The numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges
In the paper the procedure of solving integral equations with the use of auxiliary charges is used. The electrostatic field potential is presented in the form of superposition of the fields of point sources located out of the investigated region at some distance from it. The values of N charges are...
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| Дата: | 2001 |
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| Формат: | Стаття |
| Мова: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Назва видання: | Вопросы атомной науки и техники |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges / S.S. Tishkin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 121-123. — Бібліогр.: 4 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-792512015-03-31T03:02:02Z The numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges Tishkin, S.S. In the paper the procedure of solving integral equations with the use of auxiliary charges is used. The electrostatic field potential is presented in the form of superposition of the fields of point sources located out of the investigated region at some distance from it. The values of N charges are determined from the boundary conditions at N points of the boundary. The definition of the point sources is equivalent to using some function of the charge density at the boundary of the region. This approach was used for the calculation of structures with the space-uniform quadruple focusing with sinusoidal and trapezoidal modulation, and also a section of the transverse matching of the “funnel”. 2001 Article The numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges / S.S. Tishkin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 121-123. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS numbers: 29.17.+w http://dspace.nbuv.gov.ua/handle/123456789/79251 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
In the paper the procedure of solving integral equations with the use of auxiliary charges is used. The electrostatic field potential is presented in the form of superposition of the fields of point sources located out of the investigated region at some distance from it. The values of N charges are determined from the boundary conditions at N points of the boundary. The definition of the point sources is equivalent to using some function of the charge density at the boundary of the region. This approach was used for the calculation of structures with the space-uniform quadruple focusing with sinusoidal and trapezoidal modulation, and also a section of the transverse matching of the “funnel”. |
| format |
Article |
| author |
Tishkin, S.S. |
| spellingShingle |
Tishkin, S.S. The numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges Вопросы атомной науки и техники |
| author_facet |
Tishkin, S.S. |
| author_sort |
Tishkin, S.S. |
| title |
The numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges |
| title_short |
The numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges |
| title_full |
The numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges |
| title_fullStr |
The numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges |
| title_full_unstemmed |
The numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges |
| title_sort |
numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79251 |
| citation_txt |
The numerical field calculation in structures with the space-uniform quadruple focusing by the method of auxiliary charges / S.S. Tishkin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 121-123. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT tishkinss thenumericalfieldcalculationinstructureswiththespaceuniformquadruplefocusingbythemethodofauxiliarycharges AT tishkinss numericalfieldcalculationinstructureswiththespaceuniformquadruplefocusingbythemethodofauxiliarycharges |
| first_indexed |
2025-07-06T03:17:45Z |
| last_indexed |
2025-07-06T03:17:45Z |
| _version_ |
1836865935758065664 |
| fulltext |
THE NUMERICAL FIELD CALCULATION IN STRUCTURES
WITH THE SPACE-UNIFORM QUADRUPLE FOCUSING BY THE
METHOD OF AUXILIARY CHARGES
S.S. Tishkin
NSC "Kharkov Institute of Physics and Technology"
1 Akademicheskaya Str., Kharkov, 61108, Ukraine
In the paper the procedure of solving integral equations with the use of auxiliary charges is used. The electrostatic
field potential is presented in the form of superposition of the fields of point sources located out of the investigated
region at some distance from it. The values of N charges are determined from the boundary conditions at N points of
the boundary. The definition of the point sources is equivalent to using some function of the charge density at the
boundary of the region.
This approach was used for the calculation of structures with the space-uniform quadruple focusing with sinusoidal
and trapezoidal modulation, and also a section of the transverse matching of the “funnel”.
PACS numbers: 29.17.+w
Presently as the initial section of the ion accelerators
the structures with the space-uniform quadruple focus-
ing are widely used [1].
The main parameters necessary for calculation of the
accelerating and focusing canal of such type are: effi-
ciency of acceleration, T, efficiency of focusing, κ,
maximum electric field strength at the electrode surface,
Es [2,3].
In this paper for calculation of these values a version
of integral equation technique is used - the method of
auxiliary charges (MAC) [4].
According to this method the electrostatic field po-
tential can be represented as a superposition of the point
charge fields located outside the examined area and at a
certain distance from it. Values of N charges are found
from the boundary conditions at N points. The defini-
tion of the point charges is equivalent to application of
the certain charge density at the electrode surface.
The computational procedure is used in the
RFQFLD code. The code gives a possibility to calculate
fields in the RFQ structures with semicircular electrodes
modulated in the sinusoidal and trapezoidal law and also
the fields in the section of the transverse matching
("funnel").
The boundary conditions for the electrostatic field
potential ϕo(x,y,z) was chosen with taking into account
the symmetry
ϕo(x,y,z) =ϕo(x,-y,z)= ϕo(-x,y,z)
In the accelerating structure of the RFQ type there
are 3 types of the boundary conditions:
1. For the section of the transverse matching the
boundary conditions are:
ϕî(z = 0) = 0,
ϕî(z = Lp) = 0,
where z = 0 is the longitudinal coordinate of the cavity
wall, Lp is the length of the section of the transverse
matching.
2. For the bunching section when it is necessary to
take into account the accelerating structure irregu-
larity. When the boundary conditions are setting it
is necessary to take into account the influence of
the adjoining cells then
ϕî(z = 0) = 0, ϕî(z = Lp) = 0.
Here z = 0 is the cross-section at the point of the
maximum electrode deviation from the axis. Ls = Li-1 +
+ Li + Li+1; where Li i is the length of the accelerating
cell in which the field potential is calculated, Li-1 and
Li+1 are the lengths of the neighbor cells.
For the accelerating section where the structure ir-
regularity can be neglected
ϕî(z = 0) = 0, ϕî(z = Li) = 0.
Since the cavity walls are far from the area of inter-
est their influence is not taken into account.
Using the specified boundary conditions the field
calculation in the RFQ structure could be restricted to
calculation in one quadrant with taking into account the
influence of the symmetry of the auxiliary charges.
The static potential calculations are carried out for
the cell with sinusoidal modulation having the following
parameters:
kRo = 1; Re/Ro = 0.7; m =1.8
where k = π/L, L - the accelerating cell length;
Re - the radius of the electrode rounding;
Ro - the average aperture radius (the distance between
the electrode and the axis at the point of the exact sym-
metry of electrodes;
m - the electrode modulation depth.
In Fig. 1 the field potential generated by the calcu-
lated auxiliary charges is shown in the cross-section at
the point of accurate RFQ electrode symmetry.
The equipotentials -1 and +1 correspond to the elec-
trode surfaces. The white circles show the arrangement
of the auxiliary charges, black ones show the points at
the surface where the boundary conditions were defined.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (38), с. 121-123.
121
Fig. 1. The field potential generated with the auxil-
iary charges in the cross-section with the accurate
RFQ electrode symmetry.
One of the important features of the MAC is the op-
timization of the auxiliary charge arrangement with re-
spect to the boundary. It is necessary that the boundary
conditions are held as accurately as possible at the all
surface points.
Fig. 2. The field potential generated with the auxil-
iary charges in the transverse cross-section of the
structure.
The preliminary calculations have shown that the
distance from the auxiliary charges to the surface must
be equal or somewhat larger than the distance between
points at the electrode surface where the boundary con-
ditions are set.
With this arrangement the charge discreteness is in-
significant at the boundary surface, and the matrix for
these charges determination remains quite specified.
In Fig. 2 the field potential generated by the auxil-
iary charges at the transverse cross-section is shown.
The +1 equipotential corresponds to the electrode con-
tour.
In Fig. 3 the longitudinal field on the axis of the
structure is presented. The field values in the middle cell
are used (the range [3.14, 6.28]).
Fig. 3. The longitudinal component distribution at
the axis of the structure with the constant modula-
tion depth of electrodes.
In Fig. 4 the factor of the field increment versus the
angle coordinate α in the cross-section with the accurate
RFQ electrode symmetry is shown.
Fig. 4. Distribution of the amplification factor in the
cross-section with RFQ symmetry.
Here Uo is the potential difference between the elec-
trodes.
The following ought to be said about this value. At
the initial part of the structure (kRo ≤ 1), χ has the maxi-
mum value in the cross-section with the maximum devi-
ation of the electrodes from the axis and depends
strongly on the modulation depth m. That leads to the
situation that extremely large m at the initial part of the
accelerating structure may cause significant loss of the
electric strength. With kRo > 1 χ has the maximum value
at the point of the accurate RFQ symmetry and depends
mainly on the radial fields.
The calculation errors were evaluated in the follow-
ing way [4]. If ϕ is a harmonic function approximating
the solution ϕ with the accuracy ε at the boundary Γ,
i.e., the condition |ϕ − ϕ| < ε is fulfilled, then the error
of the approximate solution will not exceed ε in any
point in the range under consideration. In Fig. 5 the rela-
tive error
ϕ
ϕϕδ −=
~
versus the angle coordinate α is
shown. The auxiliary charge arrangement was chosen so
that the presented dependence reflects the maximum er-
ror in the boundary conditions. As one can see from
Fig. 5, the maximum error of 1.75.10-1 occurs for
122
α ≅ 85o, however, this part of the electrode practically
does not influence the field distribution in the region of
the calculations. Therefore we can believe that the error
for the field potential does not exceed 1.10-3.
Fig. 5. The relative error of the boundary condi-
tion approximation in the cross-section with
RFQ symmetry.
Fig.6. Distribution of the longitudinal component
for the structure with alternating depth of the elec-
trode modulation.
In Fig. 6 the longitudinal field distribution on the
axis of the structure calculated with the method of the
auxiliary charges for the bunching section with taking
into account the irregularity of the accelerating structure
is shown.
In conclusion we will note that the MAC method is
extremely simple from the viewpoint of programming
and calculations.
It took 15.2 s to calculate T, κ, χs, δ with the Pen-
tium 150 MHz. The number of the auxiliary charges in
the region of calculations (without symmetric reflec-
tions) was 2 x 11 x 20 = 440.
REFERENCES
1. I.M.Kapchinskij, V.A.Tepliakov. Linear ion acceler-
ator with space uniform RFQ focusing // Pribory i
Tekhnika Ehksperimenta. 1970, # 2, p.19 (in Rus-
sian).
2. I.M.Kapchinskij. Theory of the linear resonance ac-
celerators. Moscow: Energoizdat, 1982 (in Russian).
3. A.I.Balabin. Numerical field calculation in the
space-uniform RFQ focusing for an electrode of the
specific form. Prepr. ITEF-107. Moscow, 1981.
4. V.P.Iljin. Numeraical methods for problems in elec-
trophysics. Moscow: Nauka, 1985 (in Russian).
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