Question

2(a) A typical metallic particle has a rest mass (i.e. mass when the particle is not moving) of 0.5x10-24Kg and an energy of 10 eV (Electron Volt). (i) What would be the wavelength and momentum of the associated wave for this particle? -- (5) (ii) Assuming the uncertainty principle and a spherical particle, estimate the diameter of this particle? Is it a nanoparticle ? ----------------------------------------- (5) (iii) This particle passes through an inter-lattice spacing of 1nm along the x axis. What would be the spread of the particle in the y-axis, 10microns from this interlattice spacing ------- (5) 2. (b) The primitive translation vectors of the hexagonal space lattice may be taken as z ccy a xaby a xaa     ; ) 2 () 4 3 (;) 2 () 4 3 ( (a) Calculate the volume of the primitive unit cell ------ (5) (b) What is its atomic packing fraction ? ----- (5)

Answer 1

2 (a) rest mass of the particle, m₀ = 0.5 x 10^-24 kg

(i) The wavelength will be given as :

using an equation,     E = h c /                                                                 { eq.1 }

where, h = planck's constant = 6.63 x 10^-34 J.s

E = energy = 10 eV

c = speed of light = 3 x 10⁸ m/s

inserting these values in above eq.

(10 eV) = (6.63 x 10^-34 J.s) (3 x 10⁸ m/s) /

= (19.9 x 10^-26 J.m) / (10 eV) (1.6 x 10^-19 J/eV)

= 12.43 x 10^-8 m

= 124.3 nm

And momentum of the associated wave for this particle which is given as :

using an equation,    = h / p                                                                    { eq.2 }

then, we have

p = (6.63 x 10^-34 J.s) / (12.43 x 10^-8 m)

p = 5.34 x 10^-27 kg.m/s