RADIATION BY MOVING CHARGES

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

3.2.12 RADIATION BY MOVING CHARGES

Before concluding this chapter it is necessary to point out the fact, that the accelerated electrons in the X-ray tube emit electromagnetic radiation as they progress from the cathode to the anode. Of particular interest is the total radiation emitted by the accelerated particles as well as the angular distribution of their radiation and the frequency spectrum [15]. For nonrelativistic motion the radiation is described by the Larmor result [15],

where P

=

the radiative power of the accelerated charge (Watts);

e

=

the charge of an electron (1.6x 10-¹⁹ C);

•

v

=

the acceleration of the charge (m s -2 ); c

=

the speed oflight (m s-\

The expression in equation 3.2.66 can also be expressed in terms of unit solid angle [15]:

(3.2.66)

(3.2.67)

The term sin²

e

in equation 3.2.67 exhibits the characteristic angular dependence of the radiated power. In the case of the accelerated charges undergoing relativistic motion it is the relativistic momentum prel of the particle that must be taken into consideration [15]. According to Jackson [15] the power of the radiation emitted by an accelerated charge moving along a linear path at relativistic speeds can be described by the following relation:

(3.2.68)

where dp is the momenttml of the accelerated particle. By recalling the relativistic foml of momentum for a charged particle from Table 3.2.14, which is

where me

=

the mass of an electron (9.1 x 1 0-³¹ kg);

h· h . . I J2eV(t)

v

=

the nonrelativistic velocity of the electron w IC IS eqUiva ent to . me The relativistic momentum can also be expressed as

teV(t)

me me me

Pre!

= ---r====== = --;====== = = =====

1- 2eV(t) 1-2e( -2.83 ^X 10⁷ t

+

8.5 ^X 10⁴⁾

2 2

meC meC

2e( -2.83 ^X 10⁷ t

+

8.5 ^X 10⁴⁾ me

(3.2.69)

where V(t)=-2.83xl0⁷ t + 8.5xl04 Volts represents the accelerating potential across the X-ray tube for every voltage pulse that is discharged across it (see equation 3.2.19, Section 3.2.6). Equation 3.2.69 is plotted against time in the graph in Figure 3.2.20 from tl = 0 S to t2 = 0.003 s, which is the duration of a pulse. The plot can also be described by equation 3.2.70, which is a second order polynomial approximation obtained by simple curve fitting:

2.00E-022

1.80E-022

kg m s-1 1.60E-022

-

'";'

en 1.40E-022

E

;. C') _1.20E-022

£

...

0::

e:

1.00E-022 8.00E-023

6.00E-023

4.00E-023

00000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Time (sec)

Fig.3.2.20: Plot of relativistic momentum prel versus time t.

PreJt)

=

1.8698 x 10-^{22 -} 2.6761 x 10-^{20 ( -}9.9472 x 10-^{18 (2 (3.2.70)}

To find the average relativistic momentum that a particle undergoes during one pulse across the tube from time t1 = 0 ^S to t2 = 0.003 s PreltJ in equation 3.2.70 must be integrated from t1 to t2 with respect to time and divided by the total duration of one pulse, which is 0.003 seconds, i.e.

12

f

^{(1.87 x 10-22 - 2.68 x 10-20 t - 9.95 x 10-18 t2 )dt}

( )

1 1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ , _

Prei t AVE

= - =

t2 - t1

If this value is substituted into equation 3.2.68 for the term dp over the total pulse duration dt

=

0.003 s, the average radiated power for an accelerated electron in the X-ray tube is found to be equal to 2.096x10-42 Watts. What is of interest, however, is the total average power radiated by all the accelerated charges per voltage pulse. In Section 3.1.2 (see Figure 3.1.3) it was shown, that the total average current per pulse is 6.67 rnA. Since current can also be expressed as the amount of charge dQ flowing through an area in a time dt [31], the current through the area, denoted by i, is

i = -=::.. dO

dt (3.2.72)

The total charge passing through the tube during each pulse can thus be fOlUld to be equal to the product of the current and the duration of one pulse. Therefore the charge passing through the tube per pulse can be approximated as being equal to dQ = i. dt = (0.00667 A)·(0.003 s)

=

2x10-⁵ C.

Having determined the net charge dQ flowing through an area, the number of charges or electrons moving from the cathode to the anode of the tube per pulse applied can be found by dividing dQ by e

=

1.6xlO-¹⁹ C, which is the electric charge of an electron. The number of electrons n is thus found to be approximately 1.25xl0¹⁴ It can be seen, that the total average power radiated by all the accelerated charges per voltage pulse clischarged across the tube is:

PTOTAL. AVE = n x ^PAVE =

2.618x10-^ZS W,

(3.2.71)

which is negligible compared to the overall average power of 14l.83 Watts dissipated in the X-ray tube (see Section 3.1.2). Furthermore the radiation of the accelerated electr~ns .cha~ges. as. they accelerate towards the anode. As can be seen in Figure 3.2.21 the angular distnbutIon IS tIpped forward more and more as it increases in magnitude [19]. The angle Bmax for which the intensity is a maximum is

I[

^I

(.J 2 2 )]

^Jl-v2/c2

B

=

cos - - - 1 + 15v / c - 1 ---+ ,

max 3v / c 2 ^(3.2.73)

where the last form is the limiting value for vlu ---+ 1. Even for vlu

=

0.5, corresponding to electrons of approximately 80 keY kinetic energy, Bmax

=

38.2°.

/3 ::

O.

Fig.3.2.21: Radiation pattern for a charge accelerated in its direction of motion. The two patterns are not to scale, the relativistic one having been reduced by a factor of approximately 100 for the same acceleration [15J.

For relativistic particles ^()max is very small being of the order of the ratio of the rest energy of the particle to its total energy [15].At the point at which the electrons leave the cathode of the tube and move at relatively low velocities towards the target the peak intensity of radiation will be at approximately right angles to the direction of motion of the particle. Only once Bmax approaches 0°

as the particle accelerates does the radiation of the moving charge affect the target even though the effects of that radiation on the production of X-rays at the target <lre negligible as shown earlier.

However, this does suggest that X-radiation, no matter how low in intensity, will in fact be produced by the interaction of the radiation of the accelerated electrons with the target even before the electrons themselves arrive at the anode of the X-ray tube.