Measuring energy losses of beams of charged
particles is an important diagnostic tool in both
modern condensed matter and plasma physics..

Arkhipov, Yu. V.

Ashikbayeva, A. B.

Askaruly, A.

Davletov, A. E.

Dubovtsev, D. Yu.

Tkachenko, I. M.

Nazarbayev University Repository

The 21st International Symposium on Heavy Ion Inertial Fusion | July 18-22, 2016 | Astana, Kazakhstan58 | The stopping power and straggling of strongly coupled electron fluids revisited  Yu.V. Arkhipov 1 , A.B. Ashikbayeva 1 , A. Askaruly 1 , A.E. Davletov 1 , D.Yu. Dubovtsev 1 ,  and I.M. Tkachenko 2    1 al-Farabi Kazakh National University, Almaty, 050012, Kazakhstan 2 Universidad Politécnica de Valencia, Valencia, 46022, Spain * e-mail: Yuriy.Arkhipov@kaznu.kz      Measuring energy losses of beams of charged particles is an important diagnostic tool in both modern condensed matter and plasma physics. If single-particle effects can be neglected, the gen-eral expression for polarization losses [1] sim-plifies as  (𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑)𝑝𝑝𝑝𝑝𝑝𝑝=2𝜋𝜋(𝑍𝑍𝑝𝑝𝑒𝑒𝑣𝑣)2∫𝑑𝑑𝑑𝑑𝑑𝑑∞0 ∫ 𝜔𝜔2𝐿𝐿𝑑𝑑𝑣𝑣0(𝑘𝑘, 𝜔𝜔)𝑑𝑑𝜔𝜔 . Here Zpe and v are the projectile charge and ve-locity, while 𝐿𝐿(𝑘𝑘, 𝜔𝜔) = −Im𝜀𝜀−1(𝑘𝑘, 𝜔𝜔)/𝜔𝜔 is the target plasma loss function, 𝜀𝜀(𝑘𝑘, 𝜔𝜔) being the system dielectric function (DF). Another im-portant characteristic of energy loss by the beam of projectiles is the straggling:  Ω2 =2ℏ𝜋𝜋(𝑍𝑍𝑝𝑝𝑒𝑒𝑣𝑣)2∫𝑑𝑑𝑘𝑘𝑘𝑘∞0∫𝜔𝜔3𝐿𝐿(𝑘𝑘, 𝜔𝜔)𝑑𝑑𝜔𝜔𝑡𝑡𝑡𝑡𝑡𝑡ℎ(𝛽𝛽ℏ𝜔𝜔/2)𝑑𝑑𝑣𝑣0 . Usually the polarizational stopping power and its straggling are calculated presuming the tar-get plasma to be an electronic fluid and the DF of the latter has been evaluated within the ran-dom-phase approximation (RPA), the T-matrix approach, the method of effective potentials, or using the Mermin or more sophisticated DF models, see [2] for the references. It has been shown recently in [2] that all these perturbative DF models, except for the local-field corrected RPA do not satisfy the interaction-related sum rules and thus cannot be applied for the calcula-tion of the stopping power and its straggling under the conditions of strong coupling. The approach we suggest here is based on the method of moments. It is not perturbative and the only limitation is that the electron fluid is presumed to be in the liquid phase. The mo-ments are effectively the power frequency of the loss function,  𝐶𝐶𝜈𝜈(𝑘𝑘) =1𝜋𝜋∫ 𝜔𝜔𝜈𝜈𝐿𝐿∞−∞(𝑘𝑘, 𝜔𝜔)𝑑𝑑𝜔𝜔,   𝜈𝜈 = 0,2,4 and are the sum rules. The approach permits to apply the Nevanlinna formula of the classical method of moments and to model the system DF via the Nevanlinna parameter function (NPF) q(k,z) [3]: 𝜀𝜀(𝑘𝑘, 𝜔𝜔) = 1 +𝜔𝜔𝑝𝑝2 (𝜔𝜔+𝑞𝑞)𝜔𝜔(𝜔𝜔2−𝜔𝜔22+𝜔𝜔𝑝𝑝2 )+𝑞𝑞(𝜔𝜔2−𝜔𝜔12+𝜔𝜔𝑝𝑝2 ) , where 𝜔𝜔12(𝑘𝑘) = 𝐶𝐶2 𝐶𝐶0⁄  and 𝜔𝜔22(𝑘𝑘) = 𝐶𝐶4 𝐶𝐶2⁄  and 𝜔𝜔𝑝𝑝 is the plasma frequency. We model the NPF here as 𝑞𝑞(𝑘𝑘, 𝜔𝜔) = 𝑞𝑞(𝑘𝑘, 0) = 𝑖𝑖 𝜔𝜔22(𝑘𝑘) √2⁄ 𝜔𝜔1(𝑘𝑘), which is justified by the specific behavior of the one-component plasma (OCP) dynamic structure factor near the zero frequency [4]. Thus the calculation of the OCP plasma stopping power and starggling is reduced to the knowledge of the system static structure factor. The latter can be either calculated within a modified or local-field corrected RPA, or computed using the MD method with an adequate effective potential [5].   The results of our calculations are to be com-pared to the available experimental or simula-tion data.  References [1] J. Ortner and I. M. Tkachenko, 2001 Phys. Rev. E 63, 026403. [2] Yu.V. Arkhipov et al 2015 Phys. Rev. E, 91, 019903.  [3] I.M. Tkachenko, Yu.V. Arkhipov, A. Askaruky, The Method of Moments and its Applications in Plasma Physics (Lambert, 2012) and references therein. [4] Yu.V. Arkhipov et al 2016 Phys. Rev. Lett., to be published.  [5] Yu.V. Arkhipov et al 2016 Contrib. Plasma Phys. DOI 10.1002/ctpp.201500129 . 

The stopping power and straggling of strongly coupled electron fluids revisited

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The stopping power and straggling of strongly coupled electron fluids revisited

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