Consider a tetragonal phase of which the lattice parameters satisfy c = 2a. Diffusivities of the phase were measured in two different directions as:
6.0x10-12 cm2/sec along [101] and 8.0x10-12 cm2/sec along [001].
(a) Determine the diffusivity tensor (Dij) expressed with respect to the coordinate system consisting of crystallographic axes [100], [010] and [001].
(b) Calculate the diffusivity along [201].
To determine the diffusivity tensor (Dij), we need to express the diffusivities measured in different directions in the coordinate system consisting of crystallographic axes [100], [010], and [001].
First, let's define the coordinate system:
[100] axis aligns with the a-axis.
[010] axis aligns with the b-axis.
[001] axis aligns with the c-axis.
Next, let's write down the diffusivity tensor (Dij) in matrix form:
| D11 D12 D13 |
D = | D21 D22 D23 |
| D31 D32 D33 |
Since we are given diffusivities along [101] ( a_1 vector) and [001] (c_1 vector), we need to determine the corresponding components of the Dij matrix.
(a) Determining the diffusivity tensor (Dij):
The diffusivity along [101] ( a_1 vector, parallel to [100] + [001]) is given as: D(a_1) = 6.0x10^(-12) cm^2/sec.
We can express the a_1 vector in terms of the crystallographic axes as:
a_1 = (1/√2) * [100] + (1/√2) * [001]
= (1/√2) * a + (1/√2) * c
Similarly, the diffusivity along [001] ( c_1 vector) is given as: D(c_1) = 8.0x10^(-12) cm^2/sec.
We can express the c_1 vector in terms of the crystallographic axes as:
c_1 = [001] (already aligned with the c-axis)
To find the individual components, we project D(a_1) and D(c_1) onto the crystallographic axes.
D(a_1) is projected as:
D(a_1) = (1/√2) * D11 + (1/√2) * D13
Comparing the coefficient of D11 and D13, we get:
1/√2 * D11 = 6.0x10^(-12) cm^2/sec
1/√2 * D13 = 6.0x10^(-12) cm^2/sec
Therefore, D11 = D13 = 6.0x10^(-12) * √2 cm^2/sec
D(c_1) is projected as:
D(c_1) = D33
Comparing the coefficient of D33, we get:
D33 = 8.0x10^(-12) cm^2/sec
Finally, we can determine the remaining components of the Dij matrix by substitution:
D = | D11 D12 D13 |
| D21 D22 D23 |
| D31 D32 D33 |
= | 6.0x10^(-12) * √2 cm^2/sec D12 6.0x10^(-12) * √2 cm^2/sec |
| D21 D22 D23 |
| 0 0 8.0x10^(-12) cm^2/sec |
(b) Calculating the diffusivity along [201]:
To calculate the diffusivity along [201] ( a vector parallel to [010] + 2[001]), we need to express it in terms of the crystallographic axes.
[201] = [010] + 2[001]
= b + 2c
Since we have D12 and D23 missing in the Dij matrix, we cannot directly calculate the diffusivity along [201]. The diffusivity tensor Dij does not contain the required information to determine the diffusivity along [201].
In order to calculate the diffusivity along [201], we would need additional information, such as the values of D12 and D23, or any other directional diffusivities in the [010] and [001] directions.