物质结构
（ 1） 完美晶体的结构描述：晶体学对象：原子排列方式 （ 简化；理想；几何 ）
=,周期 。
晶体结构＝结构单元 （ 单胞 ） ＋单胞周期平移二、晶体结构对称操作：
使图形保持不变 （ 完全复原 ） 的操作；
在对称操作中始终保持不变的轴,平面,或点称为对称元素 。
单胞内的原子位置由对称性操作联系 。
对称操作分点对称和平移两种 。
点对称操作,1,2,3,4,6,m,1
平移对称性,14种 Bravais点阵单胞位置由平移对称性联系 。
对称性阵点 （ 几何点代替结构单元 ） 和点阵 （ 阵点的分布总体 ）
注意与晶体结构（ =点阵 +结构单元）的区别。
点阵与晶体结构点阵与晶体结构：例子
a-Fe,bcc
Steps to reach lattice
1,determine the basic unit
2,regard the unit as a point
3,the geometry of the points = lattice
a-Fe
1,the basic unit,one Fe atom
2,regard the unit as a point
3,the geometry of the points =
Body centered cubic lattice
a b
c
点阵与晶体结构：例子
CsCl,simple cubic
Steps to reach lattice
1,determine the basic unit
2,regard the unit as a point
3,the geometry of the points = lattice
CsCl
1,the basic unit,one Cs atom + one Cl
2,regard the unit Cs + Cl as a point
3,the geometry of the points =
simple cubic lattice
Cs
Cl
点阵与晶体结构：例子
g-Fe,fcc
Cu3Au,simple cubic a
b
c
点阵与晶体结构：例子
g-Fe,fcc
CuAu,tetragonal a
b
c
晶体结构例子
a
b
c
x
y
z
g-Fe a-Fe
非晶结构与晶体结构比较二维准晶模型,Penrose 拼图
Al-Cu-Fe单晶外形
3-fold
2-fold
5-fold
T2Fe二十面体准晶八次准晶的八次轴十次准晶的十次轴
g-Fe,fcc
Cu3Au,simple cubic
CuAu,tetragonal
有序化
a-Fe,bcc CsCl,simple cubic
Cs
Cl
36.87o/001
S=5
i
j
i’j’
k
j
i
M
k
j
i
R
'
'
'
缺点：不连续
S=2.5
Composition rules based on clusters in
quasicrystals and metallic glasses
Chuang DONG
Dalian University of Technology,China
Both quasicrystals and metallic glasses are multi-
component compounds with narrow composition ranges:
Example for stable quasicrystals
– i-Al62.5Cu24.5Fe13,i-Al69Pd22Mn9,d-Al67.5Pd14Mn18.5,
i-Zr45Ti38Ni17 …
Example for,good” metallic glasses:
– Binary,Zr-Ni (Zr76Ni34),Zr-Cu…
– Ternary,Zr60Al20Ni20,Zr65Cu27.5Al7.5,
– Quaternary,Zr65Al7.5Ni10Cu17.5 (Inoue alloy),
Ti34Cu47Zr11Ni8
– More,Zr52.5Ti5Cu17.9Ni14.6Al10,
Zr41.2Ti13.8Cu12.5Ni10Be22.5,Zr58.5Nb2.8Cu15.6Ni12.8Al10.3
Basic rule:
alloying
100
90
80
70
60
50
40
30
20
10
100
90
80
70
60
50
40
30
20
10
100 90 80 70 60 50 40 30 20 10
B
A C
AxBy
AuBvCw
Composition rules in ternary quasicrystals
b
(Cu)Al4Cu9Al
7Cu2Fe
Al10Cu10Fe
IQC
θ
ζ 2
Cu (+1)Al (+3)
Fe（ -2）
1.62.4
0.7
0.4
0.1
-0.2
-0.5
-0.8
-1.1
-1.4
-1.7-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
2.62.8 2.0 1.8 1.4 1.22.2
Al84Fe16
e/a = 1.86
The e/a-constant line
in ternary phase
diagram:
Criterion 1:
e/a-constant line
b
Al13Fe4
(Cu)Al4Cu9Al
7Cu2Fe Al
10Cu9F
e
IQC e/a-constant
e/a=1.86α
θ
ζ 2
Cu(+1)Al (+3)
Fe（ -2）
1.62.4
0.7
0.4
0.1
-0.2
-0.5
-0.8
-1.1
-1.4
-1.7-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Al-Cu-Fe e/a diagram
2.62.8 2.0 1.8 1.4 1.22.2
Al5Fe2
b
Al13Fe4
(Cu)Al4Cu9Al
7Cu2Fe Al
10Cu9F
e
IQC e/a-constant
e/a=1.86α
θ
ζ 2
Cu(+1)Al (+3)
Fe（ -2）
1.62.4
0.7
0.4
0.1
-0.2
-0.5
-0.8
-1.1
-1.4
-1.7-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Al-Cu-Fe e/a diagram
2.62.8 2.0 1.8 1.4 1.22.2
Al5Fe2
e/a-variant line,binary QC,ternary QC,and the 3rd element.
e/a-variant line
Criterion 2:
e/a-variant line
Al-Pd-Mn成分相图 （ at.%）
Al-Pd-Mn：
Ico-Al6Mn
Dec-Al4Mn
b
a
c
Al-Ni-Fe,Exception?
Qiang et al,J,Mater,Res.,2001,16; 2653-2660,
Al84Fe16
Al4Ni
Controversial reports on Decagonal quasicrystal compositions
a,Al75-70Fe9-21Ni16-9,e/a = 2.07,1.89,1.82,1.68 (typical values)
Saito,Tanaka,Tsai,Inoue,Masumoto,Jpn J,Appl,Phys.,1992
b,Al75Fe22.9Ni3.1,e/a = 1.79
,Dunlop,Phil Mag B,1993
c,Al71Fe5Ni24,stable,e/a = 2.03 (TOO LARGE!)
Lemmerz,Grushko,Freiburg,Jansen,Phil,Mag,Lett.,1994)
c
Al-Ni-Fe：
Qiang et al,J,Mater,Res.,2001,16; 2653-2660,
0
5
10
15
20
25
0 10 20 30 40 50 60 70
Ni (at.%)
Fe (at.%)
Al3Ni5Al
3Ni
Al13Fe4
D-Al84Fe16
Al71Fe5Ni21
e/a =1.86 e/a variant line Al86Fe14 -Ni
Al5FeNi
Al3Ni2
Al4Ni3
Al2FeNi
Al10Fe3Ni
Al6Fe
Al (at.%) Ni (at.%)
D-Al80Ni20
e/a variant line Al80Ni20-
Fe
D D’
AlNi
Formation rule of ternary
quasicrystals
1,e/a-constant
Fermi surface and Brillouin zone interaction
3,e/a-variant
quantitative criteria
Multi-component system and its subsystems are inter-related
Explanation of the e/a-variant line
Phases in Ni-Ti-Zr system
Kelton’s group
A stable Bergman-type QC Ni17Ti38Zr45
A Bergman phase W-Ni12.7Ti61.4Zr25.9
The Bergman phase structure
xyz xyz xyz x
yz
xyz xyz a
b
c xyz
0 2 4 6 8 10
1,2 0
1,2 5
1,3 0
1,3 5
1,4 0
1,4 5
1,5 0
1,5 5
1,6 0
1,6 5
C lu st e r rad iu s
She ll rad iu s
C lu st e r rad iu s
a
t
o
m
ic
rad
iu
s
o
f
She
ll
1,2 0
1,2 5
1,3 0
1,3 5
1,4 0
1,4 5
1,5 0
1,5 5
1,6 0
1,6 5
a
t
o
m
ic
rad
iu
s
o
f
C
lu
st
e
r
Shell inflation growth rule
Atomic size,in small-middle-large cycles
and ending at middle-size (~equal to
average).
Three types of composition inflations,
small-Ni,middle-Ti,large-Zr.
Does shell composition inflates
along the e/a-variant lines?
Phase diagram
100
9 0
8 0
7 0
6 0
5 0
4 0
3 0
2 0
1 0
100
9 0
8 0
7 0
6 0
5 0
4 0
3 0
2 0
1 0
100 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0
Ti
Ni Zr
c1 -T i 1 2 N i
c2 -Z r2 0 T i 1 2 N i 1
c3 -Z r2 0 T i 1 2 N i 1 3
c4 -Z r2 6 T i 6 6 N i 1 3
c6 -Z r4 8,4 T i 7 5,6 N i
c7 -Z r7 8,4 T i 1 0 5,6 N i 3 7
Eu t e c-N i 2 4 T i 7 6
Eu t e c-N i 2 4 Z r7 6
Eu t e c-N i 3 6 Z r6 4
Eu t e ct i c N i 2 Z r9 8
Eu t e ct i c N i 6 1 T i 4 9
Eu t e ct i c N i 6 4 Z r3 6
Eu t e ct i c N i 8 3,5 T i 1 6,5
Eu t e ct i c N i 9 1,2 Z r8,8
Eu t e ct o i d N i 5 T i 9 5
N i 9 T i 2 Z r
N i T i Z r
p h a s e a v
Q C -1 7 -3 8 -4 5
Ti-④ -W-⑦ -QC-NiZr2 define a special e/a-variant line.
NiTi2-QC-Zr define another one.
Composition rules in ternary quasicrystals:
1,The e/a-constant line is a common property of
ternary QC phase diagram; it reflects the Hume-
Rothery mechanism in QCs and approximants.
2,The e/a-variant line is another common property,
which reflects the shell growth path of a common
cluster shared by the phases along the line.
3,Ternary QCs are at the intersections of the
specific e/a-constant and e/a-variant lines.
Composition rules in bulk metallic glasses
Formation rules for bulk metallic glasses
Multi-component
Atomic size difference>12％
Negative heat of mixing
Large DTx
Tg/Tm>0.6
Eutectic
Nagel-Tauc rule (Hume-Rothery)…
No quantitative criterion!
Binary systems,deep eutectic
Ni24Zr76
Ternary systems,example of Al-Ni-Zr
Ni24Zr76
Ni36Zr64
Ni24Zr76
Zr60Al20Ni20
Criterion 1,e/a-variant line
Ni24Zr76
Ni36Zr64
Ni24Zr76
Zr60Al20Ni20
Criterion 2,e/a-constant
Ni24Zr76
Ni36Zr64
Ni24Zr76
e/a=1.5
Zr60Al20Ni20
20 30 40 50 60 70 80
e/ a = 1,57
e/ a = 1,55
e/ a = 1,53
e/ a = 1,50
e/ a = 1,47
e/ a = 1,42
e/ a = 1,37
e/ a = 1,34
In
te
n
sity
(
a
r
b
it.u
n
its
)
2 q ( )
Samples along the e/a-variant line
1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 1.52 1.54
24.0
24.2
24.4
24.6
24.8
25.0
25.2
25.4
25.6
25.8
26.0
2Kf
Qp
2K
f&
K
p(
nm
-1
)
Kp - 2Kf decreases with increasing e/a.
e/a
Largest e/a? largest Fermi sphere
550 600 650 700 750 800 850
e / a = 1,3 7
e / a = 1,4 2
e / a = 1,4 7
e / a = 1,5 0
e / a = 1,5 3
T
x
T
g
E
xot
h
er
m
ic
(
a
r
b
it
,un
it
)
T ( K )DSC,thermal stability increases with increasing e/a.
the ideal Zr-Al-Ni BMG is a Hume-Rothery
phase with e/a=1.5,satisfying the Nagel-Tauc rule.
It is located at the crossing point of the e/a-
constant and e/a-variant lines.
DSC
600 650 700 750 800 850
Zr
63.8
Al
11.4
Ni
17.2
Cu
7.6
Zr
64.5
Al
9.2
Ni
13.2
Cu
13.1
Zr
64.8
Al
8.3
Ni
11.4
Cu
15.5
Zr
65
Al
7.5
Ni
10
Cu
17.5
Zr
65.3
Al
6.5
Ni
8.2
Cu
20
Zr
65.5
Al
5.6
Ni
6.5
Cu
22.4
0.67K/s
Tx
Tg
Ex
oth
er
mi
c(a
.u)
Temperature,T/K
①
②
③
④
⑤
⑥
Zr-Al-Ni-Cu system
①
②
③
④
⑤
⑥
DTA curves
400 600 800 1000 1200 1400
Tm
Tl
0.67K/s
1
2
3
4
5
6
Temperature,T/K
Ex
oth
erm
ic(a
.u
)
0 1 2 3 4 5 6
Sam p le n u m b e r
T
g
/T
l
T
g
/T
m
T
g
/T
l
T
g
/T
m
0,5 2
0,5 4
0,5 6
0,5 8
0,6 0
0,6 2
0,5 2
0,5 4
0,5 6
0,5 8
0,6 0
0,6 2
Re du ced gl ass t rans i t i on
t em perat ures
1-AlNiZr,Fe2P-type (trigonal prism),Cu2Mg-type (Bergman)
0 2 4 6
1,2 0
1,2 5
1,3 0
1,3 5
1,4 0
1,4 5
1,5 0
1,5 5
1,6 0
1,6 5
She ll rad iu s
C lu st e r rad iu s
d ist a n ce t o cen t e r (A )
She
ll
rad
iu
s
1,2 0
1,2 5
1,3 0
1,3 5
1,4 0
1,4 5
1,5 0
1,5 5
1,6 0
1,6 5
clu
st
e
r
rad
iu
s
a
b
c
a
b
c
x
y
z
3rd-shell cluster
2nd-shell cluster
2nd-shell
3rd-shell
The tricapped trigonal cluster and easy-amorphous-forming composition
100
90
80
70
60
50
40
30
20
10
100
90
80
70
60
50
40
30
20
10
100 90 80 70 60 50 40 30 20 10
Ni
Al Zr
r
p
j
k
o
m
n
*
l
q
8
t
u
v
w
x
y
z
{
|
Al2NiZr6(In2Mg)j
Al3NiZr2(Cu2Mg)k
Al3Zr2*
Al4Ni2Zrl
Al5Ni2Zrm
Al5Ni3Zr2(MnTh6)n
Al5NiZr2(AuCu3)o
AlN4Zr6q
AlNi2Zr (CsCl,BiF3)p
AlNiZr (Cu2Mg,Fe2P)r
BMG8
c1-0-1-0t
c2-3-1-0u
c3-3-1-6v
c4-3-3-6w
c5-3-6-6x
c6-9-6-6y
c7-9-12-6z
c8-12-12-6{
c9-12-12-12|
Cu2Mg-type,Bergman-cluster-based phase
a
b
c
x
y
z
a
b
c
Composition rules in quasicrystals
and in metallic glasses:
Combined influences from both
electronic structure (e/a-constant line)
and atomic structure requirement (e/a-
variant line),quantitative criteria.
The e/a-variant lines in ternary phase
diagrams reflect cluster-to-phase
composition inflation rules.
（ 1） 完美晶体的结构描述：晶体学对象：原子排列方式 （ 简化；理想；几何 ）
=,周期 。
晶体结构＝结构单元 （ 单胞 ） ＋单胞周期平移二、晶体结构对称操作：
使图形保持不变 （ 完全复原 ） 的操作；
在对称操作中始终保持不变的轴,平面,或点称为对称元素 。
单胞内的原子位置由对称性操作联系 。
对称操作分点对称和平移两种 。
点对称操作,1,2,3,4,6,m,1
平移对称性,14种 Bravais点阵单胞位置由平移对称性联系 。
对称性阵点 （ 几何点代替结构单元 ） 和点阵 （ 阵点的分布总体 ）
注意与晶体结构（ =点阵 +结构单元）的区别。
点阵与晶体结构点阵与晶体结构：例子
a-Fe,bcc
Steps to reach lattice
1,determine the basic unit
2,regard the unit as a point
3,the geometry of the points = lattice
a-Fe
1,the basic unit,one Fe atom
2,regard the unit as a point
3,the geometry of the points =
Body centered cubic lattice
a b
c
点阵与晶体结构：例子
CsCl,simple cubic
Steps to reach lattice
1,determine the basic unit
2,regard the unit as a point
3,the geometry of the points = lattice
CsCl
1,the basic unit,one Cs atom + one Cl
2,regard the unit Cs + Cl as a point
3,the geometry of the points =
simple cubic lattice
Cs
Cl
点阵与晶体结构：例子
g-Fe,fcc
Cu3Au,simple cubic a
b
c
点阵与晶体结构：例子
g-Fe,fcc
CuAu,tetragonal a
b
c
晶体结构例子
a
b
c
x
y
z
g-Fe a-Fe
非晶结构与晶体结构比较二维准晶模型,Penrose 拼图
Al-Cu-Fe单晶外形
3-fold
2-fold
5-fold
T2Fe二十面体准晶八次准晶的八次轴十次准晶的十次轴
g-Fe,fcc
Cu3Au,simple cubic
CuAu,tetragonal
有序化
a-Fe,bcc CsCl,simple cubic
Cs
Cl
36.87o/001
S=5
i
j
i’j’
k
j
i
M
k
j
i
R
'
'
'
缺点：不连续
S=2.5
Composition rules based on clusters in
quasicrystals and metallic glasses
Chuang DONG
Dalian University of Technology,China
Both quasicrystals and metallic glasses are multi-
component compounds with narrow composition ranges:
Example for stable quasicrystals
– i-Al62.5Cu24.5Fe13,i-Al69Pd22Mn9,d-Al67.5Pd14Mn18.5,
i-Zr45Ti38Ni17 …
Example for,good” metallic glasses:
– Binary,Zr-Ni (Zr76Ni34),Zr-Cu…
– Ternary,Zr60Al20Ni20,Zr65Cu27.5Al7.5,
– Quaternary,Zr65Al7.5Ni10Cu17.5 (Inoue alloy),
Ti34Cu47Zr11Ni8
– More,Zr52.5Ti5Cu17.9Ni14.6Al10,
Zr41.2Ti13.8Cu12.5Ni10Be22.5,Zr58.5Nb2.8Cu15.6Ni12.8Al10.3
Basic rule:
alloying
100
90
80
70
60
50
40
30
20
10
100
90
80
70
60
50
40
30
20
10
100 90 80 70 60 50 40 30 20 10
B
A C
AxBy
AuBvCw
Composition rules in ternary quasicrystals
b
(Cu)Al4Cu9Al
7Cu2Fe
Al10Cu10Fe
IQC
θ
ζ 2
Cu (+1)Al (+3)
Fe（ -2）
1.62.4
0.7
0.4
0.1
-0.2
-0.5
-0.8
-1.1
-1.4
-1.7-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
2.62.8 2.0 1.8 1.4 1.22.2
Al84Fe16
e/a = 1.86
The e/a-constant line
in ternary phase
diagram:
Criterion 1:
e/a-constant line
b
Al13Fe4
(Cu)Al4Cu9Al
7Cu2Fe Al
10Cu9F
e
IQC e/a-constant
e/a=1.86α
θ
ζ 2
Cu(+1)Al (+3)
Fe（ -2）
1.62.4
0.7
0.4
0.1
-0.2
-0.5
-0.8
-1.1
-1.4
-1.7-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Al-Cu-Fe e/a diagram
2.62.8 2.0 1.8 1.4 1.22.2
Al5Fe2
b
Al13Fe4
(Cu)Al4Cu9Al
7Cu2Fe Al
10Cu9F
e
IQC e/a-constant
e/a=1.86α
θ
ζ 2
Cu(+1)Al (+3)
Fe（ -2）
1.62.4
0.7
0.4
0.1
-0.2
-0.5
-0.8
-1.1
-1.4
-1.7-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Al-Cu-Fe e/a diagram
2.62.8 2.0 1.8 1.4 1.22.2
Al5Fe2
e/a-variant line,binary QC,ternary QC,and the 3rd element.
e/a-variant line
Criterion 2:
e/a-variant line
Al-Pd-Mn成分相图 （ at.%）
Al-Pd-Mn：
Ico-Al6Mn
Dec-Al4Mn
b
a
c
Al-Ni-Fe,Exception?
Qiang et al,J,Mater,Res.,2001,16; 2653-2660,
Al84Fe16
Al4Ni
Controversial reports on Decagonal quasicrystal compositions
a,Al75-70Fe9-21Ni16-9,e/a = 2.07,1.89,1.82,1.68 (typical values)
Saito,Tanaka,Tsai,Inoue,Masumoto,Jpn J,Appl,Phys.,1992
b,Al75Fe22.9Ni3.1,e/a = 1.79
,Dunlop,Phil Mag B,1993
c,Al71Fe5Ni24,stable,e/a = 2.03 (TOO LARGE!)
Lemmerz,Grushko,Freiburg,Jansen,Phil,Mag,Lett.,1994)
c
Al-Ni-Fe：
Qiang et al,J,Mater,Res.,2001,16; 2653-2660,
0
5
10
15
20
25
0 10 20 30 40 50 60 70
Ni (at.%)
Fe (at.%)
Al3Ni5Al
3Ni
Al13Fe4
D-Al84Fe16
Al71Fe5Ni21
e/a =1.86 e/a variant line Al86Fe14 -Ni
Al5FeNi
Al3Ni2
Al4Ni3
Al2FeNi
Al10Fe3Ni
Al6Fe
Al (at.%) Ni (at.%)
D-Al80Ni20
e/a variant line Al80Ni20-
Fe
D D’
AlNi
Formation rule of ternary
quasicrystals
1,e/a-constant
Fermi surface and Brillouin zone interaction
3,e/a-variant
quantitative criteria
Multi-component system and its subsystems are inter-related
Explanation of the e/a-variant line
Phases in Ni-Ti-Zr system
Kelton’s group
A stable Bergman-type QC Ni17Ti38Zr45
A Bergman phase W-Ni12.7Ti61.4Zr25.9
The Bergman phase structure
xyz xyz xyz x
yz
xyz xyz a
b
c xyz
0 2 4 6 8 10
1,2 0
1,2 5
1,3 0
1,3 5
1,4 0
1,4 5
1,5 0
1,5 5
1,6 0
1,6 5
C lu st e r rad iu s
She ll rad iu s
C lu st e r rad iu s
a
t
o
m
ic
rad
iu
s
o
f
She
ll
1,2 0
1,2 5
1,3 0
1,3 5
1,4 0
1,4 5
1,5 0
1,5 5
1,6 0
1,6 5
a
t
o
m
ic
rad
iu
s
o
f
C
lu
st
e
r
Shell inflation growth rule
Atomic size,in small-middle-large cycles
and ending at middle-size (~equal to
average).
Three types of composition inflations,
small-Ni,middle-Ti,large-Zr.
Does shell composition inflates
along the e/a-variant lines?
Phase diagram
100
9 0
8 0
7 0
6 0
5 0
4 0
3 0
2 0
1 0
100
9 0
8 0
7 0
6 0
5 0
4 0
3 0
2 0
1 0
100 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0
Ti
Ni Zr
c1 -T i 1 2 N i
c2 -Z r2 0 T i 1 2 N i 1
c3 -Z r2 0 T i 1 2 N i 1 3
c4 -Z r2 6 T i 6 6 N i 1 3
c6 -Z r4 8,4 T i 7 5,6 N i
c7 -Z r7 8,4 T i 1 0 5,6 N i 3 7
Eu t e c-N i 2 4 T i 7 6
Eu t e c-N i 2 4 Z r7 6
Eu t e c-N i 3 6 Z r6 4
Eu t e ct i c N i 2 Z r9 8
Eu t e ct i c N i 6 1 T i 4 9
Eu t e ct i c N i 6 4 Z r3 6
Eu t e ct i c N i 8 3,5 T i 1 6,5
Eu t e ct i c N i 9 1,2 Z r8,8
Eu t e ct o i d N i 5 T i 9 5
N i 9 T i 2 Z r
N i T i Z r
p h a s e a v
Q C -1 7 -3 8 -4 5
Ti-④ -W-⑦ -QC-NiZr2 define a special e/a-variant line.
NiTi2-QC-Zr define another one.
Composition rules in ternary quasicrystals:
1,The e/a-constant line is a common property of
ternary QC phase diagram; it reflects the Hume-
Rothery mechanism in QCs and approximants.
2,The e/a-variant line is another common property,
which reflects the shell growth path of a common
cluster shared by the phases along the line.
3,Ternary QCs are at the intersections of the
specific e/a-constant and e/a-variant lines.
Composition rules in bulk metallic glasses
Formation rules for bulk metallic glasses
Multi-component
Atomic size difference>12％
Negative heat of mixing
Large DTx
Tg/Tm>0.6
Eutectic
Nagel-Tauc rule (Hume-Rothery)…
No quantitative criterion!
Binary systems,deep eutectic
Ni24Zr76
Ternary systems,example of Al-Ni-Zr
Ni24Zr76
Ni36Zr64
Ni24Zr76
Zr60Al20Ni20
Criterion 1,e/a-variant line
Ni24Zr76
Ni36Zr64
Ni24Zr76
Zr60Al20Ni20
Criterion 2,e/a-constant
Ni24Zr76
Ni36Zr64
Ni24Zr76
e/a=1.5
Zr60Al20Ni20
20 30 40 50 60 70 80
e/ a = 1,57
e/ a = 1,55
e/ a = 1,53
e/ a = 1,50
e/ a = 1,47
e/ a = 1,42
e/ a = 1,37
e/ a = 1,34
In
te
n
sity
(
a
r
b
it.u
n
its
)
2 q ( )
Samples along the e/a-variant line
1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 1.52 1.54
24.0
24.2
24.4
24.6
24.8
25.0
25.2
25.4
25.6
25.8
26.0
2Kf
Qp
2K
f&
K
p(
nm
-1
)
Kp - 2Kf decreases with increasing e/a.
e/a
Largest e/a? largest Fermi sphere
550 600 650 700 750 800 850
e / a = 1,3 7
e / a = 1,4 2
e / a = 1,4 7
e / a = 1,5 0
e / a = 1,5 3
T
x
T
g
E
xot
h
er
m
ic
(
a
r
b
it
,un
it
)
T ( K )DSC,thermal stability increases with increasing e/a.
the ideal Zr-Al-Ni BMG is a Hume-Rothery
phase with e/a=1.5,satisfying the Nagel-Tauc rule.
It is located at the crossing point of the e/a-
constant and e/a-variant lines.
DSC
600 650 700 750 800 850
Zr
63.8
Al
11.4
Ni
17.2
Cu
7.6
Zr
64.5
Al
9.2
Ni
13.2
Cu
13.1
Zr
64.8
Al
8.3
Ni
11.4
Cu
15.5
Zr
65
Al
7.5
Ni
10
Cu
17.5
Zr
65.3
Al
6.5
Ni
8.2
Cu
20
Zr
65.5
Al
5.6
Ni
6.5
Cu
22.4
0.67K/s
Tx
Tg
Ex
oth
er
mi
c(a
.u)
Temperature,T/K
①
②
③
④
⑤
⑥
Zr-Al-Ni-Cu system
①
②
③
④
⑤
⑥
DTA curves
400 600 800 1000 1200 1400
Tm
Tl
0.67K/s
1
2
3
4
5
6
Temperature,T/K
Ex
oth
erm
ic(a
.u
)
0 1 2 3 4 5 6
Sam p le n u m b e r
T
g
/T
l
T
g
/T
m
T
g
/T
l
T
g
/T
m
0,5 2
0,5 4
0,5 6
0,5 8
0,6 0
0,6 2
0,5 2
0,5 4
0,5 6
0,5 8
0,6 0
0,6 2
Re du ced gl ass t rans i t i on
t em perat ures
1-AlNiZr,Fe2P-type (trigonal prism),Cu2Mg-type (Bergman)
0 2 4 6
1,2 0
1,2 5
1,3 0
1,3 5
1,4 0
1,4 5
1,5 0
1,5 5
1,6 0
1,6 5
She ll rad iu s
C lu st e r rad iu s
d ist a n ce t o cen t e r (A )
She
ll
rad
iu
s
1,2 0
1,2 5
1,3 0
1,3 5
1,4 0
1,4 5
1,5 0
1,5 5
1,6 0
1,6 5
clu
st
e
r
rad
iu
s
a
b
c
a
b
c
x
y
z
3rd-shell cluster
2nd-shell cluster
2nd-shell
3rd-shell
The tricapped trigonal cluster and easy-amorphous-forming composition
100
90
80
70
60
50
40
30
20
10
100
90
80
70
60
50
40
30
20
10
100 90 80 70 60 50 40 30 20 10
Ni
Al Zr
r
p
j
k
o
m
n
*
l
q
8
t
u
v
w
x
y
z
{
|
Al2NiZr6(In2Mg)j
Al3NiZr2(Cu2Mg)k
Al3Zr2*
Al4Ni2Zrl
Al5Ni2Zrm
Al5Ni3Zr2(MnTh6)n
Al5NiZr2(AuCu3)o
AlN4Zr6q
AlNi2Zr (CsCl,BiF3)p
AlNiZr (Cu2Mg,Fe2P)r
BMG8
c1-0-1-0t
c2-3-1-0u
c3-3-1-6v
c4-3-3-6w
c5-3-6-6x
c6-9-6-6y
c7-9-12-6z
c8-12-12-6{
c9-12-12-12|
Cu2Mg-type,Bergman-cluster-based phase
a
b
c
x
y
z
a
b
c
Composition rules in quasicrystals
and in metallic glasses:
Combined influences from both
electronic structure (e/a-constant line)
and atomic structure requirement (e/a-
variant line),quantitative criteria.
The e/a-variant lines in ternary phase
diagrams reflect cluster-to-phase
composition inflation rules.
