Effects of lithium, indium, and zinc on the lattice parameters
of magnesium
A. Becerra
Hatch Associates, Montreal H3B 2G2, Quebec, Canada
M. Pekguleryuza)
McGill University, Montreal H3A 2B2, Quebec, Canada
(Received 1 July 2008; accepted 28 August 2008)
The lattice parameters of magnesium solid-solution alloys with lithium, indium, and/or
zinc have been determined via x-ray diffraction (XRD). Li decreased the axial ratio
(c/a) of Mg from 1.624 to 1.6068 within 0–16 at.% Li. Indium increased the c/a of Mg
to 1.6261 with increasing In toward 3.3 at.% while Zn showed no effect on c/a in the
0.2–0.7 at.% range. The effects were explained by electron overlap through the first
Brillouin zone and by Vegard’s Law. A relationship was determined between electron
concentration (e/a) and c/a as c/a ⳱ −15.6(e/a)2 + 60(e/a) − 55.8.
I. INTRODUCTION
The interest of automotive companies in weight reduction has led to new research in wrought magnesium sheet
alloys. Studies conducted on the deformation behavior of
magnesium1–9 have discovered that magnesium sheet can
be formed at elevated temperatures.
Improving room-temperature formability of Mg is
more challenging and would require among other properties, enhanced slip behavior. An area little discussed is
the modification of the intrinsic formability of magnesium by changing its lattice parameters and the axial ratio
(c/a) of the hexagonal close packed (HCP) crystal so that
more slip systems can be activated at lower temperatures.
This paper reports on partial results of an ongoing study
investigating the effects of solute elements on the potential formability of magnesium. As such, the present paper
focuses on the effects of Li, In, and Zn on the axial ratio
(c/a) of magnesium.
II. BACKGROUND
A. Crystallographic parameters affecting
formability10,11
It is generally known that cubic metals with bodycentered and face-centered cubic structures (BCC and
FCC) are more formable than hexagonal close packed
metals such as magnesium since they can provide at least
five independent slip systems required by the von Mises
criterion for formability. HCP metals on the other hand
have only two independent slip systems. Theoretically, if
magnesium were made either BCC or FCC, it would be
possible to improve its formability.
An important factor in influencing the formability of
HCP metals is the axial ratio.10 Within the HCP metals,
there are variations in room temperature formability,
which have been related to the axial ratio (c/a) as summarized in Table I. This effect was attributed to the
change in the critical resolved shear stress (CRSS) on the
basal plane with varying axial ratio.10 As c/a decreases to
1.59 (Ti, Be), CRSS for basal slip increases and other slip
systems of HCP (prismatic and/or pyramidal slip systems) are activated. The initiation of pyramidal type-II
slip system with (c+a) dislocations would produce five
independent slip systems making magnesium formable
(Table II). Lattice parameters of HCP Mg give a c/a ratio
of 1.62354 at 25 °C. This c/a ratio, as in Cd and Zn,
results in low CRSS on the basal plane favoring basal
slip. Ti and Be on the other hand have a much lower c/a
ratio than Mg, Cd, and Zn, they have high basal CRSS
(Table II) but low prismatic CRSS. The change in the
deformation mechanism with a reduced c/a ratio is attributed to the related change in interplanar spacing, d,
since the shear stress required to move dislocations is
given by Peierles stress,
(1)
␴ = P ⭈ e兵−2␲d Ⲑ 关b共1−␮兲兴其 ,
where P is a factor depending on shear modulus, G, and
Poisson’s ratio, ␮; b is the magnitude of Burger’s vector
of the dislocation; and d is the interplanar spacing.
B. Effects of solutes on crystallographic
parameters
1. Effect of solutes on crystal structure
a)
Address all correspondence to this author.
e-mail: mihriban.pekguleryuz@mcgill.ca
DOI: 10.1557/JMR.2008.0414
J. Mater. Res., Vol. 23, No. 12, Dec 2008
Hume-Rothery has shown, after observing many alloy
systems, that the crystal structure of metallic alloys can
© 2008 Materials Research Society
3379
A. Becerra et al.: Effects of lithium, indium, and zinc on the lattice parameters of magnesium
TABLE I. Basal CRSS and c/a for HCP metals (25 °C).10
Metal
CRSS (psi)
c/a
Cd
Mg
Zn
Ti
Be
82
63
26
16,000
5,700
1.886
1.624
1.856
1.588
1.586
ratio at different temperatures and at different degrees
of deformation to improve the formability of Mg, especially at low temperatures. Au, Ce, and Pd are among the
candidates from Table V, but Au and Pd are expensive alloying additions and Ce has very limited solubility in Mg, so many more solutes need to be researched.
a. Effect of atom-size differences
TABLE II. Slip systems in magnesium.
Slip system
Slip
direction
Slip plane
and direction
Independent
slip systems
Basal
Prismatic
Pyramidal type I
Pyramidal type II
a
a
a
c+a
{0001}⟨11-20⟩
{10-10}⟨11-20⟩
{10-11}⟨11-20⟩
{11-22}⟨11-23⟩
2
2
4
5
Solid-solution alloying where solutes occupy substitutional or interstitial spaces in the crystal lattice can alter
the lattice spacing by expanding or contracting the lattice. Vegard’s Law holds only for solute atoms of similar
valency in which case the lattice parameter change is
purely based on size effects expressed in the form of
dm = ndB + 共1 − n兲dA
be related to the valence (s, p) electron-to-atom ratio
(e/a) as summarized in Table III. An explanation proposed for the tendency for HCP structure formation is the
minimization of electron pair-pair repulsion energy,
while for FCC it is the minimization of the energy of
Coulomb repulsions between nuclei. Both of these structures are favored by enthalpy. The BCC structure on the
other hand (based on the study of various structures) is
favored by high entropy.12 In magnesium, it is well
known that lithium (monovalent solute) additions transform the structure from HCP to BCC at an e/a ratio of 1.7
(30 at.%). At an e/a ratio of 2.1 an FCC structure may
also be expected for magnesium as seen from Table III.
For three-valent solute additions, the change would occur
at 10 at.%. This would be possible with indium, which
has extended solid-solubility in magnesium based on the
Mg-In phase diagram.13
2. Effect on axial ratio10,11
The effect of solutes on the lattice spacing and axial
ratio of magnesium can be regarded as the sum of many
effects: (i) effect of atom-size differences (change of
atomic volume); (ii) effect due to valency differences
(development of shear strain); (iii) effect of temperature
on lattice expansion and electron overlap; and (iv) effect
of strain. Using these principles it is possible to select a
combination of solute elements that can affect the c/a
TABLE III. Relationship between electron concentration and crystal
structure.
3380
e/a
Crystal structure
<1.5
1.7 < e/a < 2.1
2.5 < e/a < 3
e/a > 4
Body-centered cubic (BCC)
Hexagonal close packed (HCP)
Face-centered cubic (FCC)
Diamond
,
(2)
where dm is the mean interatomic distance in the solid
solution, dA and dB are the interatomic distances in the
pure components, and n is the atom fraction of B atoms.
Deviation from this law occurs as soon as the valency
difference between solvent and solute exists. Vegard’s
Law would hold for Mg for two-valent solutes. To have
a major effect, n (atom fraction B) has to be high and this
requires an extensive solid solution range, which is only
possible with elements of similar crystal structure, i.e.,
Cd. This limits the choice of alloying elements.
b. Effect due to valency differences (development of
shear strain)
Lattice spacing also depends on the e/a ratio which is
changed by solute valency. This is understood in terms of
the electronic structure of Mg, which (the first Brillouin
zone) is shown in Fig. 1(a). Brillouin zones depict polyhedra (bounded by crystallographic planes) of energy
states in k space (k ⳱ reciprocal space) which valency
electrons in a metal can occupy. An electron inside the
zone cannot change its energy to the next zone without
receiving a major increase in energy. Unlike insulators,
Brillouin zones in metals overlap; electrons may enter
the next Brillouin zone before the first one is filled and
conduction becomes possible [Fig. 1(b)].
Understanding how electron overlap occurs sheds light
to the alloying behavior of Mg, the effects of solutes on
lattice spacing and the difference from HCP Zn and Cd
which are both two-valent as well. As seen in Table IV,
in magnesium, the energy at the center of {11̄00} prismatic planes, EA, is lower than the energy at the center of
basal {0002} planes, EB. This means that electron-jump
in Mg would first occur perpendicular to the c-axis across
the {11̄00} planes, rather than parallel to it. If solutes of
higher valency than Mg are added, the electrons would
J. Mater. Res., Vol. 23, No. 12, Dec 2008
A. Becerra et al.: Effects of lithium, indium, and zinc on the lattice parameters of magnesium
overlap occurs, the expansion of the c spacing is pronounced and overriding since the resistance to expansion
parallel to the c axis is low.
Studies on the lattice spacing of magnesium alloys
confirm the electronic interpretation of changes in lattice
spacing. General trends indicate the effects of various
solutes shown in Table V.10 These studies evidently were
not exhaustive. It is known that there are other elements,
such as Li, which decrease the c/a ratio of Mg. Lithium
is known to have extensive effects on the lattice parameters, ductility, and formability of magnesium. Due to
its size factor, Li also has extensive solid solubility
(5.5 wt%) in Mg. It reduces the c/a ratio in the ␣ phase
from 1.624 to 1.61 and, correspondingly, the CRSS on
the prismatic slip and the strain-hardening rate.10
c. Effect of temperature on lattice expansion and
electron overlap
FIG. 1. (a) First Brillouin zone for Mg formed by planes {0002},
{11̄00}, and {101̄1} and adjusted for energy discontinuity. (b) Brillouin zone overlap in Mg. e/a ratio: 1.74 in the first zone.10
TABLE IV. Differences in first Brillouin zone energies of Mg, Zn,
Cd.10
Element
Approximate
c/a
n (# electrons
per atom in
the zone)
EA, energy
at center of
{11̄00} planes
EB, energy
at center of
{0002} planes
Mg
Zn
Cd
1.624
1.861
1.890
1.74
1.8
1.8
4.88
7.13
5.67
5.55
6.17
4.77
jump to the second zone resulting in a small fraction of
electrons existing outside of the Brillouin Zone. These
external electrons produce an internal stress in the crystal
causing lattice distortion. An increase in the number of
overlapping electrons pushes the k space plane toward
the center and hence the real lattice a space expands.
When solutes of lower valency are added, the opposite
happens and the a spacing contracts. On the other hand,
the c spacing dependence on alloying is more complicated. Initial alloying does not affect c spacing, until
the critical value is reached where the overlap across
the {0002} planes occurs. This is expected at around
0.75 at.% with three-valent solutes and at 0.375 at.%
with four-valent solutes. When the onset of electron
Temperature alone and in combination with alloying
can also influence the axial ratio. Temperature causes the
c spacing to expand more than the a spacing, resulting in
an increase in c/a. This is the underlying cause of the
increased formability of magnesium at temperatures
above ∼250 °C. Temperature also causes, through thermal excitation, electron overlap perpendicular to the c
axis at a critical temperature. Alloying, which changes
the e/a ratio, would cause a corresponding change in the
critical temperature for the onset of electron overlap. The
temperature for transition to non-basal slip is expected to
depend on solid-solution alloying since this affects the
electron overlap and the differential change in lattice
spacing. While these changes are difficult to predict, it is
possible to observe the overlap in damping capacity or
impact energy experiments. It is also possible to observe
these changes using in situ neutron diffraction or x-ray
diffraction analyses at high temperatures.
d. Effect of strain
Deformation also causes changes in axial ratio. Tensile
stress in the c direction may cause an increase in the c
spacing.10 It is possible that when the alloy is deformed
some of the energy received is used up to affect electronic changes such that after deformation less energy
would remain stored in the lattice than would be expected
TABLE V. Effects of certain solutes elements on a spacing and axial
ratio of Mg.10
Effect
Element
Decrease a spacing of Mg
Increase the a spacing of Mg
Decrease c/a
Increase c/a
No measurable effect on either a or c/a
Au, Mn, Rh, Zr
As, Ba, Ce, La, Ni, Pd
Au, Ce, Pd
Ba, Ir, Pt, Rh, Ti, Zr
Ca, Cu, Sb, Si, Te, W
J. Mater. Res., Vol. 23, No. 12, Dec 2008
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A. Becerra et al.: Effects of lithium, indium, and zinc on the lattice parameters of magnesium
TABLE VI. Compositions of magnesium alloys (at.%).
IV. RESULTS AND DISCUSSION
Mg-2.25Li
Mg-6.4Li
Mg-12.6Li
Mg-16.2Li
A. Effects of solutes on lattice parameters
Mg-0.11In Mg-2.82In Mg-0.190Zn Mg-6.2Li-0.34Zn
Mg-0.21In Mg-3.28In Mg-0.325Zn Mg-6.2Li-0.36Zn-0.2In
Mg-0.84In Mg-10In Mg-0.349Zn Mg-0.35Zn-0.19In
Mg-2.21In
Mg-0.70Zn
in alloys with less critical composition (where electronic
change does not take place).
III. EXPERIMENTAL PROCEDURE
A. Alloy synthesis
A number of binary and multi-element solid solution
and dilute magnesium alloys were synthesized using Mg,
Li, In, and Zn. The alloys were prepared from pure metals: magnesium (99.98 wt%) (Timminco Metals, Haley,
Ontario, Canada) and cerium (99.91 wt%) (Hefa Rare
Earths Canada Co. Ltd., Richmond, BC, Canada). Pure
lithium (99.98 wt%) and indium (99.999 wt%) were from
the same source (Alfa Aesar). The alloys were made by
adding the elements to a pure magnesium melt in a Lindberg electric resistance furnace at 680 °C. All alloys were
synthesized under CO2-0.5%SF6 gas cover except for the
Li-containing alloys which were prepared under argon.
Alloys were cast into copper disc molds of 40 mm diameter. The castings were homogenized under argon at
400 °C for 8 h to eliminate microsegregation. Chemical
compositions were determined via Inductively-Coupled
Plasma Atomic-Emission Spectroscopy (ICP-AES)
(Genitest Inc., Montreal). The compositions of the alloys
are shown in Table VI.
B. Characterization
X-ray diffraction (XRD) was carried out on fine
(65 mesh) annealed powder samples with a Phillips 1710
diffractometer with Cu-k␣ radiation, an accelerating
voltage of 40 kV, beam current of 20 mA, and a scan rate
of 0.025 deg/s in a 2␪ range of 25–120°. Fine powder
was obtained by filing the alloys. The powders were
encapsulated in a quartz tube, sealed under argon, and
annealed for 16 h at 275 °C. Pure Si was added to all
powders to correct for sample displacement. Raw XRD
data were refined and analyzed via the Rietveld method
using GSASTM software. The measurements were repeated on three different powder samples and the results
were averaged.
1. Magnesium-lithium system
The change in the lattice parameters and axial ratio of
Mg with Li additions is shown in Table VII. It can be
seen that a contraction of both c and a spacing and a
reduction in unit-cell volume occurs. Since the reduction
in c spacing is more significant, a net reduction in c/a is
observed (Fig. 2).
Vegard’s law explains the change in a spacing well
with the alloying of the small Li atom (rLi ⳱ 1.56 Å;
rMg ⳱ 1.602 Å). However, the change in c spacing
seems to be more substantial than the a spacing shift.
This may be explained by the valency effect. One-valent
lithium decreases the e/a ratio of the alloy from 2 to 1.83
as the Li concentration increases to 16 at.%. Since the
electron overlap in Mg occurs across the B faces (Fig. 1)
of the Brillouin zone, when lower valency solute is added
and e/a decreases, electrons from the second Brillouin
jump into first Brillouin zone. These produce a stress that
causes lattice distortion that pushes the B faces outward
in reciprocal space. Hence, a contraction of the c spacing
in real space occurs. This explains the decrease in the c/a
ratio as well as of the volume.
The change in the interplanar spacing of basal, prismatic, and pyramidal planes in Mg-Li alloys is shown in
Table VIII. A larger reduction in the spacing of basal
planes is observed than in the spacing for non-basal
planes. Since Peierls stress for dislocation motion increases as the interplanar spacing decreases, the activation of basal slip would be challenged and the yield stress
would increase. Using Eq. (1) and neglecting all other
changes except the interplanar spacing d, it can be understood that prismatic slip would set in when
dbasal = c Ⲑ 2 = dprismatic = a公3 Ⲑ 2
,
(3)
or at a c/a ratio ⳱ 1.73 (see Ref. 10, p. 269). Experimentally, the activation of prismatic slip has been seen to
set in the range 1.624–1.610. Extensive prismatic slip has
been observed at Mg-16 at.% Li alloys (see Ref 10, pp.
268–269).
Pyramidal slip with c + a dislocations (i.e., {11-22 <
11-23}system) is expected to be activated when the
TABLE VII. Effects of Li on lattice parameters of Mg.
Alloy
wt% Li
at.% Li
e/a
a (Å)
sd a
sd c
c/a
sd c/a
Volume (Å)3
Mg
Mg-2.3Li
Mg-6.4Li
Mg-13Li
Mg- 16Li
0.00
0.65
1.92
3.97
5.24
0.00
2.25
6.41
12.64
16.22
2.000
1.978
1.936
1.874
1.838
3.2088
3.2076
3.2034
3.1964
3.1930
0.0002
0.0006
0.0004
0.0004
0.0004
0.0004
0.0006
0.0002
0.0005
0.0004
1.6240
1.6227
1.6200
1.6129
1.6068
0.0001
0.0002
0.0002
0.0002
0.0001
46.47
46.38
46.12
45.62
45.30
3382
J. Mater. Res., Vol. 23, No. 12, Dec 2008
A. Becerra et al.: Effects of lithium, indium, and zinc on the lattice parameters of magnesium
slip activation at room temperature. However, in polycrystalline material, the change in d spacings may potentially alter the balance of deformation systems and positively influence room temperature formability.
2. Magnesium-indium system
The effects of three-valent In on the lattice parameters
of Mg are shown in Fig. 3 and Tables IX and X. Figure
3(a) shows that the effect of In on the Mg unit cell is
complicated. At low In concentrations (below 0.5 at.%
In) an expansion is seen in lattice parameters that is
followed by a contraction of both a and c spacing. While
the a spacing continues to contract, the c spacing then
shows a reversal and expands again after 2.2 at.%. The
initial increase in both a and c without any change in the
c/a ratio with In can be explained by Vegard’s law due to
the higher atomic size of In (rIn ⳱ 1.66 Å).
The decrease in lattice spacing cannot be explained by
the electron overlap of a three-valent solute such as indium. That effect would have resulted in at least an increase in c spacing. A decrease may be attributed to the
change of bonding from metallic to covalent, resulting in
FIG. 2. Effect of lithium on (a) the lattice parameters and (b) axial
ratio of Mg.
TABLE VIII. Effect of Li on interplanar spacing.
Interplanar spacing (Å)
Alloy
{0002}
{101̄0}
{112̄2}
(c/2a)(c2 + a2)1/2
Mg
Mg-2.3 Li
Mg-6.4Li
Mg-13Li
Mg-16Li
2.6056
2.6026
2.5948
2.6778
2.5653
2.7789
2.7779
2.7742
2.7682
2.7752
1.3662
1.3654
1.3629
1.3583
1.3554
4.9693
4.9608
4.9398
4.8921
4.8549
parameter d/b in Eq. (1) for the {11-22 < 11-23} system
equals that for the basal {0001}⟨11-20⟩ system. That
means pyramidal slip could be activated when
dbasal Ⲑ a = c Ⲑ 共2a兲 = dpyramidal Ⲑ 共bc+a兲 ,
(4)
or when
dpyramidal = 共c Ⲑ 2a兲共c2 + a2兲1 Ⲑ 2
.
(5)
As seen in Table VII, the pyramidal spacing and the
Burger’s vectors do not reach the values for pyramidal
FIG. 3. Effect of indium on (a) the lattice parameters and (b) axial
ratio of Mg.
J. Mater. Res., Vol. 23, No. 12, Dec 2008
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A. Becerra et al.: Effects of lithium, indium, and zinc on the lattice parameters of magnesium
TABLE IX. Effects of In on lattice parameters of Mg.
Alloy
wt% In
at.% In
e/a
a (Å)
sd a
c (Å)
sd c
c/a
sd c/a
Volume (Å)3
Mg
Mg-0.1In
Mg-0.2In
Mg-0.8In
Mg-2.2In
Mg-2.8In
Mg-3.3In
0.00
0.51
0.97
3.84
9.63
12.05
13.80
0.00
0.11
0.21
0.84
2.21
2.82
3.28
2.000
2.001
2.002
2.008
2.022
2.028
2.033
3.2088
3.2095
3.2094
3.2084
3.2062
3.2054
3.2049
0.0002
0.0000
0.0003
0.0001
0.0002
0.0003
0.0001
5.2111
5.2123
5.2123
5.2117
5.2103
5.2110
5.2113
0.0004
0.0003
0.0003
0.0002
0.0002
0.0001
0.0004
1.6240
1.6240
1.6240
1.6244
1.6251
1.6257
1.6261
0.0001
0.0001
0.0002
0.0000
0.0001
0.0001
0.0001
46.47
46.50
46.50
46.46
46.39
46.37
46.35
TABLE X. Effect of In on interplanar spacing.
Interplanar spacing (Å)
Alloy
{0002}
{101̄0}
{112̄2}
(c/2a)(c2 + a2)1/2
Mg
Mg-0.1In
Mg-0.2In
Mg-0.8In
Mg-2.2In
Mg-2.8In
Mg-3.3In
2.6056
2.6062
2.6062
2.6059
2.6052
2.6055
2.6056
2.7789
2.7795
2.7794
2.7786
2.7767
2.7760
2.7755
1.3662
1.3665
1.3664
1.3661
1.3653
1.3651
1.3650
4.9693
4.9705
4.9706
4.9707
4.9709
4.9729
4.9740
a decrease of indium atom size which in turn results in a
contraction of both a and c parameters. This agrees with
previous work10 performed on Mg-In alloys where lattice
parameters were seen to contract due to the smaller
atomic size of indium. The final increase in lattice spacing above 2.2 at.% In can be explained by the electron
overlap effect across the basal planes which results in a
c spacing increase. Theoretically, an expansion in c spacing for a three-valent solute is expected at 0.75 at.%. The
fact that the expansion is observed at higher at.% (2.2%)
can be attributed to the contribution from the atom size
effect. The 2.2 at.% limit for the onset of the increase in
c spacing agrees with the previous work of Batchelder
et al.14
The change in interplanar spacing (Table X) follows
the lattice parameter changes. It can be observed that a
possibility for prismatic or pyramidal slip activation at
room temperature is not likely based on the experimentally obtained c/a ratio or d spacings. However, the balance of deformation systems may be altered in a polycrystalline material for very small changes, and these
need to be investigated experimentally.
3. Magnesium-zinc system
Zinc additions in the range 0.2–0.7 at.% Zn contracts
both c and a spacing as well as the volume (Table XI).
Since Zn is two-valent, the effect can be explained by
atomic size difference (rZn ⳱ 1.39 Å). Changes to both
lattice parameters shown in Fig. 4 indicate that the contraction in both parameters is to the same degree with no
effect on the axial ratio.
Relationships between solute content and lattice parameters were developed for Li, In, and Zn additions
to Mg as aalloy ⳱ aMg + ma x; calloy ⳱ cMg + mc x;
(c/a)alloy ⳱ (c/a)Mg + mc/a x; volalloy ⳱ volMg–mvol x
where ma, mc, mc/a, and mvol are the slopes of the lattice
parameter versus composition curves experimentally determined and x is at.% solute. The relationships for Li, In,
and Zn additions are shown in Table XII.
4. Multi-component alloys
Multi-component alloys were prepared with combined
additions of Li, Zn, and In. The axial ratio of these alloys
is given in Table XIII. For the Li-containing multicomponent alloys, it is noted that the major effect on
axial ratio of these alloys is contributed by the Li addition.
5. Relationship between axial ratio and electron
concentration of magnesium alloys
Since the effect of combined additions on lattice parameters would be complex, the effect of changing electron ratio (e/a) as a result of solid-solution alloying was
determined and related to lattice parameters. The axial
ratio (c/a) of solid-solution alloys with single and combined additions of alloying elements was related to their
valence electron concentration (e/a). The results are summarized in Table XIII. An empirical relationship [Eq.
(6)] was found between the experimentally determined
axial ratio (c/a) and the valence electron concentration
(e/a) of binary and multi-component alloys [Fig. 5(a)].
c Ⲑ a = −0.32共e Ⲑ a兲2 + 1.34共e Ⲑ a兲 + 0.25
.
(6)
TABLE XI. Effects of Zn on lattice parameters of Mg.
Nominal
wt% Zn
at.% Zn
e/a
a (Å)
sd a
c (Å)
sd c
c/a
sd c/a
Volume (Å)3
Mg
Mg-0.2Zn
Mg-3Zn
Mg-0.7Zn
0.00
0.51
0.87
1.86
0.00
0.19
0.32
0.7
2.000
2.000
2.000
2.000
3.2088
3.2084
3.2076
3.2054
0.0003
0.0005
0.0002
0.0003
5.2111
5.2107
5.2091
5.2049
0.0004
0.0002
0.0005
0.0003
1.6240
1.6241
1.6240
1.6238
0.0001
0.0003
0.0001
0.0002
46.47
46.45
46.41
46.31
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A. Becerra et al.: Effects of lithium, indium, and zinc on the lattice parameters of magnesium
FIG. 4. Effect of zinc on the lattice parameters of Mg.
TABLE XII. Lattice parameter versus solute content.
Solute
addition
Li
In
Zn
a
C
c/a
Volume
3.21 − 0.0059x 5.22 − 0.0053x 1.624 + 0.0011x 46.5 − 0.049x
3.21 − 0.0015x 5.21 − 0.0004x 1.624 + 0.0006x 46.5 − 0.049x
3.21 − 0.0011x 5.21 − 0.0113x
—
46.5 − 0.272x
TABLE XIII. c/a versus e/a for single and combined solute additions
to Mg.
Alloy (at.%)
c/a
e/a
Alloy (at.%)
c/a
e/a
Mg
Mg-2.25Li
Mg-6.4Li
Mg-12.6Li
Mg-0.11In
Mg-0.21In
Mg-0.84In
Mg-2.21In
Mg-2.82In
1.6240
1.6227
1.6200
1.6129
1.6240
1.6240
1.6244
1.6251
1.6257
2
1.98
1.94
1.87
2.001
2.002
2.008
2.025
2.028
—
Mg-3.28In
Mg-0.190Zn
Mg-0.325Zn
Mg-0.349Zn
Mg-0.70Zn
Mg-6.2Li-0.34Zn
Mg-6.2Li-0.36Zn-0.2In
Mg-0.35Zn-0.19In
—
1.6261
1.6241
1.6240
1.6240
1.6238
1.6219
1.6219
1.6241
—
2.033
2.0
2.0
2.0
2.0
1.9381
1.9396
2.0019
Atomic radius (Å): Mg: 1.602; In: 1.663; Li: 1.562; Zn; 1.394 (metallic
bonding as per Ref. 15).
Using Hume-Rothery rules and the data for Mg-Li from
the phase diagram,13 as well as the experimental values
of this study, a more extensive c/a versus e/a graph was
plotted [Fig. 5(b)]. The axial ratio versus electron concentration relationship here is given as
c Ⲑ a = −15.6共e Ⲑ a兲2 + 60共e Ⲑ a兲 − 55.8
.
(7)
From Fig. 5(b), it is interesting to note that an e/a of
∼1.83 and 2.03 would give an axial ratio of 1.58, similar
to Ti and Be which are known to have non-basal slip
activation at room temperature. Mg would have this c/a
value for a Li concentration of 20 at.% and, hypothetically, at In solute concentrations of ∼3–5 at.%. An
interesting study would be the determination of the deformation behavior and slip systems as the Mg structure approaches a c/a of around 1.58–1.59 or an e/a of
FIG. 5. (a) c/a versus e/a for binary, ternary and quaternary Mg solidsolution alloys and (b) experimental data of (a) plotted together with
the two theoretical points (BCC, FCC) for cubic structures.
∼2.03–2.04. Mg with 5 at.% In can also give an e/a of
2.05 and should be studied in the future for possible
non-basal slip activation. It is also possible to make calculations for other multi-component alloys that could result in electron concentrations in the range 1.7–1.8 which
would change the unit cell of Mg toward BCC structure,
and in the range 2.03–2.1 which could result in FCC
structure. Investigation of Mg with 10 at.% In and the
design of multi-component Mg solid-solution alloy to
give an e/a ratio of 2.1 to see if a near-FCC structure
develops are also worthy of study in the future.
B. Synopsis
The axial ratio (c/a) of magnesium can be effectively reduced by Li additions. Furthermore, this study
shows that the axial ratio of magnesium can be designed by developing solid-solution alloys with specific electron-to-atom (e/a) ratios using the relationship
c/a ⳱ −15.6(e/a)2 + 60(e/a) − 55.8.
J. Mater. Res., Vol. 23, No. 12, Dec 2008
3385
A. Becerra et al.: Effects of lithium, indium, and zinc on the lattice parameters of magnesium
V. CONCLUSIONS
(1) The one-valent lithium decreased the axial ratio (c/a) of magnesium from 1.624 to 1.6068 within the
0–16 at.% Li range. This was attributed to the decrease in
the e/a ratio causing electron overlap from the second
Brillouin zone to the first Brillouin which leads to a
contraction of the c spacing in real space and a decrease
in the c/a ratio.
(2) The three-valent indium increased the c/a ratio
of magnesium to 1.6261 as indium increased toward
3.3 at.%. The changes in a and c spacings were related to
changes in the e/a ratio as well as the atom size effect
through Vegard’s law.
(3) The two-valent zinc showed no effect on the c/a
ratio in the 0.2–0.7 at.% range, since, as explained by
Vegard’s law, the atom size caused similar change in
both a and c parameters.
(4) A relationship was determined between the e/a
and c/a ratios as c/a ⳱ −15.6(e/a)2 + 60(e/a) − 55.8.
ACKNOWLEDGMENTS
3.
4.
5.
6.
7.
8.
9.
10.
The authors gratefully acknowledge the financial support of Natural Sciences and Engineering Research
Council (NSERC) of Canada, General Motors (GM) of
Canada, and McGill University and thank Pierre Vermette for assisting in the casting experiments.
11.
12.
13.
14.
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