ppt on crystal structure

Lecture on CRYSTALLINE SOLIDS

 SPACE LATTICE AND UNIT CELLS

 Space Lattice -- atoms arranged in a pattern

that repeats itself in three dimensions. 

Unit cell -- smallest grouping which can be translated in three dimensions to recreate the

space lattice.  

CRYSTAL SYSTEMS AND BRAVAIS LATTICES

  Seven crystal systems are each described by

the shape of the unit cell which can be translated to fill space.

Bravais lattices -- fourteen simple and complex lattices within the seven crystal

systems. The complex lattices have atoms centered

either in the center of a "primitive" unit cell or in the center of two/or more of the unit cell

faces. 

PRINCIPAL METALLIC CRYSTAL STRUCTURES

 We will concentrate on three of the more densely packed crystal structures, BCC - body centered cubic, FCC - face centered

cubic, and HCP - hexagonal close packed.

 BCC - 2 atoms per unit cell CN = 8 

0.68 = V

V = APF 3

R 4 = a

cell unit

atomsBCC

FCC - 4 atoms per unit cell CN = 12 

HCP - 6 atoms per unit cell CN = 12

0.74 = V

V = APF 2

R 4 = a

cell unit

atomsFCCFCC

0.74 = V

V = APF

; 1.633 = a

c ; R 3.266 = c ; R 2 = a

cell unit

atomsHCP

HCP

HCPHCP

BCC FCC HCP

EQUIVALENT SITES (ATOMIC POSITIONS) IN CUBIC UNIT CELLS

 Simple Cubic, SC - one per unit cell - corner

atoms only  (0,0,0) (1,0,0) (1,1,0) (0,1,0)

(0,0,1) (1,0,1) (1,1,1) (0,1,1) Body Centered Cubic, BCC - two per unit cell - corner atoms as above, plus (1/2, 1/2, 1/2) Face Centered Cubic, FCC - four per unit cell -

corner atoms as above plus (1/2, 1/2, 0) (1/2, 0, 1/2) ( 0, 1/2, 1/2)

(1, 1/2, 1/2) (1/2, 1, 1/2) (1/2, 1/2, 1) 

BCC FCC HCP

Lattice Sites in an Orthogonal Coordinate Systemi.e. Simple Cubic

DIRECTIONS IN CUBIC LATTICES 1 . Vector components of the direction are resolved along each of the coordinate axes

and reduced to the smallest integers. 2. All parallel directions have the same

direction indices.  3. Equivalent directions have the same atom

spacing.4. The cosine of the angle between two

directions is given by 

) l+ k+ h()l+k+h(

ll+kk+hh =

222222

cos

Indices of a Family or Form 

00]1[],1[000],1[0[001],[010],[100], >100<

negativessixtheplus

][],[],[],[],[],[ >< 110101011101011110110

][],[],[],[

][],[],[],[ ><

111111111111

,111111111111111

MILLER INDICES FOR CRYSTALLOGRAPHIC PLANES

 Definition: Miller Indices are the

reciprocals of the fractional intercepts (with fractions cleared) which the plane makes with

the crystallographic x,y,z axes of the three nonparallel edges of the cubic unit cell.

 ! cleared fractions with

c

1,

b

1,

a

1 = Indices Miller

Spacing between planes in a cubic crystal    

where dhkl = inter-planar spacing between

planes with Miller indices h,k,and l.

a = lattice constant (edge of the cube)h, k, l = Miller indices of cubic planes

being considered.

l + k + h

a = d

222hkl

CRYSTALLOGRAPHIC PLANES AND DIRECTIONS IN HEXAGONAL UNIT

CELLS  

Miller-Bravais indices -- same as Miller indices for cubic crystals except that there are

3 basal plane axes and 1 vertical axis.

Basal plane -- close packed plane similar to the (1 1 1) FCC plane.

contains 3 axes 120o apart.

Miller Bravais indices are h,k,i,l

with i = -(h+k). Basal plane indices (0 0 0 1) 

Prism planes -- {1 0 0} family  

Direction Indices in HCP Unit Cells -- [hkil] where h+k = -i

COMPARISON OF FCC, HCP, AND BCC CRYSTAL STRUCTURES

 Both FCC and HCP structures are close

packed APF = 0.74.  

The closed packed planes are the {111} family for FCC and the (0001) plane for HCP.Stacking sequence is ABCABCABC in FCC

and ABABAB in HCP.

BCC is not close packed, APF = 0.68. Most densely packed planes are the {110} family.

VOLUME, PLANAR, AND LINEAR DENSITY

 Volume density --

Planar density --   

Linear Atomic density --   

cell tvolume/uni

cell mass/unit = = metal ofdensity Volume v

plane of area selected

dintersecte centers atom # = =density atomic Planar p

line of length selected

dintersecte diameters atom # = =density atomic Linear l

X-ray Diffraction and Braggs Law

 n = order of reflection, whole number of

reflectionsλ = x-ray wavelength

dhkl = spacing between planes with indices

(hkl)θ = angle between incident x-ray beam and

crystal planes (hkl)

sin2 hkld = n

Figure 3.29

Figure 3.28

X-ray Diffraction and Braggs Law

 n = order of reflection, whole number of

reflectionsλ = x-ray wavelength

dhkl = spacing between planes with indices

(hkl)θ = angle between incident x-ray beam and

crystal planes (hkl)

sin2 hkld = n

For cubic crystals

l + k + h

a = d

222hkl

sin2154.0222 lkh

a = nm

sin2 hkld = n

1n nmKCu 154.0,

Selection Rules for Observing X-ray Peaks  

FCC : (h k l) must all be either odd or even

  BCC : sum h + k + l must be even

 (Otherwise, an in between plane will cancel

the reflection)

POLYMORPHISM OR ALLOTROPY  

Existence of more than one equilibrium crystallographic form for elements or compounds at different conditions of

temperature and pressure.  

Example: 

Iron liquid above 1539 C.δ-iron (BCC) between 1394 and

1539 C.γ-iron (FCC) between 912 and

1394 C.α-iron (BCC) between -273 and

912 C.