InterviewSolution
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InterviewSolution
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(a) Obtain the de-Broglie wavelength of a neutron of kinetic energy 150 eV. As you have seen in previous problem 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. Given `m_(n)=1.675xx10^(-27)kg`. (b) Obtain the de-Broglie wavelength associated with thermal neutrons at room temperature `(27^(@)C)`. Hence explain why a fast neutrons beam needs to be thermalised with the environment before it can be used for neutron diffraction experiments. |
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Answer» De Broglie wavelength `=2.327xx10^(-12)m`, neutron is not suitable for the diffraction exeriment Kinetic energy of the neutron, `K=150eV =150xx1.6xx10^(-19)` `2.4xx10^(-17)J` Mass of a neutron, `M_(n)=1.675xx10^(-27)Kg` The kinetic energy of the neutron is given by the relatin : `K=(1)/(2)m_(n)v^(2)` `m_(n)v=sqrt(2Km_(n))` Where, v=Velocity of the neutron `m_(n)v`=Momentum of the neutron De-Broglie wavelength of the neutron is given as : `lambda =(h)/(m_(n)v)=(h)/sqrt(2Km_(n))` It is clear that wavelength is inversely proportional to the square root of mass. Hence, wavelength decreases with increase is mass and vice versa. `therefore lambda =(6.6xx10^(-34))/sqrt(2xx2.4xx10^(-17)xx1.675xx10^(-27))` =`2.327xx10^(-12)m` It is given in the previous problem that the inter-atomic spacing of a crystal is about `1Å`, i.e., `10^(-10)`m.Hence, the inter- atomic spacing is about a hundred times greater, Hence a neutron beam of energy 150 eV is not suitable for diffraction experiments. (b) De Broglie wavelegth =`1.447xx10^(-10)m` Room temperature ,`T=27^(@)C=27+273=300K` The average kinetic energy of the neutron is given as : `E=(3)/(2)kT` Where, K= Boltzmann constant `=1.38xx10^(-23)JMol^(-1)K^(-1)` The wavelength of the neutron is given as : `lambda=(h)/sqrt(2M_(n)E)=(h)/sqrt(3M_(n)kT)` `=(6.6xx10^(-34))/sqrt(3xx1.675xx10^(-27)xx1.38xx10^(-23)xx300)` `=1.447xx10^(-10)m` This wavelength is comparable to the inter-atmomic spacing of a crystal. Hence, the high-energy neutron beam should first be thermalised, before using it for diffraction. |
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