Animated Solution for Physics - Dual Nature of Matter and Radiation: Comprehension Passage
Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere constructively (see figure).
Question 1:
Electrons accelerated by potential V are diffracted from a crystal. If d=1A˚ and i=30∘, V should be about (h=6.6×10−34 J-s, me=9.1×10−31 kg, e=1.6×10−19 C)
Select Answer:
Visualized Solution
Visualizing the Setup
Electron diffraction from crystal planes.
Path Difference
Path difference for constructive interference:
Δx=2dcosi=nλ
De Broglie Wavelength
De Broglie wavelength of an electron accelerated by potential V:
How does this affect the resolving power of an electron microscope?
00:00 / 00:00
The Sigma Insight: Matter Waves and de Broglie Relation
Solution Diagram
The phenomenon of electron diffraction is a beautiful demonstration of the dual nature of matter. When electrons are accelerated and fired at a crystal, the regular arrangement of atoms acts like a three-dimensional diffraction grating. Let's dive into the mechanics of this process and see how we can determine the accelerating potential from the diffraction pattern.
Analyzing the Setup
Imagine an electron beam striking a crystal surface. The crystal consists of parallel planes of atoms separated by a distance d. The incoming beam makes an angle i with the normal to the surface.
When the beam hits the crystal, it reflects off the different atomic planes. For these reflected waves to produce a strong, detectable signal, they must interfere constructively. This means the extra distance traveled by the wave reflecting off the second plane must be an integer multiple of the electron's wavelength, λ.
By looking at the geometry, the extra path length for the beam hitting the lower plane is dcosi on the way in, and another dcosi on the way out. Therefore, the total path difference is 2dcosi. According to Bragg's Law for constructive interference, we can write:
2dcosi=nλ
The Master Equation
To connect this macroscopic geometry to the quantum properties of the electron, we use the de Broglie wavelength. An electron accelerated through a potential V gains kinetic energy equal to eV. Its momentum p is given by 2meeV.
The de Broglie wavelength is then:
λ=ph=2meeVh
Substituting this into our interference condition (and assuming first-order diffraction, n=1), we get our master equation:
2dcosi=2meeVh
Final Calculation
Our goal is to find the accelerating potential V. To make the algebra easier, let's square both sides of the equation to eliminate the square root:
4d2cos2i=2meeVh2
Rearranging to solve for V:
V=2mee(4d2cos2i)h2
Now, we carefully substitute the given values. The plane spacing d=1A˚=10−10 m, and the angle i=30∘, which means cos30∘=23.
First, let's evaluate the geometric term in the denominator:
4d2cos2i=4(10−10)2(23)2=4×10−20×43=3×10−20 m2
Next, we plug in the physical constants:
- h2=(6.6×10−34)2=43.56×10−68
- 2mee=2(9.1×10−31)(1.6×10−19)=29.12×10−50
Putting it all together:
V=(29.12×10−50)×(3×10−20)43.56×10−68
V=87.36×10−7043.56×10−68
V=87.3643.56×100≈0.4986×100≈50 V
The required accelerating potential is approximately 50 V. This elegant calculation shows how quantum mechanics perfectly predicts the behavior of particles in a macroscopic crystal lattice!
