材料科学基础
Fundamental of Materials
Prof,Tian Min Bo
Tel,62795426, 62772851
E-mail,tmb@mail.tsinghua.edu.cn
Department of Material Science and Engineering
Tsinghua University,Beijing 100084
Chapter Ⅴ
Imperfections in Crystals
Lesson seventeen
§ 5.1 Introduction
1,Classification:
Ideal or perfect crystal
real or defect crystal
According to dimension:
? point defect,vacancies,interstitial atoms
? line defect,dislocation
? planar defect,stacking fault
? volume defect,mosaic structure
2,Effects:
? physical properties (point defect)
? mechanical properties (dislocation)
? metallurgical properties (all of the defects)
§ 5.2 Point defects
? Ⅰ.Classification,
1,Schottky defect,2,Frenkel defect:
only vacancies pairs of vacancy and
interstitial atom.
? Ⅱ.Equilibrium concentration of
point defects.
Equilibrium concentration of point defects in
metal
)( mvvv SSTHG ???????
Suppose we introduce n vacancies into a crystal
containing N atom sites.
Determine,
N
nCv ?
)(P mvv SSTVUSTHG ?????????????
mS? —— Entropy of mixing
)]1l n ()1(ln[
)!(!
!
lnlnln
vvvv
n
Nm
CCCCNk
nNn
N
kCkkS
?????
???????
?
???? ?
)]1l n ()1(ln[ vvvvvv CCCCT N kUNCG ???????
U? —— Energy of formation of vacancy
0]
1
1)1l n ([ l n
0
?
?
????????
?
?
???
C
CC
C
CCN k TUN
C
G
GGG vv
? RT STHRT U
C
C v
v
v )(
1ln
????????
?
G
G0 C
C
mST??
(for FCC A≈1)or
?
?
?
nv,the number of vacancies per cm3.
Nv,the number of atoms per cm3.
For ionic crystals:
???, 正离子空位
,负离子空位
,正离子间隙
,负离子间隙
???
?i
?i
)e x p ()e x p ( S][][][][ RTURT UUCC ???????? ????
)e x p ()e x p ( F][][ RTURT UUCC ii ???????? ??? ?
Schottky
Frenkel
Example
1,Determine the number of vacancies needed for a
BCC iron crystal to have a density of 7.87g/cm3,The
lattice parameter of the iron is 2.866× 10-8cm.
Solution:
3
2338 cm/g8814.7)m o l/a t o m s1002.6()cm10866.2(
)m o l/g847.55)(c e l l/a t o m s( ?
??? ?
X?
9971.1847.55 )1002.6()10866.2)(87.7(c e l l/a t o m s
2338
?????
?
X
So,there should be 2.00－ 1.9971=0.0029 vacancies per unit cell.
20
38
3 1023.1
)cm10866.2(
c e l l/v a c a n c i e s0 0 2 9.0cm/V a c a n c i e s ??
??? ?
2,?Calculate the concentration of vacancies in copper
at room temperature (25℃ ).
? What temperature will be needed to heat treat
copper such that the concentration of vacancies
produced will be 1000 times more than the
equilibrium concentration of vacancies at room
temperature? Assume that 20,000 cal are required
to produce a mole of vacancies in copper.
Solution:
? The lattice parameter of FCC copper is 0.36151nm.
322
38 a t o m / c mc o p p e r1047.8)cm106 1 5 1.3(
c e l l/a t o m s4 ??
?? ?vN
38
3
22
cm/v a c a n c i e s10815.1
K298
Km ol
cal
987.1
m ol
cal
000,20
e x p)
cm
a t om s
1047.8(
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
??
?? RTU
vv
eNn
?
)987.1/(000,2022
/11
)1047.8(
108 15.1
T
RTU
vv
e
eNn
??
??
??
????
87.261047.8 108 1 5.1ln9 8 7.1 0 0 0,20 22
11
???????? T
C102K37587.26987.1 000,20 ?????T
3,In FCC iron,carbon atoms are located at octahedral
sites at the center of each edge of the unit cell (1/2,0,0)
and at the center of the unit cell (1/2,1/2,1/2),In BCC
iron,carbon atoms enter tetrahedral sites,such as
(1/4,1/2,0),The lattice parameter is 0.3571 nm for FCC
iron and 0.2866 nm for BCC iron,Assume that carbon
atoms have a radius of 0.071 nm.
? Would we expect a greater distortion of the crystal
by an interstitial carbon atom in FCC or BCC iron?
? What would be the atomic percentage of carbon in
each type of iron if all the interstitial sites were filled?
Solution:
?For BCC iron:
nm1 2 4 1.04 )2 8 6 6.0)(3(43 0BCC ??? aR
)()41()21( B C CB C C2020 Rraa ???
02567.02866.03125.03125.0)( 20BCCBCC ????? aRr
For FCC iron:
nm1263.04 )3571.0)(2(42 0FC C ??? aR
0F C CF C C )(2 aRr ??
nm0361.01241.002567.0BCC ???r
nm0 5 2 2.02 1 2 6 3.023 5 7 1.0FC C ????r
F C CB C C rr ?
Carbon distorts the BCC crystal structure more than the FCC crystal,
As a result,fewer carbon atoms are expected to enter interstitial positions
in BCC iron than in FCC iron.
?For BCC iron:
c e l lu n i tp e rs i t e sali n t e r s t i t i12)21)(s i t e s24( ?
If all of the interstitial sites were filled:
%86100a t o m sFe2a t o m sC12 a t o m sC12C%at ???? ％
For FCC iron:
c e l lu n i tp e rs i t e sali n t e r s t i t i4)41)(d e d g e s12( ?
If all of the octahedral interstitial sites were filled:
%50100a t o m sFe4a t o m sC4 a t o m sC4C%at ???? ％
? Ⅲ.Methods for obtaining extra -
equilibrium point defects
? Quenching
? Cool-working
? Bombardment by high energy particles
? Ⅳ,Experimental Determination of U?
cDCTBTA ???? 2?
??
??
TTT
?? ?
? ?? ??? ??
测
淬火
测 1
11 RT
U
DeDC
??
?????? ?? ???
1
lnln RTUD ???? ?
Examples and Discussions
Exercise
Thank you !
17
Fundamental of Materials
Prof,Tian Min Bo
Tel,62795426, 62772851
E-mail,tmb@mail.tsinghua.edu.cn
Department of Material Science and Engineering
Tsinghua University,Beijing 100084
Chapter Ⅴ
Imperfections in Crystals
Lesson seventeen
§ 5.1 Introduction
1,Classification:
Ideal or perfect crystal
real or defect crystal
According to dimension:
? point defect,vacancies,interstitial atoms
? line defect,dislocation
? planar defect,stacking fault
? volume defect,mosaic structure
2,Effects:
? physical properties (point defect)
? mechanical properties (dislocation)
? metallurgical properties (all of the defects)
§ 5.2 Point defects
? Ⅰ.Classification,
1,Schottky defect,2,Frenkel defect:
only vacancies pairs of vacancy and
interstitial atom.
? Ⅱ.Equilibrium concentration of
point defects.
Equilibrium concentration of point defects in
metal
)( mvvv SSTHG ???????
Suppose we introduce n vacancies into a crystal
containing N atom sites.
Determine,
N
nCv ?
)(P mvv SSTVUSTHG ?????????????
mS? —— Entropy of mixing
)]1l n ()1(ln[
)!(!
!
lnlnln
vvvv
n
Nm
CCCCNk
nNn
N
kCkkS
?????
???????
?
???? ?
)]1l n ()1(ln[ vvvvvv CCCCT N kUNCG ???????
U? —— Energy of formation of vacancy
0]
1
1)1l n ([ l n
0
?
?
????????
?
?
???
C
CC
C
CCN k TUN
C
G
GGG vv
? RT STHRT U
C
C v
v
v )(
1ln
????????
?
G
G0 C
C
mST??
(for FCC A≈1)or
?
?
?
nv,the number of vacancies per cm3.
Nv,the number of atoms per cm3.
For ionic crystals:
???, 正离子空位
,负离子空位
,正离子间隙
,负离子间隙
???
?i
?i
)e x p ()e x p ( S][][][][ RTURT UUCC ???????? ????
)e x p ()e x p ( F][][ RTURT UUCC ii ???????? ??? ?
Schottky
Frenkel
Example
1,Determine the number of vacancies needed for a
BCC iron crystal to have a density of 7.87g/cm3,The
lattice parameter of the iron is 2.866× 10-8cm.
Solution:
3
2338 cm/g8814.7)m o l/a t o m s1002.6()cm10866.2(
)m o l/g847.55)(c e l l/a t o m s( ?
??? ?
X?
9971.1847.55 )1002.6()10866.2)(87.7(c e l l/a t o m s
2338
?????
?
X
So,there should be 2.00－ 1.9971=0.0029 vacancies per unit cell.
20
38
3 1023.1
)cm10866.2(
c e l l/v a c a n c i e s0 0 2 9.0cm/V a c a n c i e s ??
??? ?
2,?Calculate the concentration of vacancies in copper
at room temperature (25℃ ).
? What temperature will be needed to heat treat
copper such that the concentration of vacancies
produced will be 1000 times more than the
equilibrium concentration of vacancies at room
temperature? Assume that 20,000 cal are required
to produce a mole of vacancies in copper.
Solution:
? The lattice parameter of FCC copper is 0.36151nm.
322
38 a t o m / c mc o p p e r1047.8)cm106 1 5 1.3(
c e l l/a t o m s4 ??
?? ?vN
38
3
22
cm/v a c a n c i e s10815.1
K298
Km ol
cal
987.1
m ol
cal
000,20
e x p)
cm
a t om s
1047.8(
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
??
?? RTU
vv
eNn
?
)987.1/(000,2022
/11
)1047.8(
108 15.1
T
RTU
vv
e
eNn
??
??
??
????
87.261047.8 108 1 5.1ln9 8 7.1 0 0 0,20 22
11
???????? T
C102K37587.26987.1 000,20 ?????T
3,In FCC iron,carbon atoms are located at octahedral
sites at the center of each edge of the unit cell (1/2,0,0)
and at the center of the unit cell (1/2,1/2,1/2),In BCC
iron,carbon atoms enter tetrahedral sites,such as
(1/4,1/2,0),The lattice parameter is 0.3571 nm for FCC
iron and 0.2866 nm for BCC iron,Assume that carbon
atoms have a radius of 0.071 nm.
? Would we expect a greater distortion of the crystal
by an interstitial carbon atom in FCC or BCC iron?
? What would be the atomic percentage of carbon in
each type of iron if all the interstitial sites were filled?
Solution:
?For BCC iron:
nm1 2 4 1.04 )2 8 6 6.0)(3(43 0BCC ??? aR
)()41()21( B C CB C C2020 Rraa ???
02567.02866.03125.03125.0)( 20BCCBCC ????? aRr
For FCC iron:
nm1263.04 )3571.0)(2(42 0FC C ??? aR
0F C CF C C )(2 aRr ??
nm0361.01241.002567.0BCC ???r
nm0 5 2 2.02 1 2 6 3.023 5 7 1.0FC C ????r
F C CB C C rr ?
Carbon distorts the BCC crystal structure more than the FCC crystal,
As a result,fewer carbon atoms are expected to enter interstitial positions
in BCC iron than in FCC iron.
?For BCC iron:
c e l lu n i tp e rs i t e sali n t e r s t i t i12)21)(s i t e s24( ?
If all of the interstitial sites were filled:
%86100a t o m sFe2a t o m sC12 a t o m sC12C%at ???? ％
For FCC iron:
c e l lu n i tp e rs i t e sali n t e r s t i t i4)41)(d e d g e s12( ?
If all of the octahedral interstitial sites were filled:
%50100a t o m sFe4a t o m sC4 a t o m sC4C%at ???? ％
? Ⅲ.Methods for obtaining extra -
equilibrium point defects
? Quenching
? Cool-working
? Bombardment by high energy particles
? Ⅳ,Experimental Determination of U?
cDCTBTA ???? 2?
??
??
TTT
?? ?
? ?? ??? ??
测
淬火
测 1
11 RT
U
DeDC
??
?????? ?? ???
1
lnln RTUD ???? ?
Examples and Discussions
Exercise
Thank you !
17
