![Page 1: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/1.jpg)
Atomic Bonding and
Crystal Structure
![Page 2: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/2.jpg)
Metallic Bond
���������� ��� �����������������������������
������������ ��������������� !��!�ก����# ����$$
Electron ��-�ก�� ���������#������$�!��.�������ก����� ��# /�� � 0 1 � � �� 1 � $ $ � � Electron Charge Cloud
![Page 3: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/3.jpg)
.#�� ���8����# ก�$�����# ����# �!��.������ �����-��ก�ก���ก!�01�� ����� ��8���9�� �ก�� :��ก�� �!��.����/����1���;�����ก��-�0ก��ก!�01�� ����� ��8���9�� �ก�� �� ����� ก�;#����ก�!���ก����
�� ��8 � . � � �# � � � �# �!��.������ ��.#.#���� /��/<���0���� ���</���;� ������8�� ����ก�# =Interatomic DistanceC
Ra 20 =
![Page 4: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/4.jpg)
Bonding Energy = E0 - Emin.# Bonding Energy /���� ��8� ��:�/<����;���������� ������ ��8�
![Page 5: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/5.jpg)
Crystal Structure
![Page 6: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/6.jpg)
![Page 7: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/7.jpg)
![Page 8: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/8.jpg)
Principle Metallic Crystal Structures
Body Centered Cubic
BCCFace Centered Cubic
FCCHexagonal Closed Pack
HCP
![Page 9: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/9.jpg)
Body Centered Cubic
![Page 10: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/10.jpg)
a
a
a
aaa
2
2 222
=
=+
a
a2
( )a
aaaa
3
22 2222
=
+=+
![Page 11: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/11.jpg)
![Page 12: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/12.jpg)
Atomic Packing Factor (APF)
APF = total sphere volume
total unit cell volume
=
![Page 13: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/13.jpg)
Ra 43 =
( ) ( )
3
33
373.8
3
48
8
1
3
41
R
RRVatom
=
+
= ππ
3aVunit =
68.032.12
373.83
3
===R
R
V
VAPF
unit
atom
3
3
3
3
33
32.12
33
64
3
4
3
4
R
R
Ra
Ra
=
=
=
=
![Page 14: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/14.jpg)
Exercises
Lithium at 20 °C is BCC and has a lattice constant of 0.35092 nm. Calculate a
value for the atomic radius of lithium in nanometers.
![Page 15: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/15.jpg)
Face Centered Cubic
![Page 16: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/16.jpg)
![Page 17: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/17.jpg)
( ) ( )
3
33
3
16
3
4
8
18
3
4
2
16
R
RRVatom
π
ππ
=
+
=
3aVunit =
74.0216
3
16
3
3
===R
R
V
VAPF
unit
atom
π
Ra 42 =
3
3
3
3
3
33
216
2
2
2
32
22
64
2
4
2
4
R
R
R
Ra
Ra
=
=
=
=
=
![Page 18: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/18.jpg)
Exercises
![Page 19: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/19.jpg)
Hexagonal Close-Packed
![Page 20: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/20.jpg)
����������� !"#$"�%&�'&(��)�*ก,-- HCP
![Page 21: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/21.jpg)
caVcell 60sin3 2=
![Page 22: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/22.jpg)
Exercises
![Page 23: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/23.jpg)
),,( zyx −
./( comma %�#0 1233(����4�5-
Atomic Position in Cubic Unit Cell
6(��73�-.�(�734%&8#���-3(�0�0(�
![Page 24: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/24.jpg)
1. 09�:;3<7ก�3=���9�,�0��.0'�$$7�7 x, y ,� z $�4="�0.0��4�5-��$�9�3�- x,y,z
2. �3�ก9��03.�(,��� 3(�0=�� Cubic $"%��$���4!��ก�- 1
3. ��ก�9�,�0��$"%��0(��ก��� 1 .�(4="�0.0&>14?@'��0
Atomic Position in Cubic Unit Cell
�7A"ก�&
![Page 25: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/25.jpg)
Atomic Position in Cubic Unit Cell
![Page 26: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/26.jpg)
][ zyx
B$�./( comma %�#0 1233(�� Square Bracket
6(��73�-./(="33(�0-0
Direction in Cubic Unit Cell
![Page 27: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/27.jpg)
Direction in Cubic Unit Cell
1. 09�:;3<7ก�3 Destination ��C��-3(��:;3<7ก�3=�� Origin
2. 4$8#�B3()�����,�(�!9����4�=.�(41D0:9�0�04�5$ (ก9�:�34?@'��03(��ก�&09����'��0$�%>F)
3. 09�%��!"#B3($�4="�0.0 Square Bracket 4&"����$�9�3�- xyz �3�B$�$" comma
4. ��ก���4�=�73�-.�(4="�04%&8#���$���-3(�0-0=�����4�=
�7A"ก�&
![Page 28: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/28.jpg)
Direction in Cubic Unit Cell
![Page 29: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/29.jpg)
Direction in Cubic Unit Cell
]121[
],1,[2
],1,[
]0,01,0[)0,0,0(),1,(
21
21
21
21
21
21
21
21
=
×=
=
−−−=−
Example
![Page 30: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/30.jpg)
Direction in Cubic Unit Cell
]312[
],1,[3
],1,[)0,0,0(),1,(
31
32
31
32
31
32
=
−×=
−=−−
Example
![Page 31: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/31.jpg)
Direction in Cubic Unit Cell
]123)[
]101)[
]112)[
]110[]100)[
d
c
b
anda
Exercises
Draw the following direction vectors in cubic unit cells
![Page 32: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/32.jpg)
![Page 33: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/33.jpg)
[???] [???]
![Page 34: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/34.jpg)
Crystallographic Plane in Cubic Unit Cells
![Page 35: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/35.jpg)
1. ��:;3��3=��& 0�-!"#,ก0 xyz
2. ก��-4?@'��0=��:;3��3!�C�'�$3. ก9�:�34?@'��0,� !�04�=.�(�#9�!"#';34. ��ก���4�=�73�-.�(4="�04%&8#���$���-3(�0-0
=�����4�=5. 4="�04&"����$�9�3�-H��.0��4�5- �3�B$�$"
comma %�#0
�7A"ก�&
Crystallographic Plane in Cubic Unit Cells
![Page 36: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/36.jpg)
Crystallographic Plane in Cubic Unit Cells
)100()0,0,1(),,1( →→∞∞
)110()0,1,1(),1,1( →→∞
)111()1,1,1()1,1,1( →→
![Page 37: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/37.jpg)
)632()1,,3()1,,(23
32
31 →→
![Page 38: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/38.jpg)
)110)(
)221)(
)011)(
)101)(
d
c
b
a
Exercises
Draw the following crystallographic planes in cubic unit cells
Crystallographic Plane in Cubic Unit Cells
![Page 39: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/39.jpg)
![Page 40: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/40.jpg)
Home Work
![Page 41: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/41.jpg)
222 lkh
adhkl
++=
Interplanar spacing
![Page 42: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/42.jpg)
nm
lkh
adhkl
128.0
)0()2()2(
361.0
222
222
=
++=
++=
Interplanar spacing
ExampleCopper has an FCC crystal structure and a unit cell with a lattice constant of 0.361 nm. What is its interplanar spacing d220?
![Page 43: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/43.jpg)
Crystallographic Planes and Directions in Hexagonal Unit Cells
$" 4 ,ก0 %8� a1, a2, a3 ,� c./(���4�= 4 ���.0ก�&& -; direction ,� plane
![Page 44: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/44.jpg)
Crystallographic Plane in Hexagonal Unit Cell
![Page 45: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/45.jpg)
)1110(
)1,1,0,1(
)1,1,,1(
1
1
1
3
2
1
−
−∞
=
−=
∞=
=
c
a
a
a
![Page 46: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/46.jpg)
)0022(
)0,0,2,2(
),,,(21
21
3
21
2
21
1
−
∞∞−
∞=
∞=
−=
=
c
a
a
a
![Page 47: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/47.jpg)
∞=
−=
=
−=
∞−−
−−
4
3
21
2
1
21
1
1
),1,,1(
)0,1,2,1(
)0121(
a
a
a
a
![Page 48: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/48.jpg)
Crystallographic Direction in Hexagonal Unit Cell
![Page 49: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/49.jpg)
Home Work
Determine Miller-Bravais indices of
cratal plane in figure on the left.
![Page 50: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/50.jpg)
Family of Direction and Plane
]100[],010[],001[],001[],010[],100[
Some planes and directions are “Crystallographic equivalent”
><100
)001(),010(),100( }100{
Directions
Planes
![Page 51: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/51.jpg)
Comparison of FCC and HCP
![Page 52: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/52.jpg)
![Page 53: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/53.jpg)
Volume and Planar Density Calculations
Volume Density
3/
/
/
cmg
cellunitvolume
cellunitmassv
=
=ρ
![Page 54: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/54.jpg)
Volume and Planar Density Calculations
Cu has FCC crystal structure and an atomic radius of 0.1278 nm. Calculate a theoretical density of copper in g/cm3. The atomic mass of copper is 63.54 g/mol
gmolatoms
molgatomm
v
m
nmR
a
Ra
v
22
231022.4
/1002.6
)/54.63)(4(
361.02
)1278.0)(4(
2
4
42
−×=×
=
=
===
=
ρ
![Page 55: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/55.jpg)
3323
22
323
32939
3
98.81070.4
1022.4
1070.4
1070.4)10361.0(
cm
g
cm
g
v
m
cm
mm
av
v =××
==
×=
×=×=
=
−
−
−
−−
ρ
![Page 56: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/56.jpg)
Volume and Planar Density Calculations
Planar Density
2/
.
mmatoms
areaselected
atomofcenterofnoequivalentp
=
=ρ
![Page 57: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/57.jpg)
Volume and Planar Density Calculations
Calculate the planar atomic density on the (110) plane of α iron in atoms/mm2. The lattice constant of α iron is 0.287 nm.
( )( )
2
12
22
2
41
102.17
2.17)287.0(2
2
2
2
)(2
41
mm
atoms
nm
atoms
aaap
×=
==
=×+
=ρ
![Page 58: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/58.jpg)
Crystal Structure Analysis
![Page 59: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/59.jpg)
![Page 60: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/60.jpg)
Off-Phase
In-Phase
![Page 61: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/61.jpg)
θλ
θλλ
sin2
sin2
hkl
hkl
d
dn
PNMPn
=
=
+=
![Page 62: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/62.jpg)
X-Ray Diffractometer
![Page 63: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/63.jpg)
![Page 64: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/64.jpg)
![Page 65: 01 Crystal Structure - eng.sut.ac.theng.sut.ac.th/metal/images/stories/pdf/01_Crystal Structure.pdf · Copper has an FCC crystal structure and a unit cell with a lattice constant](https://reader031.vdocuments.pub/reader031/viewer/2022021612/5b93e24609d3f29e348c1ffb/html5/thumbnails/65.jpg)
Example
A sample of BCC iron was placed in an x-ray diffractrometer using incoming x-rays with a wavelength 0.1541 nm. Diffraction from the {110} planes was
obtained at 2θ = 44.704°. Calculate a value for the lattice constant a of BCC iron. (Assume first order diffraction with n = 1)
nmnm
lkhda
nmnm
d
d
hkl
hkl
hkl
287.0)414.1)(2026.0(
0112026.0
2026.0)3803.0(2
1541.0
sin2
sin2
35.22
704.442
222
222
==
++=
++=
==
=
=
=
=
θλ
θλθ
θo
o