Theoretical study of the effect of the size of a high-energy proton beam of the Large Hadron Collider on the formation and propagation of shock waves in copper irradiated by 450-GeV proton beams

The interaction of 450GeV protons with copper, which is the material of the collimators of the Large Hadron Collider, has been theoretically studied. A theoretical model for the formation and propagation of shock waves has been proposed on the basis of the anal ysis of the energy released by a proton beam in the electronic subsystem of the material owing to the deceleration of secondary particles appearing in nuclear reactions induced by this beam on the electronic subsy stem of the material. The subsequent transfer of the energy from the excited electronic subsystem to the crystal lattice through the electron–phonon interaction has been described within the thermal spike model [I.M. Lifshitz, M.I. Kaganov, and L.V. Tanatarov, Sov. Phys. JETP 4 , 173 (1957); I.M. Lifshitz, M.I. Kaganov, and L.V. Tanatarov, At. Energ. 6 , 391 (1959); K. Yasui, Nucl. Instrum. Methods Phys. Res., Sect. B 90 , 409 (1994)]. The model of the formation of shock waves involves energy exchange processes between excited electronic and ionic subsystems of the irradiated material and is based on the hydrodynamic approxim ation proposed by Zel’dovich [Ya.B. Zel’dovich and Yu.P. Raizer, Physics of Shock Waves and HighTemperature Hydrodynamic Phenomena (Nauka, Moscow, 1966; Dover, New York, 2002)]. This model makes it po ssible to obtain the space–time distributions of the main physical characteristics (temperatures of the ionic and electronic subsystems, density, pressure, etc.) in materials irradiated by highenergy proton beams and to analyze the formation and propagation of shock waves in them. The nonlinear differential equations describing the conservation laws of mass, energy, and momentum of electrons and ions in the Euler variables in the case of the propagation of shock waves has been solved with the Godunov scheme [S. K. Godunov, A.V. Zabrodin, M.Ya. Ivanov, A.N. Kraiko, and G.P. Prokopov, Numerical Solution of Multidimen sional Problems in Gas Dynamics (Nauka, Moscow, 1976) [in Russian]]. The temperature, pressure, and specific internal energy have been calculated independently and separately for the ionic and electronic subsystem s of the irradiated material. The formation of shock waves in copper has been simulated in the cylindrical ge ometry for various sizes and various types of the beam: for a needle beam (the distribution of protons in the beam is described by the Dirac delta function) and beams with a finite radius (0.3, 3, and 5 mm) with a Gaussian distribution of protons near the axis of the beam. The effect of the size and geometry of the proton beam, as well as of the form of the chosen boundary conditions, on the intensity of shock waves formed in copper and on change in their physicomechanical properties in the space and time has been revealed in the numerical simulation results.