Moseley's Law And Its Application

Moseley's Law:

Physicist Henry Moseley undertaking a systematic study of characteristic spectra of different elements and the frequencies of the emitted lines. Taking different elements as the target, the frequency of emitted x-ray is measured by Moseley. He tried to develop a relation between the frequency and the atomic number of the target but not any remarkable result could achieved.

When the square root of frequency is plotted against the atomic number of the target, a linear relation is found as shown in figure.

On the basis of these experiment, Moseley formulate a law called Moseley law.

Statement: The square root of frequency of emitted x-rays during the characteristic x-ray emission is directly proportional to the atomic number of the target.

If $\nu$ be the frequency of emitted x-ray and z be the atomic number of the target atom, then

$\sqrt{\nu}$ $\propto$ z

or, $\sqrt{\nu}$ = a(z-b) ..... (1)

where , a is proportionality constant and b is screening constant.

Theoritical explanation of Moseley law:

As we have know, the expression of energy in the $n^{th}$ particular orbit is given by

$$E_n = \frac{-me^4z^2}{8\epsilon_\circ^2n^2h^2}$$ where the symbols have their usual meanings.

for $n_1$ and $n_2$ orbits,

$$E_{n_1} = \frac{-me^4z^2}{8\epsilon_\circ^2n_1^2h^2}\;\;...(2)$$

$$and\;\;\;E_{n_2}=\frac{-me^4z^2}{8\epsilon_\circ^2n_2^2h^2}\;\;...(3)$$ now subtracting equation (3) from (2)

$$E_{n_2}-E_{n_1}=\frac{me^4z^2}{8\epsilon_\circ^2h^2}\biggl[\frac{-1}{n_2^2}+\frac{1}{n_1^2}\biggr]$$

$$or\;\;\;h\nu=\frac{me^4z^2}{8\epsilon_\circ^2h^2}\biggl[\frac{1}{n_1^2}-\frac{1}{n_2^2}\biggr]$$

$$or\;\;\;\nu=\frac{me^4z^2}{8\epsilon_\circ^2h^3}\biggl[\frac{1}{n_1^2}-\frac{1}{n_2^2}\biggr]$$

$$or\;\;\;\nu=\frac{me^4}{8\epsilon_\circ^2ch^3}cz^2\biggl[\frac{1}{n_1^2}-\frac{1}{n_2^2}\biggr]$$

$$or\;\;\;\nu=Rcz^2\biggl[\frac{1}{n_1^2}-\frac{1}{n_2^2}\biggr]\;\;...(4)$$

$$where, R = \frac{me^4}{8\epsilon_\circ^2ch^3}$$

For particular orbit, from equation (4) we see that $\nu$$\propto$$z^2$

Therfore: $\sqrt{\nu}$ $\propto$ z

which is Moseley law.

For $k_\alpha$-line ; $n_1$=1 and $n_2$=2

therefore $\nu$ = $\frac{3}{4}$Rc$z^2$

$$a=\sqrt\frac{3}{4}Rc$$

Importance of Moseley Law:

(1) Discovery of new element :

On the basis of Moseley law, the new elements were discovered which were not discovered during the time of Mendeleev. He left the gap in the periodic table and these gap were filled up by Moseley.

(2)Arrangement of element in the periodic table on the basis of atomic number:

During the time of Mendeleev , the elements were arranged on the basis of atomic weight but Moseley proved that the atomic number is the basic characteristic property of elements , both physical and chemical,Moseley arranged the elements in the periodic table on the basis of atomic number.

For example: According to Mendeleev 19$k^{39}$ comes before 18$Ar^{40}$ similar be the case of

28Ni58.7 and 27Co58.9.But on the basis of Moseley, elements are arranged in the periodic table accourding to their atomic number such that 18Ar40 ccomes before 19K39 similar be the case of 27Co58.9 and 28Ni58.7.

Fine structure of X-ray spectra:

The characteristic x-ray spectra are not single, it consists of a number of closely packet other spectral lines called fine structure of x-ray spectra. The resolution of L-lines into 3-lines, M-line into 5-lines & N-line into 7-line is called fine structure of x-ray spectra.

To explain the fine structure of x-ray spectra, we have to consider the following two effects (1) Screening effect (2) Relativistic variation of mass with velocity

The 8$e^{-s}$ in L-shell fells that the effect of nuclear charge is reduced due to the presence of two $e^{-s}$ in k-shell. Similar be the case of $e^{-s}$ in the M-shell. The effect of reducing the nuclear charge deu to the presence of $e^{-s}$ in the innermost orbit is called screening effect. By consiering the relativistic variation of mass with velocity & screening effect, the term value for characteristic x-ray spectra is given by, $$T=\frac{R(z-a)^2}{n^2}+\frac{R\alpha^2(z-b)^4}{n^4}\biggl(\frac{n}{n_\phi}-\frac{3}{4}\biggr)$$

Where; R= Rydberg constant

$\alpha$= Fine structure constant

b= Screening constant

n= Principle quantum number

$n_{\phi}$= Azimuthal quantum nuber

z= Number of proton in the nucleus.

From equation (1) we see that for particular value of n, $n_{\phi}$ takes the values from 0 to n-1. This explains the fine structure of x-ray spectra.

Figure here:

From above figure we see that l-lline is splitted into 3lines & M-line is splitted nto s-lines which is in the agrement of fine structure of x-ray spectra.

References:

Adhikari, P.B, Daya Nidhi Chhatkuli and Iswar Prasad Koirala. A Textbook of Physics. Vol. II. Kathmandu: Sukunda Pustak Bhawan, 2012.

Jenkins, F.A and H.E White. Fundamental of optics. New York (USA): McGraw-Hill Book Co, 1976.

wood, R.W. Physical Optics. New York (USA): Dover Publication , 1934.