For possible application of a silicon bent crystal as a beam collimator in the Large Hadron Collider, the thermal stability of a strip-like crystal has been studied. The thermal power generated by the beam inside the crystal has been evaluated through a simulation and used as a source in the heat equation. The equation has been solved numerically with the semi-analytical method of lines to show the thermal stability of crystal-assisted collimation with the parameters currently being considered for its exploitation. The method we presented appears to be useful for calculation of the thermal load concerning any situation involving Crystal channeling

Giberti, A.

Guidi, V.

Munerato, M.

English

For possible application of a silicon bent crystal as a beam collimator in the Large Hadron Collider, the thermal stability of a strip-like crystal has been studied. The thermal power generated by the beam inside the crystal has been evaluated through a simulation and used as a source in the heat equation. The equation has been solved numerically with the semi-analytical method of lines to show the thermal stability of crystal-assisted collimation with the parameters currently being considered for its exploitation. The method we presented appears to be useful for calculation of the thermal load concerning any situation involving crystal channeling. © Società Italiana di Fisica

GIBERTI, Alessio

GUIDI, Vincenzo

MUNERATO, Marco

Archivio istituzionale della ricerca - Università di Ferrara

Thermal stability of a strip crystal for channeling of halo particles in the Large Hadron Collider

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2011-10909-7Colloquia: Channeling 2010IL NUOVO CIMENTO Vol. 34 C, N. 4 Luglio-Agosto 2011Thermal stability of a strip crystal for channeling of halo particlesin the Large Hadron ColliderA. Giberti(∗), V. Guidi and M. MuneratoDipartimento di Fisica and INFN, Universita` di Ferrara - Via Saragat 1/c, 44124 Ferrara, Italy(ricevuto il 22 Dicembre 2010; pubblicato online il 7 Luglio 2011)Summary. — For possible application of a silicon bent crystal as a beam colli-mator in the Large Hadron Collider, the thermal stability of a strip-like crystal hasbeen studied. The thermal power generated by the beam inside the crystal has beenevaluated through a simulation and used as a source in the heat equation. Theequation has been solved numerically with the semi-analytical method of lines toshow the thermal stability of crystal-assisted collimation with the parameters cur-rently being considered for its exploitation. The method we presented appears to beuseful for calculation of the thermal load concerning any situation involving crystalchanneling.PACS 44.05.+e – Analytical and numerical techniques.PACS 61.85.+p – Channeling phenomena (blocking, energy loss, etc.).1. – IntroductionThe possibility to perform halo collimation in hadron colliders through channelingof particles in a bent crystal has been investigated since the nineties [1]. Since then,significant progress has been made since the first appearance of the idea [2-6]. Crystalcollimation could be a valid scheme for halo collimation [7], allowing the deflection ofhalo particles with high efficiency thanks to multi-turn extraction mechanism [8], or canbe used as a precision diagnostics to aid the traditional collimation systems [9].In particular, the application of a silicon bent crystal at the Large Hadron Collidercould serve to kick most external particles in the particle beam, directing them onto asecondary collimator [10]. Among the many challenges concerned with implementationof a crystal-based collimation scheme, such as alignment with beam, compensation ofaberrations in crystal bending, radiation hardness of the crystal, the problem of crystalheating has never been addressed. Yet, although the crystal is intended to intercept halo(∗) E-mail: giberti@fe.infn.itc© Societa` Italiana di Fisica 431432 A. GIBERTI, V. GUIDI and M. MUNERATOFig. 1. – Sketch of the channeling crystal, with the details of the curvatures.particles, i.e., to be lodged at the periphery of the beam, the beam circulating in theLHC will store the largest amount of energy at the highest energy (up to 7TeV) thanany other accelerator ever. Thus the problem of thermal stability of the crystal exposedto the beam is an important issue to be addressed to ensure proper working conditions.In this work, the behaviour of a strip-like crystal, shown in fig. 1, has been studiedfrom a thermal point of view, during its operation as channeling device for the LHCproton beam. Beam parameters are those reported in ref. [11], the most relevant ofwhich to our purpose are summarized in table I.The crystal, during its operation, is hit by high-energy protons, up to 7TeV. Beamparticles lose part of their energy during the motion under channeling, acting as a heatsource for the crystal, while heat dissipation occurs through radiation and conductiononly because convection is prevented due to ultra-high-vacuum condition.2. – Evaluation of the heat sourceThe energy lost by a single proton inside a silicon crystal has been evaluated using aGEANT4 framework, with a custom-made geometry description, inserting a silicon layerwith a thickness of 1 cm. Since GEANT4 is not equipped with a routine to simulate thedynamics of particle motion, the interaction of particles with the crystal was treated asit were an amorphous material with the same density as for a Si crystal. The dynamicsof particles channelled in a crystal is subject to the planar potential, i.e., the particlesoscillate between neighbouring crystalline planes while following the global curvature ofthe crystal. It means that interaction with nuclei is minimal and the density of theelectron cloud experienced by channelled particles is always lower than for interactionwith an equivalent amorphous medium. Thus, the heat rate evaluated through GEANT4is to be regarded as an overestimate of the current heating.Table I. – LHC beam parameters.No. of bunches Protons/bunch Total intensity Bunch length Bunch spacing2808 1.15× 1011 3.23× 1014 7.55 cm 25 nsTHERMAL STABILITY OF A STRIP CRYSTAL FOR CHANNELING OF HALO PARTICLES ETC. 433For the simulation, 105 protons were hit onto the crystal at 7TeV energy, then wedetermined the energy deposited per unit length with the following strategy: during thereconstruction of the particle’s track, we summed up the deposited energy along the pathof an individual particle, and we divided it by the length of interaction with the material.In particular, GEANT4 renders a lot of useful information about the energy depositedand the length of the interaction path of the particle with the material.The thermal power transferred to the material can be calculated by taking into ac-count beam parameters, in particular the number of protons per bunch and the length ofthe bunch, both reported in table I. Beam intensity, i.e. the number of protons per unitarea and time, decreases with the radial coordinate r (the distance from the central axis),and could be written as F (r) = ρ(r)v, where ρ(r) is the volumetric density of protonsand v is the velocity of the particles (v ∼= c). Since the radial profile of the concentrationof protons can be modelled with a Gaussian function with r.m.s. σ = 0.1mm, beamintensity can be written asF (r) = ρ0c e−r2/2σ2 ,where ρ0 is the density on the centre of the beam. By imposing that the integral of ρover the whole bunch must be equal to N , it readily holdsρ0 =N2πσ2L.If a single proton looses an energy per unit length equal to dE/dx, the power transferredto the material per unit volume as a function of r is S(r) = dE/dxF (r), therefore(1) S(r) = S0 e−r2/2σ2 , S0 =dEdxNc2πσ2L.Numerically, with a beam energy of 7TeV, it results S0 = 1.33× 1017 W/m3.3. – Heat equation: steady stateIn order to solve the thermal problem, the two-dimensional heat equation with theGaussian-like heat source of eq. (1) has been applied to the strip-like crystal. In thissection we will approach the problem with the thermal power source of eq. (1) as it wasdue to interaction with a coasting beam with the same density as that in the bunch forthe real case; then, interaction with a bunched beam will be worked out in sect. 4.3.1. Geometry of the problem. – The geometry of the problem is sketched in fig. 2,which is a cross-section orthogonal to the nominal axis of the beam. The length of thelonger side is L = 70mm, whereas the shorter one is l = 2mm. We choose a Cartesiancoordinate system, with the origin in the left edge on the lower side of the crystal.For symmetry reasons, we can reduce the problem to a half crystal, within the domain[0, L/2]× [0, l].Once the coordinate system and its origin have been fixed, heat equation reduces tothe Poisson equation:(2) λ∇2u + S0 e−r2/2σ2 = 0,434 A. GIBERTI, V. GUIDI and M. MUNERATOFig. 2. – Sketch of the geometry of the problem. The origin of the coordinate system coincideswith the left edge on the lower side of the crystal. Therefore, the centre of the beam is a pointof coordinates x = L/2, y = −6σ.where u = u(x, y) is the local temperature of the crystal and λ = 148W/mK is theconduction coefficient for Si. This equation has been numerically solved with the semi-analytical method of lines [12].3.2. Solution with the semi-analytical method of lines. – This numerical techniqueis very powerful to solve elliptic partial differential equations in 2 variables, becauseit requires a much smaller number of divisions of the domain than finite differencesmethods. It consists in the discretization of only one variable, leading to a system ofcoupled ordinary differential equations in the other variable; thus, the solution is givenas a set of functions. Equation (2) in Cartesian coordinates is written as∂2u∂x2+∂2u∂y2= −f(x, y),(3)f(x, y) =S0λexp[− (x− L/2)2 + (y + 6σ)22σ2].As a first step, we consider the thermal dissipation through radiation to be negligible,therefore the crystal transfers thermal power only to the metallic support, which providesto hold and bend it. Since heat capacity of the holder is much larger than that of thecrystal, the holder can be regarded as a heat reservoir, i.e., it is kept at constant roomtemperature T0. With this statement, the conditions for the differential equation (3) atthe boarder of the domain are the following:– x = 0: a Dirichlet condition which fixes the value of the temperature of the leftside;– y = 0 and y = l Neumann conditions which fix the heat transfer along y;– x = L/2: a Neumann condition which equals to zero the partial derivative of uwith respect to x, for symmetry reasons.Therefore, the boundary conditions for eq. (3) are(4) u(0, y) = T0 uy(x, l) = 0uy(x, 0) = 0 ux(L/2, y) = 0,where the subscript x and y indicate the partial derivatives with respect to x and y,respectively.THERMAL STABILITY OF A STRIP CRYSTAL FOR CHANNELING OF HALO PARTICLES ETC. 435Fig. 3. – Discretization of the horizontal variable x.The discretization of the horizontal direction must be applied so that the convergenceof the solution is achieved with a relatively small number of divisions. Being the charac-teristic length, σ, of the heat source much smaller than the total length to discretize L/2,a convenient choice is to discretize only a zone near the centre of the crystal, where thesource f(x, y) of eq. (3) attains its maximum (then vanishes rapidly due to their Gaus-sian character). Thereby, we chose to discretize only a zone of length Γ starting from thecentre of the crystal (see fig. 3). The advantage of this choice is that the intensity of heatsource in the left zone can be considered zero, so that the solution is linear, as indicatedby the Green’s function of eq. (3). As a consequence, the first of the boundary conditionsof eqs. (4) must be changed in a condition set at x = L/2 − Γ: a Robin condition thatstates that the x-derivative must be equal to the slope of the solution in the first zone,u′(L/2− Γ, y) = u(L/2− Γ, y)− T0L/2− Γ .Now we introduce n+1 points x0, . . . , xn+1 (where x0 = L/2−Γ and xn+1 = L/2), whichgenerate n + 1 intervals of width h = Γ/(n + 1), as shown in fig. 3 (the additional pointxn+2 is a “virtual point” useful for a better statement of the condition on the partialderivative of u with respect to x in the point xn+1).Approximating the partial derivative of u with respect to x with the finite differences,eq. (3), with the proper boundary conditions discussed above, becomes a system of n+1ordinary differential equations in the variable y which can be written in the matrixnotation:d2Udy2=1h2PU − F,withU =⎛⎜⎝u1...un+1⎞⎟⎠ P =⎛⎜⎜⎜⎜⎜⎝L/2−Γ+2hL/2−Γ+h −1 0−1 2 −10. . . . . .00 −1 2 −10 −2 2⎞⎟⎟⎟⎟⎟⎠(5)F =⎛⎜⎜⎜⎝T0h(L/2−Γ+h) + f1f2...fn+1⎞⎟⎟⎟⎠ ,436 A. GIBERTI, V. GUIDI and M. MUNERATOFig. 4. – (a) Plot of all 13 solutions as a function of y (u1 is for x = L/2−Γ, u13 is for x = L/2);(b) 3D interpolation of all the ui(y).where ui(y) = u(xi, y) and fi(y) = f(xi, y). The solutions of eqs. (5) can be easilyobtained with a change of base that diagonalizes matrix P . Indeed, after diagonalization,we have n + 1 independent ordinary differential equations, which can be easily solvednumerically. Then, the vector U can be obtained returning to the old base.3.3. Results of the calculation. – Convergence to the second decimal in the solutionhas been achieved with a number of divisions n = 12, and with a discretization lengthΓ = 1.3mm. The results of the calculation are shown in fig. 4. In fig. 4a all ui(y)functions are shown, while fig. 4b illustrates a 3D interpolation of the ui(y), which clearlyhighlights a Gaussian-like shape. It is apparent that the increase in temperature of thecrystal during operation is negligible inasmuch as the hottest point exhibits an increaseof only 0.5K with respect to room temperature.The correction due to the thermal power dissipated through radiation is thereforenegligible, because the achieved temperatures are too low. This has been confirmed byiterating the solution as follows: each iteration is obtained imposing, in the boundaryconditions for the derivatives in y = 0 and y = l, the Stefan-Boltzmann radiation lawwith the temperature values obtained in the previous iteration (the first iteration is thesolution without radiation). The process converges immediately, indeed every iterationyields the same solution, identical to the one obtained in the absence of radiated power.We considered the case with beam suffering trajectory perturbation or erroneousinsertion of the crystal too deep into the core of the beam. Therefore, the same calculationhas been carried out with various distances of the centre of the beam. The results forthe hottest point of the crystal, (x = L/2, y = 0), are reported in table II. The crystalnever reaches high temperatures, even in the case of insertion into the beam by σ/2.Table II. – Maximum temperature of the crystal as a function of the distance from the beamcentre.Distance (σ) 6.0 5.9 5.8 5.7 5.6 5.5u13(0) (K) 300.5 301.0 301.8 303.2 305.8 310.2THERMAL STABILITY OF A STRIP CRYSTAL FOR CHANNELING OF HALO PARTICLES ETC. 437Fig. 5. – Time-dependent solution u13(t) if the time dependence of the heat source were aHeaviside function. The time to reach a steady state is about 2.5 s.4. – Heat equation: time dependenceIn the previous section we worked out with the heat source (1) as it were continuousin time. A more realistic condition is to account for the bunched structure of the beamin the LHC. Each bunch is 250 ps long, with a time spacing between consecutive bunchesof 25 ns. In terms of channeling time, a proton spends 0.25 ns every 25 ns inside thecrystal, therefore the heat source for the time-dependent heat equation is multiplied bya periodic step function which is 0 for a time 2 orders of magnitude longer than the timein which is 1.In order to solve also the time-dependent equation with the semi-analytical method oflines, we considered the crystal as it were one-dimensional, a reasonable approximationsince the x-length is 35 times greater than the y-length. The equation is written asfollows:(6) ρCV∂u∂t= λ∂2u∂x2+ f(x, t); f(x, t) = g(t)S0 exp[− (x− L/2)2 + (6σ)22σ2],where g(t) is the periodic step function discussed above. Applying the discretization onthe spatial variable, we obtainρCVduidt= λui+1 − 2ui + ui−1h2+ fi(t) i = 1, . . . , n + 1,where ρ = 2330 kg/m3 is the density of silicon, CV = 710 J/kgK the specific heat ca-pacity, and fi(t) = f(xi, t). It is a system of first-order ordinary differential equations,which is easily solvable after a diagonalization of the matrix of coefficients. The nu-merical solution is the constant temperature T0 of the metallic support with which is incontact. Indeed, the thermal inertia of the crystal is such that the time during whichthe heat source is “on” is not high enough to increase the temperature of the crystal.A confirmation of that can be found if one evaluates the time constant τ of the thermalbehaviour of the system. It can be easily obtained by setting g(t) in eq. (6) equal to theHeaviside function and plot the solution of the equations system as a function of time;the result for τ is of the order of 1 s, as shown in fig. 5.438 A. GIBERTI, V. GUIDI and M. MUNERATO5. – ConclusionsA model to address the problem of a strip-like crystal heating under channeling condi-tion has been addressed in full generality. The method relies on the numerical solution ofheat equation, obtained with the semi-analytical method of lines, which implies the dis-cretization of one variable, in order to obtain a system of ordinary differential equations.In particular, the case of thermal behaviour of a silicon bent crystal during channeling ofLHC protons was studied. Particular attention was paid to the spatial domain, since theheat source is negligible within a large part of the crystal. It has been found that crys-tal heating is never a problem for usage of the crystal as a collimator under channelingcondition.REFERENCES[1] Afonin A. G., Arkhipenko A. A., Baranov V. I., Biryukov V. M., ChesnokovYu. A., Gavrilushkin V. A., Gres V. N., Kotov V. I., Maisheev V. A., MinchenkoA. V., Terekhov V. I., Troyanov E. F., Zelenov V. A., Chunin B. A., DenisovA. S., Gordeeva M. G., Ivanov Yu. M., Petrunin A. A. and Skorobogatov V. V.,Phys. Lett. B, 435 (1998) 240.[2] Biryukov V. 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Thermal stability of a strip crystal for channeling of halo particles in the Large Hadron Collider

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