The energies for an electron in the $K$, $L$, and $M$ shells of the tungsten atom are $-69,500$ eV, $-12,000$ eV, and $-2,200$ eV, respectively. Calculate the wavelengths of the $K_{\alpha }$ and $K_{\beta }$ x rays of tungsten.

In an atom, electrons occupy specific energy levels or shells, denoted as K, L, M, etc. Each shell has a distinct energy associated with it, which is negative due to the attractive force between the negatively charged electrons and the positively charged nucleus. The energy levels determine the electron's binding energy and influence the emission of photons when electrons transition between these levels.

X-rays are produced when high-energy electrons collide with a target atom, causing inner-shell electrons to be ejected. When an electron from a higher energy level falls into the vacancy left by the ejected electron, it emits energy in the form of X-rays. The Kα and Kβ X-rays correspond to transitions from the L shell to the K shell and from the M shell to the K shell, respectively, each with specific energy differences.

The wavelength of emitted X-rays can be calculated using the energy of the emitted photon, which is related to the energy difference between the two electron shells involved in the transition. The relationship is given by the equation E = hc/λ, where E is the energy of the photon, h is Planck's constant, c is the speed of light, and λ is the wavelength. By rearranging this equation, one can determine the wavelength from the energy difference.