[earthnation112]

Had a question regarding crystal structures, I had gone through quite a few so far but one keeps troubling me which is Nickel Arsenide.

A image of the structure has been attached.
 
My calculations so far for the number of spheres in unit cell are as follows:

Red Spheres = 6 x 1 = 6 (inside unit cell)
Blue Spheres = (18 x 1/6) + (2 x 1/2 ) + 1 (right in centre) = 5
OR
Blue Spheres = (18 x 1/6) + (3 x 1/3 )  = 4
Coordination number of R:B IS 6:6

Yes I know naming of the colours is a bit off 

Are my attempts right? Any input would be greatly appreciated. Thanks   

[mjc123]

Consider the blue spheres half way down the edges. How many unit cells are they shared between?

[AWK]

(18 x 1/6)

- wrong

[earthnation112]

The blue spheres half way down the edges are shared by 6 unit cells right as its 1/6 for each one.

The reason I think its 1/6 is because the unit cell ZnS (Wurtzite) also has a similar structure with 6 spheres at the top edges with one in the centre between these 6 and each one is 1/6. Why wouldn't the same apply for Nickel Arsenide? I have attached a image below of ZnS (Wurtzite).

[AWK]

The blue spheres half way down the edges are shared by 6 unit cells right as its 1/6 for each one.

At these edges angles are 120 degrees.

[earthnation112]

Since the edges are 120 degrees would this equal 1/2?

I have used google books and found the following image regarding Nickel Arsenide (attached below).

Would the following then follow:
Red Spheres = (3 x 3/4) + (3 x 1/4) = 3 (inside unit cell)
Blue Spheres = (14 x 1/2) + 1 (in centre)= 8

[AWK]

Do not compare hexagonal arrangement with trigonal one. Moreover, Structure NiAs and ZnS are different. Only common points are atoms at both bases of prism and atoms on the edges at z=1/2 for NiAs and z=1/3 for ZnS.
On your scheme for ZnS you have 6x(1/3) (3x120 degree = 360). This concerns also NiAs

[earthnation112]

I do apologise if it seems very simple but I keep asking, I’m just finding it really hard to grasp as I find it hard visualising it all.

My latest attempt:

Red Spheres = (6 x 1) = 1 (inside unit cell)
Blue Spheres = (18 x 1/3) + (3 x 1/2) = 4.5

[AWK]

Stil the same problem.
Split 18 into 12 and 6 with different occupancy - this is evident from your picture.
Red spheres was correct at the beginning - not now (printing error?)
Look ar your wurcite structure (atoms from base)
z=0 black atoms 6x1/6 + 1x1/2
z=1/3 white 6x1/3 + 1x1
z=1/2 black 3x1
z=2/3 white 3x1
z=1 black 6x1/6 + 1x1/2

Do the same for NiAs

[earthnation112]

Thank you for the input it helps a lot.

Red Spheres = (3 x 3/4) + (3 x 1/4) = Total 3 (inside unit cell).

The blue spheres are:
(12x1/6) + (1x1/2) = 2.5 This is for the bottom base and top base as they are the same.
For the middle section it is = (6 x 1/3) + 1/2 = 2.5

I'm I any closer?

[AWK]

Red Spheres = (3 x 3/4) + (3 x 1/4)

You should multiply by occupancy, but not by position on z. Still wrong. Sum of red spheres was correct at the beginning. All atoms inside prism always shows occupancy equal to 1.

(12x1/6) + (1x1/2)

Two bases of hexagonal prism are the same.

(6x1/3)+1/2

Where you see half an atom at z=1/2 inside the hexagonal prism?
Note, your scheme for unit cell of nickeline (third scheme) is wrong. Should be as below.





Summation of atoms
As 8x1/8 + 4x1/4=2
Ni 2x1=2
Hexagonal prism is formed by rotation this unit cell by triad axis.


see: http://www.benjamin-mills.com/chemistry/structures/NiAs/

[earthnation112]

You should multiply by occupancy, but not by position on z. Still wrong. Sum of red spheres was correct at the beginning. All atoms inside prism always shows occupancy equal to 1.

At the begining i mentioned that red spheres was: 6 x 1 = 6

The confusion i have with the blue spheres is that you mentioned:

As 8x1/8 + 4x1/4=2

There are in total 21 blue spheres since there is 1 in the centre of each base and 6 around the edges for each base, based on the image I attached for Nickel Arsenide. How do I possibly reach that number with "8x1/8 + 4x1/4"? Since that only accounts for 12 spheres.

I know it sounds basic, but is that a valid question, if it makes sense. I know not each sphere actually represents 1 atom but to get the right answer i need a summation of 21 blue spheres right?

[AWK]

At the begining i mentioned that red spheres was: 6 x 1 = 6

In my first post I questioned only 18x1/6 ! The rest was correct.

There are in total 21 blue spheres since there is 1 in the centre of each base and 6 around the edges for each base, based on the image I attached for Nickel Arsenide. How do I possibly reach that number with "8x1/8 + 4x1/4"? Since that only accounts for 12 spheres.

Your first crystallographic scheme (hexagonal prism contain 3 unit cells). The second scheme for NiAs is for 1 unit cell and calculations should be different. To 3 unit cells belong 6 As and 6 Ni; to 1 unit cell belongs 2 Ni and 2 As

summation of 21 blue spheres right?

from 21 blue spheres
1 belongs completely to hexagonal prism
2 are splited into 2 prisms
6 are splited into 3 prisms
12 are splited into 6 prisms. These give 6 blue spheres in 1 hexagonal prism.


In (my) scheme for 1 unit cell you have z position for these spheres 0, 1/2 and 1 (default not showed).
From these spheres that of positions 0 and 1, each sphere belongs to 8 unit cells, and that of positions 1/2 belongs to 4 unit cells. Sum is 8x1/8+4x1/4.

In my summation for correct unit cell of NiAs

Summation of atoms
As 8x1/8 + 4x1/4=2
Ni 2x1=2

there is an error - As and Ni should be interchanged (ie Ni=8x1/8+ 4x1/2 and As 2x1) My fault.

[earthnation112]

Thank You for all the replies my last and final attempt is:

Blue spheres  =  1 + (2 x 1/2) + (6 x 1/3) + (12 x 1/6) = 6 unit cells in total
Red spheres  = 6 x 1 = 6 unit cells in total
C.N = 6,6
Unit Cell = NiAs Nickel Arsenide

Is my calculation for the total number of unit cells for blue spheres correct? If not, il just keep going over the replies on this thread as im sure you have gave more than enough information to figure it out or il re read the chapter on crsytal strucres again and get a better understanding before attempting to get an answer 

Really appreciate all the help, thank you.

[AWK]

OK, but this should be done in a few minutes.
Note - hexagonal prism contains 3 unit cells.
Hexagonal and trigonal structures are thought as rather difficult. Trigonal structure was the first that I solved at the beginning of my crystallographic life (1975), and the only one so far.