# Crystal Axial Ratio

1. calculate the axial ratio of scheelite (tetragonal) based on the following inter facial (polar) angles. If the answers differ from the two measurements use the average(0,1.1) angle (0,-1,1) =130.37 degree

(0,1,3) angle (0,-1,3) = 71.53 degree

2. Olivine crystallizes in the orthorhombic system and in its crystal class there is a mirror plane perpendicular to each of the three axes. It commonly shows the following faces and inter facial angles

(1,1,0) angle (1,-1,0) = 50 degree

(0,2,1) angle (0,-2,10 = 99 degree

(1,0,1) angle (-1,0,1) = 103 degree

a) Calculate the axial ratio a to b to c based on the inter facial angles. Put in the for a to 1 to c. Use the third angle to check your answer and to get an average value for a and c.

b) What is the polar angle between the faces (1,2,0) and (1,0,0) ?

Answers

1. Scheelite has a tetragonal structure having as x axis the a axis, as y axis a b axis

(=a) and as z axis a c axis. If you draw on a paper the planes (011) and (0,-1,1) (see attached picture) you will figure that the axes perpendicular to these two planes form an isosceles triangle having as base the segment 2a and as height the segment c.

therefore tan(130.37)/2 =a/c

2.1627 = a/c

the axial ratio is a to c is 1 to 2.1627

From the second isosceles triangle formed having as base the segment 2a and as height the segment 3c one can write

tan(71.53) = a/3c

0.7203 =a/3c

the axial ratio is a to c 1 to 3*0.7203 =2.1609

the average axial ratio is a to c is 1 to 2.1618

2. Olivine has a orthorhombic structure , x axis is a, y axis is b, z axis is c

If you draw on a paper the planes (110) and (1,-1,0) you will observe that the axes perpendicular to these planes for an isosceles triangle having the height a and the base 2b.

therefore tan(50/2)= b/a

b =0.466*a (1)

Similar for planes (0,2,1) and (0,-2,1) see the triangle in the second picture

tan(99/2) = 2b/c

2b =1.171*c

b = 0.585 c (2)

The axial ratio is a to b to c is 0.466 to 1 to 0.585

For planes (101) and (-1,0,10 the isosceles triangle formed by the axis perpendicular to the faces has base 2a and height c

Hence tan(103/2) =a/c (3)

a =1.257*c

and from the first relation b =0.466*a one gets b =0.585*c which verifies the second relation ratio

b) For the triangle formed between the axes perpendicular to the faces see attached picture 3

tan (alpha1) =2b/a =4.289

alpha1= 76.87 degree

tan(alpha2) =a/a =1

alpha2 =45 degree

total angle is alpha = alpha1+alpha2 =76.87+45 =121.87 degree