Respuesta :
A) The linear density expressions for BCC [110] and [111] directions in terms of the atomic radius R are;
i) LD_110 = √3/(4R√2)
ii) LD_111 = 1/(2R)
B) The linear density values for these same two directions for iron (Fe) are;
i) LD_110 = 2.4 × 10^(9) m^(-1)
ii) LD_111 = 4 × 10^(9) m^(-1)
Calculating Linear Density of Crystalline Structures
A) i) To find linear density expression for BCC 110, first of all we will calculate the length of the vector using the length of the unit cell which is 4R/√3 and the cell edge length which is 4R. Thus, the vector length can now be calculated from this expression;
√((4R)² - (4R/√3)²)
This reduces to; 4R√(1 - 1/3) = 4R√(2/3)
Now, the expression for the linear density of this direction is;
LD_110 =
Number of atoms centered on (110) direction/vector length of 110 direction
In this case, there is only one atom centered on the 110 direction. Thus;
LD_110 = 1/(4R√(2/3))
LD_110 = √3/(4R√2)
ii) The length of the vector for the direction 111 is equal to 4R, since
all of the atoms whose centers the vector passes through touch one another. In addition, the vector passes through an equivalent of 2 complete atoms. Thus, the linear density is;
LD_111 = 2/(4R) = 1/(2R)
B)i) From tables, the atomic radius for iron is 0.124 nm or 0.124 x 10^(-9) m. Therefore, the linear density for the [110] direction is;
LD_110 = √3/(4R√2) = √3/(4*0.124*10^(-9)(√2))
LD_110 = 2.4 × 10^(9) m^(-1)
ii) for the 111 direction, we have;
LD_111 = 1/(2R) = 1/(2*0.124*10^(-9))
LD_111 = 4 × 10^(9) m^(-1)
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