- where α is the fine structure constant and c the velocity of light. - Energy Loss of Ions in Matter. - When a charged particle travels through condensed matter, it loses its kinetic energy gradually by transferring it to the electrons of the medium. - In this chapter we evaluate the energy loss of the particle as a function of its mass and its charge, by studying the modifications that the state of an atom undergoes when a charged particle passes in its vicinity. - The electric potential created by the moving particle appears as a time- dependent perturbation in the atom’s Hamiltonian. - The nucleus and the internal electrons will be treated globally as a core of charge +q, infinitely massive and, therefore, fixed in space. - We consider the x, y plane defined by the trajectory of the particle and the center of gravity of the atom, which is chosen to be the origin, as shown on Fig. - Let R (t) be the position of the particle at time t and r = (x, y, z) the coordinates of the electron of the atom. - The impact parameter is b and the notation is specified in Fig. - The time at which the particle passes nearest to the atom, i.e. - We write E n and | n for the energy levels and corresponding eigenstates of the atom in the absence of an external perturbation.. - 10.1 Energy Absorbed by One Atom. - Write the expression for the time-dependent perturbing potential V ˆ (t) due to the presence of the charged particle.. - 88 10 Energy Loss of Ions in Matter. - |R| and express the result in terms of the coordinates x and y of the electron, and of b, v and t.. - the atom is in a state | i of energy E i . - Using first order time-dependent perturbation theory, write the probability amplitude γ if to find the atom in the final state | f of energy E f after the charged particle has passed (t. - ln z for z 1, and K 0 (z) 2π/z e − z for z 1. - Under what condition on the parameters ω fi , b and v is the transition probability P if. - Give the physical interpretation of the condition derived above. - Show that, given the parameters of the atom, the crucial parameter is the effective interaction time , and give a simple explanation of this effect.. - 10.2 Energy Loss in Matter. - We assume in the following that the Hamiltonian of the atom is of the form. - Definition of the coordinates.. - Using the Thomas–Reiche–Kuhn sum rule, calculate the expectation value δE of the energy loss of the incident particle when it interacts with the atom.. - Let E be the energy of the particle before the interaction. - When these ions traverse condensed matter, they interact with many atoms of the medium, and their energy loss implies some averaging over the random impact parameter b. - where the constant k depends on the nature of the medium.. - Semiconductor detectors used for the identification of the nuclei in nu- clear reactions are based on this result. - In the following example, the ions to be identified are the final state products of a reaction induced by 113 MeV nitrogen ions impinging on a target of silver atoms.. - 10.2 each point represents an event, i.e. - the energy E and energy loss δE of an ion when it crosses a silicon detector. - (a) Calculate the constant k and the theoretical prediction for the energy loss at 60 and 70 MeV. - 90 10 Energy Loss of Ions in Matter. - Energy loss δE versus energy E through a silicon detector, of the final products of a reaction corresponding to 113 MeV nitrogen ions impinging on a target of silver atoms. - what nuclei are effectively produced in the reaction? Justify your answers by putting the points corresponding to E = 50 MeV and E = 70 MeV on the figure.. - 10.3 Solutions. - Section 10.1: Energy Absorbed by One Atom. - The interaction potential between the particle and the atom is the sum of the Coulomb interactions between the particle and the core, and those between the particle and the outer electron:. - To first order in ˆ V , the probability amplitude is γ if = 1. - The condition ω fi τ 1 means that the interaction time τ must be much smaller than the Bohr period ∼ 1/ω fi of the atom. - 92 10 Energy Loss of Ions in Matter. - In the opposite limiting case, where the perturbation is infinitely slow, the atom is not excited.. - the second integral is then zero for symmetry reasons and the first one is easily evaluated, and gives the desired result.. - Section 10.2: Energy Loss in Matter 10.2.1. - The expectation value δE of the energy transferred to the atom is δE. - Making use of the Thomas–Reiche–Kuhn sum rule, we obtain δE = 2Z 1 2 e 4. - where m is the electron mass. - where we see that the product E δE does not depend on the energy of the incident particle, but is proportional to its mass and to the square of its charge.. - We have put the calculated points of the various isotopes on Fig. - Interpretation of the data of Fig. - 94 10 Energy Loss of Ions in Matter. - 10.4 Comments. - In order to calculate the energy loss of an ion in matter, one must integrate the above results over the impact parameter. - In practice, taking everything into account, one ends up with the following formula, due to Hans Bethe and Felix Bloch, for the rate of energy loss per unit length:. - β where β = v/c, K is a constant, N is the number density of atoms in the medium and I is the mean excitation energy of the medium (I ∼ 11.5 eV).. - The cases of protons or heavy ions is of great interest and, in comparatively recent years, it has allowed a major improvement in the medical treatment of tumors in the eyes (proton therapy) and in the brain (ion therapy. - Owing to the factor 1/β 2 , or equivalently 1/v 2 , in (10.1), practically all the energy is deposited in a very localized region near the stopping point. - Figure (10.4) shows the comparison between the effect of ion beams and photons. - Energy loss of ions (left) and survival rate of cells (right) as a function of the penetration depth
Xem thử không khả dụng, vui lòng xem tại trang nguồn hoặc xem
Tóm tắt