Pulsed intense electron beam emittance measurement

Pulsed intense electron beam emittance measurement 1 corresp first-author Corresponding author, zhangzhuo96@tsinghua.org.cn Corresponding author, zhangzhuo96@tsinghua.org.cn zhangzhuo96@tsinghua.org.cn 1 Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang621900, China Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang621900, China Recently we measured with the Modified Three Gradient Method (MTGM) [ 1 1 , 2 2 ] the beam emittance of an injector constructed in 2012, which was designed to provide a 2.4 kA, 2.6 MeV electron beam. The MTGM is a non-intercept indirect method, which is based on the three gradient type measurements of beam profiles and subsequent data processing which helps to get the least square solution to the beam emittance. Beam profiles under different currents of guiding coil are measured using Cerenkov radiation given off by a piece of quartz glass in the beam tube, which is recorded with a CCD camera. MTGM Code is developed to realize the data fitting as well as beam transport simulation, in which both the σ matrix method and the numerical solution of root-mean-square beam envelope equation are used. The error is also analyzed. Emittance measurements High current beam Modified three gradient method I. INTRODUCTION 1 In most applications, emittance and brightness are the main figures of merit of particle beams. The horizontal emittance and vertical emittance are related to the beam brightness, which is defined as <math id="M1"><mrow><mi>B</mi><mo>=</mo><mi>I</mi><mo>/</mo><msup><mi>π</mi><mn>2</mn></msup><msub><mi>ε</mi><mi>x</mi></msub><msub><mi>ε</mi><mi>y</mi></msub></mrow></math> . Here, the horizontal (vertical) emittance is usually defined by considering the x and x’ phase space, as the area (optionally divided by π) of the ellipse containing 95% (or an arbitrary number) of all particles in its interior. Low beam emittance is key to achieving the required spot size at the output focus of the beam line. The Liouville theorem [ 1 1 ] states that volumes in phase space are invariant for a Hamiltonian system. In linear accelerators, where the particle energy is varied, the emittance is not invariant. Instead, one defines the so-called normalized emittance, ε n = βγε . The normalized emittance is conserved during acceleration. Conventional measurements of emittance include the pepper pot method [ 2 2 ], the three gradient method [ 3 3 ], the MTGM[ 4 4 , 5 5 ], etc. Here we use the MTGM for its advantages, such as the online feasibility for non-destructive procedure. The main concept of MTGM is shown in Fig. 1 Fig. 1 . It is based on the three gradient type measurements of the beam radius and a data fitting to the measured cross-over curve. Under a certain current of guiding coil, supposing that ε and initial conditions ( R 0 , <math id="M2"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>'</mo></msubsup></mrow></math> ) are known, the root-mean-square beam envelope of an axially symmetric beam is described by the following equation [ 6 6 ]: <math display="block" id="M3"><mrow><msubsup><mi>R</mi><mrow><mi>r</mi><mi>m</mi><mi>s</mi></mrow><mrow><mo>'</mo><mo>'</mo></mrow></msubsup><mo>+</mo><msup><mi>k</mi><mn>2</mn></msup><msub><mi>R</mi><mrow><mtext>rms</mtext></mrow></msub><mo>−</mo><mfrac><mi>K</mi><mrow><mn>2</mn><msub><mi>R</mi><mrow><mtext>rms</mtext></mrow></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msubsup><mi>ε</mi><mrow><mtext>rms</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>R</mi><mrow><mtext>rms</mtext></mrow><mn>3</mn></msubsup></mrow></mfrac><mo>=</mo><mn>0</mn><mn>,</mn></mrow></math> <math display="block" id="M4"><mrow><mi>k</mi><mo>=</mo><mfrac><mrow><mi>e</mi><mi>c</mi><msub><mi>B</mi><mi>z</mi></msub></mrow><mrow><mn>2</mn><mi>β</mi><mi>γ</mi><msub><mi>m</mi><mn>0</mn></msub><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac><mn>,</mn><mtext> </mtext><mi>K</mi><mo>=</mo><mfrac><mrow><mn>2</mn><msub><mi>I</mi><mi>b</mi></msub></mrow><mrow><mn>17045</mn><msup><mi>β</mi><mn>3</mn></msup><msup><mi>γ</mi><mn>3</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> where R rms is the beam radius and ε rms is the beam emittance normalized to the beam energy. It is found that ε rms remains a constant during non-accelerating transport procession. The foot rms denotes root-mean-square, which will be dropped for the remainder of this paper for the sake of concession. Note that all ε in the following is edge emittance normalized to energy. k is one half the cyclotron wave numbers with B z , the axial magnetic field determined by focusing the current of the guiding coil. K is the general diversion coefficient with I b , the beam current, measured by the CVRs. β and γ are relativistic factors and can be calculated from beam energy, which is measured with a capacitive probe mounted in the gap between the accelerator cells. Equation 1 can be solved with the numerical method by converting the problem to a 1-order differential equation group, which can be solved with the Runge-Kutta method [ 7 7 ]. Solving the equation under a different magnetic field will generate a set of beam radii data, from which a cross-over curve revealing the relation between beam radius at the analysis plane with magnetic field can be constructed. Reversely, if we can get the cross-over curve through experiments, ( ε , R 0 , <math id="M5"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>'</mo></msubsup></mrow></math> ) can be deduced by the data fitting method. In our work, the σ matrix method is used as an aid to find the least square fitting to the cross-over curve. II. MEASUREMENTS 1 Beam profiles are measured by making use of Cerenkov Radiation (CR) [ 8 8 ], which is given off by charged particles traversing a transparent dielectric medium in which its velocity exceeds that of light. It is preferred in experiments profiting from its definite direction, rapid response time, and the proportional property of the yield of photons to the number of electrons. Given off at the analysis plane, which is located at a distance of 574.5 mm downstream from the center of the guiding coil, the CR light is reflected by a reflector and accepted by a CCD camera. Through a fiber, it is transmitted on to an online computer. The experimental sketch is shown in Fig. 2 Fig. 2 . The parameters of the solenoid coils on the experimental beam line are listed in Table 1 Table 1 . 1 1 Solenoid No. 1 1 Length (m) 1 1 R O (m) 1 1 R I (m) 1 1 Excitation (A×n) 1 1 Position (m) 1 1 LC1 1 1 0.172 1 1 0.186 1 1 0.164 1 1 31476 1 1 0.356 1 1 LC2 1 1 0.217 1 1 0.186 1 1 0.164 1 1 0 1 1 0.774 1 1 IC01 1 1 0.373 1 1 0.248 1 1 0.230 1 1 10560 1 1 1.386 1 1 IC02 1 1 0.373 1 1 0.248 1 1 0.230 1 1 13200 1 1 1.859 1 1 IC03 1 1 0.373 1 1 0.248 1 1 0.230 1 1 17160 1 1 2.332 1 1 IC04 1 1 0.373 1 1 0.248 1 1 0.230 1 1 15840 1 1 2.805 1 1 IC05 1 1 0.373 1 1 0.248 1 1 0.230 1 1 17160 1 1 3.278 1 1 VC 1 1 0.373 1 1 0.248 1 1 0.230 1 1 31680 1 1 3.740 1 1 MC01 1 1 0.338 1 1 0.121 1 1 0.093 1 1 0-61440 1 1 4.253 The images of beam profiles under different focusing currents are shown in Fig. 3 Fig. 3 . Scan the images in the radial direction and draw its gray scale curve, from which the beam radii are read out (see Table 3 Table 3 in Sec. III). 1 1 NO. 1 1 I (A) 1 1 R i (mm) 1 1 R ( I i ) (mm) 1 1 δ i (mm) 1 1 1 1 1 0 1 1 24.9 1 1 25.73 1 1 0.83 1 1 2 1 1 2 1 1 24.0 1 1 25.49 1 1 1.49 1 1 3 1 1 4 1 1 22.0 1 1 24.79 1 1 2.79 1 1 4 1 1 6 1 1 21.0 1 1 23.65 1 1 2.65 1 1 5 1 1 8 1 1 19.9 1 1 22.08 1 1 2.18 1 1 6 1 1 10 1 1 18.4 1 1 20.14 1 1 1.74 1 1 7 1 1 12 1 1 16.5 1 1 17.86 1 1 1.36 1 1 8 1 1 15 1 1 13.4 1 1 14.08 1 1 0.68 1 1 9 1 1 17 1 1 10.9 1 1 11.41 1 1 0.51 1 1 10 1 1 20 1 1 7.2 1 1 7.56 1 1 0.36 1 1 11 1 1 24 1 1 3.7 1 1 4.13 1 1 0.43 1 1 12 1 1 28 1 1 5.9 1 1 5.75 1 1 -0.15 1 1 13 1 1 32 1 1 7.8 1 1 9.39 1 1 1.59 1 1 14 1 1 35 1 1 12.0 1 1 11.32 1 1 -0.68 1 1 15 1 1 38 1 1 15.5 1 1 12.09 1 1 -3.41 III. SIMULATION RESULTS AND DISCUSSION 1 MTGM was applied to measure the emittance of a space charge dominated electron beam and has been tested to be effective. The magnetic field under different current supply is calculated. Fig. 4 Fig. 4 plots the magnetic field curve from z 0 =3 m to z 0 =4.642 m. The plot reveals the proportional property of the magnetic field to the current supply. The start point of calculation is selected to be z =3 m, a position where the magnetic field of MC01 is trivial and the initial conditions are relatively constant during different measurements times. MTGM CODE is designed to aid in searching for ( ε , R 0 , <math id="M6"><mrow><msub><mi>R</mi><mn>0</mn></msub><mo>’</mo></mrow></math> ) by fitting the least square to the measured ( R i , I i ). The Beam envelope equation is a second order derivative function with a varying coefficient, whose explicit solution is difficult to get by integration directly [ 9 9 ]. However, the σ matrix method provides a concise way to find the direct relation between R and initial conditions. It makes it possible to use the data fitting least square method [ 10 10 ]. As a matter of fact, the σ matrix method is a beam transport solving method which can have an independent effect even though it plays the role of a complement to the beam envelope method in this way. After a great deal of manipulation, we obtained <math display="block" id="M7"><mrow><mi>R</mi><mo>=</mo><msqrt><mrow><msubsup><mi>M</mi><mrow><mn>11</mn></mrow><mn>2</mn></msubsup><msubsup><mi>R</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>M</mi><mrow><mn>11</mn></mrow></msub><msub><mi>M</mi><mrow><mn>12</mn></mrow></msub><mrow><mo>(</mo><mrow><mo>±</mo><msqrt><mrow><msubsup><mi>R</mi><mn>0</mn><mn>2</mn></msubsup><msup><mrow><mover accent="true"><mrow><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub></mrow><mo stretchy="true">˜</mo></mover></mrow><mn>2</mn></msup></mrow></msqrt><mo>−</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>+</mo><msubsup><mi>M</mi><mrow><mn>12</mn></mrow><mn>2</mn></msubsup><msup><mrow><mover accent="true"><mrow><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub></mrow><mo stretchy="true">˜</mo></mover></mrow><mn>2</mn></msup></mrow></msqrt><mn>.</mn></mrow></math> here, M is the transport matrix of the beam line as a whole [ 11 11 ]. Special attention should be paid to the <math id="M8"><mrow><mover accent="true"><mrow><mi>R</mi><msub><mo>’</mo><mn>0</mn></msub></mrow><mo stretchy="true">˜</mo></mover></mrow></math> associated with the σ matrix in this function, since it is not the derivative of R over z , but the beam angular envelope, which is the maximum of <math id="M9"><mrow><mi>r</mi><msub><mo>’</mo><mn>0</mn></msub></mrow></math> in the phase space [ 12 12 ] and can be calculated by solving the ellipse function in phase space. As a matter of fact, the ( ε , R 0 , <math id="M10"><mrow><mover accent="true"><mrow><mi>R</mi><msub><mo>’</mo><mn>0</mn></msub></mrow><mo stretchy="true">˜</mo></mover></mrow></math> ) here is a set of initial conditions of the σ matrix method, which is equivalent to the ( ε , R 0 , <math id="M11"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>'</mo></msubsup></mrow></math> ) of the beam envelope equation. Both sets can be derived from one another by taking the derivative over z of Eq. (3) at z = z 0 . The sign of the second term is determined by the direction of the ellipse in phase space, negative for converging procession, while positive for diverging procession. The least square method demands the minimum of <math display="block" id="M12"><mrow><mo>∥</mo><mi>δ</mi><msubsup><mo>∥</mo><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover></mstyle><msup><mrow><mrow><mo>[</mo><mrow><mi>R</mi><mrow><mo>(</mo><mrow><msub><mi>i</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><mo>−</mo><msub><mi>R</mi><mi>i</mi></msub></mrow><mo>]</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><munder><mstyle mathsize="140%" displaystyle="true"><mrow><mi>min</mi></mrow></mstyle><mrow><msub><mi>R</mi><mn>0</mn></msub><mn>,</mn><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub><mn>,</mn><mi>ε</mi></mrow></munder><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover></mstyle><msup><mrow><mrow><mo>[</mo><mrow><mi>R</mi><mrow><mo>(</mo><mrow><msub><mi>i</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><mo>−</mo><msub><mi>R</mi><mi>i</mi></msub></mrow><mo>]</mo></mrow></mrow><mn>2</mn></msup><mn>.</mn></mrow></math> This requires the zero derivatives of <math id="M13"><mrow><mo>∥</mo><mi>δ</mi><msubsup><mo>∥</mo><mn>2</mn><mn>2</mn></msubsup></mrow></math> to ε , R 0 , <math id="M14"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>'</mo></msubsup></mrow></math> : <math display="block" id="M15"><mrow><mfrac><mrow><mo>∂</mo><mo>∥</mo><mi>δ</mi><msubsup><mo>∥</mo><mn>2</mn><mn>2</mn></msubsup></mrow><mrow><mo>∂</mo><mi>ε</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mn>,</mn><mtext> </mtext><mfrac><mrow><mo>∂</mo><mo>∥</mo><mi>δ</mi><msubsup><mo>∥</mo><mn>2</mn><mn>2</mn></msubsup></mrow><mrow><mo>∂</mo><msub><mi>R</mi><mn>0</mn></msub></mrow></mfrac><mo>=</mo><mn>0</mn><mn>,</mn><mtext> </mtext><mfrac><mrow><mo>∂</mo><mo>∥</mo><mi>δ</mi><msubsup><mo>∥</mo><mn>2</mn><mn>2</mn></msubsup></mrow><mrow><mo>∂</mo><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub></mrow></mfrac><mo>=</mo><mn>0</mn><mn>.</mn></mrow></math> A three dimension equations set on R 0 , <math id="M16"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>'</mo></msubsup></mrow></math> , ε can be obtained by substituting the expression of R and <math id="M17"><mrow><mo>∥</mo><mi>δ</mi><msubsup><mo>∥</mo><mn>2</mn><mn>2</mn></msubsup></mrow></math> into the functions above. By solving the equation set, we can get the solution of <math id="M18"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup></mrow></math> , <math id="M19"><mrow><msubsup><mi>R</mi><mn>0</mn><mrow><mo>'</mo><mo>*</mo></mrow></msubsup></mrow></math> , ε * proscribing by the least square approximation to the measured data. The emittance and the corresponding initial conditions ( ε * , <math id="M20"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup></mrow></math> , <math id="M21"><mrow><msubsup><mi>R</mi><mn>0</mn><mrow><mo>'</mo><mo>*</mo></mrow></msubsup></mrow></math> ) are listed in Table 2 Table 2 . 1 1 Parameters 1 1 ε * ( π ·mm·mrad) 1 1 <math id="M23"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup></mrow></math> (mm) 1 1 <math id="M24"><mrow><msubsup><mi>R</mi><msup><mn>0</mn><mo>′</mo></msup><mo>*</mo></msubsup></mrow></math> (mrad) 1 1 Values 1 1 1120 1 1 9 1 1 18 Using the ( ε * , <math id="M25"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup></mrow></math> , <math id="M26"><mrow><msubsup><mi>R</mi><msup><mn>0</mn><mo>′</mo></msup><mo>*</mo></msubsup></mrow></math> ) above, the numerical solution of the beam envelope equation is obtained with the forth order Runge-Kutta method. The measured R – I data and calculated data is listed in Table 3 Table 3 . The cross-over curve corresponding to ( ε * , <math id="M27"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup></mrow></math> , <math id="M28"><mrow><msubsup><mi>R</mi><msup><mn>0</mn><mo>′</mo></msup><mo>*</mo></msubsup></mrow></math> ) is plotted in Fig. 5 Fig. 5 . The beam envelopes are shown in Fig. 6 Fig. 6 . Precision of the method requires sufficient data acquisition within a wide range. This can be realized by carefully choosing of the distance between the center of guiding coil and the analysis plane, as well as their position within the beam envelope evolution. Now let’s write equation (1) in the following form <math display="block" id="M29"><mrow><msubsup><mi>R</mi><mrow><mi>r</mi><mi>m</mi><mi>s</mi></mrow><mrow><mo>'</mo><mo>'</mo></mrow></msubsup><mo>=</mo><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><msub><mi>R</mi><mrow><mtext>rms</mtext></mrow></msub><mo>+</mo><mfrac><mi>K</mi><mrow><mn>2</mn><msub><mi>R</mi><mrow><mtext>rms</mtext></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msubsup><mi>ε</mi><mrow><mtext>rms</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>R</mi><mrow><mtext>rms</mtext></mrow><mn>3</mn></msubsup></mrow></mfrac><mn>.</mn></mrow></math> Here, the term on the left is a kind of force on the beam envelope, while the terms on the right side of the equation represent contributions of different forces, including the focusing force of the axial magnetic field B z and defocusing forces from the space charge effect and beam emittance. This will be clearer if we deal with the three functions individually. The focusing effect of the axial magnetic field over a small step can be described by <math display="block" id="M30"><mrow><mi>R</mi><mo>=</mo><mfrac><mrow><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><msub><mi>R</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msubsup><mi>R</mi><mn>0</mn><mrow><mo>'</mo><mn>2</mn></mrow></msubsup></mrow></msqrt></mrow><mrow><mi>k</mi><msub><mi>R</mi><mn>0</mn></msub></mrow></mfrac><mi>sin</mi><mrow><mo>[</mo><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>arctan</mi><mrow><mo>(</mo><mrow><mo>−</mo><mi>k</mi><mfrac><mrow><msub><mi>R</mi><mn>0</mn></msub></mrow><mrow><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>.</mn></mrow></math> The effective emittance force describes the beam transport in drift space <math display="block" id="M31"><mrow><mi>R</mi><mo>=</mo><msqrt><mrow><mrow><mo>[</mo><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>R</mi><mn>0</mn></msub><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow><mrow><msubsup><mi>R</mi><mn>0</mn><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>]</mo></mrow><msup><mrow><mrow><mo>[</mo><mrow><mi>z</mi><mo>−</mo><msub><mi>z</mi><mn>0</mn></msub><mo>+</mo><mfrac><mrow><msubsup><mi>R</mi><mn>0</mn><mn>3</mn></msubsup><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>R</mi><mn>0</mn></msub><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>]</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mfrac><mrow><msup><mi>ε</mi><mn>2</mn></msup><msubsup><mi>R</mi><mn>0</mn><mn>2</mn></msubsup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>R</mi><mn>0</mn></msub><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow></mfrac></mrow></msqrt><mn>.</mn></mrow></math> This equation defines a hyperbolic evolution of the beam envelope under the effect of emittance, restricted by which the minimum of beam radius is <math display="block" id="M32"><mrow><msub><mi>R</mi><mrow><mi>e</mi><mn>,</mn><mi>min</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>ε</mi><msub><mi>R</mi><mn>0</mn></msub></mrow><mrow><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>R</mi><mn>0</mn></msub><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mn>.</mn></mrow></math> The effect of space charge can not be expressed explicitly. However, through integrating the equation over a small step, we can get <math display="block" id="M33"><mrow><msup><mi>R</mi><mo>′</mo></msup><mo>=</mo><mo>±</mo><msqrt><mrow><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub><mo>+</mo><mi>K</mi><mi>ln</mi><mfrac><mi>R</mi><mrow><msub><mi>R</mi><mn>0</mn></msub></mrow></mfrac></mrow></msqrt><mn>.</mn></mrow></math> Or written in the form <math display="block" id="M34"><mrow><mi>R</mi><mo>=</mo><msub><mi>R</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><msup><msup><mi>R</mi><mo>′</mo></msup><mn>2</mn></msup><mo>−</mo><msubsup><mi>R</mi><msup><mn>0</mn><mo>′</mo></msup><mn>2</mn></msubsup></mrow><mi>K</mi></mfrac></mrow></msup><mn>.</mn></mrow></math> This equation representing the expanding effect of space charge force restricted by the minimum of beam radius is <math display="block" id="M35"><mrow><msub><mi>R</mi><mrow><mtext>sc</mtext><mn>,</mn><mi>min</mi></mrow></msub><mo>=</mo><msub><mi>R</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><msubsup><mi>R</mi><mn>0</mn><mrow><mo>'</mo><mn>2</mn></mrow></msubsup></mrow><mi>K</mi></mfrac></mrow></msup><mn>.</mn></mrow></math> When R is a big number, or at positions far away from the bottom of the cross over curve, where the magnetic field focusing force dominates, R is sensitive to variation in magnetic field. However, the situation changed at the bottom of the cross over curve, where R is a small number and is mainly determined by emittance and space charge force. Therefore, finding the exact minimum spot of the crossover curve is crucial to determining of beam emittance. A feasible measure is to get as much data as possible in the vicinity of the bottom of the cross-over curve by shortening the steps of the guiding current. Repeated measurements at the same guiding current also help to eliminate the intrinsic uncertainty of beam size. The error is mainly introduced by: (1) error of beam radius measurements; (2) error of beam energy and beam current; (3) error of the MTGMCODE. Elaborate error analysis has been done elsewhere [ 13 13 ]. It is found that the influence of (1) affects the precision while (2) and (3) are relatively trivial. We simulated the transport of beams with different emittances, increasing from 120 to 1720 in an arithmetical progression, as is shown in Fig. 7 Fig. 7 . Each step between two adjacent curves is 200π mm mrad. By taking partial derivatives of Eq. (3) over ε , R 0 , <math id="M36"><mrow><msubsup><mover accent="true"><mi>R</mi><mo>˜</mo></mover><mn>0</mn><mo>'</mo></msubsup></mrow></math> , we can get the error formula <math display="block" id="M37"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>Δ</mi><mi>R</mi><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>∂</mo><mi>R</mi></mrow><mrow><mo>∂</mo><mi>ε</mi></mrow></mfrac><mi>Δ</mi><mi>ε</mi><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>R</mi></mrow><mrow><mo>∂</mo><msub><mi>R</mi><mn>0</mn></msub></mrow></mfrac><mi>Δ</mi><msub><mi>R</mi><mn>0</mn></msub><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>R</mi></mrow><mrow><mo>∂</mo><msub><msup><mi>R</mi><mo>′</mo></msup><mn>0</mn></msub></mrow></mfrac><mi>Δ</mi><mover accent="true"><mi>R</mi><mo>˜</mo></mover><msub><mo>'</mo><mn>0</mn></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>∓</mo><mfrac><mrow><mi>ε</mi><mi>Δ</mi><mi>ε</mi></mrow><mi>R</mi></mfrac><mo>⋅</mo><mfrac><mrow><msub><mi>M</mi><mrow><mn>11</mn></mrow></msub><msub><mi>M</mi><mrow><mn>12</mn></mrow></msub></mrow><mrow><msqrt><mrow><msubsup><mi>R</mi><mn>0</mn><mn>2</mn></msubsup><msubsup><mi>R</mi><mn>0</mn><mrow><mo>'</mo><mn>2</mn></mrow></msubsup><mo>−</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mfrac><mrow><msub><mi>R</mi><mn>0</mn></msub><mi>Δ</mi><msub><mi>R</mi><mn>0</mn></msub></mrow><mi>R</mi></mfrac><mrow><mo>(</mo><mrow><msubsup><mi>M</mi><mrow><mn>11</mn></mrow><mn>2</mn></msubsup><mo>±</mo><mfrac><mrow><msub><mi>M</mi><mrow><mn>11</mn></mrow></msub><msub><mi>M</mi><mrow><mn>12</mn></mrow></msub></mrow><mrow><msqrt><mrow><msubsup><mi>R</mi><mn>0</mn><mn>2</mn></msubsup><msubsup><mi>R</mi><mn>0</mn><mrow><mo>'</mo><mn>2</mn></mrow></msubsup><mo>−</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mo>⋅</mo><msubsup><mover accent="true"><mi>R</mi><mo>˜</mo></mover><mn>0</mn><mrow><mo>'</mo><mn>2</mn></mrow></msubsup></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mfrac><mrow><mover accent="true"><mi>R</mi><mo>˜</mo></mover><mo>'</mo><mi>Δ</mi><mover accent="true"><mi>R</mi><mo>˜</mo></mover><msub><mo>'</mo><mn>0</mn></msub></mrow><mi>R</mi></mfrac><mrow><mo>(</mo><mrow><msubsup><mi>M</mi><mrow><mn>12</mn></mrow><mn>2</mn></msubsup><mo>±</mo><mfrac><mrow><msub><mi>M</mi><mrow><mn>11</mn></mrow></msub><msub><mi>M</mi><mrow><mn>12</mn></mrow></msub></mrow><mrow><msqrt><mrow><msubsup><mi>R</mi><mn>0</mn><mn>2</mn></msubsup><msubsup><mi>R</mi><mn>0</mn><mrow><mo>'</mo><mn>2</mn></mrow></msubsup><mo>−</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mo>⋅</mo><msubsup><mi>R</mi><mn>0</mn><mn>2</mn></msubsup></mrow><mo>)</mo></mrow><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Beam radius measurement uncertainty will result in errors in the discrimination of R 0 , <math id="M38"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>'</mo></msubsup></mrow></math> , and ε . Substituting R 0 , ε in the formula above with <math id="M39"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup></mrow></math> , ε * , and <math id="M40"><mrow><mover accent="true"><mi>R</mi><mo>˜</mo></mover><msub><mo>'</mo><mn>0</mn></msub></mrow></math> calculated from <math id="M41"><mrow><msubsup><mi>R</mi><mn>0</mn><mo>'</mo></msubsup></mrow></math> by taking the derivative over z of Eq. (3) at z = z 0 . For a given transport system, M 11 and M 12 are functions of z determined by its structure and parameters. Write the error formula in the form <math display="block" id="M42"><mrow><mi>Δ</mi><mi>R</mi><mo>=</mo><mi>α</mi><mfrac><mrow><mi>ε</mi><mi>Δ</mi><mi>ε</mi></mrow><mi>R</mi></mfrac><mo>+</mo><mi>β</mi><mfrac><mrow><msub><mi>R</mi><mn>0</mn></msub><mi>Δ</mi><msub><mi>R</mi><mn>0</mn></msub></mrow><mi>R</mi></mfrac><mo>+</mo><mi>γ</mi><mfrac><mrow><mover accent="true"><mi>R</mi><mo>˜</mo></mover><msub><mo>'</mo><mn>0</mn></msub><mi>Δ</mi><mover accent="true"><mi>R</mi><mo>˜</mo></mover><msub><mo>'</mo><mn>0</mn></msub></mrow><mi>R</mi></mfrac><mn>.</mn></mrow></math> Here, R is the result of the least square fitting to the measured data. For each point on the plot in Fig. 7 Fig. 7 , the error of ε is roughly in positive proportion to that of R . The relation between emittance and radii can be roughly formulated as <math display="block" id="M43"><mrow><mi>Δ</mi><mi>ε</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>200</mn><mo>∼</mo><mn>1600</mn><mo stretchy="false">)</mo><mi>Δ</mi><mi>R</mi><mo stretchy="false">(</mo><mi>m</mi><mi>m</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> This means that around the minimum of the cross-over curve, small radius measurement uncertainty may result in large emittance calculation errors, about 8 times that of the measurements far away from it. For example, at the bottom of the cross over curve, a 1 mm radius measurement uncertainty will cause about a 1600π mm mrad emittance calculation error, while at the upper side of the curve, the same radius measurement uncertainty will cause only about a 200π mm mrad emittance calculation error, only one eighth of the former. The relative error is <math display="block" id="M44"><mrow><mfrac><mrow><mi>Δ</mi><mi>ε</mi></mrow><mi>ε</mi></mfrac><mo>=</mo><mi>η</mi><mfrac><mrow><mi>Δ</mi><mi>R</mi></mrow><mi>R</mi></mfrac><mn>.</mn></mrow></math> Under measuring conditions, η is within the region of 6 and 10, with 10 at the lower part of the plot and 6 at the upper part. The parameters corresponding to each plot are listed in Table 4 Table 4 , including the emittance, and the initial values of the radius and the spreading angle. 1 1 NO. 1 1 ε (π mm mrad) 1 1 R 0 (mm) 1 1 <math id="M45"><mrow><msub><mi>R</mi><msup><mn>0</mn><mo>′</mo></msup></msub></mrow></math> (mm) 1 1 1 1 1 120 1 1 13.9 1 1 2.2 1 1 2 1 1 320 1 1 13.8 1 1 4.5 1 1 3 1 1 520 1 1 13.2 1 1 8.2 1 1 4 1 1 720 1 1 11.9 1 1 12.3 1 1 5 1 1 920 1 1 10.5 1 1 15.5 1 1 6 1 1 1120 1 1 9 1 1 18 1 1 7 1 1 1320 1 1 7.8 1 1 19.6 1 1 8 1 1 1520 1 1 6.6 1 1 20.2 1 1 9 1 1 1720 1 1 5.8 1 1 18.5 IV. CONCLUSION 1 Results from the Modified Three Gradient Method (MTGM), applied to a pulsed high intensity electron source, are presented. The method, experimental set-up, and experimental results referring to the non-destructive beam emittance measurements are presented. The MTGM makes possible the non-destructive determination of beam emittance in a space charge presence. The method is based on an integration of both beam cross-section measurements, realized in the three-gradient arrangement, and the beam envelope equation for an axially symmetric configuration. The experimental data is processed with a numerical matching program to determine the emittance and diameter. In the program, the σ matrix method was used to find the least square fitting to the measured cross-over curve to find the emittance; the envelope equation was solved with the numerical method of the 4 th order of the Runge-Kutta method to obtain beam radius. The measurement error is also analyzed. 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Optics of charged particle beams . Beijing (CHN) : Senior Education Publishing House , 2004 , 1 - 78 . (in Chinese) Optics of charged particle beams Zhang Z . Master Thesis , China Academy of Engineering Physics , 2004 . (in Chinese) Master Thesis The authors would like to thank the members of the testing and operating team for their contribution to this work. 10.13538/j.1001-8042/nst.25.060201 2014-01-22 2014-05-19 2014-12-20 2014-12-20 2021-04-11 2014 Nuclear Science and Techniques 6 25