Which 3-D packing has the maximum empty space in its crystal lattice?
A
Hexagonal closed packing (hcp)
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
Body centred cubic close packing (bcc)
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
Simple cubic close packing (scp)
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D
Cubic close packing (ccp)
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution
The correct option is C Simple cubic close packing (scp) Packing Efficiency of Hexagonal Close Packing :
Total no. of spheres in a hcp unit cell =(12×16)+(2×12)+3=6
where, (12×16) is for corners atoms of two hexagonal layers (2×12) is for the centre atoms of two hexagonal layers
3 is for the atoms of middle layer Total volume occupied=6×43×π×r3=8πr3 Volume of hcp unit cell (hexagon)=Base area×Height
Base area of a regular hexagon=6×√34(2r)2=6×√3r2
Height of Unit Cell (h)=4r×√23
Volume of hexagon (V)=(6×√3r2)×(4r×√23)V=24√2r3 Packing fraction=8πr324√2r3=π3√2P.F≈0.74
Fraction empty space=1−0.74=0.26
For body centred cubic arrangement, Zeff=2
4r=√3a
where,
r is radius of atom
a is edge length of cube
Thus, a=4r√3 Volume of cell =a3 Volume of cell =(4r√3)3 Packing fraction =Total volume occupied by sphereVolume of unit cell Packing fraction =Z×43πr3a3 =2×434r3(4r√2)3=√3π8=0.68 Fraction empty space =1−0.68=0.32
For simple cubic (sc):
Zeff for sc =1
So,
The total volume occupied by sphere V=1×43×πr3 Volume of the cube=a3=(2r)3 Packing fraction (P.F)=1×43×πr3(2r)3P.F=π6=0.52 Fraction empty space =1−0.52=0.48
In ccp lattice:
Packing efficiency (P.F) of ccp:
P.F=No of spheres in one ccp unit cell ×Volume of each sphereTotal Volume of the Unit cell
ccp unit cell have FCC arrangement
Here, √2×a=4Ra=2√2×R
Zeff for FCC=(8×18)+(6×12)=4
So,
The total volume occupied by sphere V=4×43×πr3 Volume of the cube=a3=(2√2R)3