DETERMINATION OF Zn COMPOSITION OF α-BRASS USING DIFFRACTION METHOD: A COMPARISON WITH VEGARD’S LAW - e-Repository BATAN
MATERIALS SCIENCE and TECHNOLOGY
Edited by Evvy Kartini et.al.
DETERMINATION OF Zn COMPOSITION OF α-BRASS USING
DIFFRACTION METHOD: A COMPARISON WITH VEGARD’S LAW
Tri Hardi P1, Mirza Wibisono2
1
Center for Technology of Nuclear Industry Materials. BATAN Puspiptek, Serpong,
Tangerang 15314, Indonesia
2
Center for Industrial Technology Process, BPPT, Jln. M.H. Thamrin, Jakarta, Indonesia
e-mail: thardi@batan.go.id
P
P
P
P
P
P
P
7TU
U7T
ABSTRACT
For a single phase binary alloy such as α brass, it is also possible to determine the alloy
composition by measuring the lattice parameter of the crystal structure. This technique makes
use of Vegard’s law, which states that the lattice parameter of a single phase binary alloys
varies linearly with the addition of the solute element. By the use of the neutron diffraction
and x-ray diffraction methods, lattice parameters of α-brass were calculated. Neutron
diffraction data was obtained by using DN2 neutron diffractometer, and the lattice parameter
ao=3.6782± 0.0006Å is obtained with a reliability factor of Rwp =19.33%, and the measured
density is 4.8197 gr/cm3. From X-ray diffraction, the lattice parameter ao=3.6782 ± 0.0005 Å
with Rwp= 27.78% is obtained. Diffraction data was analyzed using the MAUD application
package software. From the lattice parameters and by the use of Vegard’s law the % weight
of Zn composition obtained from neutron and x-ray diffraction turns out to be the same and is
equal to 32.35%. Chemical composition measurement shows that the % weight of Zn
composition for Al weight > 0.03% is 30.18515 %. Maximum deviation of the Zn % weight
calculated from lattice parameter values and the chemical composition measurement is less
than 10%.
Keywords : α-brass, neutron diffraction, Vegard’s law, MAUD
INTRODUCTION
Vegard’s law has been used extensively in mineralogy, metallurgy and materials science for the
past six decades. According to the law, unit cell parameters should vary linearly with composition
for a continuous substitutional solid solution in which atoms or ions that substitute for each other
are randomly distributed. Although the law was postulated on empirical evidence, several cases
of both positive and negative deviations from this law have been documented. Its theoretical
foundations have not been critically explored. The law should be reclassified as an approximation
valid only under specific conditions. The approximation is valid for ideal solutions when the
lattice parameters of the pure components differ by less than 5 %. For solid solutions with
positive deviations from the ideal conditions, there will always be positive deviations from
Vegard’s law. For solid solutions with moderately negative deviations from ideal conditions,
positive deviation from linearity of lattice parameters caused by size mismatch can be
compensated for by the attractive interaction between the components, resulting in compliance
with Vegard’s law[1].
Materials Science and Technology
Some applications of Vegard law was used for a single phase binary alloy such as α
brass. Values of lattice parameter versus composition for the Cu/Zn system were found and
are reported in the literature [2]. The Vegard’s Law is also applied in the investigation of split
interstitial defect of N-N split in GaAs1-xNx alloys[3], band gap bowing parameter of zinc
blende InxGa1-x N [4].
By using the neutron diffraction- and x-ray diffraction method lattice parameters of αbrass with fcc crystal structure, space group Fm3m were calculated. Applications of a
density-functional theory of nonuniform fluid mixtures to the fluid-solid transition of simple
binary mixtures of hard spheres demonstrates the importance of relative atomic sizes in
determining the lattice constants and suggests that for sufficiently small disparities in atomic
size, Vegard's law may also hold along the fluid-solid coexistence curve[5].
The aim of this paper is to determine % weight Zn using diffraction method with
Vegard’s law as a tool to convert from lattice parameter to % weight Zn. The result is then
compared with the result obtained from chemical composition measurement.
Determination of Composition by Vegard’s Law
Vegard's law is an approximate empirical rule which holds that at constant
temperature a linear relation exists between the crystal lattice constant of an alloy and the
concentrations of the constituent elements. It is applied for determination of composition of
binary and ternary compounds. For binary compounds, consider two elements A and B and
its alloy AxB1-x, where x is alloy composition or alloy mole fraction, Assume that A has
lattice constant aA and B has lattice constant aB, then the lattice constant of the alloy is given
by :
a AB (x) xaA (1 x)a B
(1)
This equation is called Vegard’s law. For ternary compound, assuming that the lattice
constants of the ternary compounds can be expressed as a linear combination of the lattice
constants of the two forming binary compounds, and the physical properties of the ternary
compounds are usually investigated based on Vegard’s law.
For a single phase binary alloy such as α brass, it is also possible to determine the
alloy composition from the lattice parameter. Values of lattice parameter versus composition
for the Cu/Zn system were found in the literature and are reported in Table 1. Vegard’s Law
is strictly applicable to composition by atomic percent, however since the atomic weights of
Cu and Zn are close, the difference between atomic and weight percent is negligible. Weight
percent is reported in this study for comparison to conventional composition reporting style
in the literature.[2]
Table 1: Brass lattice parameters and composition
at. % Zn
3.64
5.96
7.32
11.8
14.78
19.89
24.45
34.25
138
wt.% Zn
3.74
6.12
7.52
12.1
15.14
20.35
24.98
34.9
Lattice Parameter
Å (angstroms)
3.6145
3.6194
3.6224
3.6318
3.6382
3.6505
3.6611
3.6849
Determination Of Zn Composition Of α-Brass Using……
The values in Table 1 have been plotted and a linear fit line applied as seen in Figure
1. In this figure, the lattice parameter has been converted to the d-spacing for the (111) plane
of the α brass standards
d hkl
a
h k2 l2
2
(2)
where a is the lattice parameter, h, k, and l are the Miller indices of the plane and dhkl is the dspacing for the plane.
Figure 1: (up) Relation between weight% Zn and lattice parameter converted from d111 spacing, and
(bottom) The copper-zinc phase diagram
Converting from lattice parameter to d-spacing facilitates easy comparison between
the measured results from x ray and neutron diffraction in this study to the values reported in
139
Materials Science and Technology
the literature which act as standard values for the Vegard’s Law analysis. In this study the
(111) peak was chosen because it is the most intense peak in the diffraction patterns. The
equation fitted to the data of Figure 1(a) can then be used to solve for the Zn composition of
the sample as:
wt. % alpha = 443.1 d (111) - 1597
(3)
where wt.% alpha is the Zn composition (wt.%) of the sample, and d111 is the d-spacing (in
angstroms) of the (111) index measured by XRD and neutron diffraction. The error in the
w_alpha values is of the order of 4 wt. % due to the extreme sensitivity of the composition to
the lattice parameter measurement. It is important to clarify that compositional analysis by
Vegard’s law is only pertinent to the α-brass. Figure 1(b) shows the phase diagram of Cu-Zn.
EXPERIMENTAL METHOD
Utilization of both the neutron diffraction and x-ray diffraction methods, lattice parameter of
α-brass with fcc crystal structure with space group Fm3m were calculated. Neutron
diffraction data of α-brass was obtained by using Texture-Diffractometer (DN2) at neutron
wavelength of 1.2793Å obtained from bent Si(311) monochromator. The Bragg peaks of
diffraction pattern were fitted using a Materials Analysis Using Diffraction (MAUD)
software package, giving the indices of (h k l) plane, peak location, error, and full-width at
half-maximum (FWHM) and Rwp of the diffraction pattern. Once the diffraction patterns have
been indexed, it is a simple task to solve for the α brass phase. The α brass (111) peak was
chosen for the measurement of zinc composition because it has the highest intensity of the
three α brass peaks present in the data. Then from d-spacing of the (111) plane and by using
the vegard’s law chart shown in Figure 1, the zinc composition can be determined.
RESULTS AND DISCUSSION
a. Chemical composition
Based on the chemical composition of the two Brass samples, the main constituent
elements of brass such as Cu, Zn, Ni and Fe are already in compliance with the industrial
standard specifications. However aluminium has two kind of weight composition, the first is
0,00685% and the second is 0,16112% which are indexed as being under and above the
standard composition of 0.03%, respectively. As shown in Table 2. Aluminium is known as a
precursor for the β phase formation in brass alloys having BCC crystal structure. Addition of
1% aluminium is equivalent to an addition of 6% zinc. So that aluminium with a 0.16112%
weight composition differs by only 0.13112% from the standard composition of 0.03%. In
this case, only an additional 0,007867% of zinc is needed to raise the weight composition
from 30,18515% to 30,19302%, and it is still far removed from the 33.6% maximum range of
α-brass. It is still in the α-phase.
b. Neutron and X-ray diffraction.
Neutron and X-ray diffraction data were analyzed using the MAUD software package.
The neutron diffraction pattern shows the three Bragg peaks; the (111), (200) and (220) peak
appear in the intensity pattern and from the refinement results, the lattice parameter value
ao=3.6782± 0.0006Å is obtained with Rwp equals 19.33%. The density is 8. 48197 gr/cm3, and
from x-ray diffraction analysis lattice, parameter ao=3.6782 ± 0.0005 Å is determined with a
reliability factor of Rwp equals 27.78%. It is found that in - brass with 31.28% Zn, the lattice
parameter is 3.680 ± 0.005 Å [6]
140
Determination Of Zn Composition Of α-Brass Using……
Table 2: Chemical composition of -brass.
Aluminium
(> 0,03%)
(% wt)
69,42995
30,18515
0,16112
0,10787
Indonesian
standard.
(% wt)
69,5-72
Remainder
Max 0,03
Max 0,05
No
Elements
1
2
3
4
Cu
Zn
Al
Fe
Aluminium
(≤ 0,03%)
(% wt)
69,75858
30,05447
0,00685
0,05761
5
Ni
0,08965
0,07765
Max 0,20
6
Sn
0,01608
0,01691
Max 0,03
7
Pb
0,00993
0,01063
Max 0,05
8
P
0,00172
0,00355
---
9
Si
---
---
---
10
Mn
---
---
---
11
S
0,00283
0,0033
---
12
Sb
0,00517
0,0063
Max 0,01
13
Mg
0,00033
0,00033
---
Figure 2. (a) Neutron diffraction and (b) x-ray diffraction of α-brass
141
Materials Science and Technology
CONCLUSION
In conclusion, the authors have shown that Vegard’s law can be used to determine the %
weight of Zn. Comparison between the neutron diffraction and x-ray diffraction methods and
the chemical composition measurement, has shown that by using the lattice parameter values
and by the use of Vegard’s law, the %weight of Zn composition obtained from both the
neutron and x-ray diffraction are the same, and is equal to 32.35%,. Chemical composition
measurement shows that %weight of Zn composition for Al>0.03% weight is 30.18515%.
Even though the % weight of Zn calculated from the diffraction method data is larger than the
value obtained from the chemical composition measurement, it is still in the range of -brass
with a deviation of less than 10%
REFERENCES
[1] K. T. Jacob, Shubhra Raj, L. Rannesh, Vegard’s law: a fundamental relation or an
approximation ? Department of Materials Engineering, Indian Institute of Science
Bangalore, India, (2007)
[2] W. B. Pearson, Handbook of Lattice Spacing and Structures of Metals , Pergamon,
New York, Vol. I, (1964), 620.
[3] Pierre Carrier, Su-Huai Wei, S. B. Zhang, and Sarah Kurtz, Physical Review B, 71,
(2005), 165212
[4] Yen-Kuang Kuo, Bo-Ting Liou, Sheng-Horng Yen, Han-Yi Chu, Optics
Communications 237, (2004) 363–369
[5] Denton, A. R.; Ashcroft, N. W.,. Physical Review A, 43, Issue 6, March 15, (1991).,
pp.3161-3164
[6] Koji Hashimoto, Shiro Ogawa and Saburo Shimodaira, Trans JIM, 4 (1963) 42-45
142
Edited by Evvy Kartini et.al.
DETERMINATION OF Zn COMPOSITION OF α-BRASS USING
DIFFRACTION METHOD: A COMPARISON WITH VEGARD’S LAW
Tri Hardi P1, Mirza Wibisono2
1
Center for Technology of Nuclear Industry Materials. BATAN Puspiptek, Serpong,
Tangerang 15314, Indonesia
2
Center for Industrial Technology Process, BPPT, Jln. M.H. Thamrin, Jakarta, Indonesia
e-mail: thardi@batan.go.id
P
P
P
P
P
P
P
7TU
U7T
ABSTRACT
For a single phase binary alloy such as α brass, it is also possible to determine the alloy
composition by measuring the lattice parameter of the crystal structure. This technique makes
use of Vegard’s law, which states that the lattice parameter of a single phase binary alloys
varies linearly with the addition of the solute element. By the use of the neutron diffraction
and x-ray diffraction methods, lattice parameters of α-brass were calculated. Neutron
diffraction data was obtained by using DN2 neutron diffractometer, and the lattice parameter
ao=3.6782± 0.0006Å is obtained with a reliability factor of Rwp =19.33%, and the measured
density is 4.8197 gr/cm3. From X-ray diffraction, the lattice parameter ao=3.6782 ± 0.0005 Å
with Rwp= 27.78% is obtained. Diffraction data was analyzed using the MAUD application
package software. From the lattice parameters and by the use of Vegard’s law the % weight
of Zn composition obtained from neutron and x-ray diffraction turns out to be the same and is
equal to 32.35%. Chemical composition measurement shows that the % weight of Zn
composition for Al weight > 0.03% is 30.18515 %. Maximum deviation of the Zn % weight
calculated from lattice parameter values and the chemical composition measurement is less
than 10%.
Keywords : α-brass, neutron diffraction, Vegard’s law, MAUD
INTRODUCTION
Vegard’s law has been used extensively in mineralogy, metallurgy and materials science for the
past six decades. According to the law, unit cell parameters should vary linearly with composition
for a continuous substitutional solid solution in which atoms or ions that substitute for each other
are randomly distributed. Although the law was postulated on empirical evidence, several cases
of both positive and negative deviations from this law have been documented. Its theoretical
foundations have not been critically explored. The law should be reclassified as an approximation
valid only under specific conditions. The approximation is valid for ideal solutions when the
lattice parameters of the pure components differ by less than 5 %. For solid solutions with
positive deviations from the ideal conditions, there will always be positive deviations from
Vegard’s law. For solid solutions with moderately negative deviations from ideal conditions,
positive deviation from linearity of lattice parameters caused by size mismatch can be
compensated for by the attractive interaction between the components, resulting in compliance
with Vegard’s law[1].
Materials Science and Technology
Some applications of Vegard law was used for a single phase binary alloy such as α
brass. Values of lattice parameter versus composition for the Cu/Zn system were found and
are reported in the literature [2]. The Vegard’s Law is also applied in the investigation of split
interstitial defect of N-N split in GaAs1-xNx alloys[3], band gap bowing parameter of zinc
blende InxGa1-x N [4].
By using the neutron diffraction- and x-ray diffraction method lattice parameters of αbrass with fcc crystal structure, space group Fm3m were calculated. Applications of a
density-functional theory of nonuniform fluid mixtures to the fluid-solid transition of simple
binary mixtures of hard spheres demonstrates the importance of relative atomic sizes in
determining the lattice constants and suggests that for sufficiently small disparities in atomic
size, Vegard's law may also hold along the fluid-solid coexistence curve[5].
The aim of this paper is to determine % weight Zn using diffraction method with
Vegard’s law as a tool to convert from lattice parameter to % weight Zn. The result is then
compared with the result obtained from chemical composition measurement.
Determination of Composition by Vegard’s Law
Vegard's law is an approximate empirical rule which holds that at constant
temperature a linear relation exists between the crystal lattice constant of an alloy and the
concentrations of the constituent elements. It is applied for determination of composition of
binary and ternary compounds. For binary compounds, consider two elements A and B and
its alloy AxB1-x, where x is alloy composition or alloy mole fraction, Assume that A has
lattice constant aA and B has lattice constant aB, then the lattice constant of the alloy is given
by :
a AB (x) xaA (1 x)a B
(1)
This equation is called Vegard’s law. For ternary compound, assuming that the lattice
constants of the ternary compounds can be expressed as a linear combination of the lattice
constants of the two forming binary compounds, and the physical properties of the ternary
compounds are usually investigated based on Vegard’s law.
For a single phase binary alloy such as α brass, it is also possible to determine the
alloy composition from the lattice parameter. Values of lattice parameter versus composition
for the Cu/Zn system were found in the literature and are reported in Table 1. Vegard’s Law
is strictly applicable to composition by atomic percent, however since the atomic weights of
Cu and Zn are close, the difference between atomic and weight percent is negligible. Weight
percent is reported in this study for comparison to conventional composition reporting style
in the literature.[2]
Table 1: Brass lattice parameters and composition
at. % Zn
3.64
5.96
7.32
11.8
14.78
19.89
24.45
34.25
138
wt.% Zn
3.74
6.12
7.52
12.1
15.14
20.35
24.98
34.9
Lattice Parameter
Å (angstroms)
3.6145
3.6194
3.6224
3.6318
3.6382
3.6505
3.6611
3.6849
Determination Of Zn Composition Of α-Brass Using……
The values in Table 1 have been plotted and a linear fit line applied as seen in Figure
1. In this figure, the lattice parameter has been converted to the d-spacing for the (111) plane
of the α brass standards
d hkl
a
h k2 l2
2
(2)
where a is the lattice parameter, h, k, and l are the Miller indices of the plane and dhkl is the dspacing for the plane.
Figure 1: (up) Relation between weight% Zn and lattice parameter converted from d111 spacing, and
(bottom) The copper-zinc phase diagram
Converting from lattice parameter to d-spacing facilitates easy comparison between
the measured results from x ray and neutron diffraction in this study to the values reported in
139
Materials Science and Technology
the literature which act as standard values for the Vegard’s Law analysis. In this study the
(111) peak was chosen because it is the most intense peak in the diffraction patterns. The
equation fitted to the data of Figure 1(a) can then be used to solve for the Zn composition of
the sample as:
wt. % alpha = 443.1 d (111) - 1597
(3)
where wt.% alpha is the Zn composition (wt.%) of the sample, and d111 is the d-spacing (in
angstroms) of the (111) index measured by XRD and neutron diffraction. The error in the
w_alpha values is of the order of 4 wt. % due to the extreme sensitivity of the composition to
the lattice parameter measurement. It is important to clarify that compositional analysis by
Vegard’s law is only pertinent to the α-brass. Figure 1(b) shows the phase diagram of Cu-Zn.
EXPERIMENTAL METHOD
Utilization of both the neutron diffraction and x-ray diffraction methods, lattice parameter of
α-brass with fcc crystal structure with space group Fm3m were calculated. Neutron
diffraction data of α-brass was obtained by using Texture-Diffractometer (DN2) at neutron
wavelength of 1.2793Å obtained from bent Si(311) monochromator. The Bragg peaks of
diffraction pattern were fitted using a Materials Analysis Using Diffraction (MAUD)
software package, giving the indices of (h k l) plane, peak location, error, and full-width at
half-maximum (FWHM) and Rwp of the diffraction pattern. Once the diffraction patterns have
been indexed, it is a simple task to solve for the α brass phase. The α brass (111) peak was
chosen for the measurement of zinc composition because it has the highest intensity of the
three α brass peaks present in the data. Then from d-spacing of the (111) plane and by using
the vegard’s law chart shown in Figure 1, the zinc composition can be determined.
RESULTS AND DISCUSSION
a. Chemical composition
Based on the chemical composition of the two Brass samples, the main constituent
elements of brass such as Cu, Zn, Ni and Fe are already in compliance with the industrial
standard specifications. However aluminium has two kind of weight composition, the first is
0,00685% and the second is 0,16112% which are indexed as being under and above the
standard composition of 0.03%, respectively. As shown in Table 2. Aluminium is known as a
precursor for the β phase formation in brass alloys having BCC crystal structure. Addition of
1% aluminium is equivalent to an addition of 6% zinc. So that aluminium with a 0.16112%
weight composition differs by only 0.13112% from the standard composition of 0.03%. In
this case, only an additional 0,007867% of zinc is needed to raise the weight composition
from 30,18515% to 30,19302%, and it is still far removed from the 33.6% maximum range of
α-brass. It is still in the α-phase.
b. Neutron and X-ray diffraction.
Neutron and X-ray diffraction data were analyzed using the MAUD software package.
The neutron diffraction pattern shows the three Bragg peaks; the (111), (200) and (220) peak
appear in the intensity pattern and from the refinement results, the lattice parameter value
ao=3.6782± 0.0006Å is obtained with Rwp equals 19.33%. The density is 8. 48197 gr/cm3, and
from x-ray diffraction analysis lattice, parameter ao=3.6782 ± 0.0005 Å is determined with a
reliability factor of Rwp equals 27.78%. It is found that in - brass with 31.28% Zn, the lattice
parameter is 3.680 ± 0.005 Å [6]
140
Determination Of Zn Composition Of α-Brass Using……
Table 2: Chemical composition of -brass.
Aluminium
(> 0,03%)
(% wt)
69,42995
30,18515
0,16112
0,10787
Indonesian
standard.
(% wt)
69,5-72
Remainder
Max 0,03
Max 0,05
No
Elements
1
2
3
4
Cu
Zn
Al
Fe
Aluminium
(≤ 0,03%)
(% wt)
69,75858
30,05447
0,00685
0,05761
5
Ni
0,08965
0,07765
Max 0,20
6
Sn
0,01608
0,01691
Max 0,03
7
Pb
0,00993
0,01063
Max 0,05
8
P
0,00172
0,00355
---
9
Si
---
---
---
10
Mn
---
---
---
11
S
0,00283
0,0033
---
12
Sb
0,00517
0,0063
Max 0,01
13
Mg
0,00033
0,00033
---
Figure 2. (a) Neutron diffraction and (b) x-ray diffraction of α-brass
141
Materials Science and Technology
CONCLUSION
In conclusion, the authors have shown that Vegard’s law can be used to determine the %
weight of Zn. Comparison between the neutron diffraction and x-ray diffraction methods and
the chemical composition measurement, has shown that by using the lattice parameter values
and by the use of Vegard’s law, the %weight of Zn composition obtained from both the
neutron and x-ray diffraction are the same, and is equal to 32.35%,. Chemical composition
measurement shows that %weight of Zn composition for Al>0.03% weight is 30.18515%.
Even though the % weight of Zn calculated from the diffraction method data is larger than the
value obtained from the chemical composition measurement, it is still in the range of -brass
with a deviation of less than 10%
REFERENCES
[1] K. T. Jacob, Shubhra Raj, L. Rannesh, Vegard’s law: a fundamental relation or an
approximation ? Department of Materials Engineering, Indian Institute of Science
Bangalore, India, (2007)
[2] W. B. Pearson, Handbook of Lattice Spacing and Structures of Metals , Pergamon,
New York, Vol. I, (1964), 620.
[3] Pierre Carrier, Su-Huai Wei, S. B. Zhang, and Sarah Kurtz, Physical Review B, 71,
(2005), 165212
[4] Yen-Kuang Kuo, Bo-Ting Liou, Sheng-Horng Yen, Han-Yi Chu, Optics
Communications 237, (2004) 363–369
[5] Denton, A. R.; Ashcroft, N. W.,. Physical Review A, 43, Issue 6, March 15, (1991).,
pp.3161-3164
[6] Koji Hashimoto, Shiro Ogawa and Saburo Shimodaira, Trans JIM, 4 (1963) 42-45
142