FOCUSING OF THE X-RAY TUBE ELECTRON BEAM

3. The Design of the X-ray Tube and the MgO Cold Cathode

3.1.4. FOCUSING OF THE X-RAY TUBE ELECTRON BEAM

In or?e~ to produce X-radiation of ~aximum intensity by considering all limitations of the X-ray tube It IS necessary for the force actmg on the accelerated electrons, that collide with the target to

produce X-ray photons, to be as large as possible. The Newtonian and the relativistic forms of the relations expressing the force F in terms of the electron mass

me

and its acceleration a are shown in equations 3.1.1Sa and 3.1.1Sb respectively.

F=m _e ·a ^(3.1.1Sa)

(3.1.1Sa)

From the above two expressions it can be seen, that the force F is directly proportional to the acceleration a . The kinetic energy E K of an electron accelerated by a potential V IS

1 2

E =eV=-m·v _{K 2}

e (3.1.16)

where v is the velocity of the charged particle. The velocity v is directly proportional to the accelerating potential V of the tube. If one takes V , the potential of the anode with respect to the cathode, as having a constant value, then the velocity of the electron at the instant of its impact on the target will be constant as well. The acceleration of a particle is the rate at which its velocity changes:

a = -dv

dt (3.1.17)

If dv is to be constant in value the time d t for an electron to travel from the cathode to the anode of the tube will have to be as short as possible in order to produce greater acceleration of the electron and hence a larger force F acting on it.

In this way X-radiation of optimum intensity can be produced at the target of the X-ray tube. For an electron to travel from the cathode to the anode within the shortest possible time it will have to traverse the shortest possible distance d as well. The force F acting on an electron can also be expressed as the product of its charge and the electric field present:

F=eE=e·-V

d (3.1.18)

Once again it can be seen that by decreasing the distance through which the electron travels the force will increase provided that the term e V in equation 3.1.18 is kept constant. In the proposed X-ray tube design an electron will traverse the shortest distance, if it moves in a direction normal to the surfaces of the cathode and the target (see Figure 3.1.8).

Cathode

Target/Anode

Fig. 3.1.8: The direction of an electron travelling' from the cathode to the anode of the X- ray tube.

By taking into account the situation illustrated in Figure 3.1.8 above it is evident, that parallel or rectilinear electron flow will produce X-radiation of optimal intensity. When dealing with electron beams the following two basic problems have to be dealt with [37]:

1. The electrostatic repulsion forces between electrons tend

to

cause the beam to diverge.

2. The current density required in the electron beam is normally far greater than the emission density of the cathode.

In dealing with the repulsion force problem one should examine the problem of attempting to cause the electrons to travel in parallel paths as they leave the cathode as indicated in Figure 3.1.9.

y

x

Electron beam

Beam ex lerHJ$ lo

/ f.X) i z direction

---~

Fig. 3.1.9: Parallel electron flow from a planar cathode [37].

Electrode

~t Cathode Potential "

Electron

TraJectorle$

Equipotential Profiles

Anode

Fig. 3.1.10: Equipotential profiles without (dashed lines) and with (solid lines) electrons [37].

Electrode al Cathode Potential

Cathode

Electron Trajectories

/ ./

I ... I~'

.- £qurpr.)tf:!nlial

PrQtll~s

Anode

Fig. 3.1.11: Straightening of equipotential profiles in an electron beam (37J.

Firstly the equipotential profiles in a parallel-plane diode containing no electrons are parallel and spaced equal distances apart for equal voltage increments. This is shown by the dashed lines in Figure 3.1.10. When electrons are present, however, the equipotential profiles are deflected to the right as indicated by the solid lines. Electrons tend to move perpendicular to the equipotential profiles, so the electron trajectories diverge as the electrons move away from the cathode. This is simply another way of saying that the electrons are pushing on each other and causing their trajectories to diverge [37]. By introducing an electrode arrangement that could straighten the equipotential profiles even though electron space charge forces were present, the electrons could travel along parallel paths. Intuitively it can be seen that if the electrodes adjacent to the cathode are tipped towards the anode, as indicated in Figure 3.l.11, then the equipotential profiles are straightened.

In the absence of electrons the profiles curve toward the cathode. The presence of electron charges causes them to move toward the anode. The mathematical solution that leads to a definition of the electrode configuration for parallel electron flow is similar to the solution for the problem of space-charge-lirnited flow discussed in Section 3.1.3. The main difference now is that the equations must be solved inside and outside the electron stream, and then the solutions must match at the edge of the beam. The resulting electrode and equipotential profiles for parallel electron flow are given in Figure 3.l.12 [38].

fffl~~iJ~~~ Edge of beam

2 3 4 !) 6 1 8 9 10

z Distance from cathode

To '

Cathode radius

Fig. 3.1.12: Electrodes for obtaining axially symmetrical electron flow of uniform diameter [38J.

Figure 3.l.12 shows electrode shapes which were derived by Pier~e [36][38] f?r parallel flow from a space-charge-limited cathode in a cylindrical beam accordlllg to equatlOns 3.1.19 and 3.l.20 [36][38]:

4

V

=

^j(x)

=

^Ax)

A=

⁹ⁱ ₁

( 2e) 2 4&0 me

where V

=

the accelerating potential in Volts;

2 3

x

=

the distance travelled by an electron in metres;

i

=

the tube current in Amperes;

&0

=

the permittivity of vacuum (8.854xl0·¹² F/m);

e

=

the charge of an electron (l.602xl0·¹⁹ C);

me

=

the mass ofan electron (9.11xlO·³¹ kg);

(3.1.19)

(3.1.20)

The shape of the equipotentials in Figure 3.l.12 is independent of the absolute magnitude of the potentials involved and of the units in which distance is measured. Therefore, in Figure 3.1.12 potential is indicated in terms of an arbitrary potential, V

=

^CPo' assigned to one equipotential potential surface. The potential of the cathode is taken as zero and the distance is measured in arbitrary units, with the origin located at the edge ofthe cathode. Pierce [38] showed that the zero potential surface is a plane which meets the edge of the cathode (x = y = 0) making an angle of 67.5° with the normal to the cathode (the x axis). The equipotentials that are remote from the

4

cathode are those which give V

=

Ax³ along the axis [38]. In Section 3.1.5 and Chapter 3.2 a cathode radius ro of 7 mm is chosen. In Figure 3.1.12 above the term

!...

denotes the ratio

ro

between the electrode-cathode distance and the cathode or beam radius. Obviously the electrode, which is placed at V

=

CPo' is in fact the anode of the tube. Hence with z

=

8 and ro

=

7 mm we

ro obtain

z=56mm,

where z, which is actually equivalent to x, is the distance between the cathode and the anode of the X-ray tube.

We have now not only determined the shape of the focusing electrode, which will be positioned at an angle of 67.5° to the edge of the electron beam at the cathode, but also what the spacing between the cathode and the anode should be in order to achieve rectilinear electron flow. By choosing a ceramic sleeving for the X-ray tube of 25 mm inside diameter we will choose a lens that has an outer diameter of 22 mm, which means that the distance from the axis of the beam to

,

the edge of the lens, (see Figure 3.1.12) is 11 mm. Therefore the ratio - will be

'0

~ =

11 mm

~

1.57

'0 7mm

From the graph in Figure 3.1.12 we can see, that at that value the anode will have virtually no curvature.

In Figure 3.1.13 the relative positions of the focusing electrode, the anode and the cathode with respect to each other are illustrated [56]. To complete the synthesis of the X-ray tube it is now necessary to proceed with the design of the cathode, the anode and the window.

Electron Beam Ceramic Sleeve

/

Fig. 3.1.13: Positions of the focusing lens, the cathode and the anode relative to each other [56J.