Question

The \(K_{a}\) line in copper is a very common one to use in \(X\) -ray ciystallography. To produce it, electrons are accelerated through a potential difference and smashed into a copper target. Section 7.8 gives the energies in a hydrogenlike atom as \(Z^{2}\left(-13.6 \mathrm{eV} / \mathrm{n}^{2}\right)\). Making the reasonable approximation that an \(n=1\) electron in copper orbils the nucleus and half of its fellow \(n=1\) electron. being unaff ected by the roughly spherical cloud of other electrons around it, estimate the minimum accelerating potential needed to make a hole in copper's \(K\) shell.

Short answer

The minimum accelerating potential needed to make a hole in copper's K shell is approximately \(119.34 V\).

Step-by-step solution

Find the Energy of the K Shell

Using the formula given with Z=29/2 and n=1 (the K shell, based on the Bohr's model), the energy (E) can be calculated as follows: \(E = -Z^{2}(13.6eV/(n^{2})) = -(29/2)^{2}(-13.6eV/(1^{2})) = -119.34eV.\) The negative sign indicates the bound state of the electron.

Find the Accelerating Potential

The energy E is given by \(E = qV\), where q is the charge of the electron and V is voltage. Therefore, the accelerating potential \(V = E/q\). Note: The charge on an electron, \(q = 1.6 \times 10^{-19} C\). This gives \(V = -119.34 eV \times 1.6 \times 10^{-19} C = -1.91 \times 10^{-17} C V\), again the negative sign indicates the bound state of the electron.

Convert units from Coulomb Volt to Volt

Express your answer in volts, which is the standard unit of electric potential. The conversion factor is \(1 eV = 1.6 \times 10^{-19} C\). So, divide our result by the conversion factor to get the answer in volts: \(-1.91 \times 10^{-17} C V / (1.6 \times 10^{-19} C) = 119.34 V.\)

Key concepts

Electron Acceleration in X-ray Crystallography

Understanding electron acceleration is central to grasping the fundamentals of X-ray crystallography. When electrons are propelled through a potential difference, they gain kinetic energy that is proportional to the voltage of the potential applied. This process is crucial in generating X-rays for crystallography.

Electrons are accelerated towards a target material, such as copper, and upon collision, their sudden deceleration leads to the emission of X-rays. The potential difference must be high enough so that when these electrons hit the target, they can knock inner shell electrons out of their orbit, creating vacancies that result in the emission of X-ray photons as other electrons drop into these lower energy states to fill the gaps.

In simple terms, imagine pulling back a slingshot - the further you pull, the more potential energy you have, and when you let go, that potential energy turns into kinetic energy as the stone flies. Similarly, a high voltage acts like a stronger pull, giving the electrons more energy to cause the necessary collisions for X-rays to be produced.

Copper K Shell and Its Role in X-ray Production

The Copper K shell refers to the innermost electron shell of a copper atom, according to the Bohr model. When we talk about making a 'hole' in the K shell, we mean ejecting one of these tightly held inner electrons, which requires a substantial amount of energy.

The energy levels of electrons in different shells are described by quantum mechanics, and in the context of copper for X-ray generation, when an electron from a higher energy level falls into the K shell, it emits an X-ray photon with energy characteristic of the copper atom - this is the basis of the method called X-ray fluorescence.

The exercise provided uses the hydrogen-like energy formula for approximation, because even in multi-electron atoms like copper, the innermost electrons feel an effective nuclear charge very similar to the actual atomic number, allowing us to treat them like hydrogen for this calculation. This simplification is the key for estimating the minimum accelerating potential needed to disrupt the K shell and generate X-ray photons for crystallography.

Bohr's Model and Its Application in the Exercise

Niels Bohr's model gives us a way to visualize an atom with discrete energy levels, which is particularly helpful when attempting to understand phenomena like X-ray generation at a basic level.

According to Bohr's model, electrons move in fixed orbits around the nucleus and can jump between these orbits by absorbing or emitting energy. The energy of an electron in a particular orbit can be calculated using the formula provided in the exercise. For copper, with an effective nuclear charge of half its atomic number due to shielding effects, we plug in the values into the Bohr's model formula to get the energy of an electron in the K shell.

This understanding allows for the calculation of the exact amount of energy required to accelerate an electron enough to knock another electron out of the K shell, a necessary step in producing the X-rays used in crystallography. The calculated energy directly correlates to the minimum voltage needed in the equipment used for this purpose, demonstrating the practical application of Bohr's theoretical model in modern scientific techniques.