Let's solve each part step by step. For Miller indices, **directions** are written as \([uvw]\) and **planes** as \((hkl)\). ### 1. **Assign coordinates to each point** Since it is a cubic unit cell, assume: - \(A = (, , )\) - \(B = (1, , )\) - \(C = (1, 1, )\) - \(D = (, 1, )\) - \(E = (, 1, 1)\) - \(F = (1, 1, 1)\) - \(G = (, , 1)\) - \(H = (, .5, )\) - \(I = (1, .5, 1)\) - \(J = (1, 1, 1)\) - \(K = (, .5, .5)\) - \(L = (, , 1)\) - \(M = (.5, 1, 1)\) - \(N = (1, 1, .5)\) - \(O = (.5, .5, )\) - \(P = (1, .5, 1)\) - \(Q = (1, 1, 1)\) - \(R = (1, 1, 1)\) - \(S = (, 1, 1)\) - \(T = (, , 1)\) - \(F = (1, 1, 1)\) - \(C = (1, 1, )\) - \(N = (1, 1, .5)\) - \(R = (1, 1, 1)\) _Note: You may need to adjust coordinates for some points if a clearer version of the diagram is available, but the typical assignment for a cubic cell follows this logic._ --- ### a. **Direction OI** - \(O = (.5, .5, )\) - \(I = (1, .5, 1)\) **Step 1: Find the vector OI** \[ \text{OI} = (1 - .5, .5 - .5, 1 - ) = (.5, , 1) \] **Step 2: Normalize to smallest integer values** \[ (.5, , 1) \rightarrow (1, , 2) \] **Step 3: Miller index format** \[ \boxed{[102]} \] --- ### b. **Direction DM** - \(D = (, 1, )\) - \(M = (.5, 1, 1)\) \[ \text{DM} = (.5 - , 1 - 1, 1 - ) = (.5, , 1) \] \[ (.5, , 1) \rightarrow (1, , 2) \] \[ \boxed{[102]} \] --- ### c. **Direction FT** - \(F = (1, 1, 1)\) - \(T = (, , 1)\) \[ \text{FT} = ( - 1, - 1, 1 - 1) = (-1, -1, ) \] \[ \boxed{[\bar{1}\bar{1}]} \] --- ### d. **Plane GFS** - \(G = (, , 1)\) - \(F = (1, 1, 1)\) - \(S = (, 1, 1)\) **Step 1: Find intercepts with axes** - x-intercept: \(F\) (x = 1, y = 1, z = 1) → x=1 - y-intercept: \(S\) (x = , y = 1, z = 1) → y=1 - z-intercept: \(G\) (x = , y = , z = 1) → z=1 **Step 2: Miller indices** - Intercepts: (1, 1, 1) - Reciprocals: (1, 1, 1) \[ \boxed{(111)} \] --- ### e. **Plane OBMS** - \(O = (.5, .5, )\) - \(B = (1, , )\) - \(M = (.5, 1, 1)\) - \(S = (, 1, 1)\) Find intercepts: - x: From \(B (1,,)\), intercept at x=1 - y: From \(M (.5,1,1)\) and \(S (,1,1)\), y=1 - z: From \(M (.5,1,1)\) and \(S (,1,1)\), z=1 Plane passes through O (.5,.5,), so it cuts z=. Thus, intercepts: x=1, y=1, z= Reciprocals: 1/1, 1/1, 1/ → (1,1,∞) → ignore ∞ and use Miller notation: (1,1,) \[ \boxed{(110)} \] --- ### f. **Plane FCNR** - \(F = (1, 1, 1)\) - \(C = (1, 1, )\) - \(N = (1, 1, .5)\) - \(R = (1, 1, 1)\) All points have x=1, y=1 (so the plane is parallel to z-axis at x=1, y=1). Intercepts: x=1, y=1, z=∞ (never cuts z-axis) Reciprocals: (1,1,) \[ \boxed{(110)} \] --- ## **Summary Table** | Part | Object | Miller Indices | |------|--------------|----------------| | a | Direction OI | [102] | | b | Direction DM | [102] | | c | Direction FT | [\(\overline{1}\)\(\overline{1}\)] | | d | Plane GFS | (111) | | e | Plane OBMS | (110) | | f | Plane FCNR | (110) | --- If you need further clarification for any particular part or for a specific point's coordinates, let me know!