KAHIBARO
Table of Contents
Seeing the Inner Structure of Atoms
X rays are very energetic electromagnetic waves. In atomic physics, x ray spectra are produced when electrons in atoms change between very tightly bound inner energy levels, or when fast electrons are suddenly slowed down in matter. The result is radiation with much shorter wavelength and much higher energy than visible light.
An x ray spectrum is a graph showing how much x ray radiation is emitted at different wavelengths or frequencies or energies. This spectrum is not always a single sharp line. In many cases it contains both a smooth background and sharp peaks. These different parts come from different physical processes.
Two Main Parts of an X Ray Spectrum
When x rays are produced in an x ray tube, the spectrum usually has two components. One is a continuous spectrum, spread over a range of energies. The other is a line spectrum, made of sharp peaks at specific energies.
The continuous part appears because fast electrons strike a metal target and lose energy in many different amounts. The sharp lines appear because electrons inside the atoms of the target jump between definite atomic energy levels.
| Part of spectrum | Cause | Appearance |
|---|---|---|
| Continuous spectrum | Fast electrons decelerate in the target | Smooth range of x ray energies |
| Characteristic spectrum | Inner-shell electron transitions in atoms | Sharp lines at fixed energies |
Continuous X Ray Spectrum
In an x ray tube, electrons are accelerated through a potential difference and hit a metal target. As they pass close to atomic nuclei in the target, they are deflected and slowed down. A charged particle that accelerates or decelerates emits electromagnetic radiation. This radiation is called bremsstrahlung, which means braking radiation.
Because the electrons can lose different amounts of energy in different collisions, the emitted x ray photons can have many possible energies. That is why this part of the spectrum is continuous.
If an electron gives up all its kinetic energy in a single event, the emitted photon has the maximum possible energy. If the accelerating voltage is $V$, then the maximum photon energy is
$$
E_{\max} = eV
$$
where $e$ is the elementary charge.
Since photon energy is also
$$
E = hf = \frac{hc}{\lambda}
$$
the minimum wavelength in the continuous spectrum is
$$
\lambda_{\min} = \frac{hc}{eV}
$$
This is called the Duane-Hunt law.
For x rays produced by electrons accelerated through a voltage $V$,
$$
E_{\max} = eV
$$
and
$$
\lambda_{\min} = \frac{hc}{eV}
$$
A higher accelerating voltage gives x rays of higher maximum energy and shorter minimum wavelength.
Characteristic X Ray Spectrum
The sharp lines in an x ray spectrum are called characteristic x rays because their energies depend on the type of atom in the target. Each element has its own set of inner electron energy levels, so each element produces its own pattern of x ray lines.
The process begins when a fast incoming electron knocks out an inner-shell electron, often from the $K$ shell, which is the closest shell to the nucleus. This creates a vacancy. An electron from a higher shell then falls into the lower-energy empty place. The energy difference is emitted as an x ray photon.
If an electron falls from the $L$ shell to the $K$ shell, the emitted line is called a $K_\alpha$ x ray. If it falls from the $M$ shell to the $K$ shell, it is called a $K_\beta$ x ray.
| Transition | Name of x ray line |
|---|---|
| $L \to K$ | $K_\alpha$ |
| $M \to K$ | $K_\beta$ |
| $M \to L$ | $L_\alpha$ |
The energy of the emitted x ray is given by the difference between the two atomic energy levels:
$$
E_{\gamma} = E_{\text{upper}} - E_{\text{lower}}
$$
Because atomic energy levels are quantized, these x ray energies are also quantized. That is why the spectrum contains sharp lines.
Shells and Line Naming
Inner electronic shells are traditionally named $K$, $L$, $M$, $N$, and so on. These correspond roughly to principal quantum numbers $n = 1, 2, 3, 4, \dots$.
A vacancy in a lower shell can be filled by an electron from a higher shell, producing characteristic radiation. The line name tells both the shell that receives the electron and roughly where the electron came from.
Why the Lines Depend on the Element
Inner electrons feel a strong attraction from the nucleus. If the nucleus has more protons, the inner electrons are generally more tightly bound. Therefore the energy differences between inner shells are larger for heavier elements. This means heavier elements tend to emit characteristic x rays with higher energies.
This is why x ray spectroscopy can identify elements. If you measure the energies of the emitted x ray lines, you can compare them with known values and determine which element is present.
Moseley's Law
A very important result in atomic physics is Moseley's law. It shows that the frequency of characteristic x rays depends in a systematic way on the atomic number $Z$ of the element.
For many characteristic lines, especially $K$ lines, the frequency approximately obeys
$$
\sqrt{f} \propto (Z - b)
$$
where $b$ is a shielding constant that accounts for the fact that inner electrons partially screen the nuclear charge.
This relation showed that atomic number, not atomic mass, is the fundamental quantity that orders the elements. It was a major confirmation that the positive charge of the nucleus determines atomic structure.
Moseley's law showed that characteristic x ray frequencies are determined mainly by atomic number $Z$, not by atomic mass.
This was strong evidence that the nuclear charge controls atomic energy levels.
Absorption and Emission Spectra of X Rays
An x ray spectrum can be observed in emission, when a substance gives off x rays, or in absorption, when x rays pass through a substance and some energies are absorbed more strongly than others.
Absorption often shows sudden changes called absorption edges. These occur when the x ray photons have just enough energy to remove an electron from an inner shell. For example, a $K$ absorption edge appears when photon energy reaches the binding energy of the $K$ shell.
This is closely related to characteristic emission. If a photon can knock out a $K$ electron, then later transitions into that empty $K$ state can produce characteristic $K$ x rays.
A Simple Energy Picture
The central idea is energy conservation. An incident fast electron can transfer enough energy to remove an inner electron. Then the atom, now in an excited ionized state, moves toward a lower energy state by rearranging its electrons. The lost atomic energy appears as an x ray photon.
If the energy levels involved are $E_K$, $E_L$, and $E_M$, then for example
$$
E_{K_\alpha} = E_L - E_K
$$
and
$$
E_{K_\beta} = E_M - E_K
$$
Here the symbols represent binding energies or level energies in a consistent convention. The important point is that the photon energy equals the difference between the two levels.
What an X Ray Tube Spectrum Looks Like
A typical x ray tube spectrum has a broad continuous curve starting at a minimum wavelength cutoff, with sharp peaks rising above the curve. The cutoff is set by the accelerating voltage. The sharp peaks are set by the target material.
If you increase the tube voltage, the minimum wavelength becomes shorter and the continuous spectrum extends to higher energies. If you change the target metal, the positions of the sharp characteristic lines change.
Importance of X Ray Spectra
X ray spectra are important because they reveal the inner electronic structure of atoms. They provided key evidence for quantized atomic energy levels and for the role of atomic number in organizing the periodic table. They are also widely used in practical analysis of materials, because each element has its own characteristic x ray fingerprint.
Key Ideas to Remember
X ray spectra usually contain a continuous part and a characteristic line part. The continuous part comes from bremsstrahlung, produced when fast electrons are decelerated. The characteristic lines come from transitions between inner atomic shells. Their energies are fixed by the element and can be used to identify it. The minimum wavelength of the continuous spectrum is determined by the accelerating voltage.
Essential facts about x ray spectra:
The continuous spectrum is due to bremsstrahlung.
The sharp lines are due to inner-shell electronic transitions.
Characteristic x ray energies depend on the element.
The shortest wavelength satisfies
$$
\lambda_{\min} = \frac{hc}{eV}
$$