To calculate beam divergence and beam emittance in that case, the equation (42) for an axial symmetrical nonrelitiavistiq beam could be applied. Let we discuss a beam propagating in vacuum (no compensation of space charge occurs) in drift region. Then the differential equation (42) could be rewritten
3 2
32 0 2
2
2 1
2
8 4 U R R
R I U B dz
R
d
(88)
Design of High Brightness Welding Electron Guns and Characterization… 57 where I is the beam current; 0 is the dielectric permittivity; U is the acceleration voltage; B is the axial magnetic field; is the electron charge-to-mass-ratio; is the beam emittance.
Numerical solution of eq. (88) give the beam envelope R dependence (i.e. an evaluation of
"some averaged" beam radius due to the distributed on R beam current) on the magnetic lens field intensity (or on the distance lens-focal plane or most usable focusing coil current) at constant U and z. Here, the concept of the rms (root mean square) emittance could be used, if the corresponding values of the beam divergence and the beam radius r are defined as second moments.
The measurements are described on Figure 31 and Figure 32 at use of an static measuring plate of refractory material with two perpendicular narrow slits. Beam envelop diameter are calculated statistically for all scans, presented as bitmap lines on Figure 32 during the variation of beam focus position from f0-f to f0+f . This measurement is able to determine and correct beam astigmatism and beam misalignment additionally. After scan the two orthogonal slits and measuring line integrals the beam jumps back to the starting point of x scan on the first slit very fast. A small increment of focusing coil changes the beam focus position. Than x and y scans are fulfilled again. As a result a number of line integrals of beam intensity are collected as this is shown on Figure 32. There bright shows a high power density, dark present a low intensity. The generated bitmap consists of two beam profiles, which represents the beam dimension in x and y directions as function of the focal lengths.
The recording of such a bitmap with a resolution of 400X400 Pixel could be realized for about 100 ms. During bigger part of the measuring time the beam is defocused. Only for the short time when the focal length is coincident with the central focal length f0 (of order of 1-2 ms) high power intensity is deposited on the measuring sensor. Thus his destruction can be avoided.
The analysis starts with the determination of the beam diameter for every single focal length. According to ISO 11146 one has to find the centroid of the intensity distribution first.
Knowing X,Y of the distribution centre, the beam diameter can be calculated the second
Note, that it is important to subtract the background noise very carefully, because in (89) the term (x-X)2 overemphasizes small signals located far away from the centre of the intensity distribution. With the calculated position of the centre X and Y and the related beam and exact focal position. Beams with power up to 2 kW can be measured continuously. Then it is possible to correct astigmatism and misalignment of the beam automatically by using the obtained data to control the corrector coils of the EB-gun. The beam can be focussed exactly on the surface of the work piece by determining the position of the minimum beam diameter.
To calculate the beam divergence and the beam emittance a more advanced analysis of the data is necessary. The propagation of a charge carrying particle beam is described by the following equation
x2 ; I is the beam current; 0 is dielectric constant; U is beam acceleration voltage; is charge-to-mass-ratio for the electron; : rms beam emittance; B: axial magnetic field.
Here, the concept of the rms (root mean square) emittance is introduced (See Figure 14, and eq. (61) The divergence and beam radius r values are defined as second moments (see e.g. equation (77)).
Beam parameters in a not very powerful EB welding machine is typically in a range, that the influence of the space-charge on beam propagation could be neglected (UA > 50 kV; Ib <
200 mA). Therefore the dominating effect on the beam envelope beside external electric and magnetic fields is the emittance.
In that case and for field-free space, equation (88) and (90) are reduced to
3
Figure 43. Measured (dots) and calculated (solid line) beam radius on the sensor at different current through the magnetic lens. Determined emittance is 3.0 mm.mrad, evaluated divergence of the beam is 10.1 mrad
Design of High Brightness Welding Electron Guns and Characterization… 59
It is possible to solve this differential equation for a converging beam (focussing with a magnetic lens). If the radius R is determined at a fixed position z0, while beam divergence 0, starting radius R0 and emittance are parameters, the solution of (91) has the following form:
1 1 1 )
( ) (
2 2
0 0 0 2
0
T T
T z R z
R
(92)with T1 = 2+2/R0
2; and , , R0 defined as rms values.
With the right choice of , , R0 the graph of (92) can be fitted to the measured beam diameter (see Figure 43). Thus divergence and emittance of the studied beam is given. The big amount of the measured beam diameters (several hundreds) leads to a very reliable result.
Figure 44. Continued
Figure 44. Integral current densities at beam focus (at distance 320 mm from the focal winding) at different angles:
a) = 0; b) = 51; c) = 102; d) = 153; e) = 204; f) = 255; g) = 306.1 – Experimental; 2 – Approximated
Figure 45. Continued
Design of High Brightness Welding Electron Guns and Characterization… 61
Figure 45. Reconstructed radial current density distributions [mA] depending on x and y [mm]
coordinates in five cross-sections of the beam at different distances from the focal winding: a) z = 170 mm; b) z = 207.5 mm;c) z = 245 mm; d) z = 282.5 mm; e) z = 320 mm (focus)