Question

Neutron diffraction is used in determining the structures of molecules. a. Calculate the de Broglie wavelength of a neutron moving at 1.00\(\%\) of the speed of light. b. Calculate the velocity of a neutron with a wavelength of 75 \(\mathrm{pm}\left(1 \mathrm{pm}=10^{-12} \mathrm{m}\right)\)

Short answer

a. The de Broglie wavelength of a neutron moving at 1% of the speed of light is approximately \(1.32 \times 10^{-15} \ \text{m}\). b. The velocity of a neutron with a wavelength of 75 pm is approximately \(5.30 \times 10^5 \ \text{m/s}\).

Step-by-step solution

Part A: Calculate the de Broglie Wavelength

We are given that the neutron is moving at 1% of the speed of light. The speed of light is \(3.00 \times 10^8 \ \text{m/s}\). Thus, the velocity of the neutron is: \[v = 0.01 \times 3.00 \times 10^8 \ \text{m/s} = 3.00 \times 10^6 \ \text{m/s}\] The mass of a neutron is approximately \(1.67 \times 10^{-27} \ \text{kg}\). Using the de Broglie wavelength equation, we can find the wavelength: \[\lambda = \frac{h}{mv}\] The Planck's constant, \(h\), is equal to \(6.63 \times 10^{-34} \ \text{Js}\) Substituting values, we get: \[\lambda = \frac{6.63 \times 10^{-34} \ \text{Js}}{(1.67 \times 10^{-27} \ \text{kg})(3.00 \times 10^6 \ \text{m/s})}\]

Part A: Calculate the Result

Now, calculate the wavelength: \[\lambda = \frac{6.63 \times 10^{-34} \ \text{Js}}{(1.67 \times 10^{-27} \ \text{kg})(3.00 \times 10^6 \ \text{m/s})} \approx 1.32 \times 10^{-15} \ \text{m}\] So, the de Broglie wavelength of the neutron is approximately \(1.32 \times 10^{-15} \ \text{m}\).

Part B: Calculate the Velocity of the Neutron

We are given the wavelength of the neutron as 75 pm, which can be converted into meters: \[75 \ \text{pm} = 75 \times 10^{-12} \ \text{m}\] We'll use the de Broglie wavelength equation again. We are given the wavelength and the mass of a neutron. We need to solve the equation for the velocity, v: \[v = \frac{h}{m \lambda}\]

Part B: Calculate the Result

Substituting values, we get: \[v = \frac{6.63 \times 10^{-34} \ \text{Js}}{(1.67 \times 10^{-27} \ \text{kg})(75 \times 10^{-12} \ \text{m})} \approx 5.30 \times 10^5 \ \text{m/s}\] So, the velocity of a neutron with a wavelength of 75 pm is approximately \(5.30 \times 10^5 \ \text{m/s}\).

Key concepts

Neutron Diffraction

Neutron diffraction is a powerful technique used to determine the crystal structure of materials at the atomic level. This method is similar to X-ray diffraction but specifically uses neutrons.
Neutrons are ideal for diffraction studies because they are neutral particles, which means they can easily penetrate deep into materials without being significantly absorbed or scattered by electrons.
This property allows neutrons to provide information on the position of light atoms such as hydrogen, which is often challenging with other forms of radiation. Understanding neutron diffraction involves recognizing:

• It operates on the principle of wave-particle duality, where particles like neutrons exhibit wave-like properties.
• When a beam of neutrons is directed at a crystalline material, the atomic lattice causes the neutrons to scatter in distinct patterns. These patterns, when analyzed, reveal the arrangement of atoms within the structure.

Understanding this pattern through the analysis of neutron diffraction is crucial for developments in fields like chemistry, biology, and materials science. It helps scientists develop new materials, understand complex biological molecules, and improve industrial processes.

Planck's Constant

Planck's constant, denoted by \( h \), is a fundamental constant in physics that plays a crucial role in quantum mechanics. It was first introduced by Max Planck in the early 20th century as he worked to explain black-body radiation.
Planck's constant is central to the concept of quantization in quantum theory, underpinning the idea that energy is not continuous, but rather comes in discrete units or "quanta."The value of Planck's constant is approximately \( 6.63 \times 10^{-34} \) Joule seconds (Js).
This small number indicates the granularity of the energy levels, which becomes significant at atomic and subatomic scales.In the context of the de Broglie wavelength, Planck's constant is used in the formula \( \lambda = \frac{h}{mv} \), where it links the mass and velocity of a particle to its wavelength. Understanding Planck's constant allows us to:

• Appreciate the scale at which quantum effects become important.
• Comprehend how concepts like wave-particle duality emerge from quantum physics.
• Calculate properties related to the motion and interaction of particles, such as the wavelength of neutrons in diffraction studies.

Planck's constant is fundamental in all quantum mechanical equations and crucial for comprehending the behavior of particles at microscopic scales.

Particle Velocity

Particle velocity refers to the speed and direction in which a particle moves. In physics, understanding how to calculate and interpret the velocity of particles is essential, especially in quantum mechanics where particle behavior can be both particle-like and wave-like.
Velocity is defined as the rate of change of an object's position with respect to time and is given in units of meters per second (m/s).When discussing the particle velocity of a neutron, it is often explored in the context of de Broglie's hypothesis. According to de Broglie's equation, the velocity \( v \) of a particle is related to its wavelength \( \lambda \) by the expression \( v = \frac{h}{m \lambda} \), which allows us to derive the velocity of a particle if its wavelength and mass are known. From this relationship, we learn:

• Velocity is inversely proportional to the wavelength; as the wavelength increases, the velocity decreases, assuming a constant mass.
• In neutron diffraction, determining the velocity of neutrons can provide insights into their interactions with other particles or fields.
• Calculating particle velocity is fundamental when interpreting results from diffraction experiments as it directly influences the scattering patterns observed.

Ultimately, understanding particle velocity aids in predicting and analyzing the dynamics of particles within various physical systems.