|Nov29-08, 07:45 AM||#1|
volume of HCP unit cell from radius + c/a ratio
1. The problem statement, all variables and given/known data
Cobalt has an HCP (hexagonal close packed) crystal structure, an atomic radius of 0.1253 nm and a c/a ratio of 1.623. Compute the volume of the unit cell for cobalt.
2. Relevant equations
area of hexagon = apothem*perimeter/2
3. The attempt at a solution
calculate 1 of the 6 sides (2 atomic radii per side):
2*.1253 = 2.506e-1
2.506e-1*6 = 1.5036e+0
2.506e-1*1.623 = 4.06724e-1
calculate hexagon area (apothem = 2 radii):
1.5036e+0*2.506e-1 / 2 = 1.88401e-1
1.88401e-1*4.06724e-1 = 7.66272e-2
|Nov29-08, 05:11 PM||#2|
mistook the apothem for the hexagon radius.
|Sep11-11, 03:19 PM||#3|
Hi... I'm currently studying for my first test on Solid State Physics and looking for some info I found your question... I agree that considering the right expression for the apothem one gets to the answer proposed... Nevertheless, I think there is a consideration to make in the problem...
I think they are asking for six times the volume of the parallelipiped formed by the vectors
a1 = (a,0,0)
a2 = (a/2,a*sqr(3)/2,0)
b3 = a1/3 + a2/3 + (0,0,c/2)
. In other words, just the volume proposed by the previous answers, 3*sqr(3)*a*a*c/2.
But, due to the geometry of the hcp the distance to the nearest neighbors depends on c/a (It is not always a). One finds that this distance is the minimum between a and sqr(a*a/3 +c*c/4) (This last value is just the norm of the vector b3 I used before). Then if c/a > sqr(8/3) the closest neighbors are just the 6 ones in the xy planes (the hexagon around the point). But if c/a < sqr(8/3) (around 1.63299) then the closest neighbors are 6 different points (3 in the upper plane and 3 in the lower plane). So, for this case the radius given for the "sphere-atom" wouldn't be a/2 but 0.5*sqr(a*a/3 +c*c/4) (in order to avoid overlapping). Using this fact and the value of c/a you find a and then replacing in the volume you'll get a slightly different answer (6.7179e-2 nm^3). Am I making a mistake? If I am, I apologize for any slip either in the solution or in my English... it's my first post. Good luck.
|Similar Threads for: volume of HCP unit cell from radius + c/a ratio|
|1x1 unit cell dimensions||Atomic, Solid State, Comp. Physics||0|
|HCP unit cell||Advanced Physics Homework||1|
|crystal lattice and unit cell||Biology, Chemistry & Other Homework||5|
|ratio of nuclear to atomic radius||Introductory Physics Homework||1|
|radius ratio||Introductory Physics Homework||4|