Question

Q:

(i) Neutron wavelength: Calculate the wavelength of thermal neutrons at (a) room temperature and (b) cooled to 4 K (liquid He temperature). How do these compare with 400 keV electrons and to typical Brehmmstrahlung radiation from an x-ray tube

(ii) Thermal velocities, de Broglie wavelength: A thermal neutron has a kinetic energy of 3/2 kT, where T is room temperature, 300 K. i.e. these neutrons are in thermal equilibrium with their surroundings. (a) Calculate the thermal velocity of such a neutron. Compare this with the thermal velocity of an electron at 300 K. (b) Calculate the DeBroglie wavelength of the neutron and the electron of part (a). (c) Would either of these be useful in resolving atomic positions?

Answer 1

Q )

i )

a )

room temperature T = 27^oC

= 27 + 273

T = 300 K

the thermal energy is KE_T = 3 k_B T / 2

= 3 x 1.38 x 10^-23 x 300 / 2

KE_T = 6.21 x 10^-21 J or 0.04 eV

the KE = m v² / 2

P = m v = ( 2 m KE )^1/2  

= h / P

= 6.627 x 10^-31 / 4.56 x 10^-24

= 1.45 x 10^-10 m

= 0.145 nm

b )

Helium at 4 K

KE_T = 3 x 1.38 x 10^-23 x 4 / 2

= 8.28 x 10^-23 J

using equation

= 12.26 / V^1/2 Å

= 1226.39 / ( 400 x 10³ + 0.97845 x 10^-6 )^1/2

= 1.644 pm

the minimum wavelength for Brehmsstrahlung in a 30 kV is

_min = 0.041 nm

it says that the 3 orders of smaller than of 4 K neutron. it is between neutron and electron.