... | ... | @@ -25,33 +25,47 @@ The Energy Loss |
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The energy loss is simulated using the Bethe-Bloch equation.
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latexmath:[\[\label{eq:dEdx}
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-\frac{\mathrm{d} E}{\mathrm{d} x}=\frac{K z^2 Z}{A \beta^2}\left[\frac{1}{2} \ln{\frac{2 m_e c^2\beta^2 \gamma^2 Tmax}{I^2}}-\beta^2 \right],\]]
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where latexmath:[$Z$] is the atomic number of absorber, latexmath:[$A$]
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is the atomic mass of absorber, latexmath:[$m_e$] is the electron mass,
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latexmath:[$z$] is the charge number of the incident particle,
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latexmath:[$K=4\pi N_Ar_e^2m_ec^2$], latexmath:[$r_e$] is the classical
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electron radius, latexmath:[$N_A$] is the Avogadro’s number,
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latexmath:[$I$] is the mean excitation energy. latexmath:[$\beta$] and
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latexmath:[$\gamma$] are kinematic variables. latexmath:[$T_{max}$] is
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[latexmath]
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++++
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-\frac{\mathrm{d} E}{\mathrm{d} x}=\frac{K z^2 Z}{A \beta^2}\left[\frac{1}{2} \ln{\frac{2 m_e c^2\beta^2 \gamma^2 Tmax}{I^2}}-\beta^2 \right],
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++++
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where latexmath:[Z] is the atomic number of absorber, latexmath:[A]
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is the atomic mass of absorber, latexmath:[m_e] is the electron mass,
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latexmath:[z] is the charge number of the incident particle,
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latexmath:[K=4\pi N_Ar_e^2m_ec^2], latexmath:[r_e] is the classical
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electron radius, latexmath:[N_A] is the Avogadro’s number,
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latexmath:[I] is the mean excitation energy. latexmath:[\beta] and
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latexmath:[\gamma] are kinematic variables. latexmath:[T_{max}] is
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the maximum kinetic energy which can be imparted to a free electron in a
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single collision.
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latexmath:[\[T_{max}=\frac{2m_ec^2\beta^2\gamma^2}{1+2\gamma m_e/M+(m_e/M)^2},\]]
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where latexmath:[$M$] is the incident particle mass.
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[latexmath]
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++++
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T_{max}=\frac{2m_ec^2\beta^2\gamma^2}{1+2\gamma m_e/M+(m_e/M)^2},
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++++
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where latexmath:[M] is the incident particle mass.
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The stopping power is compared with PSTAR program of NIST in
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Figure [dEdx].
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Figure 1.
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image:figures/partmatter/dEdx.png[The comparison of stopping power with
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PSTAR. ,scaledwidth=50.0%]
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.Figure 1 : The comparison of stopping power with PSTAR.
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image:figures/partmatter/dEdx.png[width=500]
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Energy straggling: For relatively thick absorbers such that the number
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of collisions is large, the energy loss distribution is shown to be
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Gaussian in form. For non-relativistic heavy particles the spread
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latexmath:[$\sigma_0$] of the Gaussian distribution is calculated by:
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latexmath:[\[gma_0^2=4\pi N_Ar_e^2(m_ec^2)^2\rho\frac{Z}{A}\Delta s,\]]
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where latexmath:[$\rho$] is the density, latexmath:[$\Delta s$] is the
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latexmath:[\sigma_0] of the Gaussian distribution is calculated by:
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[latexmath]
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++++
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\sigma_0^2=4\pi N_Ar_e^2(m_ec^2)^2\rho\frac{Z}{A}\Delta s,
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++++
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where latexmath:[\rho] is the density, latexmath:[\Delta s] is the
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thickness.
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[[the-coulomb-scattering]]
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... | ... | @@ -62,74 +76,121 @@ The Coulomb scattering is treated as two independent events: the |
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multiple Coulomb scattering and the large angle Rutherford scattering. +
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Using the distribution given in Classical Electrodynamics, by J. D.
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Jackson, the multiple- and single-scattering distributions can be
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written: latexmath:[\[\label{eq:PM}
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P_M(\alpha) \;\mathrm{d} \alpha=\frac{1}{\sqrt{\pi}}e^{-\alpha^2}\;\mathrm{d}\alpha,\]]
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latexmath:[\[\label{eq:Ps}
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P_S(\alpha) \;\mathrm{d} \alpha=\frac{1}{8 \ln(204 Z^{-1/3})} \frac{1}{\alpha^3}\;\mathrm{d}\alpha,\]]
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written:
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[latexmath]
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++++
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P_M(\alpha) \;\mathrm{d} \alpha=\frac{1}{\sqrt{\pi}}e^{-\alpha^2}\;\mathrm{d}\alpha,
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++++
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[latexmath]
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++++
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P_S(\alpha) \;\mathrm{d} \alpha=\frac{1}{8 \ln(204 Z^{-1/3})} \frac{1}{\alpha^3}\;\mathrm{d}\alpha,
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++++
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where
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latexmath:[$\alpha=\frac{\theta}{<\Theta^2>^{1/2}}=\frac{\theta}{\sqrt 2 \theta_0}$].
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latexmath:[\alpha=\frac{\theta}{<\Theta^2>^{1/2}}=\frac{\theta}{\sqrt 2 \theta_0}].
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The transition point is
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latexmath:[$\theta=2.5 \sqrt 2 \theta_0\approx3.5 \theta_0$],
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latexmath:[\[\label{eq:Multiple}
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\theta_0=\frac{{13.6}{MeV}}{\beta c p} z \sqrt{\Delta s/X_0} [1+0.038 \ln(\Delta s/X_0)],\]]
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where latexmath:[$p$] is the momentum, latexmath:[$\Delta s$] is the
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step size, and latexmath:[$X_0$] is the radiation length.
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latexmath:[\theta=2.5 \sqrt 2 \theta_0\approx3.5 \theta_0],
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[latexmath]
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++++
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\theta_0=\frac{{13.6}{MeV}}{\beta c p} z \sqrt{\Delta s/X_0} [1+0.038 \ln(\Delta s/X_0)],
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++++
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where latexmath:[p] is the momentum, latexmath:[\Delta s] is the
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step size, and latexmath:[X_0] is the radiation length.
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[[multiple-coulomb-scattering]]
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Multiple Coulomb Scattering
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^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Generate two independent Gaussian random variables with mean zero and
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variance one: latexmath:[$z_1$] and latexmath:[$z_2$]. If
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latexmath:[$z_2 \theta_0>3.5 \theta_0$], start over. Otherwise,
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latexmath:[\[\label{eq:Multiplex}
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x=x+\Delta s p_x+z_1 \Delta s \theta_0/\sqrt{12}+z_2 \Delta s \theta_0/2,\]]
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latexmath:[\[\label{eq:Multiplepx}
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p_x=p_x+z_2 \theta_0.\]] Generate two independent Gaussian random
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variables with mean zero and variance one: latexmath:[$z_3$] and
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latexmath:[$z_4$]. If latexmath:[$z_4 \theta_0>3.5 \theta_0$], start
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over. Otherwise, latexmath:[\[\label{eq:Multipley}
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y=y+\Delta s p_y+z_3 \Delta s \theta_0/\sqrt{12}+z_4 \Delta s \theta_0/2,\]]
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latexmath:[\[\label{eq:Multiplepy}
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p_y=p_y+z_4 \theta_0.\]]
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variance one: latexmath:[z_1] and latexmath:[z_2]. If
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latexmath:[z_2 \theta_0>3.5 \theta_0], start over. Otherwise,
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[latexmath]
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++++
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x=x+\Delta s p_x+z_1 \Delta s \theta_0/\sqrt{12}+z_2 \Delta s \theta_0/2,
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++++
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[latexmath]
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++++
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p_x=p_x+z_2 \theta_0.
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++++
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Generate two independent Gaussian random
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variables with mean zero and variance one: latexmath:[z_3] and
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latexmath:[z_4]. If latexmath:[z_4 \theta_0>3.5 \theta_0], start
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over. Otherwise,
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[latexmath]
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++++
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y=y+\Delta s p_y+z_3 \Delta s \theta_0/\sqrt{12}+z_4 \Delta s \theta_0/2,
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++++
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[latexmath]
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++++
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p_y=p_y+z_4 \theta_0.
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++++
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[[large-angle-rutherford-scattering]]
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Large Angle Rutherford Scattering
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Generate a random number latexmath:[$\xi_1$], _if_
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latexmath:[$\xi_1<\frac{\int_{2.5}^\infty P_S(\alpha)d\alpha}{\int_0^{2.5} P_M(\alpha)\;\mathrm{d}\alpha+\int_{2.5}^\infty P_S(\alpha)\;\mathrm{d}\alpha}=0.0047$],
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Generate a random number latexmath:[\xi_1], _if_
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latexmath:[\xi_1<\frac{\int_{2.5}^\infty P_S(\alpha)d\alpha}{\int_0^{2.5} P_M(\alpha)\;\mathrm{d}\alpha+\int_{2.5}^\infty P_S(\alpha)\;\mathrm{d}\alpha}=0.0047],
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sampling the large angle Rutherford scattering. +
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The cumulative distribution function of the large angle Rutherford
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scattering is latexmath:[\[\label{eq:Fa}
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F(\alpha)=\frac{\int_{2.5}^\alpha P_S(\alpha) \;\mathrm{d} \alpha}{\int_{2.5}^\infty P_S(\alpha) \;\mathrm{d} \alpha}=\xi,\]]
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where latexmath:[$\xi$] is a random variable. So
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latexmath:[\[\label{eq:alpha}
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\alpha=\pm 2.5 \sqrt{\frac{1}{1-\xi}}=\pm 2.5 \sqrt{\frac{1}{\xi}}.\]]
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Generate a random variable latexmath:[$P_3$], +
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_if_ latexmath:[$P_3>0.5$]
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latexmath:[\[\theta_{Ru}=2.5 \sqrt{\frac{1}{\xi}} \sqrt{2}\theta_0,\]]
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scattering is
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[latexmath]
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++++
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F(\alpha)=\frac{\int_{2.5}^\alpha P_S(\alpha) \;\mathrm{d} \alpha}{\int_{2.5}^\infty P_S(\alpha) \;\mathrm{d} \alpha}=\xi,
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++++
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where latexmath:[\xi] is a random variable. So
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[latexmath]
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++++
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\alpha=\pm 2.5 \sqrt{\frac{1}{1-\xi}}=\pm 2.5 \sqrt{\frac{1}{\xi}}.
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++++
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Generate a random variable latexmath:[P_3], +
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_if_ latexmath:[P_3>0.5]
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[latexmath]
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++++
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\theta_{Ru}=2.5 \sqrt{\frac{1}{\xi}} \sqrt{2}\theta_0,
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++++
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_else_
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latexmath:[\[\theta_{Ru}=-2.5 \sqrt{\frac{1}{\xi}} \sqrt{2}\theta_0.\]]
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[latexmath]
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++++
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\theta_{Ru}=-2.5 \sqrt{\frac{1}{\xi}} \sqrt{2}\theta_0.
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++++
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The angle distribution after Coulomb scattering is shown in
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Figure [Coulomb]. The line is from Jackson’s formula, and the points are
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simulations with Matlab. For a thickness of latexmath:[$\Delta s=1e-4$]
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latexmath:[$m$], latexmath:[$\theta=0.5349 \alpha$] (in degree).
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Figure 2. The line is from Jackson’s formula, and the points are
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simulations with Matlab. For a thickness of latexmath:[\Delta s=1e-4]
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latexmath:[m], latexmath:[\theta=0.5349 \alpha] (in degree).
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image:figures/partmatter/10steps.png[The comparison of Coulomb
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scattering with Jackson’s book. ]
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.Figure 2: The comparison of Coulomb scattering with Jackson’s book.
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image:figures/partmatter/10steps.png[]
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[[the-flow-diagram-of-collimatorphysics-class-in-opal]]
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The Flow Diagram of _CollimatorPhysics_ Class in OPAL
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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image:figures/partmatter/diagram.png[The diagram of CollimatorPhysics in
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_OPAL_. ,scaledwidth=80.0%]
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.Figure 3: The diagram of CollimatorPhysics in _OPAL_.
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image:figures/partmatter/diagram.png[]
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image:figures/partmatter/Diagram2.png[The diagram of CollimatorPhysics
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in _OPAL_ (continued). ,scaledwidth=60.0%]
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.Figure 4: The diagram of CollimatorPhysics in _OPAL_ (continued).
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image:figures/partmatter/Diagram2.png[]
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[[the-substeps]]
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The Substeps
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... | ... | @@ -139,9 +200,9 @@ Small step is needed in the routine of CollimatorPhysics. |
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If a large step is given in the main input file, in the file
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_CollimatorPhysics.cpp_, it is divided by a integer number
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latexmath:[$n$] to make the step size using for the calculation of
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latexmath:[n] to make the step size using for the calculation of
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collimator physics less than 1.01e-12 s. As shown by
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Figure [diagram,diagram2] in the previous section, first we track one
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Figures 3 and 4 in the previous section, first we track one
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step for the particles already in the collimator and the newcomers, then
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another (n-1) steps to make sure the particles in the collimator
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experience the same time as the ones in the main bunch.
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... | ... | @@ -157,8 +218,8 @@ Available Materials in _OPAL_ |
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.List of materials with their parameters implemented in _OPAL_.
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[cols="^,^,^,^,^,^,^,^,^,^",options="header",]
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|=======================================================================
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|Material |Z |A |latexmath:[$\rho$] [latexmath:[$g/cm^3$]] |X0
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[latexmath:[$g/cm^2$]] |A2 |A3 |A4 |A5 |_OPAL_ Name
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|Material |Z |A |latexmath:[\rho] [latexmath:[g/cm^3]] |X0
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[latexmath:[g/cm^2]] |A2 |A3 |A4 |A5 |_OPAL_ Name
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|Aluminum |13 |26.98 |2.7 |24.01 |4.739 |2766 |164.5 |2.023E-02
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|`Aluminum`
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... | ... | @@ -196,6 +257,10 @@ Available Materials in _OPAL_ |
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Example of an Input File
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~~~~~~~~~~~~~~~~~~~~~~~~
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....
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include::examples/particlematterinteraction.in[]
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....
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FX5 is a slit in x direction, the `APERTURE` is *POSITIVE*, the first
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value in `APERTURE` is the left part, the second value is the right
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part. FX16 is a slit in y direction, the `APERTURE` is *NEGATIVE*, the
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... | ... | @@ -206,16 +271,16 @@ part. |
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A Simple Test
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~~~~~~~~~~~~~
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A cold Gaussian beam with latexmath:[$\sigma_x=gma_y=5$] mm. The
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A cold Gaussian beam with latexmath:[\sigma_x=\sigma_y=5] mm. The
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position of the collimator is from 0.01 m to 0.1 m, the half aperture in
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y direction is latexmath:[$3$] mm. Figure [longcoll] shows the
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y direction is latexmath:[3] mm. Figure 5 shows the
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trajectory of particles which are either absorbed or deflected by a
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copper slit. As a benchmark of the collimator model in _OPAL_,
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Figure [Espectrum] shows the energy spectrum and angle deviation at
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Figure 6 shows the energy spectrum and angle deviation at
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z=0.1 m after an elliptic collimator.
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image:figures/partmatter/longcoll6.png[The passage of protons through
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the collimator. ,scaledwidth=80.0%]
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.Figure 5: The passage of protons through the collimator.
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image:figures/partmatter/longcoll6.png[]
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image:figures/partmatter/spectandscatter.png[The energy spectrum and
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scattering angle at z=0.1 m,scaledwidth=80.0%] |
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.Figure 6: The energy spectrum and scattering angle at z=0.1 m
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image:figures/partmatter/spectandscatter.png[] |