7 X-rays, their sources & interaction with matter

Prof. P. N. Kotru

7.1  X-Rays.

7.1.1 Sources of X-Rays.

The underlying principle of generating X-rays is to bombard a solid target by fast moving electrons. According to classical electrodynamics, a moving charged particle emits electromagnetic radiation when it is accelerated. When an electron is suddenly stopped it results into a pulse of radiation which is in the form of X -rays. X-rays may be produced in a cathode-ray tube which is filled with gas at a low pressure. A metal anticathode is placed opposite the cathode; the former being used as a target which receives high speed electrons emitted by the former and thus serves as the source of X-rays. X-ray sources of increasing efficiency have been devised over a period of time. X-rays are produced by sources like gas tube, the Coolidge tube, the betatron and others.

Different means of producing X-rays may be summarily described as follows:

7.1.2  The Gas Tube .

The gas tube is essentially a cathode ray discharge tube as shown in a schematic diagram of figure 7.1

Figure 7.1 : The gas tube

C represents concave shaped cathode made of aluminium metal. A is anode which is connected to the target T. The anode is placed inside the tube as shown in the figure. The target T is so placed as to have its flat surface inclined at 45⁰ to the axis of the cathode ray stream and positioned at exactly the focus of the concave cathode C. The target T is made of metal having high melting point like tungsten, molybdenum and platinum. The gas pressure inside the tube is maintained at about 0.001 mm. Induction coil is used to obtain the high tension as required to the tune of 30,000 ─ 50,000 volts. Arrangements are made to cool the target by a stream of water. The electrons constituting the cathode ray stream are produced as a result of ionisation of the gas inside the tube. The production of X-rays by this technique suffers from certain limitations which are as follows:

(i) It is difficult to maintain the constant intensity and quality of the emitted X-rays for a prolonged period.

(ii)The walls of the tube gradually absorb the residual gas and as such the intensity of the cathode ray stream

gets reduced as a result of which it becomes harder to operate the tube at the required tension.

7.1.2  The Coolidge Tube .

Most of the drawbacks of the above described gas tube were overcome by Dr. Coolidge who introduced a new type of tube , a modified one , which is now known by his name as the Coolidge tube. It is shown in a schematic diagram of figure 7.2

Figure 7.2 : The Coolidge Tube

Filament of tungsten acts as a cathode which is heated by a battery placed outside the tube at B. The filament emits thermionic electrons which constitute the cathode ray stream. These electrons get accelerated towards the target T on maintaining high potential difference between the filament and the target T. A very high vacuum of more than 0.0001 mm is maintained inside the tube. The highly accelerated electrons strike the target T and produce hard X-rays. A copper rod which is fixed to the target T and gets projected outside the tube is water cooled. The high tension required is ordinarily supplied by a step-up transformer.

The Coolidge tube has certain advantages over the above described gas tube. Those advantages may be summarized as follows:

(i)   The intensity of the X-rays produced is easily controlled by accordingly maintaining the filament

temperature.

(ii) Quality of the X-rays produced can be conveniently controlled by designing the tubes which can be

operated at high voltages ranging from a few hundred to one million volts and, (iii) Operation of Coolidge

tube is more stable in comparison to the gas filled tube.

There is, however, one disadvantage of the Coolidge tube and that is the focussing of the electrons is not sharp because almost the whole surface of the target has the tendency of becoming a source of X-rays. This disadvantage was overcome to some extent by making use of rotating anode tube. If different substances are required to be used as targets, detachable anode tubes are employed and put in to operation. The modernized hot cathode Coolidge tube has been very useful to researchers on account of the fact that there is control over intensity and quality of the X-rays thus produced.

7.1.3 The Betatron

Beams of energetic electrons are required for the production of X-rays. Betatron is an electron accelerator in which magnetic induction is used to accelerate the electrons. X-rays are produced by allowing highly energetic electrons strike the target by using a high potential difference between the filament and the target. There is another type of X-ray tube known as betatron in which electrons acquire high energy by using principle of “Resonance Acceleration”. The principle is also applied in cyclotrons.

Figure 7.3 shows an arrangement for accelerating electrons in a Betatron. M-M represent the pole pieces of an electromagnet connected to the a.c. source of high voltage. The a.c. source is adjustable to the desired frequency. V-V is a vacuum tube in the shape of a ring which is placed between the pole pieces of the electromagnet. S-S represents the solenoids wound around specially designed pole pieces of the electromagnet. The electron injector consists of heated filament F which acts as a source of electrons, Grid G which is used for focussing electrons and plate T which is used as injector shield. High potential difference is maintained between T and F so as to send electrons at very high speed into the vacuum tube V-V. The magnetic field between the pole pieces of the electromagnet acts parallel to the axis of the vacuum tube. The electrons injected into the vacuum tube experience an electromotive force tangential to the electron orbit caused by the changing magnetic flux which provides an additional energy to the electrons.

Figure 7.3: Accelerating chamber in a Betatron

The electrons also experience a radial force because of the action of the magnetic field which acts at right angles to the motion of the electrons. These two factors maintain the motion of electrons in a circular path. With the help of solenoids S-S the magnetic field at the centre is so adjusted as to allow the electrons to move in a stable orbit of constant radius. The electrons are made to go through several thousands of revolutions in their fixed orbit, while the alternating magnetic field increases in intensity from zero to a maximum value which occurs during onequarter of a cycle. Each of these revolutions provides additional energy to the electrons. The intensity of the alternating magnetic field gets maximum at the end of the quarter cycle. There are two coils of wire one of which is placed above the stable orbit and the other below it. These coils are connected to a condenser which is discharged at the end of the quarter cycle in a manner that the electrons get switched out of their circular orbit so that they strike the back of the injector shield T resulting into production of energetic x-ray beam . The shield thus acts as the X-ray target.

Relationship between the linking flux and the magnetic field is derived as follows: As said above the electrons are made to move in a circular orbit of constant radius in the vacuum chamber which gains energy by induction caused by the change with time of the magnetic flux Φ linking the orbit.

The induced voltage per turn = dΦ ⁄dt,

The electric field (i.e., voltage per unit length) = E = (1/2πR) .dΦ /dt……………..7.1 Where, R represents

radius of the stable orbit.

The electron experiences force as given by e.E.

The equation of motion for the electron is given by:

d/dt (m v) = e E =( e/2πR ). dΦ/dt            ………………………..    7.2

In order that the motion of electron in a circular orbit of radius R is maintained, the magnetic field H at the orbit has to be increased as the energy of electron increases

H e v = mv2/R

Or, mv = e R H …………….………………….7.3

d/dt (mv) = e R dH/dt

In order to obtain acceleration of electrons by magnetic induction at constant radius R, equations 7.2 and 7.3 should be equal.

So, dΦ/dt = 2πR2.dH/dt = 2.d/dt (πR2H)                               …………………………7.4

The equation (7.4) states that in any time interval the linking flux Φ is required to change at a rate twice that which would occur if the central magnetic field were uniform and equal to the field at the orbit. This is the condition for a betatron to function. In order that this condition is met the betatron is required to have a

central iron core with high flux density inside the orbit. The cycle of acceleration in a betatron is shown in figure 7.4 .Electron gun is used to inject the electrons into the chamber at an instant when the magnetic field just rises from zero in the first quarter of cycle as illustrated in the figure 7.4. During this period the magnetic field is increasing which induces a potential within the vacuum chamber resulting into increase in the electron energy. Being in a magnetic field, the electrons move in a circular orbit which is maintained stable by the increasing magnetic field. The field strength keeps on increasing till it reaches the maximum value. After attainment of maximum value, the strength of the field decreases and the direction of the induced electromagnetic force changes because of which the electrons get slowed down. At this stage the electrons are pulled out of their stable orbit by discharging the condenser because of which they either emerge out from the apparatus or strike the target T resulting into generation of X-rays.

Figure 7.4: The acceleration cycle in a betatron

The betatron , however, suffers from a disadvantage . It needs a very large magnet so as to be able to provide variable flux which is necessary for accelerating the electrons. This disadvantage is overcome by employing electron synchrotrons. One such electron synchrotron facility at Massachusetts has a 50 ton magnet and a radio frequency of 46.5 megacycles. It produces electron energies of the order of 330 M eV.

7.1.4 Detection of X─ Rays.

X-rays have gained a lot of importance on account of their characteristics which include the following:

(i)  X-rays ionise gases through which they are passed

(ii) X-rays penetrate through matter

(iii)   X-rays affect photographic plates

(iv) X-rays produce excitation and produce fluorescence in some substances

These characteristics of X-rays become helpful in their detection.

7.1.5 Ionisation of Gases.

On passing x-rays through a gas, secondary corpuscular radiations are emitted by the gas. The radiations so emitted are composed of fast moving electrons. These high speed electrons on striking with the atoms or molecules of the gas produce positive and negative ions thus making the gas conducting which leads to an ionisation current. The ionisation current so produced depends on the rate at which ions are formed. Greater the intensity of X-rays, higher is the rate of ion formation. Therefore, measurement of ionisation current can be used to quantitatively measure the intensity of X-rays. Ionisation chambers are used for the measurement of ionisation current.

The ionisation chamber I-C is in the form of a hollow metallic cylinder which is closed at both the ends except for a small window W, as is shown in figure 7.5. A sheet of aluminium is placed over the window W. YZ is the central electrode in the form of a rod inside the chamber.

Figure 7.5: Schematic diagram of an ionisation chamber.

The central electrode located on the axis of the chamber is supported by a suitable material that keeps it insulated from it. Quadrant electrometer or a very sensitive gold leaf electroscope (Q-E) is connected to the suitably insulated rod Y-Z for measurement of ionisation current. B is a battery whose one terminal is connected to the chamber (hollow cylinder) and the other is earthed. In order that there is no leakage from the walls of the chamber to the central electrode, guard ring G is used. The guard ring is connected to the earth as shown in the schematic diagram of figure 7.5

Suitable gas, usually at atmospheric pressure, is filled in the chamber. Suitable gases used for this purpose could be any of these that include air, ether vapour, hydrogen, air, carbon dioxide, sulphur dioxide or methyle bromide. For soft X-rays sulphur dioxide gas is considered to be suitable while for hard X-rays methyl bromide is found to be the suitable one.

X-rays are made to enter the gas chamber through the window as shown in the above figure. The X-rays ionise the gas and it becomes conducting. Since there is an electric field established between the chamber and the central electrode, the latter gets electrically charged at a particular rate which is measured by the electrometer Q-E. The rate of charging depends on the intensity of the X-ray beam. This principle is applied in the measurement of X-ray intensity. The rate of charging can be determined by measuring the deflection of the electrometer needle for a given time. The ionisation current is usually of a very small intensity of the order of 10─13 amperes. Very delicate instruments are used to measure it. Sensitive electrometers are available which can measure the ionisation current of this intensity.

7.2 Interaction between X-rays and Matter.

X-rays are characterised by their strong penetrating power. There are two types of X-rays; soft X-rays and hard X-rays. Soft X-rays have small penetrating power and so can be easily absorbed by matter whereas the hard X-rays have higher penetrating power.

Let Io represent the observed intensity of an X-ray beam incident normally on the absorbing material and I the intensity after the beam has travelled through a thickness , say x , in the absorber. The intensity of the beam on travelling through this thickness will decrease. Let dI be the decrease in the intensity of the beam while travelling through a small thickness dx of the absorber, then -dI/dx is the rate of decrease of the intensity.

It may be assumed that dI/dx is proportional to I, then:

-dI/dx = µ I , where µ is a constant , called the absorption co-efficient.

or     dI/I     = -µ dx,

On integration it gives:

Loge I = – µ x + C

When x = 0 , I = I0 which gives constant of integration C = loge I0

or      LogeI = – µ x + logeI0,                              or,

I = I0 e─µx

It suggests that the absorption phenomenon of x-rays appears to obey exponential law, i.e., as x increases I diminishes exponentially.

The absorption coefficient µ of any given material is determined by finding that thickness x which reduces the intensity to half of the initial value.

i.e., I/I0 = e─µx  = ½

Or, µ = loge 2/x

If x is measured, its value can be used to calculate the absorption coefficient µ.

If µ is small, the rays are hard and have great penetrating power. If µ is large, the rays are soft and as such are readily absorbed.

The absorption coefficient µ depends on the absorbing material and on the radiation but is independent of the initial intensity of the X-rays. The above equation is true only for homogeneous X-rays.

Radiations coming out of ordinary X-ray tube are heterogeneous because with increase in the thickness of the absorber the value of µ decreases and so does not remain constant. As such, the above said equation does not hold for heterogeneous X-rays. Heterogeneous X-rays are those which consist of both hard and soft X-rays.

If a beam of primary X-rays is made to strike a plate, some part of it goes through the plate and the remaining is consumed into generation of heat or any other form of radiation. The question arises as to what becomes of the radiation which is removed from the incident beam. The absorbed portion gives rise to a complex phenomenon, known as secondary radiations. The secondary radiation coming out from the plate is comprised of characteristic X-rays, scattered β-rays and characteristic β-rays. Figure 7.6 shows various types of radiation coming out of the plate when primary X-ray beam strikes the plate.

Figure 7.6 :Transmitted X-rays and secondary radiation out of primary X-ray beam when it strikes the plate.

The characteristic X-rays are so named because these depend upon the material of which the plate is made. Besides being homogeneous, these rays exhibit the same properties for a given element, irrespective of the hardness of primary beam. The elements, in general, exhibit two kinds of characteristic radiation which differ in the value of their absorption co-efficients. These are called K ─ and L ─ radiations. The K ─ radiation is relatively harder as compared to L ─ radiation; the aborption coefficient of the former being much smaller than that of the latter. The hardness or softness of the X-radiation is determined by the value of absorption coefficient; smaller the value of µ, harder is the radiation. This holds true for both K─ as well as L ─ radiations. The characteristic radiation of the elements show increase in hardness with increase in their atomic weight.

7.2 Diffraction of X ─ rays

Diffraction of X-rays is known to be historically the first technique which made study of internal structure of crystals possible. X-rays are a radiation of smaller wavelengths as compared to visible light, a wavelength of nearly 1Å being X-ray wavelength. The spacing between the atoms within a crystal is almost this order of magnitude. The radius of an atom and the separation between two neighbouring atoms on an average is also of this order of magnitude. Real crystal is a three dimensional array of atoms and on exposure to X-rays, one finds diffracted X-rays coming out in all directions from every atom. It is because of this complication that one gets involved in threedimensional geometry in an attempt to work out interference conditions. The only few particular and specific directions from which to view the crystal so that the diffracted X-rays are in phase and add up to give a strong beam so as to get recorded in the form of spot on a photographic film , plate or with the help of ionisation chamber or counters . These specific directions will depend on (a) the arrangement of atoms in the lattice and (b) the wavelength of X-rays. In a crystal there are atoms arranged in rows along the directions of its three crystallographic axes. It is this characteristic of Xrays that made it a very useful technique in the study of the internal structure of crystals.

In 1912, Von Laue in collaboration with Friedrich and Knipping discovered that crystal act as gratings suitable for the diffraction of X-rays. They allowed a beam of X-rays from an X-ray tube to pass through a thin crystal of ZnS (Zinc blende). The transmitted beam was allowed to fall on a photographic plate P. It was observed that the photographic plate got affected and in addition to a central black spot at C where the direct beam struck the plate, there were several other spots in a regular pattern, suggesting that the X-ray beam incident on the crystal had got diffracted by the crystal planes. These spots were given the name “Laue spots”. The experimental set-up is shown in a schematic diagram of figure 7.7

Figure 7.7 : Laue’s experimental set-up for diffraction of X-rays

The Laue‘s experiment proved that:

(i)    The X-rays are electromagnetic radiation of definite wavelength,

(ii)  The atoms of a crystal are arranged in a regular pattern

7.3       Bragg’s Law

Laue’s work on X─ rays showed that crystals act as gratings for diffraction of X-rays .The diffraction does not take place because the wavelength of ordinary X-rays is somewhere between 10 ─8 and 10─9 cm, while the average interatomic distance in a crystalline solid ranges somewhere between 10 ─7 and 10─8 cm. Also, the atoms or molecules are arranged in a regular repeating order because of which some crystal symmetry is observed. We are familiar with diffraction of visible light by optical grating. Here too, diffraction of X -rays occurs when they strike a crystal

The crystal acts as series of parallel reflecting planes .We , therefore , consider a series of parallel rows , in which the atoms are regularly arranged and each row representing planes of the crystal are spaced at a distance d apart. That means the interplanar spacing is‘d’. Planes with interplanar spacing d are shown in a schematic diagram of figure 7.8

Figure 7.8 : Bragg picture of X-ray diffraction as in –phase reflections from crystal planes of interplanar spacing ‘d’

Let a train of monochromatic X-rays of wavelength λ strike the crystal such as to be incident on the plane at a glancing angle Ө. When X-ray beam falls, the atoms become centres of disturbance and spherical wavelets whose envelope results into a reflected wave front. Since X-rays have a large penetrating power it is necessary to consider reflection from several planes. So, we have reflected wave fronts from each of the parallel planes. If these reflected wave fronts are all in the same phase, they will reinforce each other and the reflection gets exceptionally strong. The reflections are said to be in phase. However, if the incident angle Ө does not satisfy the condition required for different reflections to be in phase, the latter interfere with one another and the

resulting beam is weak. The required condition for the reflected wave fronts to be all in the same phase is that the path difference between the reflected wave from one plane and that from the next should be either an exact wavelength or an integral multiple of it. This is what we know from optics. We shall consider here two parallel rays AB and DE which get reflected by two atoms B and E in the adjacent planes along BC and EF; E being just vertically below B. The ray DEF has a longer path to travel ray ABC. The reinforcement of reflected rays with each other will and produces a bright spot only if the path difference

GE + EH = nλ ,

GB and HB are perpendiculars drawn from B on DE and EF respectively.

But, GE = EH = BE sin Ө

= dsin Ө

So,   nλ  = 2d sin Ө

This is popularly known as Bragg’s law. n is an integer.

Suppose difference in path length is equal to one wavelength, and then there will be a reflected

beam at the position Ө 1 which satisfies the Bragg condition:       λ = 2 d Sin Ө 1

This beam gives the reflection or spectrum of the first order. However, if the difference in path length is 2 , 3 , 4 …….., r wavelengths, then reflections of the second , third , …… rth orders will be observed at the positions

Ө 2, Ө 3  , Ө 4…………… Ө r.

7.4       The Von Laue  Equations

By allowing high speed electrons to strike a metal target anode, X-rays can be produced. The wavelength of X-rays is of the order of 1.24 Å or 1.24 x 10─8 cm which is just of the order of inter atomic distances in actual crystals. According to Bragg equation nλ= 2d sin Ө, diffraction effects are observable for appropriate values of d and Ө.

Bragg assumed that a system of crystal planes could act and reflect x-rays, provided the conditions for constructive interference between reflections from successive atomic planes are satisfied. Let us consider two individual atoms, say A and B and take into consideration X-rays scattered from them. Here, A and B are the two identical scattering centres which are separated by distance r as shown in figure 7.9 (a)

Figure 7.9 (a) : Scattering of X-rays from scattering centres A & B

n0 represents the unit vector in the direction of the incident beam (assumed to be a parallel beam ) and n1 represents the unit vector in the direction of the scattered beam . Path difference between the radiation scattered at B and that scattered at A is:

Figure 7.9 (b) : Relationship of the incident and diffracted beams , scatttering normal and the scaettring plane

From figure 7.9 b it becomes clear that the magnitude of the vector N is :

N  = 2 Sin Ө…………………………………………………………………….7.2

Let the phase difference between the radiation scattered at two centres A and B be denoted by φ. The phase difference being 2π/λ times the path difference, one may put the phase difference between the radiation scattered at two points as:

φ = 2π/λ ( r . N )……………………………

7.3

Now, the condition for direction n to be a diffraction maximum is that the contribution to scattering from every atom in the crystal in that direction must differ in phase by an integral multiple of 2π radians. To satisfy this condition it is necessary for the radiation from atoms separated by primitive lattice vectors a , b and c to add in phase because in that case the contribution from other atoms , separated from the origin by integral combinations of these vectors will add in phase. If contributions towards scattering from the neighbouring atoms happen to differ in phase in some other form, there would then always be possible to have an atom

somewhere in the crystal which would contribute radiation π radians out of phase with the contribution from a given atom ultimately leading to cancellation of these contributions, atom by atom, giving no diffracted beam.

Thus, we will require that an integral multiple of 2π be got in equation7.3 when r equals a, b or c. In other words, it is required to simultaneously have:

2π/λ ( a . N ) = 2πnh

2π/λ ( b . N ) = 2πnk………………………………………………7.4

2π/λ ( c . N ) = 2πnl

Let α, β and γ be the angles between the scattering normal N and a , b and c axes of the crystal  respectively, then as per equation 7.2, a .N = a N cos α = 2a sin Ө cosα and so on.

Therefore, equation 7.4… is written as:

2a sin Ө cosα = nhλ

2b sin Ө cosβ= nkλ

2c sin Ө cos γ = nlλ        ………………………………………7.5

The set of equations 7.5 are called Laue equations

Equations given as number 7.5 are a set of three equations which would determine the particular value for Ө and N . That unique value of Ө and N will provide the scattering direction. Equations7.5 also suggest that the direction cosines of the scattering normal N are proportional to h/a , k/b and l/c. Considering neighbouring planes with Miller indices (hkl) and intersecting the a ,band c axes at intervals a/h , b/k and c/l , the direction cosines of the normal to the (hkl) family of planes are also proportional to h/a , k/b and c/l. The scattering normal N is , therefore, identical to the normal to the (hkl) planes which means that the (hkl) planes could be regarded as the reflecting planes of the Bragg picture.

If we set h = acosα /d, k =bcosβ /d and l = c cos γ /d , each of the Laue equation of 7.5 can be shown to lead to Bragg ‘s equation : nλ= 2dsin Ө, where d is the distance between adjacent planes of the (hkl) system.

References

1.Rajam, J.B.: “ Atomic Physics”, S.Chand&Co.,Delhi, 1960.

2. Buerger,M.J.: “ X-ray Crystallography”,1949

3. Lonsdale, K.: ” Crystals and X-rays”,Bell,London,1948.

4. Kays,G.W.C.: “ X- Rays”,1 926.

5. Compton, A.H &Allison,S.K. : ”X-rays in Theory and experiment”,1935.

6. McKelvey,J.P.: “ Solid State & Semiconductor Physics”,Harper& Row,N.Y.,1966.

7. Kaplan,I “Nuclear Physics”,Addison-wesley publishing Co.,N.Y.1969.

Suggested Reading.

1. Bragg, W.L.: “ The Crystalline State –A General Survey “,Bell, London
2. Zachariasen,W.H. “ The X-ray Diffraction In Crystals.” Wiley , N.Y.
3. Buerger,M.J. “ “ Elementary Crystallography “,Wiley, N.Y.

• * Large number of interesting books on crystals and x-rays are available for having detailed information on these topics. Most of them have been quoted in previous modules.