fix annotation in plots, add warm up Q

src/9_crystal_structure.md

@@ -142,7 +142,7 @@ latticeB = go.Scatter(visible = False, x=latB_points.T[0]+0.5,y=latB_points.T[1]
 
 # Annotate the lattice vectors
 # Button 2 
-bt2_annot = make_arrow(a1,'$a_1$') + make_arrow(a2,'$a_2$') + make_arrow(a1_alt,'$a`_1$',color='Black',text_shift=[-0.55,-0.1]) + make_arrow(a2_alt,'$a`_2$',color='Black')
+bt2_annot = make_arrow(a1,'$a_1$') + make_arrow(a2,'$a_2$',text_shift=[-0.3,-0.1]) + make_arrow(a1_alt,'$a`_1$',color='Black',text_shift=[-0.6,-0.1]) + make_arrow(a2_alt,'$a`_2$',color='Black',text_shift=[-0.35,-0.1])
 #Button 3
 bt3_annot =  make_arrow(a1,'$a_1$', vec_zero = [-0.5,0], color = 'Blue') + make_arrow(a2,'$a_2$',vec_zero = [-0.5,0], color = 'Blue')
 #Button 4
@@ -417,6 +417,8 @@ If a crystal is symmetric under $90^\circ$ rotations, then $(100)$, $(010)$ and
 1. Draw a unit cell of an FCC, a BCC, and a triangular lattice. Are the unit cells you drew primitive? 
 2. How many lattice points are there in a primitive unit cell?
 3. Suppose you found the primitive unit cell of a diatomic crystal. How many basis vectors do you minimally need to describe the crystal? Can some diatomic crystals require more basis vectors?
+4. Calculate the filling factor of a simple cubic lattice.
+5. Sketch the $(110),(1\bar{1}0),(111)$ miller planes of a simple cubic lattice.
 
 
 ### Exercise 1: Diatomic crystal