(a) Obtain the de Broglie wavelength of a neutron of kinetic energy 150 eV. As you have seen in Exercise 11.31, an electron beam of this energy is suitable for crystal diffraction experiments. Would a neutron beam of the same energy be equally suitable? Explain. (mn = 1.675 × 10–27 kg) (b) Obtain the de Broglie wavelength associated with thermal neutrons at room temperature (27 °C). Hence explain why a fast neutron beam needs to be thermalised with the environment before it can be used for neutron diffraction experiments.
a)
Given: The kinetic energy of the neutron is 150 eV and the mass of the neutron is 1.675 × 10 − 27 kg .
We know that,
1 eV = 1.6 × 10 − 19 J
The kinetic energy of the neutron is given as,
K = 1 2 m n v 2
Where, m n is the mass of neutron and v is the velocity of the neutron.
Re-arranging the above equation, we get
m n v = 2 K m n
de Broglie wavelength is given as,
λ = h p = h m n v = h 2 K m n
Where, h is the Planck’s constant, p is the momentum and λ is the de Broglie wavelength.
By substituting the given values in the above equation, we get
λ = 6.626 × 10 − 34 2 × 150 × 1.6 × 10 − 19 × 1.675 × 10 − 27 = 2.337 × 10 − 12 m
Thus, the de Broglie wavelength of neutron is 2.337 × 10 − 12 m .
In the previous problem, it is given that the inter-atomic spacing is about 1 A 0 or 10 − 10 m . This shows that inter atomic separation is 100 times more than the wavelength.
Thus, a neutron beam of energy 150 eV is not suitable for diffraction experiments because for diffraction experiment, inter atomic separation should be of same order as the wavelength.
b)
Given: The temperature of neutron beam is 27 °C .
The average kinetic energy of neutron is given as,
K = 3 2 k T
Where, k is the Boltzmann’s constant and T is the given temperature.
The de Broglie wavelength is given as,
λ = h 2 m n K = h 3 m n k T
By substituting the given values in the above equation, we get
λ = 6.626 × 10 − 34 3 × 1.675 × 10 − 27 × 1.38 × 10 − 23 × ( 273 + 27 ) = 1.45 × 10 − 10 m
Since, this De Broglie wavelength is comparable to the inter-atomic spacing of a crystal. Thus, the high energy neutron beam should be thermalised before it can be used for neutron diffraction experiments.
a)
Given: The kinetic energy of the neutron is
We know that,
The kinetic energy of the neutron is given as,
Where,
Re-arranging the above equation, we get
de Broglie wavelength is given as,
Where,
By substituting the given values in the above equation, we get
Thus, the de Broglie wavelength of neutron is
In the previous problem, it is given that the inter-atomic spacing is about
Thus, a neutron beam of energy
b)
Given: The temperature of neutron beam is
The average kinetic energy of neutron is given as,
Where,
The de Broglie wavelength is given as,
By substituting the given values in the above equation, we get
Since, this De Broglie wavelength is comparable to the inter-atomic spacing of a crystal. Thus, the high energy neutron beam should be thermalised before it can be used for neutron diffraction experiments.
