Q: Electrons accelerated by potential V are diffracted from a crystal. If d= \$1\mathring { A } \$ and \$i-30^o\$, V should be about \$(h=6.6 \times 10^{-34}Js, m_e=9.1 \times 10^{-31} kg,e=1.6 \times 10^{-19} C)\$

For constructive interference, \$2d cosi=n\lambda\$ and also wavelength, \$\lambda=\dfrac{h}{\sqrt{2meV}}\$So, \$2dcosi=n\times \dfrac{h}{\sqrt{2meV}}\$Squaring both sides, \$4d^2cos^2i=\dfrac{n^2h^2}{2meV}\$\$V=\dfrac{n^2h^2}{8d^2mecos^2i}=\dfrac{(1)^2(6.6\times 10^{-34})^2}{8(1\times 10^{-10})^2(9.1\times 10^{-31})(1.6\times 10^{-19})(cos^230)}=49.8 V\sim 50 V\$ (we assume first order interference and take \$n=1\$)

Q: Drift speed of electrons is of the order of-

Drift speed of electrons is of the order of \${10^{ - 4}}\,m/\sec \$Hence, the option \$D\$ is the correct answer.

Question 1

Describe Davisson and Germer's experiment to demonstrate the wave nature of electrons.Draw a labelled diagram of apparatus used.

Accepted Answer

Davisson and Germer Experiment: In 1927 Davisson and Germer performed a diffraction experiment with electron beam in analogy with X-ray diffraction to observe the wave nature of matter.Apparatus: It consists of three parts(i) Electron Gun: It given a fine beam of electrons, de Broglie used electron beam of energy 54eV. de Broglie wavelength associated with this beam$\lambda=\dfrac{h}{\sqrt{2mE_K}}$Here m=mass of electron =$9.1\times10^{-31}kg$E-K=Kinetic energy of electron =$54eV$$=54\times1.6\times10^{-19}joule =86.4\times10^{-19}joule$$\therefore \lambda=\dfrac{6.6\times10^{-24}}{\sqrt{2\times9.1\times10^{-31}\times86.4\times10^{-19}}}$$=1.66\times10^{-10}m=1.66A^\circ$(i) Nickel Crystal: The electron beam was directed on nickel crystal against its (iii) face. The smallest separation between nickel atoms is $0.914A^\circ$. Nickel crystal behaves as diffraction grating.(ii) Electron Detector: It measures the intensity of electron beam diffracted from nickel crystal. It may be an ionisation chamber fitted with a sensitive galvanometer. The energy of electron beam, the angle of incidence of beam on nickel crystal and the position of detector can all be varied.Method: The crystal is rotated in small steps to change the angle ($\alpha$ say) between incidence and scattered directions and the corresponding intensity (I) of scattered beam is measured. The variation of the intensity (I) of the scattered electrons with the angle of scattering $\alpha$ is obtained for different accelerating voltages.The experiment was performed by varying the accelerating voltage from 44V to 68V. It was noticed that a strong peak appeared in the intensity (I) of the scattered electron for an accelerating voltage of 54V at a scattering angle $\alpha=50^\circ$$\therefore $From Bragg's law$2dsin=\theta=n\lambda$Here n=1,d=0.914 $A^\circ$, $\theta=65 ^\circ$The measured wavelength is in close agreement with the estimated de Broglie wavelength.Thus the wave nature of electron is verified. Later on G.P. Thomson demonstrated the wave nature of fast electrons.Due to their work Davission and G.P.Thomson were awarded Nobel prize in 1937.Later on experiments showed that not only electrons but all material particles in motion(e.g., neutrons, $\alpha$ particles,protons etc.) show wave nature.

Question 2

Describe de-Broglie hypothesis. Explain how it was experimentally proved by Davisson and Germer.

Accepted Answer

de-Broglie Hypothesis and Wavelength of Matter Wave: A natural question arises: As radiation has dual (wave particle) nature, the particles of nature (the electrons, protons, etc), also exhibit wave-like character.  In $1924$, the French Physicist de-Broglie (pronounced as de brog) $(1892-1987)$  put forward the bold hypothesis that moving particles of matter should display wave like properties under suitable conditions. He reasoned that nature was symmetrical and that the two basic physical entities-matter and energy, must have symmetrical character. If radiation shows dual aspects, so should matter. de-Broglie proposed that the wavelength $ \lambda $ associated with a particle of $p$ momentum is given by the relation $ \lambda = \dfrac{h}{p} = \dfrac{h}{mv} $        &#8230;&#8230;&#8230;&#8230;&#8230;.. (1) Where $m$ is the mass of the particle and $v$ its speed. Equation (1) is known as the de-Broglie relation and the wavelength A of the matter wave is called de-Broglie wavelength. The dual aspect of matter is evident in the de-Broglie relation (1), A is the attribute of a wave while on the right hand side the momentum p is a typical attribute of a particle. Planck&#8217;s constant h relates the two attributes. Equation (1) for a material particle is basically a hypothesis whose validity can be tested only by experiment. However, it is interesting to see that it is satisfied also by a photon for a photon, as we have seen                 $ p = \dfrac{hv}{c} $  $ \therefore $             $ \dfrac{h}{p} = \dfrac{c}{v} = \lambda $ Clearly from equation (1), $ \lambda $ is smaller for a heavier particle (large $m$) or more energetic particle (large $v$). For example, the de-Broglie wavelength of a ball of mass $10\, gm$ moving with a speed of $ 2\,ms^{-1} $ is easily calculated as $ \lambda = \dfrac{h}{p} = \dfrac{h}{mv} = \dfrac{6.63 \times 10^{-34} J-s}{(10^{-2}\,kg)(2\,ms^{-1})} $     $ = 3.31 \times 10^{-34}\,m $This wavelength is so small that it is beyond any measurement. This is the reason why macroscopic objects in our daily life do not show wave-like properties, on the other hand, in the sub-atomic domain, the wave character of particles is significant and measurable.Consider an electron [mass $(m)$ charge, $(e)$] accelerated from rest through a potential $V$. The kinetic energy $K$ of the electron equal to the work done $(eV)$ on it.By the electric field              $ K = eV $ Now         $ K = \dfrac{1}{2} mv^2 = \dfrac{p^2}{2m} $ So that        $ p = \sqrt{2mK} = \sqrt{2meV} $ The de-Broglie wavelength $ \lambda $ of the electron is then ,                $ \lambda = \dfrac{h}{p} = \dfrac{h}{\sqrt{2mK}} = \dfrac{h}{\sqrt{2meV}} $      ........(2)The variation of potential with wave length

Question 3

What is the conclusion of Davison and Germer experiment on the nature of electron?

Accepted Answer

Davisson and Germer made some interesting observation in their experiment. While increasing the accelerating voltage of the electron beam definite peak intensity was observed a particular voltage. This voltage was fond to be 54 volt exactly. The peak can't be explained by momentum exchanges in electron collisions. The only logical explanation is the constructive interference of electron scattered from different layers of the crystal. Definitely a wave property. The de Broglie wavelength associated with this voltage is 0.167 nm. This was exactly the wavelength found theoretically by de Broglie for electron matters waves. This is how Davisson and Germer experiment explained the wave nature of electron and also found the wavelength associated with the electron as obtained by de Broglie relation.

Question 4

This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statements.
Statement-1 : Davisson- Germer experiment established the wave nature of electrons.
Statement-2 : If electrons have wave nature, they can interfere and show diffraction.

Accepted Answer

Davisson  Germer experiment showed that electron beams can undergo diffraction when passed through atomic crystals. This shows the wave nature of electrons as waves can exhibit interference and diffraction.

Question 5

Davisson Germer experiment explained the wave nature of electrons through

Accepted Answer

Davisson-Germer experiment was conducted by two American scientists Clinton Davisson and Lester Germer, in 1927, to verify the de Broglie hypothesis that a material particle posses wave nature. They experimentally proved the wave nature of electron through diffraction of electron beam by a nickel crystal.

Question 6

Electrons accelerated by potential V are diffracted from a crystal. If d= $1\mathring { A } $ and $i-30^o$, V should be about $(h=6.6 \times 10^{-34}Js, m_e=9.1 \times 10^{-31} kg,e=1.6 \times 10^{-19} C)$

Accepted Answer

For constructive interference, $2d cosi=n\lambda$ and also wavelength, $\lambda=\dfrac{h}{\sqrt{2meV}}$So, $2dcosi=n\times \dfrac{h}{\sqrt{2meV}}$Squaring both sides, $4d^2cos^2i=\dfrac{n^2h^2}{2meV}$$V=\dfrac{n^2h^2}{8d^2mecos^2i}=\dfrac{(1)^2(6.6\times 10^{-34})^2}{8(1\times 10^{-10})^2(9.1\times 10^{-31})(1.6\times 10^{-19})(cos^230)}=49.8 V\sim 50 V$  (we assume first order interference and take $n=1$)

Question 7

In Davisson-Germer experiment, intensity was maximum for accelerating voltage equal to

Accepted Answer

In Davisson-Germer experiment, maximum intensity of diffracted electron beam was found at different angles by varying the applied voltage to the electron gun. The highest intensity was observed at an angle $\phi =50^o$ with a voltage of $54 V$, giving the electron a kinetic energy of $54eV$.

Question 8

Drift speed of electrons is of the order of-

Accepted Answer

Drift speed of electrons is of the order of ${10^{ - 4}}\,m/\sec $Hence, the option $D$ is the correct answer.

Question 9

Match List - I (Fundamental Experiment) with List - II (its conclusion) and select the correct option from the choices given below the list

List I List II
(A) Franck-Hertz Experiment (i) Particle nature of light
(B) Photo-electric experiment (ii) Discrete energy level of atom
(C) Davison-Germer Experiment (iii) Wave nature of electron
(iv) Structure of atom

Accepted Answer

Franck-Hertz Experiment confirmed the quantum theory which predicted that electrons occupied only discrete, quantized energy states, which matches with option (ii) in list 2.Photoelectric effect experiment shown  light is also a particle.Energy comes in particle-like chunks which is basics of quantum physics, which matches with option (i) in list 2. The Davisson-Germer experiment  demonstrated the wave nature of the electron, confirming the earlier hypothesis of deBroglie. which matches with option (iii) in list 2.

Question 10

State the importance of Davisson and Germer experiment.

Accepted Answer

The Davisson and Germer experiment are probably one of the most important experiments ever since it verified the De Broglie's &#34;matter wave&#34; hypothesis applied to matter (electrons) as well as light. From this emerged modern quantum theory, the most stupendous revolution in physics of all time.

Davisson and Germer Experiment

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