Acta Materialia 52 (2004) 2889–2894
www.actamat-journals.com
The evolution of non-basal dislocations as a function
of deformation temperature in pure magnesium determined
by X-ray diffraction
a,b
, K. Nyilas a, A. Axt a, I. Dragomir-Cernatescu a, T. Ung
ar
K. M
athis
a
a,*
, P. Lukac
b
Department of General Physics, E€otv€os University Budapest, H-1117 Budapest, Pazmany P. setany, 1/A, Hungary
b
Department of Metal Physics, Charles University, CZ-121 16 Prague, Ke Karlovu, 5, Czech Republic
Received 16 October 2003; received in revised form 19 February 2004; accepted 23 February 2004
Available online 8 April 2004
Abstract
Pure magnesium is deformed up to fracture at various temperatures. The deformed samples are investigated by high resolution
X-ray diffraction peak profile analysis. The diffraction peaks are fitted by theoretical profile functions where the strain profile is
scaled for strain anisotropy by the dislocation contrast factors. The contrast factor parameters are evaluated in terms of the fundamental Burgers vector types in hexagonal crystals. The evolution of the dislocation density and the Burgers vector types with the
temperature of deformation is discussed on the basis of dislocation reactions and dynamic recovery in magnesium.
Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Pure magnesium; Tensile test; Non-basal dislocations; Line broadening; Burgers vector population
1. Introduction
Magnesium alloys, as the lightest structural materials,
are very attractive in large amount of applications. Excellent specific strength (mechanical properties) of
magnesium alloys, comparable with steel, predestines
these materials for applications in the automobile (and
aircraft) industry. On the other hand they possess low
creep resistance, low cold forming capability and low
corrosion resistance. In order to improve plasticity of
magnesium alloys it is important to understand their
deformation behaviour. However, there are limited investigations of mechanical properties, especially concerning the evolution of the dislocation structure.
The true-stress true-strain curves of hexagonal polycrystals are very similar to those for polycrystals with fcc
structure. But there are differences in the activity of slip
systems. In fcc metals there are five independent crystallographically equivalent slip systems, which are re*
Corresponding author. Tel.: +36-1-372-2801; fax: +36-1-372-2811.
E-mail addresses: ungar@metal.elte.hu, ungar@ludens.elte.hu (T.
Ung
ar).
quired for plastic deformation of polycrystals according
to von Mises [1]. In contrast, hcp metals do not possess
five independent crystallographically equivalent slip
systems. The directions for easy slip in hcp single crystals are three a type directions. The directions lie in the
basal plane and in three prismatic planes, therefore the
crystallographic slip is commonly observed to occur on
basal or prismatic slip systems. In magnesium the main
slip system is in the basal plane (0 0 0 1) with the three
close packed directions: a1 ¼ 13 ½1 1 2 0, a2 ¼ 13 ½2 1 1 0 and
a3 ¼ 13 ½1 2 1 0. These vectors are perpendicular to the c
axis (c direction) therefore, slip in this direction cannot
produce strain parallel to the c axis. It is clear that another non-basal slip system must be activated to deform
polycrystals of magnesium. The prismatic and pyramidal slip system as non-basal slip systems as well as
twinning is most probably required for deformation of
Mg polycrystals.
When a slip system with c þ a Burgers vector is operative, then the von Mises [1] criterion can be fulfilled.
Bocek et al. [2] and Lukac [3,4] have assumed that
motion of non-basal dislocations and their reaction with
basal dislocations can explain the work hardening of Zn,
1359-6454/$30.00 Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2004.02.034
K. Mathis et al. / Acta Materialia 52 (2004) 2889–2894
Cd and Mg single crystals. Among non-basal slip
systems, the second-order pyramidal slip, f1 1 2 2g1=
3h1 1 2
3i, is important. Yoo et al. [5,6] investigated the
stability of c þ a dislocations and their mobility in
hexagonal close packed metals. Identification of c þ a
dislocations in Mg was made by transmission electron
microscopy (TEM) [7–9]. Simulation analysis and texture measurements demonstrated that the c þ a dislocations play an important role in the texture evolution
of Mg alloys [9,10]. Further works are needed in order
to better understand the role of non-basal dislocations
in the deformation behaviour of hexagonal metals. The
aim of the present paper is to determine the activity of
slip systems in pure magnesium deformed in tensile tests
at various temperatures. X-ray diffraction peak profile
analysis will be used.
2. Experimental
Cylindrical specimens of pure Mg (99.9%) with a
length of 25 and 5 mm in diameter were used for tensile
tests. Specimens were deformed in an INSTRONÒ machine at a constant cross-head speed giving an initial
strain rate of 3.3 105 s1 in the temperature range
from room temperature to 300 °C [11]. Both the cross
and longitudinal section of the deformed specimens were
used for X-ray analysis. The diffraction profiles were
measured by a special double crystal diffractometer with
negligible instrumental broadening, [12,13]. A fine focus
rotating copper anode (Nonius type FR 591) has been
operated at 45 kV and 80 mA. The symmetrical (2 2 0)
reflection of a Ge monochromator is used in order to
have wavelength compensation at the position of the
detector. The Ka2 component of the Cu radiation is
eliminated by an 0.3 mm slit between the source and the
Ge crystal. The footprint of the beam on the specimen
has been 0.1 4 mm2 . The profiles were registered by a
linear position sensitive gas flow detector (OED 50
Braun, Munich). The distance between specimen and
detector was set to be 0.8 m in order to have the required
angular resolution. The profiles of the following 11
Bragg reflections were measured: 10.0, 00.2, 10.1, 10.2,
10.3, 11.0, 11.2, 20.0, 20.1, 20.2 and 21.1.
0,010
Integral breadths
FWHM
0,008
Breadths [1/nm]
2890
0,006
0,004
11.2
10.0 00.2 10.1
The diffraction profiles are evaluated by assuming
that peak broadening is caused by smallness of crystallite size and strain caused by dislocations. The Williamson–Hall plot of the full width at half maximum
(FWHM) and the integral breadths reveal a strong
strain anisotropy, as it can be seen in Fig. 1 for the
specimen deformed at 200 °C. The measured physical
profiles are fitted by theoretical profiles calculated on the
basis of well-established profile functions of size and
20.1
20.3
0,002
2
4
6
8
10
K [1/nm]
Fig. 1. The Williamson–Hall plot of the full width at half maximum (FWHM) and the integral breadths for the specimen deformed at
200 °C. The indices of the different measured reflections are also
indicated in the figure.
strain. Both, strain and strain anisotropy are accounted
for by the dislocation model of lattice distortions [14,15].
The size profile is built up by assuming that the crystallites (or coherently scattering domains) are spherical
and that the size distribution is log-normal. For details
see Eqs. (12)–(15) in [16]. The assumption of spherical
crystallites is based on TEM observations carried out on
Mg treated and deformed under the same conditions
[17]. The strain profile is constructed by assuming that
the main source of lattice distortions are dislocations.
The mean square strain, he2g;L i, (where g is the absolute
value of the diffraction vector and L is the Fourier variable) contains the dislocation contrast factors, which in
the case of hexagonal crystals requires special and
careful averaging [18]. As mentioned before, in hexagonal crystals there are three major slip systems: basal,
prismatic and pyramidal. Each of them consists of several sub-slip-systems, with fundamentally different dislocations and Burgers vectors [19–21]. When in the
different sub-slip-systems the dislocations are randomly
populated, or the specimen is polycrystalline, the mean
square strain, can be given as [18]:
he2g;L i 
2.1. Evaluation of the X-ray diffraction data
10.3
10.2
qCb2
f ðgÞ;
4p
ð1Þ
where,
Cb2 ¼
N
X
ðiÞ b2 ;
fi C
i
ð2Þ
i¼1
where N is the number of the different activated sub-slipðiÞ
systems, C is the average dislocation contrast factor
corresponding to the ith sub-slip-system and fi are the
fractions of the particular sub-slip-systems by which
they contribute to the broadening of a specific reflection.
f ðgÞ is the Wilkens function, where g ¼ L=Re and Re is
K. Mathis et al. / Acta Materialia 52 (2004) 2889–2894
the effective outer cut-off radius of dislocations [14,22].
In a texture free polycrystal or if all possible Burgers
vectors are activated in a particular slip system the dislocation contrast factors can be averaged over the permutations of the corresponding hkl indices [14,18]. In
the case of cubic crystals, usually there is only one type
of slip-systems (especially in fcc crystals it is:
h1 1 0if1 1 1g) and therefore, on the right hand side of
Eq. (2) the averaging bar would be only above the
contrast factors, C [16]. If, however, as in the case of
hexagonal crystals, there are several different types of
slip system with different Burgers vectors the averaging
has to be extended to the Burgers vectors together with
the contrast factors, as indicated in Eqs. (1) and (2). This
circumstance is the reason why the evaluation of strain
anisotropy in hexagonal crystals is far more complicated
than in cubic crystals. The average contrast factors for a
single sub-slip-system in hexagonal crystals are [18]:
C hk:l ¼ C hk:0 ½1 þ q1 x þ q2 x2 ;
ð3Þ
2
where x ¼ ð2=3Þðl=gaÞ , q1 and q2 are parameters depending on the elastic properties of the material, C hk:0 is
the average contrast factor corresponding to the hk:0
type reflections and a is the lattice constant in the basal
plane. The q1 and q2 parameters and the values of C hk:0
have been evaluated numerically and compiled for a
large number of hexagonal crystals and compounds in
[18]. The experimental values of q1 and q2 denoted as
ðmÞ
ðmÞ
q1 and q2 , are provided by the whole profile fitting
procedure, MWP fitting program, given in detail in [19].
A typical simultaneous fitting of the theoretical (solid
lines) and measured (open circles) Fourier coefficients
for eighth reflections is shown in Fig. 2.
Eq. (3) shows that the X-ray measurements provide
ðmÞ
ðmÞ
only two independent parameters: q1 and q2 (where
the superscript indicates measured values). This means
that there is no possibility to determine the activities of
each individual sub-slip-system separately. However, it
is shown that, by making some physically based assumptions or using auxiliary information on slip, the
activity or the relative fractions of the three basic
Burgers vector types: a, c or c þ a, is possible. Denote
the three Burgers vector types: b1 ¼ 1=3h2 1 1 0i, (a type),
b2 ¼ h0 0 0 1i, (c type) and b3 ¼ 1=3h21 1 3i, (c þ a type)
2
ðmÞ
and the measured value of the average of b C as: Cb2 .
For the three different Burgers vector types it follows
from Eq. (2) [18]:
b2 Chk:l
ðmÞ
¼ b21
N hai
X
i¼1
fi C ðiÞ þ b22
N hci
X
j¼1
fj C ðjÞ þ b23
NX
hcþai
fn C ðnÞ ;
10.0 00.2
where N hai, N hci and N hc þ ai are the number of subslip-systems with the Burgers vector types a, c or c þ a,
respectively. Using the schemes of Jones and Hutchinson [20] and Kuzel and Klimanek [21]: N hai ¼ 4,
10.2
10.3
11.2
20.1
20.3
1.0
A(L)
0.5
0.0
∆ L=100nm
Fig. 2. The measured (open circle) and the fitted (solid lines) Fourier
coefficients, normalised to unity, of the different profiles corresponding
to the Mg specimen deformed at 200 °C. The differences between the
measured and fitted data and the indices of the different reflections are
shown in the bottom and the top of the figure, respectively.
N hci ¼ 2 and N hc þ ai ¼ 5, see also Table 1 in [18]. It
has to be noted that, due to the structure of Eq. (4),
Chk:0 b2 is a scaling parameter for the absolute value of
the dislocation density, therefore only the q1 and q2
parameters can be used for the systematic Burgers vector analysis [18]. Once the Burgers vector types are
determined the value of Chk:0 b2 and the dislocations
density can also be calculated, for further details see [18].
ðmÞ
The measured values of these two parameters, q1 and
ðmÞ
q2 , give two equations for the evaluation. A third
equation is provided by the condition that: Rfi ¼ 1
and fi P 0, where fi are the fractions of the different
sub-slip-systems.
Since the absolute values of the Burgers vectors and
the corresponding eigen-energies of the dislocations
within a particular sub-slip-system are the same, in a
first step it is assumed that a particular Burgers vector
type has random (or uniform) distribution within each
sub-slip-system type, a, c or c þ a, respectively. With this
assumption Eq. (4) can be reduced to:
b2 Chk:l
ðmÞ
¼
3
X
ðiÞ b2 ;
hi C
i
ð5Þ
i¼1
where hi are the fractions of the three Burgers vector
types, a, c or c þ a, and C ðiÞ are the averages over the
sub-slip-systems, each corresponding to the same
Burgers vector type. Inserting Eq. (3) into Eq. (5):
n¼1
ð4Þ
10.1
2891
ðmÞ
q1 ¼
ðmÞ
q2 ¼
3
1X
ðiÞ b2 qðiÞ ;
hi C
hk:0 i 1
P i¼1
ð6Þ
3
1X
ðiÞ b2 qðiÞ ;
hi C
hk:0 i 2
P i¼1
ð7Þ
2892
K. Mathis et al. / Acta Materialia 52 (2004) 2889–2894
P3
ðiÞ b2 ¼ b2 Chk:0 ðmÞ and the fractions hi
where P ¼ i¼1 hi C
hk:0 i
P3
have to satisfy the conditions: i¼1 hi ¼ 1, and hi P 0,
for all hi fractions. A computer program has been
elaborated to determine the Burgers vector populations
as follows. The solution of Eqs. (6) and (7) with the
conditions for hi , together with the assumption that the
sub-slip-systems are equally populated within the three
Burgers vector types is not necessarily possible. In order
to solve this problem the following procedure is developed. A sub-slip-system is only activated if the corresponding Schmid factor is large enough. In a general
case it is quite plausible that not all sub-slip-systems will
be activated. Taking this into account, a numerical
procedure has been worked out in which the number of
ðmÞ
sub-slip-systems in the calculation of b2 Chk:l in Eq. (4)
is reduced systematically. This means that the numbers
N hai, N hci and N hc þ ai in the sums in Eq. (4) are reduced systematically in such a way that all possible
combinations, in the combinatoric sense, are taken into
account. This numerical procedure of solving Eqs. (6)
and (7) provides a matrix of solutions for the possible hi
fractions, satisfying the conditions below Eqs. (6) and
(7). This matrix of solutions is evaluated for the three
Burgers vector types, a, c and c þ a, respectively. The hi
fractions, which are in accordance with these conditions
are given as minimum and maximum ranges: hmin
to
i
hmax
, for the three Burgers vector types. The graphic ili
lustration of the solution matrix of the hi fractions in the
case of the as-cast specimen is shown in Fig. 3. The figure indicates that the ranges, hmin
to hmax
, for one pari
i
ticular Burgers vector type are relatively narrow: about
80–96, 0–16 and 0–6%, for a, c and c þ a, respectively.
Once the active slip systems are determined, the effective dislocation contrast factors can be calculated
according to Eq. (4). From these values Eq. (2) can be
evaluated and from Eq. (1) the average dislocation
density can be determined. For more details see
[18,23,24].
3. Results and discussion
The temperature dependence of the true stress–true
strain curves measured in tension are shown in Fig. 4. It
can be seen that the deformation behavior of magnesium polycrystals is very sensitive to temperature. Above
about 0:3Tm (Tm is the absolute melting point) a strong
decrease in the work hardening with increasing temperature is observed. Simultaneously, the ductility is
increasing. The macroscopic work hardening is a result
of the sum of hardening and softening, latter being
mainly dynamic recovery. Two main dislocation reactions can contribute to the hardening process [4]:
1=3½1 1 2 0 þ 1=3½1 1 2 3 ! ½0 0 0 1;
ð8Þ
and
1=3½1 1 2 0 þ 1=3½2 1 1 3 ! 1=3½1 2 1 3:
ð9Þ
The b ¼ ½0 0 0 1 dislocation is sessile, thus contributes
directly to strain hardening. The reaction in (9) produces
a dislocation oblique to the main, basal slip plane, thus
it is acting as a forest dislocation. The other possible
mechanism for hardening is mechanical twinning
[25,26].
Dynamic recovery processes cause strain softening
during deformation. Local cross-slip of basal dislocations and/or dislocation climb can be taken into account
as the mechanisms responsible for softening. Also the
activity of c þ a dislocations plays an important role in
dynamic recovery, the corresponding reaction can be
described as [4]:
1=3½1 1 2 3 þ 1=3½2 1 1 3 ! 1=3½1 2 1 0:
ð10Þ
200
Number of solutions
150
<c+a>
100
<a>
<c>
50
0
0
50
100
hi
[%]
Fig. 3. The bar diagram of the solution matrix of the hi fractions (see in
the text) of the three fundamental Burgers vector types, a; c and, c þ a,
in the case of the as cast specimen.
Fig. 4. The temperature dependence of the true-stress true-strain
curves of the tensile deformed Mg specimens.
K. Mathis et al. / Acta Materialia 52 (2004) 2889–2894
good correlation with the predictions of Eq. (10) according to which, when the activity of c þ a type dislocations increases dynamic recovery becomes more
efficient too.
100
<a>
<c>
60
<c+a>
hi [%]
2893
40
4. Conclusions
ρ
[10
50
-13
-2
m ]
ρ
20
0
0
0
as cast
100
200
T [˚C]
300
Fig. 5. The hi fractions (see in the text) of the three Burgers vector
types, a; c and, c þ a, as a function of the deformation temperature.
The error bars in the figure are for the ranges of the Burgers vector
types in the numerical procedure described in the text.
Screw dislocations of c þ a type can move to the next
slip planes by double cross-slip followed by dislocation
annihilation. This causes also a decrease in the flow
stress and the work hardening rate. The activity of the
pyramidal system depends strongly on temperature. The
critical resolved shear stress (CRSS) necessary for slip in
this system at room temperature is about 100 times
larger than that for basal slip [5]. This value is decreasing with increasing temperature and above 200 °C
the activation of dislocation motion in the pyramidal
system is energetically more favorable than twinning.
Therefore an increase in the density of pyramidal dislocations and the frequency of the dislocation reaction
(10) is expected at and above about 200 °C.
The mean total dislocation density q obtained by
X-ray analysis (before fraction when the work hardening rate is close to zero) as a function of the deformation
temperature is shown in Fig. 5.
The hi fractions of dislocations with the three Burgers
vector types, a, c and c þ a are also shown as a function
of the temperature of deformation in Fig. 5. The error
bars in the present figure are standing for the ranges
determined for the Burgers vector types in the numerical
procedure described above. The figure shows that at
room temperature and at 100 °C the dominant dislocation type is a or mainly basal. At higher temperatures
the fraction of a decreases, whereas the fraction of c þ a
increases. The fraction of c remains practically unchanged. It can be seen that the average dislocation
density q increases considerably upon deformation as
compared to the value in the as-cast state. However,
with increasing deformation temperature the increment
of q decreases strongly in accordance with dynamic recovery. The concomitant increase of the fraction of
c þ a dislocations and the decrease of the average dislocation density with deformation temperature is in
The dislocation types and the average dislocation
densities have been determined in the as cast and deformed states in pure Mg deformed at different temperatures between room temperature and 300 °C. High
resolution X-ray diffraction peak profile measurements
have been carried out. A numerical procedure has been
developed for evaluating experimental values of the
dislocation contrast factors in terms of the different, well
established dislocation types, a, c and c þ a. It is found
that in the as cast state the overwhelming majority of the
dislocations is of a type and the dislocation density is
about 2 1014 m2 . Due to plastic deformation up to
fracture the a type dislocations remain dominant, however, the dislocation density increases by about a factor
of three up to about 100 °C. At higher temperatures the
fraction of c þ a type dislocations is increasing on the
cost of a type dislocations and the increase of the dislocation density is strongly reduced. The results are
discussed in terms of dislocation reactions and dynamic
recovery at higher homologous temperatures.
Acknowledgements
K.M. and P.L. express their gratitude for financial
support from Grant Agency of the Academy of Sciences
of the Czech Republic under Grant A2112303. Thanks
are due to the Hungarian National Science Foundation
OTKA T-031786, T–043247 and T-046990 grants for
supporting this work.
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