Radar scattering width effects by coating ⁹⁰Sr/⁹⁰Y layers on cylindrical targets

Radar scattering width effects by coating 90 Sr/ 90 Y layers on cylindrical targets 1 first-author 1 1 1 1 1 1 1 corresp Corresponding author. E-mail address: rhuang@nju.edu.cn Corresponding author. E-mail address: rhuang@nju.edu.cn rhuang@nju.edu.cn School of Physics, Nanjing University, Nanjing 210093, China School of Physics, Nanjing University, Nanjing 210093, China In this paper, the 90 Sr/ 90 Y coating effects on scattering width ( SW ) of cylindrical conductor targets are investigated. The electron density distribution of plasma around cylindrical targets of different radiuses is simulated under different radioactivities in normal or oblique incidence. In normal incidence, the SWs are examined as functions of frequency and scattering angle; while in oblique incidence, the SW is inspected as a function of incident angle at the frequency of 1.5 GHz. The results obtained are compared with those from an ideal perfect electric conductor (PEC) cylinder. It is demonstrated that the SW decreases over a wide frequency range in the back scattering region by coating a 90 Sr/ 90 Y layer on the cylindrical target. Moreover, the reduction of bi-static SW amplitude can reach 3–20 dB, when the incident angle is smaller than 30° at 1.5 GHz. It is a significant improvement in the stealth effect. Radioactivity Plasma Cylinder Scattering width ( SW ) 1 Introduction 1 Stealth effect to electromagnetic waves (EMW) is important for the survivability of weapon systems in battlefields and for military defense systems [ 1 1 - 3 3 1-3 ]. Since the bodies of aircraft cabins or missiles can be simplified as cylindrical objects, the study of EMW scattering from a cylindrical target has attracted considerable attention from researchers [ 4 4 - 11 11 4-11 ], and enhancing the stealth effects by coating new materials has become an attractive area of research [ 12 12 - 15 15 12-15 ]. As a type of specific absorbing materials, plasma coating is of significant EMW-absorption effect [ 16 16 ]. The scattering width ( SW ) of a cylindrical target can be greatly reduced by plasma coating. Plasma can be induced by charged particles emitted from radioactive sources in the air. 90 Sr/ 90 Y is a perfect β-ray source. It emits no neutrons and ignorable γ-rays but just β-particles in maximum energy of 2.28 MeV and maximum trajectory range of 10 m in air [ 17 17 - 19 19 17-19 ]. In this paper, we investigate the absorption effect of plasma induced by 90 Sr/ 90 Y coating around a cylindrical target. The influences of 90 Sr/ 90 Y layer on mono-static and bi-static SWs of the cylindrical target at normal incidence are analyzed over the frequency range of 0.1–40 GHz; while the influences on bi-static SW at oblique incidence are analyzed at 1.5 GHz 2 Theoretical background 1 2.1. Physical behavior of plasma induced by 90 Sr 2 As 90 Y is the decay product progeny of 90 Sr, we refer 90 Sr as 90 Sr/ 90 Y. Fig. 1 Fig. 1 shows the continuous β spectrum of 90 Sr [ 20 20 ]. The emitting directions and trajectories of β-particles from 90 Sr are all random, and the average values are obtained. The average energy losses of β-particles along the radial direction of a cylinder can be calculated. The average ionization energy of air molecules in atmosphere at normal temperature is 36 eV [ 21 21 ]. The average numbers of electron-ion pairs induced by one β-particle are simulated. The emitted electrons diffuse in air [ 22 22 , 23 23 ]. The number of electrons, N , pass through a unit area in a unit time is <math display="block" id="M1"><mrow><msub><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mi mathvariant="normal">dif</mi></mrow></msub><mo>=</mo><mo>−</mo><msub><mi>D</mi><mrow><mi>d</mi><mi>i</mi><mi>f</mi></mrow></msub><mo>∇</mo><msub><mi>n</mi><mi>e</mi></msub><mo>,</mo></mrow></math> where n e is the electron density, and D dif = 3.7×10 2 cm 2 ∙s −1 is the diffusion coefficient in air, as experimentally validated. The electrons and positive ions are recombined after collision and the n e changes in the recombination process as <math display="block" id="M2"><mrow><msub><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>n</mi><mi>e</mi></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mi>r</mi><mi>e</mi><mi>c</mi></mrow></msub><mo>≈</mo><mo>−</mo><mtext>α</mtext><msub><mi>n</mi><mi>e</mi></msub><msup><mrow/><mn>2</mn></msup><mo>,</mo></mrow></math> where α = 1.5×10 −6 cm 3 ∙s −1 is the recombination coefficient, according to the experiments. The electrons are absorbed by neutral particles as a result of collision, with the formation of negative ions, which can easily be photo-detached by the sunshine. Considering the processes of ionization, diffusion, recombination, absorption and photolysis, the electron density ( N m ) distribution in the plasma induced by β-particles from 90 Sr can be estimated. For un-magnetized plasma ( μ r =1), the relative dielectric constant ε r can be derived by <math display="block" id="M3"><mrow><msub><mi>ε</mi><mtext>r</mtext></msub><mo>=</mo><mn>1</mn><mo>−</mo><msub><mi>K</mi><mn>1</mn></msub><mfrac><mrow><msubsup><mi>ω</mi><mrow><mtext>pe</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>v</mi><mrow><mtext>eff</mtext><mo>,</mo><mi>l</mi></mrow><mn>2</mn></msubsup></mrow></mfrac><mo>−</mo><mi>j</mi><msub><mi>K</mi><mn>2</mn></msub><mfrac><mi>v</mi><mrow><msub><mi>ω</mi><mn>0</mn></msub></mrow></mfrac><mfrac><mrow><msubsup><mi>ω</mi><mrow><mtext>pe</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>v</mi><mrow><mtext>eff</mtext><mo>,</mo><mi>l</mi></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow></math> where, v , v eff, l and ω pe are <math display="block" id="M4"><mrow><mi>v</mi><mo>~</mo><msub><mi>v</mi><mrow><mtext>eff, </mtext><mi>l</mi></mrow></msub><mo>≈</mo><mn>1.5</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>11</mn></mrow></msup><mfrac><mrow><msub><mi>N</mi><mtext>m</mtext></msub></mrow><mrow><mn>2.7</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>19</mn></mrow></msup></mrow></mfrac><msqrt><mrow><mfrac><mi>T</mi><mrow><mn>300</mn></mrow></mfrac></mrow></msqrt><mo>,</mo></mrow></math> <math display="block" id="M5"><mrow><msub><mi>ω</mi><mrow><mtext>pe</mtext></mrow></msub><mo>=</mo><msqrt><mrow><mfrac><mrow><msub><mi>n</mi><mtext>e</mtext></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><msub><mi>m</mi><mtext>e</mtext></msub><msub><mi>ε</mi><mn>0</mn></msub></mrow></mfrac></mrow></msqrt><mo>,</mo></mrow></math> and K 1 and K 2 are the coefficients related to the ratio of EMW frequency and the effective collision frequency [ 24 24 ]. 2.2. Electromagnetic wave scattering from a cylindrical target 2 The cylindrical PEC target surrounded by inhomogeneous plasma can be simplified as a multilayered cylinder in the scattering problem. The geometry for an obliquely incident EMW is shown in Fig. 2(a) Fig. 2(a) . The Z -axis coincides with the axis of the cylinder and X-Z is the incident plane. The tilt angle θ is the angle between the incident direction and the axis of the cylinder in the incident plane. Fig. 2(b) Fig. 2(b) shows cross-section of the multilayered cylinder model. The plasma around the cylindrical target (of r 1 in radius) is divided into J layers, with r j being the outer radius of the j th layer and m j being its complex refractive index. The scattering angle φ is defined by the scattering direction and X-axis. An eigenfunction expansion method is utilized to calculate the scattered filed for its relative simplicity in the situation of thick inhomogeneous layers coated cylinder. Based on Maxwell equations, the field in the j th layer can be written as [ 6 6 , 9 9 ] <math display="block" id="M6"><mrow><msub><mi>E</mi><mi>j</mi></msub><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>n</mi></munder><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>n</mi></msup></mrow></mstyle><mrow><mo>[</mo><mrow><msubsup><mi>N</mi><mrow><mi>n</mi><mo>,</mo><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mi>a</mi><mi>n</mi><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msubsup><msubsup><mi>N</mi><mrow><mi>n</mi><mo>,</mo><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><mi>i</mi><msubsup><mi>b</mi><mi>n</mi><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msubsup><msubsup><mi>M</mi><mrow><mi>n</mi><mo>,</mo><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo>]</mo></mrow><mo>,</mo></mrow></math> where, M n and N n are eigenvectors. For transverse magnetic (TM) incident wave, the coefficients of the scattered field, a n and b n , can be expressed as [ 9 9 ] <math display="block" id="M7"><mtable columnalign="left"><mtr><mtd><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><mo>−</mo><mi>i</mi><msub><mi>g</mi><mi>J</mi></msub><mrow><mo>(</mo><mrow><msubsup><mi>w</mi><mi>J</mi><mn>2</mn></msubsup><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>y</mi><mi>n</mi><mi>cos</mi><mi>θ</mi><msub><mi>T</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><msub><mi>L</mi><mn>1</mn></msub></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><msub><mi>L</mi><mn>2</mn></msub><msub><mi>L</mi><mn>3</mn></msub><mo>−</mo><msub><mi>g</mi><mi>J</mi></msub><msup><mi>n</mi><mn>2</mn></msup><msup><mrow><mi>cos</mi></mrow><mn>2</mn></msup><mi>θ</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msubsup><mi>w</mi><mi>J</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>n</mi></msub><mo>=</mo><msub><mi>T</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>⋅</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><msub><mi>L</mi><mn>3</mn></msub><msub><mi>L</mi><mn>4</mn></msub><mo>−</mo><msub><mi>g</mi><mi>J</mi></msub><msup><mi>n</mi><mn>2</mn></msup><msup><mrow><mi>cos</mi></mrow><mn>2</mn></msup><mi>θ</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msubsup><mi>w</mi><mi>J</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><msub><mi>L</mi><mn>2</mn></msub><msub><mi>L</mi><mn>3</mn></msub><mo>−</mo><msub><mi>g</mi><mi>J</mi></msub><msup><mi>n</mi><mn>2</mn></msup><msup><mrow><mi>cos</mi></mrow><mn>2</mn></msup><mi>θ</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msubsup><mi>w</mi><mi>J</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow></mfrac><mo>,</mo></mtd></mtr></mtable></math> where y = x J sin θ , η J = x J ( m J 2 −cos 2 θ ) 1/2 , w J = y / η J , g J = μ J / m J , h J = m J y / η J , and T n ( y )= J n ( y )/ H n (1) ( y ). For the j th layer, x j =2π r j / λ , with λ being wavelength of the incident EMW and μ j being the plasma permeability in the j th layer. L 1–4 are as follows <math display="block" id="M8"><mrow><msub><mi>L</mi><mn>1</mn></msub><mo>=</mo><msub><mi>D</mi><mi>n</mi></msub><msup><mrow/><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>−</mo><msub><mi>D</mi><mi>n</mi></msub><msup><mrow/><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><msub><mi>L</mi><mn>2</mn></msub><mo>=</mo><msub><mi>g</mi><mi>J</mi></msub><msub><mi>D</mi><mi>n</mi></msub><msup><mrow/><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>–</mo><msub><mi>h</mi><mi>J</mi></msub><msub><mi>H</mi><mi>n</mi></msub><msup><mrow><msup><mrow/><mi>b</mi></msup></mrow><mrow><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow></msup><mo>,</mo><msub><mi>L</mi><mn>3</mn></msub><mo>=</mo><msub><mi>D</mi><mi>n</mi></msub><msup><mrow/><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>–</mo><msub><mi>g</mi><mi>J</mi></msub><msub><mi>h</mi><mi>J</mi></msub><msub><mi>H</mi><mi>n</mi></msub><msup><mrow><msup><mrow/><mi>a</mi></msup></mrow><mrow><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow></msup><mo>,</mo><msub><mi>L</mi><mn>4</mn></msub><mo>=</mo><msub><mi>g</mi><mi>J</mi></msub><msub><mi>D</mi><mi>n</mi></msub><msup><mrow/><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>–</mo><msub><mi>h</mi><mi>J</mi></msub><msub><mi>H</mi><mi>n</mi></msub><msup><mrow><msup><mrow/><mi>b</mi></msup></mrow><mrow><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow></msup><mo>,</mo></mrow></math> where, D n (1) ( y )= J n '( y )/ J n ( y ), D n (2) ( y )= Y n '( y )/ Y n ( y ) and D n (3) ( y )= H n (1) '( y )/ H n (1) ( y ) are the logarithmic derivatives of the Bessel functions, that are used in computational calculation to overcome the overflow problems for high-order or large-argument cylindrical Bessel functions. A recursive method is employed to get H n a (1) and H n b (1) [ 25 25 ], expressed by the following recursive equations http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10334406&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10334403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10334406&type= where, A n (1) = B n (1) =0 and H n a (1) = H n b (1) = J n '( η j )/ J n ( η j ) are initial values. R n ( j ) is expressed as <math display="block" id="M9"><mrow><msubsup><mi>R</mi><mi>n</mi><mrow><mrow><mo>(</mo><mi>j</mi><mo>)</mo></mrow></mrow></msubsup><mo>=</mo><mrow><mo>[</mo><mrow><msub><mi>J</mi><mi>n</mi></msub><mrow><mo>(</mo><mrow><msub><mi>β</mi><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⋅</mo><msub><mi>Y</mi><mi>n</mi></msub><mrow><mo>(</mo><mrow><msub><mi>η</mi><mi>j</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mo>/</mo><mrow><mo>[</mo><mrow><msub><mi>Y</mi><mi>n</mi></msub><mrow><mo>(</mo><mrow><msub><mi>β</mi><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⋅</mo><msub><mi>J</mi><mi>n</mi></msub><mrow><mo>(</mo><mrow><msub><mi>η</mi><mi>j</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mo>.</mo></mrow></math> The D n (1) ( y ), T n ( y ) and R n ( y ) functions are calculated with the following recursive relations <math display="block" id="M10"><mtable columnalign="left"><mtr><mtd><msubsup><mi>D</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mi>ρ</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>/</mo><mi>ρ</mi><mo>−</mo><msubsup><mi>D</mi><mi>n</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo><mi>l</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3.</mn></mtd></mtr><mtr><mtd><msub><mi>T</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>T</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>.</mo><mfrac><mrow><msubsup><mi>D</mi><mi>n</mi><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mo>/</mo><mi>ρ</mi></mrow><mrow><msubsup><mi>D</mi><mi>n</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mo>/</mo><mi>ρ</mi></mrow></mfrac></mtd></mtr><mtr><mtd><msubsup><mi>R</mi><mi>n</mi><mi>j</mi></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>R</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mi>j</mi></msubsup><mo>.</mo><mfrac><mrow><msubsup><mi>D</mi><mi>n</mi><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>β</mi><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mo>/</mo><msub><mi>β</mi><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mrow><msubsup><mi>D</mi><mi>n</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>β</mi><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mo>/</mo><msub><mi>β</mi><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac><mo>.</mo><mfrac><mrow><msubsup><mi>D</mi><mi>n</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>η</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mo>/</mo><msub><mi>η</mi><mi>j</mi></msub></mrow><mrow><msubsup><mi>D</mi><mi>n</mi><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>η</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mo>/</mo><msub><mi>η</mi><mi>j</mi></msub></mrow></mfrac><mo>,</mo></mtd></mtr></mtable></math> where D n (1) ( y ) is evaluated using a downward recurrence formula. The initial value for the downward recurrence formula is generated using the method provided by Lentz [ 26 26 ]. The remaining four functions are computed using the upward recurrence formulas with initial values obtained from Matlab programs. The electric field can be divided into incident part E i and scattered part E s . So, the SW is defined as <math display="block" id="M11"><mrow><mi>σ</mi><mo>=</mo><munder><mrow><mi>lim</mi></mrow><mrow><mi>ρ</mi><mo>→</mo><mi>∞</mi></mrow></munder><mrow><mo>[</mo><mrow><mn>2</mn><mtext>π</mtext><mi>ρ</mi><mfrac><mrow><msup><mrow><mrow><mo>|</mo><mrow><msub><mi>E</mi><mtext>s</mtext></msub></mrow><mo>|</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mrow><mrow><mo>|</mo><mrow><msub><mi>E</mi><mtext>i</mtext></msub></mrow><mo>|</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow><mo>]</mo></mrow><mo>.</mo></mrow></math> The relationship between σ and the scattering coefficients is <math display="block" id="M12"><mrow><mi>σ</mi><mo>=</mo><munder><mrow><mi>lim</mi></mrow><mrow><mi>ρ</mi><mo>→</mo><mi>∞</mi></mrow></munder><mrow><mo>[</mo><mrow><mn>2</mn><mtext>π</mtext><mi>ρ</mi><mrow><mo>(</mo><mrow><msub><mi>I</mi><mn>1</mn></msub><mo>+</mo><msub><mi>I</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mo>.</mo></mrow></math> For TM waves <math display="block" id="M13"><mtable columnalign="left"><mtr><mtd><msub><mi>I</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>2</mn><mrow><msub><mi>k</mi><mn>0</mn></msub><mtext>π</mtext><mi>r</mi></mrow></mfrac><msup><mrow><mo>|</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></munderover><mrow><msub><mi>b</mi><mi>n</mi></msub><mi>cos</mi><mo stretchy="false">(</mo><mi>n</mi><mi>φ</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow><mo>|</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><msub><mi>I</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>2</mn><mrow><msub><mi>k</mi><mn>0</mn></msub><mtext>π</mtext><mi>r</mi></mrow></mfrac><msup><mrow><mo>|</mo><mrow><mn>2</mn><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></munderover><mrow><msub><mi>a</mi><mi>n</mi></msub><mi>sin</mi><mo stretchy="false">(</mo><mi>n</mi><mi>φ</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>.</mo></mtd></mtr></mtable></math> For transverse electric (TE) waves, the coefficients of the scattered field, a n and b n , are [ 9 9 ] <math display="block" id="M14"><mtable columnalign="left"><mtr><mtd><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>T</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>⋅</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><msub><mi>L</mi><mn>5</mn></msub><msub><mi>L</mi><mn>2</mn></msub><mo>−</mo><msub><mi>g</mi><mi>J</mi></msub><msup><mi>n</mi><mn>2</mn></msup><msup><mrow><mi>cos</mi></mrow><mn>2</mn></msup><mi>θ</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msubsup><mi>w</mi><mi>J</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><msub><mi>L</mi><mn>2</mn></msub><msub><mi>L</mi><mn>3</mn></msub><mo>−</mo><msub><mi>g</mi><mi>J</mi></msub><msup><mi>n</mi><mn>2</mn></msup><msup><mrow><mi>cos</mi></mrow><mn>2</mn></msup><mi>θ</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msubsup><mi>w</mi><mi>J</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><mo>−</mo><mi>i</mi><msub><mi>g</mi><mi>J</mi></msub><mrow><mo>(</mo><mrow><msubsup><mi>w</mi><mi>J</mi><mn>2</mn></msubsup><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>y</mi><mi>n</mi><mi>cos</mi><mi>θ</mi><msub><mi>T</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><msub><mi>L</mi><mn>1</mn></msub></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><msub><mi>L</mi><mn>2</mn></msub><msub><mi>L</mi><mn>3</mn></msub><mo>−</mo><msub><mi>g</mi><mi>J</mi></msub><msup><mi>n</mi><mn>2</mn></msup><msup><mrow><mi>cos</mi></mrow><mn>2</mn></msup><mi>θ</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msubsup><mi>w</mi><mi>J</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow></mfrac><mo>,</mo></mtd></mtr></mtable></math> where <math display="block" id="M15"><mrow><msub><mi>L</mi><mn>5</mn></msub><mo>=</mo><mfrac><mrow><msub><mi>J</mi><mi>n</mi></msub><msup><mrow/><mo>′</mo></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mrow><msub><mi>J</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><msub><mi>g</mi><mi>J</mi></msub><msub><mi>h</mi><mi>J</mi></msub><msubsup><mi>H</mi><mi>n</mi><mrow><mi>a</mi><mrow><mo>(</mo><mi>J</mi><mo>)</mo></mrow></mrow></msubsup><mo>,</mo></mrow></math> and <math display="block" id="M16"><mtable columnalign="left"><mtr><mtd><msub><mi>I</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>2</mn><mrow><msub><mi>k</mi><mn>0</mn></msub><mtext>π</mtext><mi>r</mi></mrow></mfrac><msup><mrow><mo>|</mo><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></munderover><mrow><msub><mi>a</mi><mi>n</mi></msub><mi>cos</mi><mo stretchy="false">(</mo><mi>n</mi><mi>φ</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow><mo>|</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><msub><mi>I</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>2</mn><mrow><msub><mi>k</mi><mn>0</mn></msub><mtext>π</mtext><mi>r</mi></mrow></mfrac><msup><mrow><mo>|</mo><mrow><mn>2</mn><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></munderover><mrow><msub><mi>b</mi><mi>n</mi></msub><mi>sin</mi><mo stretchy="false">(</mo><mi>n</mi><mi>φ</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>.</mo></mtd></mtr></mtable></math> All the SW values provided in the following sections are represented in normalized form, 10 lg( σ / λ ). 2.3. Validation of the algorithm at high frequency 2 Accuracy of the eigenfunction expansion method used in this work has always been doubted when the size parameter is of high value ( x j =2π r j / λ ) [ 27 27 ]. We compare the results of this algorithm with those of an asymptotic solution [ 28 28 ] for SWs of a dielectric layer-coated PEC cylinder in the TM case, as shown in Fig. 3. Fig. 3. The cylinder radius is 10 m and the layer is 0.08 m thick. The results of the eigenfunction expansion method and the Geometrical Optics solution (GO, an asymptotic solution in deep lit region) agree well at 40 GHz and φ > 90°; but are deviating from each other with decreasing frequencies, just because that the GO solution appears to be too simple for the issue. We note that for the TE case the two kinds of results are similar (not shown). Therefore, the calculation results of this work with r J =10 m can be reliable at high frequencies. The inaccuracy in the eigenfunction expansion results mainly originates from bad convergence of the Bessel functions with large arguments. However, as logarithmic derivatives of the Bessel functions have replaced the Bessel functions, the recursive relations with reliable initial values shall enabled us to obtain numerical values with higher orders, thereby the coefficients ( a n and b n ) can be truncated as tiny as 10 −15 in the calculation, so as to ensure accuracy of the results. The truncation order for relatively high frequency is provided in Section 3. Asymptotic solutions have been widely recognized in high frequency EMW scattering problems, but their complicity increases with the number and thickness of the coating layers. Here, the plasma around the cylinder is of 8-m thickness, divided into 100 layers. Thus, eigenfunction expansion is the right method for its unified form in the multilayer situation. 3 Results and discussion 1 3.1. Electron density distribution of the plasma 2 Figure 4 Figure 4 shows electron density distribution of the plasma induced by β-particles from the 90 Sr coating in radioactivity of 1 and 10 Ci/cm 2 , assuming a long cylinder in radiuses of 0.5 and 2 m, respectively. For the cylinder of 0.5-m radius, at 1 Ci/cm 2 , the electron density of the plasma is 1.56×10 9 cm −3 adjacent to the 90 Sr coating, while it is 6.89×10 6 cm −3 at 8 m from the coating. As the amount of electron-ion pairs in a unit time increases with the radioactivity, the electron density of the plasma increases accordingly [ 24 24 ]. At 10 Ci/cm 2 , the corresponding electron densities are 4.95×10 9 and 2.18×10 7 cm −3 , respectively. Compared to the maximum value, the electron densities are reduced by 2 orders of magnitude at 8 m from the 90 Sr coating for both radioactivities. The plasma around the cylinder of 2-m radius yields greater electron density at the same radioactivity and distance, while similar density-decreasing behavior with increasing distance up to 8 m. Therefore, the influence of plasma on the EMW propagation at 8 m from the 90 Sr coating can be ignored and 8 m is set as the thickness of the plasma surrounding the cylinder target in the calculation. 3.2. The mono-static SW 2 In order to clearly differentiate between the SWs of coated and uncoated cylindrical targets, the reduction of SW is used to display the data http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10334433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10334430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10334433&type= The mono-static d SW s of the 90 Sr-coated cylindrical targets ( r 1 =0.5 m and 2 m) verses EMW frequency (0.1–40 GHz) at normal incidence are illustrated in Fig. 5. Fig. 5. The insets show the SWs by the corresponding cylinders made with PEC. The mono-static SW by a PEC cylinder increases smoothly with the EMW frequency for TM waves and alternates for TE waves. The oscillations in the TE case arise from the creeping waves of TE-polarization. In the backscattering region (from φ =90º to φ =180º), the scattered waves are composed of creeping and reflected waves for TE-polarization, but mainly reflected waves for TM-polarization, and the phase difference between the two waves changes significantly with the EMW frequency [ 29 29 , 30 30 ]. The SW by the 90 Sr-coated cylindrical target decreases remarkably as the EMW energy is absorbed by the plasma. For the target of 0.5-m radius, the mono-static SW decreases over 0.85 dB at the radioactivity of 1 Ci/cm 2 in the frequency range of 0.5–7 GHz and more than 2.80 dB at 10 Ci/cm 2 in 0.8–4 GHz. The full-width at half-maximum (FWHM) of all d SW curves can reach 27 GHz. Having the greater plasma density, the mono-static SWs of the 90 Sr coated target of 2-m radius decreased more than those in the 0.5 m case, by 1.25 dB at 1 Ci/cm 2 in 0.6–6 GHz and 3.95 dB at10 Ci/cm 2 in 0.6–4 GHz. In addition, the FWHM can also reach 27 GHz. The truncation orders against the frequency in the TM case are shown in Fig. 6. Fig. 6. The orders for different radioactivities are close to each other for cylinders of the same radius, while they increase with the radius, but the increment is not large as the cylinder radius is much smaller than the surrounded plasma thickness. The orders for both radiuses increase linearly with the frequency, reaching 7245 for 0.5-m radius and 8534 for 2.0-m radius. The truncation order is mainly determined by the size parameter x J [ 31 31 ], and it can be predicted that it shall not exceed 20000 at 100 GHz for a cylinder of 2-m radius. The truncation orders in the TE case are identical to the TM results. 3.3. The bi-static SW at normal incidence 2 The bi-static SWs of the cylindrical target ( r 1 =0.5 m) with and without the 90 Sr coating for the EMW at normal incidence are calculated in the frequency range of 0.1–40 GHz. The data is displayed in 2D-images with scaled colors, as shown in Fig. 7. Fig. 7. For TM waves, the bi-static SW by a PEC cylinder increases steadily with the frequency and scattering angle in the back scattering region ( Fig. 7a Fig. 7a ). However, it gets complicated in the forward scattering region (from φ =0º to φ =90º). The reason is that several modes of the EMW rays are involved in that scattered filed. Similar phenomenon is observed in the TE case ( Fig. 7c Fig. 7c ). After the PEC cylinder is coated with a 10 Ci/cm 2 90 Sr layer, SW by the scatter decreases significantly in the back scattering region ( Fig. 7b Fig. 7b ). Comparing Figs. 7(b) and 7(d) Figs. 7(b) and 7(d) , the 90 Sr reduction effects in the TE and TM cases are similar. The bi-static d SW s of the 10 Ci/cm 2 90 Sr-coated cylinder are investigated as functions of scattering angle φ at four typical EMW frequencies: 0.2, 1.5, 27 and 40 GHz, as shown in Fig. 8. Fig. 8. For TM waves, the d SW s increase slightly with the scattering angle in the back scattering region. The d SW s at φ =90° are −2.87, −3.06, −1.54 and −0.96 dB, for 0.2, 1.5, 27 and 40 GHz, respectively. For TE waves, the d SW s show a similar trend but with more intense oscillations, especially at low frequencies, which can also be attributed to the phase difference between the creeping and reflected waves. 3.4. The bi-static SW at oblique incidence 2 For bi-static SWs are calculated with obliquely incident EMWs at 1.5 GHz and the cylinder of 0.5-m radius, with and without 90 Sr coating (10 Ci/cm 2 ), in the back scattering region in incident angles of 0º < θ < 90º, as depicted in Fig. 9. Fig. 9. For TM waves, the SW by the PEC cylinder does not change with the incident angle θ , while the SW by the 90 Sr-coated decreases remarkably with θ . The reason is that the propagation distance of the EMW in the plasma increases with decreasing incident angles. For TE waves, the SW by the PEC cylinder oscillates with the changes in θ and φ due to the creeping waves; while the SW by the coated one shows a downward trend for small incident angles ( θ < 50º), similar to that in the TM case. The bi-static d SW s of the 10 Ci/cm 2 90 Sr coated cylindrical target are studied as functions of θ at three typical scattering angles: 180°, 135°and 90°, as shown in Fig. 10. Fig. 10. The EMW frequency is 1.5 GHz. For TM waves, the d SW s of the 10 Ci/cm 2 90 Sr-coated cylinder decrease with θ and display close values for different scattering angles. The d SW s reach −5 ~ −30 dB when θ < 30º. For TE waves, the d SW s decrease with θ in most of the incident angles but oscillate in the range of 55º < θ < 75º, which can also be attributed to the creeping waves. Moreover, the d SW s for different scattering angles are no longer close to each other. When θ < 30º, the d SW s reach −9~ −38 dB, −6~ −31 dB and−3~ −20 dB at θ = 180°, 135°and 90°, respectively. 4 Conclusion 1 In this paper, influences of 90 Sr coating on electromagnetic scattering from a cylindrical target are investigated and evaluated. The results demonstrate that the SW of a cylindrical target with 90 Sr coating decreases effectively. The mono-static SW by the 10 Ci/cm 2 90 Sr-coated cylindrical target reduces by 2.80 dB and 3.95 dB for 0.5-m and 2-m radiuses, respectively, in the frequency range of 0.4–40 GHz. The bi-static SWs by 10 Ci/cm 2 90 Sr-coated cylindrical target of 0.5-m radius at normal incidence reduce by 2.87, 3.06, 1.54 and 0.96 dB in the back scattering region, for EMW frequencies of 0.2, 1.5, 27 and 40 GHz, respectively. In addition, the stealth effect can be significantly improved for oblique incidence of EMWs. The bi-static SW by the cylindrical target with 10 Ci/cm 2 90 Sr coating reduces by 3–20 dB when the incident angle θ is smaller than 30°. The results indicate that 90 Sr-coated cylindrical objects have significant stealth effect. References G.X. Wu , S. Liu , S.Y. Zhong . Numerical analysis of propagation characteristics of electromagnetic wave in lossy left-handed material media . Optik , 2014 , 125 : 4233 - 4237 . DOI: 10.1016/j.ijleo.2014.04.018 http://doi.org/10.1016/j.ijleo.2014.04.018 Numerical analysis of propagation characteristics of electromagnetic wave in lossy left-handed material media S. Shen , M. Chung . FDTD simulations on radar cross sections of metal cone and plasma covered metal cone . Vacuum , 2012 , 86 : 970 - 984 . DOI: 10.1016/j.vacuum.2011.08.016 http://doi.org/10.1016/j.vacuum.2011.08.016 FDTD simulations on radar cross sections of metal cone and plasma covered metal cone B.P. Li , C.G. Wang , W. Wang , et al . Electromagnetic wave absorption properties of composites with micro-sized magnetic particles dispersed in amorphous carbon . J Magn Magn Mater , 2014 , 365 : 40 - 44 . DOI: 10.1016/j.jmmm.2014.01.015 http://doi.org/10.1016/j.jmmm.2014.01.015 Electromagnetic wave absorption properties of composites with micro-sized magnetic particles dispersed in amorphous carbon R.X. Lord , On the electromagnetic theory of light . Philos Mag , 1881 , 12 : 81 - 101 . DOI: 10.1080/14786448108627074 http://doi.org/10.1080/14786448108627074 On the electromagnetic theory of light J.R Wait , Scattering of a plane wave from a circular dielectric cylinder at oblique incidence . Can J Phys , 1955 , 33 : 189 - 195 . DOI: 10.1139/p55-024 http://doi.org/10.1139/p55-024 Scattering of a plane wave from a circular dielectric cylinder at oblique incidence C.F. Bohren , D.R. Huffman , Absorbing and scattering of light by small particles . New York (U S A) : Wiley , 1983 , 194 - 199 . Absorbing and scattering of light by small particles M. Barabas , Scattering of a plane wave by a radially stratified tilted cylinder . Opt Soc Am A , 1987 , 4 : 2240 - 2248 . DOI: 10.1364/JOSAA.4.002240 http://doi.org/10.1364/JOSAA.4.002240 Scattering of a plane wave by a radially stratified tilted cylinder Z.S. Wu , Y.P. Wang , Electromagnetic scattering for multilayered sphere-recursive algorithms . Radio Sci , 1991 , 26 : 1393 - 1401 . DOI: 10.1029/91RS01192 http://doi.org/10.1029/91RS01192 Electromagnetic scattering for multilayered sphere-recursive algorithms I. Gurwich , N. Shiloah , M. Klerman , The recursive algorithm for electromagnetic scattering by tilted infinite circular multilayered cylinder . J Quant Spectrosc Ra , 1999 , 63 : 217 - 229 . DOI: 10.1016/S0022-4073(99)00017-5 http://doi.org/10.1016/S0022-4073(99)00017-5 The recursive algorithm for electromagnetic scattering by tilted infinite circular multilayered cylinder H.F. Jiang , X.E. Han , R.X. Li . Improved algorithm for electromagnetic scattering of plane waves by a radially stratified tilted cylinder and its application . Opt Commun , 2006 , 266 : 13 - 18 . DOI: 10.1016/j.optcom.2006.04.041 http://doi.org/10.1016/j.optcom.2006.04.041 Improved algorithm for electromagnetic scattering of plane waves by a radially stratified tilted cylinder and its application N. Iqbal , P.K. Choudhury , P.S. Menon , Scattering from silver metal cylinder due to L-nihility coated with conducting sheath helix embedded dielectric medium . J Electromagnet Wave , 2015 , 29 ( 10 ), 1354 - 1374 . DOI: 10.1080/09205071.2015.1044128 http://doi.org/10.1080/09205071.2015.1044128 Scattering from silver metal cylinder due to L-nihility coated with conducting sheath helix embedded dielectric medium R. Paknys , N. Wang . Creeping wave propagation constants and modal impedance for a dielectric coated cylinder . IEEE Trans , 1986 , 34 : 674 - 680 . DOI: 10.1109/TAP.1986.1143865 http://doi.org/10.1109/TAP.1986.1143865 Creeping wave propagation constants and modal impedance for a dielectric coated cylinder J. Sun , W. Sun . Directive electromagnetic radiation of a line source scattered by a conducting cylinder coated with left-handed metamaterial . Microw Opt Techn Let , 2005 , 47 ( 3 ): 274 - 279 . DOI: 10.1002/mop.21145 http://doi.org/10.1002/mop.21145 Directive electromagnetic radiation of a line source scattered by a conducting cylinder coated with left-handed metamaterial S. Arslanagic , R.W. Ziolkowski , O.B. Breinbjerg , Excitation of an electrically small metamaterial-coated cylinder by an arbitrarily located line source . Microw Opt Techn Let , 2006 , 48 ( 12 ): 2598 - 2606 . DOI: 10.1002/mop.21990 http://doi.org/10.1002/mop.21990 Excitation of an electrically small metamaterial-coated cylinder by an arbitrarily located line source A. Ghaffar , M.Z. Yaqoob , A.S. Majeed , et al . Scattering of electromagnetic wave from perfect electromagnetic conductor cylinders placed in un-magnetized isotropic plasma medium . Optik , 2014 , 125 ( 17 ): 4779 - 4783 . DOI: 10.1016/j.ijleo.2014.04.061 http://doi.org/10.1016/j.ijleo.2014.04.061 Scattering of electromagnetic wave from perfect electromagnetic conductor cylinders placed in un-magnetized isotropic plasma medium V.L. Ginzburg . The propagation of electromagnetic waves in plasmas . New York (U S A) : Pergamon Press , 1970 , 63 - 83 . The propagation of electromagnetic waves in plasmas A. Hussain , S.M. Mirza , N.M. Mirza , et al . Beta-efficiency of a typical gas-flow ionization chamber using GEANT4 Monte Carlo simulations . Nucl Technol Radiat , 2011 , 26 ( 3 ): 193 - 200 . DOI: 10.2298/NTRP1103193H http://doi.org/10.2298/NTRP1103193H Beta-efficiency of a typical gas-flow ionization chamber using GEANT4 Monte Carlo simulations S. Saghamanesh , A. Karimian , M. Abdi . Absorbed dose assessment of cardiac and other tissues around the cardiovascular system in brachytherapy with 90Sr/90Y source by Monte Carlo simulation . Radiat Prot Dosim , 2011 , 147 : 296 - 299 . DOI: 10.1093/rpd/ncr347 http://doi.org/10.1093/rpd/ncr347 Absorbed dose assessment of cardiac and other tissues around the cardiovascular system in brachytherapy with 90Sr/90Y source by Monte Carlo simulation A. Ho , S.S. Hakmana Witharana , G. Jonkmans , et al . Detection of bremsstrahlung radiation of 90Sr-90Y for emergency lung counting . Radiat Prot Dosim , 2012 , 151 : 443 - 449 . DOI: 10.1093/rpd/ncs029 http://doi.org/10.1093/rpd/ncs029 Detection of bremsstrahlung radiation of 90Sr-90Y for emergency lung counting A.A. Kriss , D.M. Hamby . Beta spectroscopy with a large-area avalanche photodiode module and a plastic scintillator . Nucl Instrum Methods , 2004 , A 525 : 553 - 559 . DOI: 10.1016/j.nima.2004.02.016 http://doi.org/10.1016/j.nima.2004.02.016 Beta spectroscopy with a large-area avalanche photodiode module and a plastic scintillator Bethesda . Average energy required to produce an ion pair, ICRU, ICRU 31 , 1997 . Average energy required to produce an ion pair, ICRU, ICRU 31 V.D. Zvorykin , A.A. Lonin , A.O. Levchenko , et al . Effects of picoseconds terawatt UV laser beam filamentation and a repetitive pulse train on creation of prolonged plasma channels in atmospheric air . Nucl Instrum Meth B , 2013 , 309 : 218 - 222 . DOI: 10.1016/j.nimb.2013.02.030 http://doi.org/10.1016/j.nimb.2013.02.030 Effects of picoseconds terawatt UV laser beam filamentation and a repetitive pulse train on creation of prolonged plasma channels in atmospheric air F. He , C.L. Liu . Studies on fluid model for numerical simulation of gas discharges in color plasma displays . Nucl Sci Tech , 2005 , 2 : 120 - 125 . Studies on fluid model for numerical simulation of gas discharges in color plasma displays W. Liu , J.Z. Zhu , C. Cui , et al . Influence of plasma induced by radionuclide layer on the radar cross section of spherical objects . Nucl Sci Tech , 2015 , 26 ( 4 ): 040502 . DOI: 10.13538/j.1001-8042/nst.26.040502 http://doi.org/10.13538/j.1001-8042/nst.26.040502 Influence of plasma induced by radionuclide layer on the radar cross section of spherical objects H. Jiang , X. Han , R. Li . Improved algorithm for electromagnetic scattering of plane waves by a radially stratified tilted cylinder and its application . Opt Commun , 2006 , 26 ( 4 ): 13 - 18 . DOI: 10.1016/j.optcom.2006.04.041 http://doi.org/10.1016/j.optcom.2006.04.041 Improved algorithm for electromagnetic scattering of plane waves by a radially stratified tilted cylinder and its application W.J. Lentz . Generating Bessel functions in Mie scattering calculations using continued fractions . Appl Optics , 1976 , 15 ( 3 ): 668 - 671 . DOI: 10.1364/AO.15.000668 http://doi.org/10.1364/AO.15.000668 Generating Bessel functions in Mie scattering calculations using continued fractions L. Kai , A. D’Alessio . Finely stratified cylinder model for radially inhomogeneous cylinders normally irradiated by electromagnetic plane waves . Appl Optics , 1995 , 34 ( 24 ): 5520 - 5530 . DOI: 10.1364/AO.34.005520 http://doi.org/10.1364/AO.34.005520 Finely stratified cylinder model for radially inhomogeneous cylinders normally irradiated by electromagnetic plane waves K. Goto , L.H. Loc . Aymptotic solutions for scattered field by a coated conducting cylinder . IEICE Electron Expr , 2013 , 403 10 ( 6 ): 1 - 10 . DOI: 10.1587/elex.10.20130139 http://doi.org/10.1587/elex.10.20130139 Aymptotic solutions for scattered field by a coated conducting cylinder G.T. Ruck , D.E. Barrick , W.D. Stuart . Radar Cross Section Handbook . Vols. 1&2 , New York (USA) : Plenum Press , 1970 . Radar Cross Section Handbook I.A. Kotelnikov , G.V. Stupakov . Electromagnetic surface waves on a conducting cylinder . Phys Lett A , 2015 , 379 : 1187 - 1195 . DOI: 10.1016/j.physleta.2015.02.013 http://doi.org/10.1016/j.physleta.2015.02.013 Electromagnetic surface waves on a conducting cylinder W.J. Wiscombe . Improved Mie scattering algorithms . Appl Optics , 1980 , 19 ( 9 ): 1505 - 1509 . DOI: 10.1364/AO.19.001505 http://doi.org/10.1364/AO.19.001505 Improved Mie scattering algorithms 10.1007/s41365-017-0196-9 This work was supported by the 863 Program through the Ministry of Science and Technology (No. 2006AA03Z458) and the National Natural Science Foundation of China (Nos. 10904061 and 50977042). 2016-01-29 2016-05-24 2016-05-29 2017-03-01 2017-04-01 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. 2017 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/s41365-017-0196-9 http://dx.doi.org/10.1007/s41365-017-0196-9 . 2017 Nuclear Science and Techniques 04 28