (~150eV)

( ) ( )eVEEM

hM

h /28.02

==

& for neutrons diffraction demandsmore less energy as 0 08eV

19

EMM nn 2 more less energy as 0.08eV

Constructive interference of theradiation reflected fromsuccessive planes occurswhenever the path difference isan integral number n ofwavelengths , where n = 1,2,3, d - interplanar spacing.

sin2 hkldQTSQn =+=

d i2 (Bragg law) nd =sin2 (Bragg law)The magnitude of the distance between two adjacent and parallel planes of atoms (i.e., the interplanar spacing dhkl) is a function of the Miller indices ( , p p g hkl)(h, k, and l) as well as the lattice parameter(s).

ad hkl = for cubic structures20

222 lkhd hkl

++for cubic structures

Evald representation

Change of wave vector at mirror reflection from plane (hkl)

For X-ray irradiation with wave length the condition of Bragg is executed for all points of reciprocal lattice those lye on Evald sphere from plane (hkl) reciprocal lattice those lye on Evald sphere with radius equal to 1/. Here direction

k r

is a possible way of diffraction.kr

- incident vector

k r

fl t d t

hklhkl Gdr

/1=

k - reflected vector

kGk hklrrr

+=' in vector form21

hkl

022 =+ hklhkl GkGrr

in scalar form

Schematic diagram of an X-ray diffractometer; ;T monochromatic X-ray source, S plate-shaped specimen, C - detector, O -the axis around which the specimen and detector rotate.

Diffraction pattern for a polycrystalline specimen of -iron. The high-intensity peaks result when the Bragg diffraction p ggcondition is satisfied by some set of crystallographic planes.

22

Theoretical density, linear and planar densities

AC NVnA

= where NA is Avogadros number 16.0221023 atoms/mol,

VC - volume of the unit cell; A - atomic weight in g/mol; n - number of atoms associated with each unit cell.

vectordirectiononcenteredatomsofnumbervectordirection oflength

vectordirection on centered atomsofnumber =LDLinear density

In fcc-lattice LD 110 = 2 atoms / 4R = 1/(2R)

planeofareaplane aon centered atoms ofnumber

=PDPlanar density

23

p

In fcc-lattice PD 110 = 2 atoms / (8R2 )= 1/(4R2 )2 2

Polymorphism and allotropy

Single crystals are materials in which the atomic order extends uninterrupted over the entirety of the specimen; under some circumstances, single crystals may have flat faces and regular geometric shapes.

The vast majority of crystalline solids, however, are polycrystalline, being composed of many small crystals or grains having different crystallographic orientations Particularly some metals may have more than one crystal structure orientations. Particularly, some metals may have more than one crystal structure, this phenomenon is known as polymorphism.

Schematic diagrams of the various stages in the solidification of a polycrystalline material; the square grids depict unit cells (a) Small the square grids depict unit cells. (a) Small crystallite nuclei. (b) Growth of the crystallites; the obstruction of some grains that are adjacent to one another is also shown (c) Upon completion of solidification shown. (c) Upon completion of solidification, grains having irregular shapes have formed. (d) The grain structure ; dark lines are the grain boundaries.

When found in pure elemental solids, polymorphism is often termed as allotropy. The prevailing crystal structure depends on both the temperature and the external pressure.p p

Tin () has 2 allotrophic modifications: white (or ) tin, having a body-centered tetragonal crystal structure at room temperature, transforms at13 2 C ( ) i hi h h l i il di d (i13.2oC to gray (or ) tin, which has a crystal structure similar to diamond (i.e.,the diamond cubic crystal structure). The rate at which this change takes placeis extremely slow; however, the lower the temperature (below 13.2oC) thefaster the rate. Accompanying this white-to-gray-tin transformation is anfaster the rate. Accompanying this white to gray tin transformation is anincrease in volume (27%), and, accordingly, a decrease in density (from 7.30g/cm3 to 5.77 g/cm3). Consequently, this volume expansion results in thedisintegration of the white tin metal into a coarse powder of the gray allotrope.