Physics > Advanced Modern Physics > 1.0 X-Rays
Advanced Modern Physics
1.0 X-Rays
2.0 Moseley's Law.
3.0 Nuclear Structure
4.0 Nuclear binding energy
4.1 Binding energy per nucleon
4.2 Variation of Binding energy per nucleon with mass number
4.3 Nuclear stability
5.0 Radioactivity
6.0 Radioactive decay law
1.2 Spectra of X-rays
4.2 Variation of Binding energy per nucleon with mass number
4.3 Nuclear stability
On the basis of the study of spectra of X-rays, it is broadly classified into two types (1) Characteristic X-ray and (2) Continuous X-ray.
Characteristic X-ray
These X-rays are called characteristic X-rays because they are particularly specific to the type of element used as target anode.There are various sharp wavelengths corresponding to discrete radiation in the characteristic spectra. As these are material specific(independent of the accelerating voltage), it helps in the identification of the same.When highly energetic electrons strike, with enough energy to penetrate into the atom(Z), they knockout electron form the inner shell ($n_1$). Then an electron from the outer shell jump ($n_2$) to fill this vacancy. And thus, in the transition process, the electron radiates energy whose frequency lies in the region of X-rays. The frequency of the emitted radiation is given by
$$\begin{equation} \begin{aligned} f = RZ_{eff}^2\left( {\frac{1}{{n_1^2}} - \frac{1}{{n_2^2}}} \right) \\ \\\end{aligned} \end{equation} $$
where,
${Z_{eff}} = Z - S$
It is the effective nuclear charge(for the electron undergoing transition from $n_1$ to $n_2$)
$S$= screening constant.
Now, another vacancy is created is created in shell $n_2$ which is filled by electron transition from subsequent higher states emitting an X-ray of different frequency.
This process carries out until the outermost shell is reached giving a series of spectral lines. The transition of electrons from various outer shells to the inner shell most $K$-shell produces a group of X-ray lines called as $K$-series.
It is further divided into ${K_\alpha },{K_\beta},{K_\gamma },...$ depending upon the outer shell from which the transition is made.
Similar things go for the other shells.The sharp peaks obtained in the graph as shown in Fig. 4 are known as characteristic X-rays because they are characteristic of the target material.
Continuous X-ray
These X-rays are independent of the target material and are continuous in nature.
When the electron projected toward the anode having a kinetic energy enters into the extremely high electric field of the nucleus of the atom of the anode, it experiences a strong eclectic force toward the nucleus and due to this, when it emerges it has a negligible energy(and velocity) in comparison to the initial energy.
So, the electron undergoes a high acceleration and hence, emits electromagnetic radiation in the form of X-rays(because accelerating charged particles emit radiation).
According to the Law of conservation of energy,
Energy of EM radiations = Loss in the KE of the electron...(i)
Let $V$ be applied accelerating voltage.
Let $m$ be the mass of the electron and ${{\lambda _{\min }}}$ be the minimum(threshold) wavelength
of X-ray emitted from the tube.Some electron which passes relatively more close to the nucleus of an atom loses more energy than other and hence, an X-ray of relatively smaller wavelength is emitted.
${{\lambda _{\min }}}$ will be emitted when electron with maximum energy loses almost all its energy while passing through the atom. so, using $(i)$, we have
$$\begin{equation} \begin{aligned} \Delta E = \frac{1}{2}m{v^2} = eV = \frac{{hc}}{{{\lambda _{\min }}}} \\ {\text{or }}{\lambda _{\min }} = \frac{{hc}}{{eV}} = \frac{{12400}}{{{V_{\left( {{\text{in volts}}} \right)}}}}{\text{ }}\mathop {\text{A}}\limits^0 \\\end{aligned} \end{equation} $$
This gives us the minimum wavelength.
The obtained X-rays thus range from $\left[ {{\lambda _{\min }},\infty } \right]$.
Now, characteristic X-rays are emitted only when a projectile electron of the atom makes a collision with another bound electron of the atom of the target material. When cathode rays pass through the target, then the probability of a projectile electron to collide with the bound electron is very less. A majority of the projectile electrons will pass through the anode without making the collision and they produce continuous X-rays. So, always both type of X-rays will be emitted. X-rays of only single type can't be emitted.
Hence the characteristic line spectra are superimposed on continuous X-ray (15.3) spectra of various intensities.
 |  |
| We can obtain X-rays in any range $\left[ {{\lambda _{\min }},\infty } \right]$ by applying an appropriate voltage across the discharge tube which will fix ${\lambda _{\min }}$ and other photons emitted from the tube will have wavelengths more than ${\lambda _{\min }}$ and ranging ranging up to $\infty $. That is why these X-rays are called continuous X-rays as shown in $Fig. 3$. As we can see in the graph, the intensity of emitted X-rays will be maximum (maximum number of photons) for a particular value of wavelength at a particular accelerating voltage across the discharge tube. At a particular voltage, the intensity of X-rays can be varied by changing the current in the circuit because the intensity of X-rays (number of photons) is proportional to the number of electrons attacking the anode. The broad continuous spectrum beyond the peak intensity is referred as "Bremsstrahlung". | Sharp peaks obtained for a material gives their characteristic X-rays which are superimposed upon the continuous X-ray spectra of the material to get the complete spectrum. |
Question 3. In $Fig. 4$ give the explanation for the positions of ${K_\alpha }$ and ${K_\beta }$.
Solution: $$\begin{equation} \begin{aligned} \Rightarrow \Delta E = \frac{{hc}}{\lambda }\; \\ \Rightarrow \;\lambda = \frac{{hc}}{{\Delta E}} \\\end{aligned} \end{equation} $$
Since energy difference of ${K_\alpha }$ is less than ${K_\beta }$.
$$\Delta {E_{{K_\beta }}} > \Delta {E_{{K_\alpha }}}$$ Therefore, $${\lambda _{{K_\beta }}} < {\lambda _{{K_\alpha }}}$$
So, peak of ${K_\beta }$ comes before ${K_\alpha }$.
