Intensive electron beams are those, in which the beam electrons have group behavior due to the perceptible interaction forces between them. The behavior of the electrons, moving in such an electron beam with high density of the particles in it, is defined to a considerable extent by the electrostatic interaction forces between them. The negative space charge influences are demonstrate mainly as a) emission of the current by a virtual cathode (current limited by the space charge) , and b) extension of the cross-section of the intensive electron beam.
With a big increase of the density of the particles in one unit volume of the beam, the energy distribution of the beam is changed due to two body interaction between neighboring electrons.
The particle's own electric field is not the only thing that affects the characteristics of the beam. Under certain circumstances (space charge compensation or relativistic electron velocities) and electrons' own magnetic field affects them. In presence of ionized particles from the residual gases or the vapors of the processed material in the technological vacuum chamber, wave movement of the electrons, plasma oscillations and beam instability are possible.
a) Current density, voltage and distance (cathode-anode) relation and limitations of the beam current by the beam space charge
The distribution of the electricity potential U in an intensive (dense) beam defines the velocity and the direction of movement of each electron, but at the same time depends on the space distribution of charges in the beam region. On account of this, instead of the Laplace equation, which is valid for beams with low density of electrons, here the distribution of the potentials is described by Poisson equation:
0 environment and is the density of the space charge. The vector of the current density j
is connected with and the velocity of the electrons
V
by:
V
j
, (21)
which in the case of electrons is:
V
j . (22)
Two other relations are also valid - the continuity equation and the conservation of energy law (the collisions between the particles of the beam and of the residual gases are neglected):
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Here e and m are the charge and the mass of the electrons, correspondingly. In such way for the distribution of the potential in intensive electron beams is obtained:
2 enough big quantity of electrons, the current is limited by their space charge. Equation (25) is easily integrated under the assumption for linear and laminar trajectories of mono-energetic beam of electrons, i.e. neglecting their initial velocities in flat parallel, coacsial cylindrical or spherical structure. For flat cathode and anode, after integration of eq. (25), the density of the current of the cathode, limited by the beam space charge is:
2
where U is the potential on a distance z from the emitting surface of the cathode. In this way, at distance z=d between the electrodes and anode voltage Ua, the equation (26), known as Child-Langmuir equation or 3/2 power law, becomes:
2
Figure 5. Correction coefficient for a cylindrical coaxial diode as a function of ra and rc
In the cases of cylindrical and spherical two-electrode systems, as well as multi-electrode systems, the coefficient 2,33.10-1 changes. For example, for cylindrical construction with length 1 m, from coaxial anode, including the cathode, the coefficient is 2,33.10-62 (when defining the density of the current on the anode). Here is Langmuir correction coefficient, which is a function from the ratio between the anode radius ra and the cathode radius rc
(Figure 5). From the figure it is seen that with the decrease of the ratio ra/rc the density of the current increases. At constant ratio between these radiuses with the decrease of ra the intensity of the field in front of the cathode increases, which leads to considerable increase of the current, obtained from the cathode by such construction.
b) Perveance
The characteristic conductivity p, called perveance, is defined as:
2 calculations of electron beams shows that the space charge influences the electron trajectories in good vacuum conditions at values of perveance p>10-7AV-3/2, and that value of the perveance can be accepted as the limit between the intensive electron beams and the beams with low density of electrons. In the nowadays technology installation for welding the beam perveance values lay between p=10-8AV-3/2 and p=2.10-5AV-3/2 (as example, a typical perveance value of EBW gun could be 5. 10-7AV-3/2). Note, that there a correction of perveance value due to higher pressure in the draft space and the action of the effect of compensation of negative space of beam electrons by generating positive ions become appreciable.
The maximum value of the perveance, and consequently of the beam current, which can be obtained after the beam formation, is also limited by the space charge of the beam electrons. Due to the negative charge of beam electrons the potential in the space, occupied by the beam, decreases. For example, in unlimitedly wide electron beam going along the axis between two perpendicular to this axis equi-potential planes, situated on a distance l from each other, the potential distribution U(z) has minimum in the middle between these planes.
From integrating eq. (26) follows that with the increase of the current density the value of the potential in the minimum decreases, reaching Ua/3 for jl2
2 / 3
Ua =18,6.10-6 [AV-3/2]. Further increasing of the current density leads to a jump of the potential in the middle point from the initial value to value, equal to 0, i.e. a virtual cathode is formed. This abrupt decrease of the potential is physically connected with slowing down of the electrons and considerable increase of the space charge. That is why with the decrease of the current density the potential in the minimum stays equal to zero until current densities corresponding to jl2
2 / 3
Ua =9,3.10 -6 [AV-3/2] are reached, then the potential in the middle between the equi-potential planes jumps to 0,75U and the normal current flow is restored.
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In the case of limited cylindrical electron beam, fully filling metal tube with potential Ua, the maximum value of the perveance is 32,4.10-6 [AV-3/2]. In this case, the potential along the axis of the tube decreases to Ua/3. Near the axis of such a beam the electrons are moving slowly, the space charge increases, and the potential abruptly decreases. That is why the current density in the border part increases, the potential decreases, and the current flow is variable. The distribution of electron according their velocities in real beams leads to smoother transition of the beam to this unstable state. Characteristics of the different types of configuration of electron optical systems affect these two values of the beam perveance (the first - described unstable and gradually decreasing current flow and the second, where the normal flow is gradually restored).
c) Extension of the beam wide, due to the space charge of the electrons
Another (second) very important effect of the space charge is the action of the electrostatic repulsion forces between the beam electrons. They lead to difficulties in the focusing and to a widening of the beam cross section. The equation describing the movement of the electron in radial direction is:
2 r 2
dt eE r
m d
. (29)Here Er is the radial intensity of the electric field created by the volumetric charge. Let us assume that outer accelerating, focusing and deflecting electric and magnetic fields are missing. Applying Ostrogradski-Gauss theorem for the field intensity vector flow through a cylinder with radius r, situated co-axially with the beam, and eq. (22), for the radial force is obtained:
Here ro is the radius to the border trajectories.
Differentiating by z in eq. (29), using
dz
eq. (30), the boundary electron trajectory equation becomes:
2 o
Again the importance of the perveance as a characteristic of the space charge in the beam is clear. Here k=6,6.10-4 [AV-3/2]. If the extending of the beam is limited by = r - ro, which
are small compared to ro, then ro in the right part of eq. (31) can be accepted as constant and after integration the following equation is obtained:
2
More precise integration of eq. (31) is proposed by Glazer.
It gives the universal relationship between the dimensionless radius ro/amin and the
This relationship is shown on Figure 6. Here amin is defined by:
is the initial angle of shrincage of the border electron trajectory,
z0
a
- the initial radius of the beam. When there is initially expanding beam, o is negative.
Figure 6. Universal relationships between the dimensionless radius ro/amin, the angle of the slope
ro/z and the dimensionless distance along the axis Z, characterizing the border trajectories in axially symmetrical electron beam
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