Question

Determine the approximate probing distance of a photon with an energy of \(2.0 \mathrm{keV}\).

Short answer

The approximate probing distance of a photon with an energy of 2.0 keV is 3.0995 x 10^-10 meters.

Step-by-step solution

Convert energy to wavelength

First, we will convert the photon energy (\(2.0 \mathrm{keV}\)) to wavelength using the relationship between energy, Planck's constant (h), and the speed of light (c): $$ E = \dfrac{hc}{\lambda} $$ where \(E = 2.0 \times 10^3 \mathrm{eV}\) \(h = 4.135 \times 10^{-15} \mathrm{eV} \cdot \mathrm{s}\) \(c = 2.998 \times 10^8 \mathrm{m} \cdot \mathrm{s^{-1}}\) We'll now solve for wavelength (\(\lambda\)): $$ \lambda = \dfrac{hc}{E} $$

Calculate the wavelength

Plug the values into the formula: $$ \lambda = \dfrac{(4.135 \times 10^{-15} \mathrm{eV} \cdot \mathrm{s}) (2.998 \times 10^8 \mathrm{m} \cdot \mathrm{s^{-1}})}{2.0 \times 10^3 \mathrm{eV}} $$ $$ \lambda = 6.199 \times 10^{-10} \mathrm{m} $$

Determine the scattering angle

Now, we will determine the scattering angle (\(\theta\)) for a maximal resolution, which occurs when \(\theta = 90^\circ\).

Calculate the approximate probing distance using Bragg's law

Next, utilize Bragg's law to determine the approximate probing distance (d): $$ 2d \sin{\theta} = n\lambda $$ Since \(n=1\) is the first order of reflection and \(\theta = 90^\circ\), the equation simplifies to: $$ d = \dfrac{\lambda}{2} $$

Calculate the approximate probing distance

Plug in the calculated wavelength: $$ d = \dfrac{6.199 \times 10^{-10} \mathrm{m}}{2} $$ $$ d = 3.0995 \times 10^{-10} \mathrm{m} $$

Answer

The approximate probing distance of a photon with an energy of \(2.0 \mathrm{keV}\) is \(3.0995 \times 10^{-10} \mathrm{m}\).

Key concepts

Photon Energy

Understanding the energy of photons is crucial in the field of physics, particularly when studying electromagnetic waves and quantum mechanics. Photons are elementary particles that constitute the smallest quanta of light and have properties of both waves and particles. The energy of a photon can be calculated using the equation:

\[ E = \dfrac{hc}{\lambda} \]
This equation reveals the direct relationship between energy (E\text{)} and both Planck's constant (\( h \)\text{)} and the speed of light (\( c \)\text{)}, inversely proportional to the wavelength (\( \lambda \)\text{)}. In other words, the shorter the wavelength of the light, the higher the energy of the photons. This concept is crucial for understanding phenomena such as the photoelectric effect and the behavior of light at atomic and subatomic scales.

Planck's Constant

Planck's constant (\( h \)\text{)} is a fundamental constant in quantum mechanics, named after the physicist Max Planck. It describes the size of the discrete units of energy that can be absorbed or emitted by an atom or molecule. Its value is very small,a: \[ h = 6.626 \times 10^{-34} \mathrm{Js} \]
However, in calculating the energy of photons, Planck's constant is often expressed in electronvolts per second (eV·s) due to the small energies involved at the quantum level. This constant is instrumental in understanding quantum effects and is a cornerstone for the quantization of energy levels in physics.

Speed of Light

The speed of light (\( c \)\text{)} in a vacuum is an essential constant in physics, serving as a fundamental speed limit of the universe. It has a value of approximately \[ c = 2.998 \times 10^8 \mathrm{m/s} \]
It is not only crucial for the calculation of photon energy, but also integral to the equations of Einstein's theory of relativity. In everyday terms, it means that light can travel around the Earth approximately 7.5 times in just one second. The speed of light plays a critical role in the synchronization of GPS satellites and is a key parameter in the propagation of electromagnetic radiation through different media.

Wavelength Calculation

The wavelength of a photon dictates its color in the visible spectrum and is related to its energy. Calculating wavelength involves rearranging the photon energy equation:\[ \lambda = \dfrac{hc}{E} \]
By inserting the known values for Planck's constant and the speed of light, and the photon's energy, you can solve for its wavelength. As illustrated in the problem's step-by-step solution, this calculation is crucial in determining the properties of light and its interactions with matter. The ability to compute the wavelength of photons is not only theoretical; it has practical applications in fields such as medical imaging, astronomy, and spectroscopy.

Bragg's Law

Bragg's law is a fundamental principle of X-ray crystallography, where it is used to analyze the atomic structure of crystals. The law relates the wavelength of electromagnetic radiation to the angles at which light is preferentially scattered by the crystal lattice:\[ 2d \sin{\theta} = n\lambda \]
Here, \( d \)\text{)} is the distance between crystal planes, \( \theta \)\text{)} is the angle of incidence, \( n \)\text{)} is the order of reflection and \( \lambda \)\text{)} is the wavelength of the incident light. According to Bragg's law, clear diffraction patterns occur at specific angles when the path difference between reflected waves from different crystal planes leads to constructive interference. This principle provides essential insights into the microscopic properties of materials and has improved the understanding of solid-state physics and materials science.