DEMO_XY / gallery 18: Asymmetrical electrostatic quadrupole. Defined in radial coordinates (hexapole without 1st and 4h element). Mesh size: 401 x 401 nodes.

Running program

Windows:
DEMO_XY.exe 18

Linux:
./DEMO_XY.run 18

Problem definition

The problem has been outlined in books devoted to electron optics (multiple symmetry lenses), e.g. Szymański, Mulak, Duda, Romanowski, "Electron Optics", 2nd edition, Wydawnictwa Naukowo Techniczne, Warsaw 1984. Szymański's book mentioned the asymmetrical distribution of electrodes.

Numeric meshes

Each pixel represents one mesh point.

colour white - vacuum
colour red   - casing     1 (U = +0.0 [V])
colour green - electrodes 2 (U = +1.0 [V])
colour blue  - electrodes 3 (U = -1.0 [V])

Visualization of mesh on picture.

Model parameters

mesh size:

number of rows     ==  401
number of columns  ==  401

x ==  -200.0 ... +200.0 [mm]
y ==  -200.0 ... +200.0 [mm]

number of iterations

>>> SUCCESS - solution V(x,y) has been found
after 25513 iterations

pi constant:
M_PI ==  +3.14...


electrode type 1 (casing, red)
boundary ==  +1
voltage  ==  +0.0   [V]
x_center ==  +0.0   [mm]
y_center ==  +0.0   [mm]
radius   >=  +190.0 [mm]


electrodes type 2 (2 elements, green)
boundary   ==  +2
voltage    ==  +1.0  [V]
radius     <=  +20.0 [mm]
N          ==  +6 /* hexapole without some elements */
alpha_zero ==  +0.0  [rad]
R          ==  +50.0 [mm]
coordinates of centres of electrodes type 2:
1/2
	n = +2	
	x_3 = R * cos(alpha_zero + 2 * n * M_PI / N);
	y_3 = R * sin(alpha_zero + 2 * n * M_PI / N);
2/2
	n = +4
	x_5 = R * cos(alpha_zero + 2 * n * M_PI / N);
	y_5 = R * sin(alpha_zero + 2 * n * M_PI / N);

	
electrodes type 3 (2 elements, blue)
boundary   ==  +3
voltage    ==  -1.0  [V]
radius     <=  +20.0 [mm]
N          ==  +6 /* hexapole without some elements */
alpha_zero ==  +0.0  [rad]
R          ==  +50.0 [mm] 
coordinates of centres of electrodes type 3:
1/2
	n = +1	
	x_2 = R * cos(alpha_zero + 2 * n * M_PI / N);
	y_2 = R * sin(alpha_zero + 2 * n * M_PI / N);
2/2	
	n = +5	
	x_6 = R * cos(alpha_zero + 2 * n * M_PI / N);
	y_6 = R * sin(alpha_zero + 2 * n * M_PI / N);


computation accuracy:
eps ==  1.0E-9 [V]

Computation results

Equipotential lines

Mapping of electrostatic potential [V] to colours

Colormap: grayscale

Colormap: grayscale inverted

Colormap: hot-to-cold

Colormap: hot-to-cold inverted

Colormap: jet

Colormap: jet inverted

Mapping of component x of electric force (E_x) [V/mm] to colours

Important: determinig of exact values of E near electrode surface is generally problematic. These maps have generally illustrative character.

Colormap: grayscale

Colormap: grayscale inverted

Colormap: hot-to-cold

Colormap: hot-to-cold inverted

Colormap: jet

Colormap: jet inverted

Mapping of component y of electric force (E_y) [V/mm] to colours

Important: determinig of exact values of E near electrode surface is generally problematic. These maps have generally illustrative character.

Colormap: grayscale

Colormap: grayscale inverted

Colormap: hot-to-cold

Colormap: hot-to-cold inverted

Colormap: jet

Colormap: jet inverted

Mapping of electric force (E) [V/mm] to colours

algorihm of computation:
E = sqrt(E_x*E_x + E_y*E_y)

Important: determinig of exact values of E near electrode surface is generally problematic. These maps have generally illustrative character.

Colormap: grayscale

Colormap: grayscale inverted

Colormap: hot-to-cold

Colormap: hot-to-cold inverted

Colormap: jet

Colormap: jet inverted

Mapping of square of electric force (E*E == E2) [V2/mm2] to colours

algorithm of computation:
E*E = E_x*E_x + E_y*E_y

Important: determinig of exact values of E near electrode surface is generally problematic. These maps have generally illustrative character.

Colormap: grayscale

Colormap: grayscale inverted

Colormap: hot-to-cold

Colormap: hot-to-cold inverted

Colormap: jet

Colormap: jet inverted

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