Crystal diffraction experiments can be performed using X-rays, or electrons accelerated through appropriate voltage. Which probe has greater energy? (For quantitative comparison, take the wavelength of the probe equal to 1 Å, which is of the order of inter-atomic spacing in the lattice) (m_e= 9.11 × 10⁻³¹ kg).

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Solution

An X-ray probe has a greater energy than an electron probe for the same wavelength.

Wavelength of light emitted from the probe, λ = 1 Å = 10⁻¹⁰ m

Mass of an electron, m_e = 9.11 × 10⁻³¹ kg

Planck’s constant, h = 6.6 × 10⁻³⁴ Js

Charge on an electron, e = 1.6 × 10⁻¹⁹ C

The kinetic energy of the electron is given as:

Where,

v = Velocity of the electron

m_ev = Momentum (p) of the electron

According to the de Broglie principle, the de Broglie wavelength is given as:

Energy of a photon,

Hence, a photon has a greater energy than an electron for the same wavelength.

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Exercise 11.31:

Crystal diffraction experiments can be performed using X-rays, or electrons accelerated through appropriate voltage. Which probe has greater energy? (For quantitative comparison, take the wave length of the probe equal to 1

A∘A∘ which is of the order of inter-atomic spacing in the lattice) (

m_{e} = 9.11 \times 10^{- 31} k g

)

Q. What will be the wavelength of the X-rays which give a diffraction angle

2 θ

equal to

{16.80}^{o}

for a crystal, if the inter planner distance in the crystal is 0.200 nm and only first order diffraction is observed

(sin {8.40}^{o} = 0.146) .

Q. An X-ray beam of wavelength

1.0 ˚ A

is incident on a crystal of lattice spacing

2.8 ˚ A

. The value of Bragg angle for first order of diffraction is :

Q. The wavelength of a probe is roughly a measure of the size of a structure that it can probe in some detail. The quark structure of protons and neutrons appears at the minute length-scale of 10–15 m or less. This structure was first probed in early 1970’s using high energy electron beams produced by a linear accelerator at Stanford, USA. Guess what might have been the order of energy of these electron beams. (Rest mass energy of electron = 0.511 MeV.)

X-rays are incident normally on a crystal of lattice constant 0.6 nm. The first order reflection of diffraction from the crystal occurs at an angle of $30^{\circ}$ . What is the wavelength of X-rays used?

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