CONCLUSIONES

The process of electron beam welding is influenced by the beam energy space distribution, being a characteristic of the beam quality. Various methods for estimation of the electron beam quality were proposed. Measuring of the current distribution of powerful mono-energetic electron beams in a transverse cross section (called also the beam profile) was proposed and applied recently [11-15]. It is clear, that for prognostication of deep penetrating welding results one need from evaluation of the ―parallelism‖ or ―laminarity‖ of the beam (namely the angular distribution of beam particles) in the same time of evaluation the current radial distributions in the studied transverse cross sections along the beam axis. It were mention that, for description of collective behavior of the beam particles one need of a knowledge of the value of the particle density in the six-dimensional phase space (x,Vx, y, Vy, z, Vz), because t is excluded in the case of continuous electron beam. There x,y,z are coordinate axes and Vx, Vy and Vz are the respective velocity components. There z is the beam axis direction. It is important to note, that the phase volume of the beam in the 6D phase space(x,y,z, Vx, Vy, Vz) termed 6D hiper emittance, as well as the related particle densities and/or these values in a 4D trace space (x,y, dx/dz, dy/dz ), involving transverse coordinates and angles, are constant along the beam axis and in time, under ideal condition of a beam, particles of which are non-interacting with short – range forces. In cases of not coupled transverse dimensions is more practical to determine the projections of beam parameters in two 2D sub-planes: (x, x'=dx/dz ) and respective ( y, y' = dy/dz ) plane. Together with the mentioned conditions - lack of collisions, which is required for conservation of volume of a non-relativistic beam phase (trace) space, is an additional requirement for excluding the frictional forces that depend on particle velocity. The thermal spread of the emitted electrons is a reason for non-zero value of the geometry emittance. Coulomb interaction lead to a

―space-charge‖ effect causing increase of the beam phase volume and emittance; the non-linear elements of beam forming system lead to distortions and wrapping of the phase volume and a quasi-expanding of the beam effective emittance.

As was mentioned, the six-dimensional description for a beam in the drift space is usually split into two-dimensional (x,x‘) and (y,y‘) subspaces and a geometry emittance is defined there as the areas, occupied by all or a chosen part of the beam particles(current) in these two-dimensional spaces, dividing to π (Figure 3). For x0x' plane:

ε^x =

 Ax

, (53)

where A^x is the area, occupied by the beam (respectively a beam part); the index x means, that parameter A and emittance are measures in the (x,x‘) sub-space. As example ε^x and ^y signed

the emittances in the (x,x‘) and (y,y‘) subspaces. Conservation of ε^x and ^y take place in the case that beam transport releases at not coupled sub-spaces, that is usual at electron beam welding optical systems. In case of characterization of part of the beam current p =

I0

I , where I is an investigated part of the total beam current I0, than a bottom index p is added to the ε^x and ε^y and





In the case of accelerating of the electrons or at describing a relativistic beam the velocity V of beam particles is changed. At increase of longitudinal component of V, the divergence of beam gets smaller. Then the geometry emittance decreases too. A scaling velocity could be c, the speed of light in vacuum, that give a independent of beam energy emittance. So is introduced normalized emittance, which is invariant in the case of acceleration regions of the electrons of studied powerful beam. At assuming the relativistic Lorenz factor equal to 1(or multiplying with him calculated value) it can be written:

εp,n

where x, x are the standard deviations of the particle coordinates and angles x and x, and r is correlation between these random quantities. At r=0 (no correlation) the probability density N could be presented by the product of two normal distributions and the boundary of the projection of phase space on xOx takes place of an ellipse in a canonical position (namely its main axes coincide with x and x axes). In the case of r=1 the ellipse becomes a straight line x=(x/_x)x.

The use of 2D normal distribution (55) leads to elliptical shapes of the boundaries of the particle distribution diagram, given in the xOx plane that coinciding to the elliptical trajectories of particles in the phase plane.

The equation of emittance ellipses could be written as:

x²+2xx+x²=p (56)

There p is the emittance for part p of the beam current, containing in respective ellipse;

α,β and γ are so called Twiss (or Courant-Snyder) parameters that obey:

β.γ-α²=1, (57)

and are given on Figure 13. Note, that (57) is just the geometrical properties of an ellipse.

Design of High Brightness Welding Electron Guns and Characterization… 29

Figure 13. Determination of emittance ellipse by Twiss parameters

Coefficient (or Twiss parameter) β characterize changes of the beam envelope. Its definition could be written in terms of second order moments of distribution function:

x x

x

 



2

. (58)

There the brackets means an average value, performed over the beam particles distribution.

Respectively  is a measure of the average declination of electron trajectories from the beam axis:

x x

x

 



'2

, (59)

and the Twiss coefficient α is determined as:

x x

x x

 

 . '

. (60)

In the case of a more complicated beam distribution the area, occupied by particle points in x,x‘ or y,y‘ planes, could have a not easily defined shape (Figure 14). The effective root-mean-square (r.m.s.) emittance , the definition of which is based on the concept of

―equivalent perfect beam‖, is applicable in that case.

Figure 14. Effective root-mean-square (r.m.s.) emittance  and the concept of ―equivalent perfect

This is taken as a definition of the effective r.m.s. emittance in general (at assumption to contain about 0.9 of the beam current).

The correlation coefficient r in eq.(55) could be defined as:

2

and the Gaussian (normal) distribution (55) can be rewritten as:

N(x,x)=

The emittance of a beam is not measured directly parameter. It can be inferred by beam current profile in the transverse cross-section (radial intensity profile) and by angular distributions of beam particles in that transverse position, evaluated or measured (see below).

A beam profile monitor placed in the beam path convert the beam flux density in a measurable signal that is a function of positions towards the beam axis. A schematic presentation of radial profile monitor is shown on Figure 15.