材料科学基础第一章习题答案
1. (P80 3-3) Calculate the atomic radius in cm for the following:
(a) BCC metal with a 0=0.3294nm and one atom per lattice point; and
(b) FCC metal with a 0=4.0862Å and one atom per lattice point.
Solution:
(a) In BCC structures, atoms touch along the body diagonal, which is 3a 0 in length. There are two atomic radii from the center atom and one atomic radius from each of the corner atoms on the body diagonal,
so 340r
a =
430a r ==0.14263nm=1.4263
810-⨯cm (b) In FCC structures, atoms touch along the face diagonal of the cube, which is
02a in length. There are four atomic radii along this length —two radii from the face-centered atom and one radius from each corner, so
240r a =, 420
a r ==1.44447 Å=1.44447810-⨯cm
2.(P80 3-4) determine the crystal structure for the following:
(a) a metal with a0=4.9489Å, r=1.75Å, and one atom per lattice point; and (b) a metal with a0=0.42906nm, r=0.1858nm, and one atom per lattice point.
Solution:
We know the relationships between atomic radii and lattice parameters are 430
a r =
in BCC and 420a r =
in FCC. (a) 420a r ==⨯=4
9489.42 1.75 so its crystal structure is FCC;
(b) 430
a r ==4
4296.03⨯=0.186nm so its crystal structure is BCC.
3. (P80 3-5) the density of potassium, which has the BCC structure and one
atom per lattice point, is 0.855g/cm 3.
the atomic weight of potassium is 39.09g/mol. Calculate
(a) the lattice parameter; and
(b) the atomic radius of potassium.
Solution
(a) For a BCC unit cell, there are two
atoms in per unit cell,
atomic mass is 39.09g/mol,
density ρ=0.855g/cm 3
Avogadro ’s number N A =6.022310⨯atoms/mol
())'()()(/snumber Avogadro cell unit of volume mass atomic cell atoms of numbers =ρ
0.855g/cm 3=
)/()1002.6()
/(09.392233mol atoms a mol g atoms ⨯⨯⨯ So a=3231002.6855.009.392⨯⨯⨯=0.53cm
710-⨯=5.3Å
(b)then r=a 4
3=0.229710-⨯cm=2.29Å
4. (P81 3-20) determine the indices for
the directions in the cubic unit cell shown in Figure 3-32.
The procedure for finding the Miller indices for directions is as follows:
ing a right-handed coordinate
system, determine the coordinates of two points, which lie on the direction.
2.Subtract the coordinates of the “tail”
point from the coordinates of the “head”point to obtain the number of
lattice parameters traveled in the direction of each axis of the coordinate system.
3.Clear fractions and/or reduce the results obtained from the subtraction to lowest integers.
4.Enclose the number in square brackets
[ ]. If a negative sign is produced, represent the negative sign with a bar over the number.
Solution
Direction A
1.Two points are 0,0,1 and 1,0,0
2.0,0,1-1,0,0=-1,0,1
3.no fraction to clear or integers to reduce
4.[]011
Direction B 1.Two points are 1,0,1 and 2
1,1,0 2.1,0,1-21,1,0=2
1,-1,1 3.2(2
1,-1,1)=1,-2,2
