Article Type : Research Article
Authors : Xu R
Keywords : Covariance; Weighted and non-weighted; Lattice constants; Al content; TiAl; TiAl-Mo; Si and V; regression linear equation
According to covariance defined in Ti-Al,
new simplified equations to cause covariance are gained. From relations of them
we find easily that in the case of lattice constants it will be precision to
compare with a. In the case of they will precision too. C*a2 is the best
precision with about 0.02%. C is better one in precision than a. c& a is
preciser than c/a & c*a2 from covariance’s. C& a is preciser 4~5 times
than c/a & c*a2 from covariance. To use the simplified formula to enter to
computer and calculation velocity is quickly more than two times compared with
original one. The regression linear equation between c/ A and Al concentration
is calculated as
Deviation with their
weight and non-weight in measurement of data can be estimated in applications.
The variance shall be used currently in applies of measuring for the error
scope. This method can be used to apply to more precision. Minimum two multiply
with weight is important to value the precision here. Covariance with weight is
another way to be used for two dependent constants. It is advanced way to value
two factors. Here we induce these two methods into new and simple equations. It
is found that these equations can be applied to simple cases. [1-3]. In TiAl
the lattice constants in binary and ternary TiAl are evaluated to observe the
precision problem. The values are induced from the X-ray diffraction [4-6]. The
lattice constants in TiAl are important factor to measure precision value of c
& a so as for their values which are analysed after measurement. This
process need to be through covariance even variance methods, it can be used to
actual. We can compare covariance with standard value to find which precision
they can reach [7]. This is the first valuation of their detail value. We can
find the detail differences among them. For the searching later we should know
whether the value size is available to scientific items. And some program are
used in laboratories made by Japan and we can observe the precision over it. We
also can judge whether they can be used publicly. In this program we use the
XRD distance of plains to be substituted into their interface and the c and a
can be calculated by two plains miller exponents. The result is good to all of
data and can apply to virtual case in accordance with it. Al content and
lattice constant of covariance has certain significance.
The binary and ternary
TiAl were produced to analyze. The 99.7wt. % sponge Ti, 99.9wt. % bulk Al and
99.9wt.%Mo,V and Si were used to produce specimen. They were melted under 99.9%
Ar gas in plasma arc furnace. For homogeneous specimen they were melted more
than two times. The lattices constant are investigated on X-ray diffraction and
lattice program. Experimental conditions are 40KV and 30mA, the scanning speed
is 10°C/min.
As to covariance
According to (1) we calculate the covariance (Figure 1).
(a) Constant c and a
(a) constant c/a and c*a2
Figure 1: Covariances with weighted & non in lattice constant c and a, here w-weight, cov-covariance.
(a) Si element
(a) Mo element
(a) V element.
Figure 2: Covariances with weight in lattice constant & c/a and c*a2 in the third element.
It can be seen from
Figure 1(a,b) that a with large error has the largest covariance under both
conditions. Under both conditions, c has the lowest covariance and the highest
accuracy. According to Figure 1, the maximum error is ca2, whose
value decreases after weighting. C /a error is the minimum, with better
accuracy (Figure 2).
The Figure 2(a~c) show
that the Si correlation with ca2 is the closest so this one’s covariance is
lowest. The second correlation is a is important factor to constant. Its most
difference value is ±0.4% whose standard deviation is 63.2%. C is third one
whose value is 0.4% too but the one isn’t changeable in 1atSi. The last is c/a
which is ±1% at the end of 0 and 2Si, it explains it is the best in 1Si worst at
the end of ones. The figure 12 shows that the deviation of certain difference
is observed with c in Mo contents. The others has linear relations with Mo.
They have low deviations comparing with 46AL. As Al and ca2 is almost same, it
explains Al and this item is very approaching both in covariance and variance
status. They have consistency with the item like in weight state. They have
dependence with the item strongly. As it is known that anisotropic lattices of
c and a cause the anistrope mechanical properties according to c and a direction
and related directions. The Figure 2(c) explains V has dependency on Al. The
values increase with V element. V can play the concentration changes in 46Al
base. Maybe it plays resolution strength. The resolution includes V content and
maintains high temperature B phase which play roles of strength result in high
strength and ductility respectively. But it has high deviation ie. Dependency
in 3V. From Figure 2 it shows that a? c and c/a is low deviation relatively
with ca2. It explains that it will affect a and c primarily and then others.
Here ca2 is the highest and then c in 1V alloy. It may shows that the good
consistent with TiAl-3V and deficiency in 1V.
The
linear regression equation between c/a and Al
It is supposed that the
function f(X,Y)=0 is a linear equation. In order to find the specific
relationship between lattice constant and Al from the physical meaning, the
reason and correlation degree of the phenomenon of inadequate ductility of
metal at room temperature were found out. So we use the least square method to
calculate this linear regression equation. According to the least square
method, set the function f(X,Y)=0, X=c/a and Y=Al% above
It has
Partial derivatives to a
there is
Partial derivatives to b
it has
Above two formula is zero
if they are minimum there is
And
Calculate a and b there
is
Due to
With these two formula it
has
a=2.06, b=-0.0025
The linear regression
equation is as below
·
C*a2
is the best precision with 0.02% approximately. C is better one in precision
than a.
·
The
standard deviation of c & a to weight is 0.2% while that of c/a & c*a2
is 0.85% approximately. C & a is preciser 4~5 times than c/a & c*a2
from covariance’s.
The anisotropic lattices of c and a cause the
anistrope mechanical properties according to c and a direction and related
directions. The linear regression equation is for binary TiAL as below.