Physics · Dual Nature Of Radiation And Matter · NEET
A free neutron in a gas moves in all three directions: x, y and z. The kinetic theory says each direction of motion carries an average energy of (1/2)kT. This is called equipartition of energy. Three directions give 3 times (1/2)kT = (3/2)kT. So the total average kinetic energy is KE = (3/2)kT. Here k is the Boltzmann constant and T is the absolute temperature in kelvin. This (3/2) is the source of the 3 that appears in the wavelength formula.
Start with lambda = h/p and p = sqrt(2 m KE). Put KE = (3/2)kT. Then 2 m KE = 2 m times (3/2)kT = 3 m k T. So p = sqrt(3 m k T) and lambda = h/sqrt(3 m k T). The factor 2 from momentum cancels the (1/2) inside the energy, leaving a clean 3 under the root.
You use the absolute temperature in kelvin. If a problem gives Celsius, first convert: T(K) = T(C) + 273. For a 'thermal' neutron the room-temperature value is about T = 300 K (roughly 27 C). Never put Celsius directly into the formula, and never leave T out; a thermal neutron's speed comes only from temperature.
An electron is charged, so we speed it up with a voltage V and its energy is KE = eV, giving lambda = h/sqrt(2 m e V). A neutron has no charge, so a voltage cannot push it. A neutron gets its speed only from heat, so its energy is (3/2)kT and lambda = h/sqrt(3 m k T). Same idea lambda = h/sqrt(2 m KE), but the source of KE is different.
Yes. Lower temperature means smaller KE, smaller momentum p, and since lambda = h/p a smaller p gives a larger lambda. A thermal neutron at 300 K has a wavelength near 0.15 to 0.18 nm, which is close to the spacing between atoms in a crystal. That is why slow neutrons diffract from crystals, just like the electrons in the Davisson-Germer experiment.
The de Broglie wavelength of a neutron in thermal equilibrium with heavy water at a temperature T (kelvin) and mass m is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
lambda = h / sqrt(3 m k T), where h is Planck's constant, m is the neutron mass, k is the Boltzmann constant and T is the absolute temperature in kelvin.
At about T = 300 K it is roughly 0.145 nm (about 1.45 angstrom). This is close to atomic spacing in crystals, which makes thermal neutrons good for diffraction.
A neutron has no electric charge, so an accelerating voltage does no work on it. Its kinetic energy comes only from thermal motion, giving KE = (3/2)kT instead of eV.
k = 1.38 x 10^-23 J/K. Use the neutron mass m = 1.67 x 10^-27 kg and h = 6.63 x 10^-34 J·s for numerical problems.
It increases. Cooling lowers T, which lowers KE and momentum p, and since lambda = h/p a smaller p gives a larger lambda. Very cold (slow) neutrons have longer wavelengths.
