Transport characteristics of space charge dominated multi- species deuterium beam in electrostatic low energy beam line

Transport characteristics of space charge dominated multi- species deuterium beam in electrostatic low energy beam line 1 2 corresp first-author Corresponding author: E-mail address: luxl@lzu.edu.cn; Corresponding author: E-mail address: luxl@lzu.edu.cn; luxl@lzu.edu.cn 1 2 1 2 1 3 1 3 1 2 corresp E-mail address: zeyao@lzu.edu.cn. E-mail address: zeyao@lzu.edu.cn. zeyao@lzu.edu.cn School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China Engineering Research Center for Neutron Application, Ministry of Education, Lanzhou University, Lanzhou 730000, China Engineering Research Center for Neutron Application, Ministry of Education, Lanzhou University, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China The transport characteristics of a space charge dominated multi-species deuterium beam consisting of <math id="M1"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , <math id="M2"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> , and <math id="M3"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> particles in an electrostatic low energy beam line are studied. First, the envelope equations of the primary <math id="M4"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> beam are derived considering the space charge effects caused by all particles. Second, the evolution of the envelope of the multi-species deuterium beam is simulated using the PIC code TRACK, with the results showing a significant effect of the unwanted beam on the transport of the primary beam. Finally, different injected beam parameters are used to study beam matching, and a new beam extraction system for the existing duoplasmatron source is designed to obtain the ideal injected beam parameters that allow a <math id="M5"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> beam of up to 50 mA to pass unobstructed through the electrostatic low energy beam transport line in the presence of an unwanted ( <math id="M6"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> , <math id="M7"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> ) beam of 20 mA; at the same time, distortions of the beam emittance and particle distributions are observed. Envelope equation Space charge effects Multi-species beam Electrostatic LEBT I. INTRODUCTION 1 High intensity accelerators such as the High Intensity Heavy Ion Accelerator Facility (HIAF), Accelerator Driven Sub-critical System (ADS), Intense Neutron Generator, etc., are required for scientific research and industrial applications. High intensity low energy beam transport (LEBT) lines are crucial for the development of high intensity accelerators; however, the associated strong space charge effects present challenges to the transport of high intensity beams in the LEBTs [ 1 1 , 2 2 ] . The LEBT design is normally based on electrostatic or magnetostatic focusing. The advantage of a magnetostatic LEBT is that the beam can be fully neutralized by the residual gas present in the line, while in an electrostatic LEBT the beam is fully un-neutralized, resulting in strong beam divergence. Therefore, magnetostatic LEBTs are usually adopted. However, electrostatic LEBTs can be very compact because focusing of a high intensity low energy beam using an electric field is more effective than using a magnetic field [ 3 3 ] . Hence, electrostatic LEBTs are very attractive for some high intensity accelerators used in education laboratories and industrial applications, such as high intensity ion implanters, intense neutron generators based on accelerators, and others [ 4 4 ] . In our laboratory at Lanzhou University, an intense DT/DD neutron generator based on a Cockcroft–Walton accelerator is under construction, and the electrostatic LEBT design is adopted. As shown in figure 1 figure 1 , the designed electrostatic LEBT can be viewed as a system of two electrostatic lenses (A1 and A2) combined with a drift tube [ 5 5 , 6 6 ] . The multi-species beam contains <math id="M8"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , <math id="M9"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> , and <math id="M10"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> particles directly injected into the electrostatic LEBT; therefore, the transport of the primary particles <math id="M11"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> is affected by the space charge effects not only from the <math id="M12"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> particles themselves but also from the unwanted <math id="M13"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> and <math id="M14"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> particles [ 7 7 ] . In this paper, the envelope equations of the primary beam in an electrostatic LEBT are deduced considering the space charge effects of both the primary and the unwanted beams. In addition, PIC simulations are performed to study the influences of the unwanted beam on the transport of the primary beam. Finally, the influence of the parameters of the beam injected from the duoplasmatron source on the transport of the multi-species beam are studied by the PIC simulations, and a new beam extraction system for the duoplasmatron source is designed. II. THEORETICAL ANALYSIS 1 A. Beam space charge field 2 We consider an axisymmetric high intensity multi-species continuous beam propagating through an axisymmetric electrostatic LEBT. We will use a cylindrical coordinate system <math id="M15"><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></math> . We consider a uniform charge density profile, which can be expressed as [ 8 8 ] <math display="block" id="M16"><mrow><msub><mi>ρ</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>λ</mi><mi>i</mi></msub></mrow><mrow><mi>π</mi><msubsup><mi>r</mi><mrow><mtext>b</mtext><mi>i</mi></mrow><mtext>2</mtext></msubsup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mfrac><mo>,</mo></mrow></math> where <math id="M17"><mrow><msub><mi>λ</mi><mi>i</mi></msub></mrow></math> is the linear charge density, <math id="M18"><mrow><msub><mi>r</mi><mrow><mtext>b</mtext><mi>i</mi></mrow></msub></mrow></math> is the beam radius, and <math id="M19"><mrow><mi>s</mi><mo>=</mo><mi>z</mi></mrow></math> is the axial coordinate along the beam direction. According to Gauss’s theorem, <math id="M20"><mrow><mo>∇</mo><mo>·</mo><mi>E</mi><mo>=</mo><mfrac><mi>ρ</mi><mrow><msub><mi>ε</mi><mn>0</mn></msub></mrow></mfrac></mrow></math> , the x -component of the space charge electric field can be expressed as [ 9 9 ] <math display="block" id="M21"><mrow><msub><mi>E</mi><mrow><mi>x</mi><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><msubsup><mrow/><mrow><mfrac><mrow><msub><mi>I</mi><mi>i</mi></msub></mrow><mrow><mn>2</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mi>i</mi></msub><mi>c</mi></mrow></mfrac><mfrac><mi>x</mi><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac><mo>,</mo><mi>x</mi><mo>></mo><msub><mi>r</mi><mrow><mtext>b</mtext><mi>i</mi></mrow></msub></mrow><mrow><mfrac><mrow><msub><mi>I</mi><mi>i</mi></msub></mrow><mrow><mn>2</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mi>i</mi></msub><mi>c</mi><msubsup><mi>r</mi><mrow><mtext>b</mtext><mi>i</mi></mrow><mn>2</mn></msubsup></mrow></mfrac><mi>x</mi><mo>,</mo><mi>x</mi><mo>≤</mo><msub><mi>r</mi><mrow><mtext>b</mtext><mi>i</mi></mrow></msub></mrow></msubsup></mrow></mrow><mo>,</mo></mrow></math> where <math id="M22"><mrow><msub><mi>I</mi><mi>i</mi></msub></mrow></math> is the beam current of the species <math id="M23"><mi>i</mi></math> , <math id="M24"><mrow><msub><mi>β</mi><mi>i</mi></msub><mi>c</mi><mo>≈</mo><msub><mi>v</mi><mrow><mi>z</mi><mi>i</mi></mrow></msub></mrow></math> is the particle axial velocity in the paraxial approximation, <math id="M25"><mrow><mi>r</mi><mo>=</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></msqrt></mrow></math> is the distance from the beam axis in the radial direction, and <math id="M26"><mi>x</mi></math> and <math id="M27"><mi>y</mi></math> are the transverse components in the horizontal and vertical directions, respectively. The constants <math id="M28"><mrow><msub><mi>ε</mi><mn>0</mn></msub></mrow></math> , <math id="M29"><mrow><msub><mi>β</mi><mi>i</mi></msub></mrow></math> , and <math id="M30"><mi>c</mi></math> are the permittivity, the relativistic parameter, and the speed of light in vacuum, respectively. B. Envelope equations of a multi-species beam 2 Assume that the multi-species beam contains particles of two species, a <math id="M31"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> beam with a radius of <math id="M32"><mrow><msub><mi>r</mi><mn>1</mn></msub></mrow></math> and <math id="M33"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> beam with a radius of <math id="M34"><mrow><msub><mi>r</mi><mn>2</mn></msub></mrow></math> . The non-relativistic transverse equation of motion describing a <math id="M35"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> particle evolving in an applied electric accelerating field and the beam space charge field can be written as [ 10 10 ] <math display="block" id="M36"><mrow><msub><mi>m</mi><mn>1</mn></msub><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mo>=</mo><mi>q</mi><mo stretchy="false">(</mo><msubsup><mi>E</mi><mi>x</mi><mrow><mtext>appl</mtext></mrow></msubsup><mo>+</mo><msubsup><mi>E</mi><mrow><msub><mi>D</mi><mn>1</mn></msub><mi>x</mi></mrow><mrow><mtext>self</mtext></mrow></msubsup><mo>+</mo><msubsup><mi>E</mi><mrow><msub><mi>D</mi><mn>2</mn></msub><mi>x</mi></mrow><mrow><mtext>self</mtext></mrow></msubsup><mo stretchy="false">)</mo></mrow></math> Here, <math id="M37"><mrow><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mo>=</mo><msubsup><mi>v</mi><mrow><msub><mrow/><mrow><mi>z</mi><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup><mi>x</mi><mo>'</mo><mo>'</mo><mo>+</mo><msub><mi>v</mi><mrow><mi>z</mi><mn>1</mn></mrow></msub><msubsup><mi>v</mi><mrow><msub><mrow/><mrow><mi>z</mi><mn>1</mn></mrow></msub></mrow><mo>'</mo></msubsup><mi>x</mi><mo>'</mo></mrow></math> ; primes denote derivatives with respect to the axial coordinate <math id="M38"><mi>z</mi></math> ; <math id="M39"><mrow><msub><mi>v</mi><mrow><mi>z</mi><mn>1</mn></mrow></msub><mo>≈</mo><msub><mi>β</mi><mn>1</mn></msub><mi>c</mi></mrow></math> is the axial velocity of the <math id="M40"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> particle; <math id="M41"><mrow><msub><mi>m</mi><mn>1</mn></msub></mrow></math> and <math id="M42"><mi>q</mi></math> are the mass and charge of the <math id="M43"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> particle, respectively; <math id="M44"><mrow><msubsup><mi>E</mi><mi>x</mi><mrow><mtext>appl</mtext></mrow></msubsup></mrow></math> , <math id="M45"><mrow><msubsup><mi>E</mi><mrow><msub><mi>D</mi><mn>1</mn></msub><mi>x</mi></mrow><mrow><mtext>self</mtext></mrow></msubsup></mrow></math> , and <math id="M46"><mrow><msubsup><mi>E</mi><mrow><msub><mi>D</mi><mn>2</mn></msub><mi>x</mi></mrow><mrow><mtext>self</mtext></mrow></msubsup></mrow></math> are the applied electric field, the space charge field of the <math id="M47"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> beam, and the space charge field of the <math id="M48"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> beam along the x-direction, respectively. In the paraxial approximation, <math id="M49"><mrow><msubsup><mi>E</mi><mi>x</mi><mrow><mtext>appl</mtext></mrow></msubsup><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>φ</mi></mrow><mrow><mo>∂</mo><msup><mi>z</mi><mn>2</mn></msup></mrow></mfrac><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mi>φ</mi><mo>'</mo><mo>'</mo></mrow><mn>2</mn></mfrac><mi>x</mi></mrow></math> , <math id="M50"><mi>ϕ</mi></math> is the normalized potential corresponding to the applied on-axis electric field, <math id="M51"><mrow><mi>q</mi><mi>ϕ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>m</mi><mn>1</mn></msub><msubsup><mi>β</mi><mn>1</mn><mn>2</mn></msubsup><msup><mi>c</mi><mn>2</mn></msup></mrow></math> , while <math id="M52"><mrow><msubsup><mi>E</mi><mrow><msub><mi>D</mi><mn>1</mn></msub><mi>x</mi></mrow><mrow><mtext>self</mtext></mrow></msubsup></mrow></math> and <math id="M53"><mrow><msubsup><mi>E</mi><mrow><msub><mi>D</mi><mn>2</mn></msub><mi>x</mi></mrow><mrow><mtext>self</mtext></mrow></msubsup></mrow></math> can be calculated with Eq.(2). Then, the equations of motion of the <math id="M54"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> particle in the x-direction can be written as [ 11 11 ] <math display="block" id="M55"><mrow><mrow><mo>{</mo><mrow><msubsup><mrow/><mrow><msup><mi>x</mi><mo>″</mo></msup><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>′</mo></msup><mrow><mn>2</mn><mi>ϕ</mi></mrow></mfrac><msup><mi>x</mi><mo>′</mo></msup><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>″</mo></msup><mrow><mn>4</mn><mi>ϕ</mi></mrow></mfrac><mi>x</mi><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>1</mn></msub><mi>c</mi></mrow></mfrac><mfrac><mi>x</mi><mrow><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>2</mn></msub><mi>c</mi></mrow></mfrac><mfrac><mi>x</mi><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn><mtable><mtr><mtd><mrow/></mtd><mtd><mrow><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>></mo><msub><mi>r</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow><mrow><msup><mi>x</mi><mo>″</mo></msup><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>′</mo></msup><mrow><mn>2</mn><mi>ϕ</mi></mrow></mfrac><msup><mi>x</mi><mo>′</mo></msup><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>″</mo></msup><mrow><mn>4</mn><mi>ϕ</mi></mrow></mfrac><mi>x</mi><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>1</mn></msub><mi>c</mi><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup></mrow></mfrac><mi>x</mi><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>2</mn></msub><mi>c</mi><msubsup><mi>r</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfrac><mi>x</mi><mo>=</mo><mn>0</mn><mtable><mtr><mtd><mrow/></mtd><mtd><mrow><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>≤</mo><msub><mi>r</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></msubsup></mrow></mrow><mo> </mo><mo>.</mo></mrow></math> In order to derive the envelope equations, we let <math id="M56"><mrow><msubsup><mi>R</mi><mi>x</mi><mn>2</mn></msubsup><mo>=</mo><msubsup><mi>X</mi><mtext>b</mtext><mtext>2</mtext></msubsup><mo>=</mo><mn>2</mn><mrow><mo>〈</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>〉</mo></mrow></mrow></math> , where <math id="M57"><mrow><msub><mi>X</mi><mtext>b</mtext></msub></mrow></math> is the beam envelope in the x-direction, and <math id="M58"><mrow><mrow><mo>〈</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>〉</mo></mrow><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>r</mi><mrow><mtext>b</mtext><mi>i</mi></mrow></msub></mrow></msubsup><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>π</mi></mrow></msubsup><mrow><msup><mi>x</mi><mn>2</mn></msup><mi>ρ</mi><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow><mo>)</mo></mrow><mi>r</mi><mtext>d</mtext><mi>r</mi><mtext>d</mtext><mi>θ</mi></mrow></mrow></mstyle></mrow></mrow></mstyle></mrow><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>r</mi><mrow><mtext>b</mtext><mi>i</mi></mrow></msub></mrow></msubsup><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>π</mi></mrow></msubsup><mrow><mi>ρ</mi><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow><mo>)</mo></mrow><mi>r</mi><mtext>d</mtext><mi>r</mi><mtext>d</mtext><mi>θ</mi></mrow></mrow></mstyle></mrow></mrow></mstyle></mrow></mfrac></mrow></math> is the transverse average of the particles [ 8 8 - 11 11 8-11 ] . Then, <math display="block" id="M59"><mrow><msub><msup><mi>X</mi><mo>″</mo></msup><mtext>b</mtext></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mrow><mo>〈</mo><mrow><mi>x</mi><msup><mi>x</mi><mo>″</mo></msup></mrow><mo>〉</mo></mrow></mrow><mrow><msub><mi>X</mi><mtext>b</mtext></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mrow><mo>〈</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>〉</mo></mrow><mrow><mo>〈</mo><mrow><msup><msup><mi>x</mi><mo>′</mo></msup><mn>2</mn></msup></mrow><mo>〉</mo></mrow><mo>−</mo><mn>4</mn><msup><mrow><mrow><mo>〈</mo><mrow><mi>x</mi><msup><mi>x</mi><mo>′</mo></msup></mrow><mo>〉</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msubsup><mi>X</mi><mtext>b</mtext><mn>3</mn></msubsup></mrow></mfrac></mrow></math> For <math id="M60"><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>≤</mo><msub><mi>r</mi><mn>2</mn></msub></mrow></math> , equation (4) can be transformed into equation (6) by multiplying by <math id="M61"><mi>x</mi></math> and averaging over the distribution of the particles, <math display="block" id="M62"><mrow><mrow><mo>〈</mo><mrow><mi>x</mi><msup><mi>x</mi><mo>″</mo></msup></mrow><mo>〉</mo></mrow><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>′</mo></msup><mrow><mn>2</mn><mi>ϕ</mi></mrow></mfrac><mrow><mo>〈</mo><mrow><mi>x</mi><msup><mi>x</mi><mo>′</mo></msup></mrow><mo>〉</mo></mrow><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>″</mo></msup><mrow><mn>4</mn><mi>ϕ</mi></mrow></mfrac><mrow><mo>〈</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>〉</mo></mrow><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>1</mn></msub><mi>c</mi><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup></mrow></mfrac><mrow><mo>〈</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>〉</mo></mrow><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>2</mn></msub><mi>c</mi><msubsup><mi>r</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfrac><mrow><mo>〈</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>〉</mo></mrow><mo>=</mo><mn>0</mn></mrow></math> Thus, combining Eq.(5) with Eq.(6), the envelope for the <math id="M63"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> species of the multi-species beam can be described as <math display="block" id="M64"><mrow><msub><msup><mi>X</mi><mo>″</mo></msup><mtext>b</mtext></msub><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>′</mo></msup><mrow><mn>2</mn><mi>ϕ</mi></mrow></mfrac><msub><msup><mi>X</mi><mo>′</mo></msup><mtext>b</mtext></msub><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>″</mo></msup><mrow><mn>4</mn><mi>ϕ</mi></mrow></mfrac><msub><mi>X</mi><mtext>b</mtext></msub><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>1</mn></msub><mi>c</mi></mrow></mfrac><mfrac><mn>1</mn><mrow><msub><mi>X</mi><mtext>b</mtext></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>2</mn></msub><mi>c</mi></mrow></mfrac><mfrac><mn>1</mn><mrow><msubsup><mi>X</mi><mrow><mtext>bd</mtext><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></mfrac><msub><mi>X</mi><mtext>b</mtext></msub><mo>−</mo><mfrac><mrow><msubsup><mi>ε</mi><mrow><mi>x</mi><mtext>d</mtext><mn>1</mn></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>X</mi><mtext>b</mtext><mn>3</mn></msubsup></mrow></mfrac><mo>=</mo><mn>0</mn><mo>,</mo></mrow></math> where <math id="M65"><mrow><msub><mi>X</mi><mrow><mtext>bd</mtext><mn>2</mn></mrow></msub></mrow></math> is the beam envelope of the <math id="M66"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> species in the x-direction, <math id="M67"><mrow><msub><mi>ε</mi><mrow><mi>x</mi><mi>d</mi><mn>1</mn></mrow></msub></mrow></math> is the beam emittance of the <math id="M68"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> species in the x-direction, and <math id="M69"><mrow><msubsup><mi>ε</mi><mrow><mi>x</mi><mtext>d</mtext><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><mn>4</mn><mrow><mo>〈</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>〉</mo></mrow><mrow><mo>〈</mo><mrow><msup><msup><mi>x</mi><mo>′</mo></msup><mn>2</mn></msup></mrow><mo>〉</mo></mrow><mo>−</mo><mn>4</mn><msup><mrow><mrow><mo>〈</mo><mrow><mi>x</mi><msup><mi>x</mi><mo>′</mo></msup></mrow><mo>〉</mo></mrow></mrow><mn>2</mn></msup></mrow></math> . Similarly, for <math id="M70"><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>></mo><msub><mi>r</mi><mn>2</mn></msub></mrow></math> , equation (5) can be written as <math display="block" id="M71"><mrow><mrow><mo>〈</mo><mrow><mi>x</mi><msup><mi>x</mi><mo>″</mo></msup></mrow><mo>〉</mo></mrow><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>′</mo></msup><mrow><mn>2</mn><mi>ϕ</mi></mrow></mfrac><mrow><mo>〈</mo><mrow><mi>x</mi><msup><mi>x</mi><mo>′</mo></msup></mrow><mo>〉</mo></mrow><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>″</mo></msup><mrow><mn>4</mn><mi>ϕ</mi></mrow></mfrac><mrow><mo>〈</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>〉</mo></mrow><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>1</mn></msub><mi>c</mi></mrow></mfrac><mfrac><mn>1</mn><mrow><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup></mrow></mfrac><mrow><mo>〈</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>〉</mo></mrow><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>2</mn></msub><mi>c</mi></mrow></mfrac><mrow><mo>〈</mo><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>〉</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></math> <math display="block" id="M72"><mrow><mtext>where </mtext><mrow><mo>〈</mo><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>〉</mo></mrow><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>r</mi><mn>2</mn></msub></mrow><mrow><msub><mi>r</mi><mn>1</mn></msub></mrow></msubsup><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>π</mi></mrow></msubsup><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac><msub><mi>ρ</mi><mn>1</mn></msub><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow><mo>)</mo></mrow><mi>r</mi><mtext>d</mtext><mi>r</mi><mtext>d</mtext><mi>θ</mi></mrow></mrow></mstyle></mrow></mrow></mstyle></mrow><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>r</mi><mn>1</mn></msub></mrow></msubsup><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>π</mi></mrow></msubsup><mrow><msub><mi>ρ</mi><mn>1</mn></msub><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow><mo>)</mo></mrow><mi>r</mi><mtext>d</mtext><mi>r</mi><mtext>d</mtext><mi>θ</mi></mrow></mrow></mstyle></mrow></mrow></mstyle></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><msubsup><mi>r</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup></mrow></mfrac><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><msubsup><mi>X</mi><mrow><mtext>bd</mtext><mn>2</mn></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>X</mi><mtext>b</mtext><mn>2</mn></msubsup></mrow></mfrac><mo stretchy="false">)</mo></mrow></math> Thus, the envelope for the <math id="M73"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> species of the multi-species beam can be described as <math display="block" id="M74"><mrow><msub><msup><mi>X</mi><mo>″</mo></msup><mtext>b</mtext></msub><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>′</mo></msup><mrow><mn>2</mn><mi>ϕ</mi></mrow></mfrac><msub><msup><mi>X</mi><mo>′</mo></msup><mtext>b</mtext></msub><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>″</mo></msup><mrow><mn>4</mn><mi>ϕ</mi></mrow></mfrac><msub><mi>X</mi><mtext>b</mtext></msub><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>1</mn></msub><mi>c</mi></mrow></mfrac><mfrac><mn>1</mn><mrow><msub><mi>X</mi><mtext>b</mtext></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>2</mn></msub><mi>c</mi></mrow></mfrac><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><msubsup><mi>X</mi><mrow><mtext>bd</mtext><mn>2</mn></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>X</mi><mtext>b</mtext><mn>2</mn></msubsup></mrow></mfrac><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><msubsup><mi>ε</mi><mrow><mi>x</mi><mtext>d</mtext><mn>1</mn></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>X</mi><mtext>b</mtext><mn>3</mn></msubsup></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></math> From the above, the x-direction envelope for the <math id="M75"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> species of the multi-species beam consisting of <math id="M76"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> and <math id="M77"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> can be expressed as <math display="block" id="M78"><mrow><mrow><mo>{</mo><mrow><msubsup><mrow/><mrow><msub><msup><mi>X</mi><mo>″</mo></msup><mtext>b</mtext></msub><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>′</mo></msup><mrow><mn>2</mn><mi>ϕ</mi></mrow></mfrac><msub><msup><mi>X</mi><mo>′</mo></msup><mtext>b</mtext></msub><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>″</mo></msup><mrow><mn>4</mn><mi>ϕ</mi></mrow></mfrac><msub><mi>X</mi><mtext>b</mtext></msub><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>1</mn></msub><mi>c</mi></mrow></mfrac><mfrac><mn>1</mn><mrow><msub><mi>X</mi><mtext>b</mtext></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>2</mn></msub><mi>c</mi></mrow></mfrac><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><msubsup><mi>X</mi><mrow><mtext>bd</mtext><mn>2</mn></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>X</mi><mtext>b</mtext><mtext>2</mtext></msubsup></mrow></mfrac><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><msubsup><mi>ε</mi><mrow><mi>x</mi><mtext>d</mtext><mn>1</mn></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>X</mi><mtext>b</mtext><mtext>3</mtext></msubsup></mrow></mfrac><mo>=</mo><mn>0</mn><mtable><mtr><mtd><mrow/></mtd><mtd><mrow><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>></mo><msub><mi>r</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow><mrow><msub><msup><mi>X</mi><mo>″</mo></msup><mtext>b</mtext></msub><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>′</mo></msup><mrow><mn>2</mn><mi>ϕ</mi></mrow></mfrac><msub><msup><mi>X</mi><mo>′</mo></msup><mtext>b</mtext></msub><mo>+</mo><mfrac><msup><mi>ϕ</mi><mo>″</mo></msup><mrow><mn>4</mn><mi>ϕ</mi></mrow></mfrac><msub><mi>X</mi><mtext>b</mtext></msub><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>1</mn></msub><mi>c</mi></mrow></mfrac><mfrac><mn>1</mn><mrow><msub><mi>X</mi><mtext>b</mtext></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mn>4</mn><mi>ϕ</mi><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>β</mi><mn>2</mn></msub><mi>c</mi></mrow></mfrac><mfrac><mn>1</mn><mrow><msubsup><mi>X</mi><mrow><mtext>bd</mtext><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></mfrac><msub><mi>X</mi><mtext>b</mtext></msub><mo>−</mo><mfrac><mrow><msubsup><mi>ε</mi><mrow><mi>x</mi><mtext>d</mtext><mn>1</mn></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>X</mi><mtext>b</mtext><mtext>3</mtext></msubsup></mrow></mfrac><mo>=</mo><mn>0</mn><mtable><mtr><mtd><mrow/></mtd><mtd><mrow><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>≤</mo><msub><mi>r</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></msubsup></mrow></mrow></mrow></math> In the y -direction, the envelope equations have a similar form. For the multi-species beam containing <math id="M79"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , <math id="M80"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> , and <math id="M81"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> , equation (3) can be written as <math display="block" id="M82"><mrow><msub><mi>m</mi><mn>1</mn></msub><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mo>=</mo><mi>q</mi><mo stretchy="false">(</mo><msubsup><mi>E</mi><mi>x</mi><mrow><mtext>appl</mtext></mrow></msubsup><mo>+</mo><msubsup><mi>E</mi><mrow><msub><mi>D</mi><mn>1</mn></msub><mi>x</mi></mrow><mrow><mtext>self</mtext></mrow></msubsup><mo>+</mo><msubsup><mi>E</mi><mrow><msub><mi>D</mi><mn>2</mn></msub><mi>x</mi></mrow><mrow><mtext>self</mtext></mrow></msubsup><mo>+</mo><msubsup><mi>E</mi><mrow><msub><mi>D</mi><mn>3</mn></msub><mi>x</mi></mrow><mrow><mtext>self</mtext></mrow></msubsup><mo stretchy="false">)</mo><mo>,</mo></mrow></math> where <math id="M83"><mrow><msubsup><mi>E</mi><mrow><msub><mi>D</mi><mn>3</mn></msub><mi>x</mi></mrow><mrow><mtext>self</mtext></mrow></msubsup></mrow></math> is the space charge field of the <math id="M84"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> beam along the x-direction. Then, the equations of motion of <math id="M85"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> in x-direction can be described as http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10328860&type= http://notExist.jpg http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10328860&type= Using the same method as that used for the derivation of Eq.(11), the x -direction envelope for the <math id="M86"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> species of the multi-species beam consisting of <math id="M87"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , <math id="M88"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> , and <math id="M89"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> can be expressed as http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10328867&type= http://notExist.jpg http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10328867&type= In the y -direction, the envelope equations for the particles of the three species have a similar form. III. SIMULATION 1 A. Beam extraction system of the duoplasmatron source 2 1. Current operational beam extraction system 3 The current operational duoplasmatron ion source has been used for decades, and it is very reliable. The particle distributions in the phase space obtained in previous experiments [ 12 12 ] are shown in Fig.2 Fig.2 . For the LEBT simulation, the following parameters of the old beam extraction system are assumed [ 5 5 ] : the normalized beam emittance is <math id="M90"><mrow><msub><mi>ε</mi><mi>x</mi></msub><mo>=</mo><msub><mi>ε</mi><mi>y</mi></msub><mo>=</mo><mn>0.71</mn></mrow></math> mm·mrad, and the Twiss parameters are <math id="M91"><mrow><msub><mi>α</mi><mi>x</mi></msub><mo>=</mo><msub><mi>α</mi><mi>y</mi></msub><mo>=</mo></mrow></math> −4 and <math id="M92"><mrow><msub><mi>β</mi><mi>x</mi></msub><mo>=</mo><msub><mi>β</mi><mi>y</mi></msub><mo>=</mo></mrow></math> 80 cm/rad, respectively. The fraction of <math id="M93"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> in the multi-species beam is approximately 75%, and <math id="M94"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> and <math id="M95"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> comprise approximately 18% and approximately 7% of the total beam, respectively. 2. New designed beam extraction system 3 As shown in figure 3 figure 3 , a new beam extraction system of the duoplasmatron source is designed by the PIC simulation method. Compared to the old beam extraction system, the size of the extraction electrode, the dip angle of the anode, and the distance between the anode and the extraction electrode are changed. In addition, two small permanent magnet lenses with a magnetic strength of 1000 Gs are added. The simulated particle distributions in the phase space are shown in Fig. 4 Fig. 4 , where the normalized beam emittance is <math id="M96"><mrow><msub><mi>ε</mi><mi>x</mi></msub><mo>=</mo><msub><mi>ε</mi><mi>y</mi></msub><mo>=</mo><mn>2.5</mn></mrow></math> mm·mrad, and the Twiss parameters are <math id="M97"><mrow><msub><mi>α</mi><mi>x</mi></msub><mo>=</mo><msub><mi>α</mi><mi>y</mi></msub><mo>=</mo></mrow></math> −3.5 and <math id="M98"><mrow><msub><mi>β</mi><mi>x</mi></msub><mo>=</mo><msub><mi>β</mi><mi>y</mi></msub><mo>=</mo><mn>57</mn></mrow></math> cm/rad, respectively. B. Transport characteristics of the multi-species beam 2 1. Effects of the unwanted beam composition on the primary beam transport 3 It is clear that using a pure beam is beneficial for improving the beam transport efficiency and decreasing the beam power. Therefore, various beam separation elements, such as dipole-bending magnets, Wien filters, or apertures combined with solenoids, are usually utilized to obtain a pure beam in the LEBT [ 3 3 ] . However, using such beam separation elements will not only have the disadvantage of significantly increasing the length of the LEBT but is also not conducive to industrial application marketing due to the increased cost. In our scheme, the multi-species beam will be injected into the electrostatic LEBT directly. In order to understand the influence of the unwanted beam on the primary beam, the evolution of the <math id="M100"><mrow><msup><mi>D</mi><mo>+</mo></msup></mrow></math> beam envelope for various ratios of species in the multi-species beam is simulated using the PIC code TRACK [ 14 14 ] . The simulation results are illustrated in figure 5 figure 5 . They show that the envelope radius of the <math id="M101"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> beam increases with the increasing unwanted beam intensity. The four envelopes corresponding to Beam (40 mA <math id="M102"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> ), Beam (30 mA <math id="M103"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , 10 mA <math id="M104"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> ), Beam (30 mA <math id="M105"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , 10 mA <math id="M106"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> ), and Beam (30 mA <math id="M107"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , 5 mA <math id="M108"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> , 5 mA <math id="M109"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> ) almost fully overlap. The two envelopes corresponding to Beam (30 mA <math id="M110"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , 5 mA <math id="M111"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> ) and Beam (30 mA <math id="M112"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , 5 mA <math id="M113"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> ) almost fully overlap as well. This illustrates that the total intensity of the unwanted beams has a significant influence on the primary beam divergence. However, the influence of each species on the primary beam envelope is not obvious. 2. Effects of the injection beam parameters on the multi-species beam transport 3 Matching the extracted beam to the subsequent electrostatic LEBT beam is important for high intensity beam transport [ 15 15 ] . Therefore, two different sets of the injected beam parameters from the old beam extraction system and the new designed beam extraction system, as described above, are used to simulate the effects of these parameters on the transport of the multi-species beam by the PIC code TRACK. The results are described below. a. Transport efficiency 4 For the LEBT simulation, we use two different sets of the initial beam parameters obtained from the new and old beam extraction systems. For the new beam extraction system, the beam envelope radius is <math id="M115"><mrow><msub><mi>X</mi><mrow><mi>D</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mi>D</mi><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mi>D</mi><mn>3</mn></mrow></msub><mo>=</mo><mn>1.7</mn></mrow></math> cm, and the beam divergence angle is <math id="M116"><mrow><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>2</mn></mrow></msub><mo>=</mo><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>3</mn></mrow></msub><mo>=</mo><mn>110</mn></mrow></math> mrad. For the old beam extraction system, the beam envelope radius is <math id="M117"><mrow><msub><mi>X</mi><mrow><mi>D</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mi>D</mi><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mi>D</mi><mn>3</mn></mrow></msub><mo>=</mo><mn>1.1</mn></mrow></math> cm, and the beam divergence angle is <math id="M118"><mrow><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>2</mn></mrow></msub><mo>=</mo><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>3</mn></mrow></msub><mo>=</mo><mn>55</mn></mrow></math> mrad. The intensity of each species and the simulation results are shown in figure 6 figure 6 . It can be seen that using the initial beam parameters obtained from the new beam extraction system reduces the beam divergence for the beam transport in the electrostatic LEBT. The beam transport efficiency of each species in the electrostatic LEBT is simulated by the PIC code. The intensities of the <math id="M120"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , <math id="M121"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> , and <math id="M122"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> beams are 50 mA, 10 mA, and 10 mA, respectively. The beam envelope radius and divergence angle are the same with those described in the previous paragraph. The simulation results are shown in figure 7 figure 7 . When the initial beam parameters obtained from the new extraction system are used for the simulation, the transport efficiency of the <math id="M123"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , <math id="M124"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> , and <math id="M125"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> beams are 100%, 94%, and 39%, respectively. For the old beam extraction system, the corresponding values are 100%, 58%, and 52%. This implies that in the new beam extraction system, the <math id="M126"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> beam will quickly hit the LEBT pipe and lose the majority of its particles. b. Emittance distortion and particle distributions 4 In the electrostatic LEBT, due to the presence of high electrostatic fields, the high intensity beam is fully un-neutralized. The space charge effect of the un-neutralized beam will cause a significant increase in emittance and strong beam filamentation. In addition, the nonlinear space charge field will cause emittance distortion [ 16 16 ] . At the same time, the beam parameters at the output of the electrostatic LEBT are important for matching the beam to the subsequent beam transport elements. Therefore, we simulate the beam emittance and particle distributions. In the simulation, the fraction of each species beam is set based on the current operational duoplasmatron source, and the intensities of the <math id="M127"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , <math id="M128"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> , and <math id="M129"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> beams are set to 30 mA, 7 mA, and 3 mA, respectively. For the new beam extraction system, the beam envelope radius is <math id="M130"><mrow><msub><mi>X</mi><mrow><mi>D</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mi>D</mi><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mi>D</mi><mn>3</mn></mrow></msub><mo>=</mo><mn>1.7</mn></mrow></math> cm, and the beam divergence angle is <math id="M131"><mrow><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>2</mn></mrow></msub><mo>=</mo><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>3</mn></mrow></msub><mo>=</mo><mn>110</mn></mrow></math> mrad. For the old beam extraction system, the beam envelope radius is <math id="M132"><mrow><msub><mi>X</mi><mrow><mi>D</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mi>D</mi><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mi>D</mi><mn>3</mn></mrow></msub><mo>=</mo><mn>1.1</mn></mrow></math> cm, and the beam divergence angle is <math id="M133"><mrow><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>2</mn></mrow></msub><mo>=</mo><msub><msup><mi>X</mi><mo>′</mo></msup><mrow><mi>D</mi><mn>3</mn></mrow></msub><mo>=</mo><mn>55</mn></mrow></math> mrad. A water-bag distribution is used. The simulation results are shown in figures 8 figures 8 , 9 9 and 10 10 . Figure 8 Figure 8 shows the simulated envelopes of the multi-species beam with different injected beam parameters, confirming that the injected beam parameters have a significant influence on the transport of the multi-species beam, and therefore it is necessary to match the beam to the electrostatic LEBT by optimizing the injected beam parameters. Figure 9 Figure 9 shows the simulated beam emittance at the output of the electrostatic LEBT for different injected beam parameters. It can be seen that, for the new designed beam extraction system ( Figs. 9(a) and 9(b) Figs. 9(a) and 9(b) ), the emittance distortion of the primary <math id="M140"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> beam at the output of the electrostatic LEBT is more severe than that for the old beam extraction system ( Figs. 9(c) and 9(d) Figs. 9(c) and 9(d) ); however, the particle distributions of the primary beam are more focused, as shown in figure 10 figure 10 . For the new designed beam extraction system, the emittance of the primary beam at the output is approximately 10.1 mm·mrad, ~4 times greater than that at the input, which is approximately 2.5 mm·mrad. However, for the old beam extraction system, the emittance of the primary beam at the output is approximately 9.1 mm·mrad, ~12 times greater than that at the input, which is approximately 0.7 mm·mrad. From Fig. 10 Fig. 10 , it can be seen that hollow beams are formed for all species ( <math id="M141"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , <math id="M142"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> , and <math id="M143"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> ) when the beam is extracted from the old beam extraction system. From Fig. 10(a) Fig. 10(a) , the hollow structure of the <math id="M144"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> beam is obvious, and exhibits a spot size of approximately 5 cm. However, in Fig.10(b) Fig.10(b) , this effect is reduced significantly, and the spot size of the beam for the new extraction system is approximately 3 cm, which is obtained considering 98% of all <math id="M145"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> particles and neglecting the distorted portion. It is found that the appropriate injected beam parameters play an important role in both the beam transport and the beam spot size in the electrostatic LEBT. This can be helpful for developing compact ion implanters and intense neutron generators. IV. CONCLUSION 1 The envelope equations of the multi-species deuterium beam in an electrostatic LEBT are derived considering the space charge effects caused by the particles of all species. The evolution of the envelopes of the multi-species beam is simulated in the presence of different ratios of the components <math id="M146"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> , <math id="M147"><mrow><msubsup><mi>D</mi><mn>2</mn><mo>+</mo></msubsup></mrow></math> , and <math id="M148"><mrow><msubsup><mi>D</mi><mn>3</mn><mo>+</mo></msubsup></mrow></math> , and for two sets of the injected beam parameters corresponding to the old and new beam extraction systems of the duoplasmatron source. The simulation results show that the envelope radius of the <math id="M149"><mrow><msubsup><mi>D</mi><mn>1</mn><mo>+</mo></msubsup></mrow></math> beam increases with the increase of the unwanted beam intensity. The new injected beam parameters improve the transport of the primary beam in the electrostatic LEBT. However, the corresponding emittance is found to be distorted at the output of the electrostatic LEBT. 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E 53 , 5358 ( 1996 ). doi: 10.1103/PhysRevE.53.5358 http://doi.org/10.1103/PhysRevE.53.5358 Conservation of space-charge-dominated beam emittance in a strong nonlinear focusing field 10.1007/s41365-018-0384-2 This work was supported by the Fundamental Research Funds for the Central Universities (No. lzujbky-2017-93), the National Natural Science Foundation of China (Nos. 11375077, 11027508, and 21327801), and the National Key Scientific Instrument and Equipment Development Projects (No. 2013YQ04086101). 2017-05-05 2017-07-05 2017-07-26 2018-03-14 2018-04-01 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2018 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-018-0384-2 http://dx.doi.org/10.1007/s41365-018-0384-2 . 2018 Nuclear Science and Techniques 04 29