Moseley’s Law

Moseley’s Law describes how the energy of characteristic x-rays emitted by an atom depends on its internal structure, especially the pull of the nucleus on its inner electrons. When an inner electron is knocked out and an outer one drops down to replace it, the x-ray released reflects the effective nuclear charge. Moseley showed that this x-ray energy increases regularly with atomic number, proving that the true identity of an element comes from the number of protons rather than its atomic mass. This discovery strengthened the modern periodic table and provided a reliable way to identify elements. Today, Moseley’s Law is widely used in x-ray spectroscopy to analyze material composition, detect unknown elements, check purity, and study atomic structure in fields like chemistry, geology, materials science, and forensics.

1.0X-Rays

It was discovered by Roentgen. The wavelength of x-rays is found between $0.1\text{ A˚}to10\text{ A˚}$. These rays are invisible to the eye. They are electromagnetic waves and have speed $c=3\times 10^8\text{ m/s in a vacuum}$. Its photons have energy around 1000 times more than the visible light.

When fast moving electrons having energy of order of several KeV strike the metallic target then x-rays are produced.

2.0Production of X-Rays by Coolidge Tube

The target material should have a high melting point, high specific heat capacity, and high atomic number. When a voltage is applied across the filament, it gets heated and emits electrons. A collimator is used to shape the electron beam. When these electrons strike the target, X-rays are produced. However, part of the electron energy is converted into heat, so the target must be able to withstand high temperatures. To aid in heat dissipation, a copper rod is attached to the target, allowing it to absorb and transfer heat, preventing the target from overheating. For more energetic electrons, the accelerating voltage is increased, while increasing the filament voltage produces a higher number of photons.

• The X-ray was analysed by mostly taking their spectrum.

•  Variation of Intensity of x-Rays with \lambda is Plotted as shown in figure:

The minimum wavelength corresponds to the maximum energy of the X-rays which in turn is equal to the maximum kinetic energy eV of the striking electrons thus

$\text{eV}=h{\nu }_{\text{max}}=\frac{hc}{{\lambda }_{\text{min}}}$

${\lambda }_{\text{min}}=\frac{hc}{\text{eV}}=\frac{12400}{V\text{ in Volts}}\text{ A˚}$

We see that cut off wavelength ${\lambda }_{\text{min}}$ depends only on accelerating voltage applied between target and filament. It does not depend upon the material of target, it is same for two different metals (Z and Z').

3.0Characteristic X-Rays

•  The sharp peaks obtained in graphs are known as characteristic x-rays because they are characteristic of target material.
• ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\dots$ = characteristic wavelength of material having atomic number Z are called characteristic X-rays and the spectrum obtained is called characteristic

spectrum. If the target of atomic number Z’ is used then peaks are shifted.

• Characteristic X-ray emission occurs when a high-energy electron strikes a target atom and ejects an inner-shell electron. The resulting vacancy is then filled by an electron from a higher energy level, leading to the emission of X-rays.
• If a vacancy in the innermost K-shell is filled by an electron from the next higher L-shell, a $K_{\alpha }$ characteristic X-ray is emitted. When the vacancy in the K-shell is filled by an electron from the M-shell, a $K_{\beta }$ line is produced. Similarly, transitions to the L-shell give rise to $\text{L}\alpha ,\text{L}\beta$ and other L lines, while transitions to the M-shell produce $\text{M}\alpha ,\text{M}\beta ,{\text{M}}_{\gamma }$ lines, and so on.

4.0Moseley’s Law

• Moseley measured the frequencies of characteristic X-rays for numerous elements and plotted the square root of these frequencies against their atomic numbers. He found that the resulting graph was nearly a straight line, which did not pass through the origin.

Moseley’s observations can be mathematically expressed as

$\sqrt{\nu }=a(Z-b)$

a and b are positive constants for one type of X-rays and for all elements (independent of Z).

Moseley’s Law can be derived on the basis of Bohr’s theory of atom, frequency of X-rays is given by

$\sqrt{\nu }=\sqrt{cR\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)}(Z-b)$

By using the formula,

$\frac{1}{\lambda }=R{Z}^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$ with modification for a multi-electron system.

$b\to$ known as screening constant or shielding effect, and (Z – b) is an effective nuclear charge.

For $K_{\alpha }$ Line

$n_1=1,n_2=2$

$\therefore \sqrt{\nu }=\sqrt{\frac{3Rc}{4}}(Z-b)$

$\sqrt{\nu }=a(Z-b)$

$a=\sqrt{\frac{3Rc}{4}},b=1\text{ for }{K}_{\alpha }\text{ Lines}$

Illustration-1.Find in $Z_1$ and $Z_2$ which one is greater.

Solution:

$\sqrt{\nu }=\sqrt{cR\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)}(Z-b)$

If Z is greater then $\nu$ will be greater,  \lambda will be less

$\therefore {\lambda }_1<{\lambda }_2$

$\therefore Z_1<Z_2$

Illustration-2.A cobalt target is bombarded with electrons and the wavelength of its characteristic spectrum are measured.A second, fainter, characteristic spectrum is also found because of an impurity in the target. The wavelengths of the $K_{\alpha }$ lines are 178.9pm (cobalt) and 143.5pm (impurity). What is impurity?

Solution:

Using Moseley’s law and putting $\sqrt{c/\lambda }$ for $\sqrt{\nu }$ (and assuming b = 1), we obtain

$\sqrt{\frac{c}{{\lambda }_{\text{Co}}}}=a(Z_{\text{Co}}-1)\text{and}\sqrt{\frac{c}{{\lambda }_x}}=a(Z_x-1)$

Dividing yields

$\frac{\sqrt{{\lambda }_{\text{Co}}}}{\sqrt{{\lambda }_x}}=\frac{Z_x-1}{Z_{\text{Co}}-1}$

Substituting gives us

$\sqrt{\frac{178.9\text{ pm}}{143.5\text{ pm}}}=\frac{Z_x-1}{27-1}$

$Z_x=30$ the impurity is Zinc