In a nuclear physics experiment, a proton is fired toward a Z = 13 nucleus with the diameter and neutron energy levels shown in Figure 40.17. The nucleus, which was initially in its ground state, subsequently emits a gamma ray with wavelength 1.73×10⁻⁴ nm. What was the minimum initial speed of the proton? Hint: Don't neglect the proton-nucleus collision.

Verified step by step guidance

1. Understand the problem: The proton collides with the nucleus, exciting it to a higher energy state. The nucleus then emits a gamma ray as it transitions back to its ground state. The goal is to find the minimum initial speed of the proton required to excite the nucleus. This involves energy conservation and quantum mechanics principles.
2. Step 1: Calculate the energy of the emitted gamma ray. Use the formula for the energy of a photon: \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \ \text{J·s} \)), \( c \) is the speed of light (\( 3.00 \times 10^8 \ \text{m/s} \)), and \( \lambda \) is the wavelength of the gamma ray (\( 1.73 \times 10^{-4} \ \text{nm} \), converted to meters).
3. Step 2: Recognize that the energy of the gamma ray corresponds to the energy difference between the excited state and the ground state of the nucleus. This means the proton must transfer at least this much energy to the nucleus during the collision to excite it.
4. Step 3: Use the principle of conservation of energy. The kinetic energy of the proton before the collision is given by \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the proton (\( 1.67 \times 10^{-27} \ \text{kg} \)) and \( v \) is its speed. Set this equal to the energy of the gamma ray to find the minimum speed of the proton: \( \frac{1}{2}mv^2 = E \).
5. Step 4: Solve for \( v \) by rearranging the equation: \( v = \sqrt{\frac{2E}{m}} \). Substitute the values for \( E \) (calculated in Step 1) and \( m \) to find the minimum initial speed of the proton.

Key Concepts

Nuclear Reactions

Nuclear reactions involve interactions between atomic nuclei, which can result in the emission of particles or radiation. In this context, the collision between the proton and the Z = 13 nucleus can lead to an excited state of the nucleus, which may subsequently decay by emitting a gamma ray. Understanding the energy changes during these reactions is crucial for determining the initial conditions required for the proton.

Energy Conservation

The principle of energy conservation states that the total energy in a closed system remains constant. In this scenario, the kinetic energy of the incoming proton must be sufficient to overcome the binding energy of the nucleus and excite it to a higher energy state. The energy associated with the emitted gamma ray can be used to calculate the minimum initial speed of the proton, as it reflects the energy lost during the transition.

Wavelength and Energy Relationship

The energy of a photon is inversely related to its wavelength, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. In this problem, the emitted gamma ray's wavelength of 1.73×10⁻⁴ nm can be converted into energy, which is essential for calculating the minimum kinetic energy required by the proton to initiate the nuclear reaction.