# Random electronic stopping power

### This page gives hints on how to perform random stopping power calculation with the ABINIT package.

##### Copyright (C) 2016-2017 ABINIT group (FB)

Mentioned in help_features#7.

### Table of content:

- 1. Introduction.
- 2. Related input variables.
- 3. Selected input files.
- 4. References.

### 1. Introduction.

The slowing down of a swift charged particle inside condensed matter has been a subject of intense interest since the advent of quantum-mechanics. The Lindhard formula [Lindhard1954] that gives the polarizability of the free electron gas has been developed specifically for this purpose. The kinetic energy lost by the impinging particle by unit of path length is named the stopping power. For large velocities, the stopping power is dominated by its electronic contribution: the arriving particle induces electronic excitations in the target. These electronic excitations in the target can be related to the inverse dielectric function ε-1(q,ω) provided that linear response theory is valid.

As a consequence, the electronic stopping power randomized over all the possible impact parameters reads

S(v) = (4π Z2/Nq Ω|v|)∑q∑G Im{-ε-1[q,v.(q+G)]} (v.(q+G)/|q+G|2),

where Z and v are respectively the charge and the velocity of the impinging particle, Ω is the unit cell volume, Nq is the number of q-points in the first Brillouin zone, and G are reciprocal lattice vectors.

Apart from an overall factor of 2, this equation is identical to the formula published [Campillo1998].

The GW module of ABINIT gives access to the full inverse dielectric function for a grid of frequencies ω. Then, the implementation of the above equation is a post-processing employing a spline interpolation of the inverse dielectric function in order to evaluate it at ω=v.(q+G). The energy cutoff on G is governed by the ecuteps, as in the GW module. The integer npvel and the cartesian vector pvelmax control the discretization of the particle velocity.

Note that the absolute convergence of the random electronic stopping power is a delicate matter that generally requires thousands of empty states together with large values of the energy cutoff.
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### 2. Related input variables.

Compulsory input variables:

... ecuteps [Energy CUT-off for EPSilon (the dielectric matrix)]
... npvel [Number of Particle VELocities]

Basic input variables:

... pvelmax [Particle VELocity MAXimum]
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### 3. Selected input files.

WARNING : as of ABINITv8.6.x, the list of input files provided in the specific section of the topics Web pages is still to be reviewed/tuned. In some cases, it will be adequate, and in other cases, it might be incomplete, or perhaps even useless.
The user can find some related example input files in the ABINIT package in the directory /tests, or on the Web:

tests/v7/Input: t16.in
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### 4. References.

[Campillo1998] I. Campillo, J. M. Pitarke and A. G. Eguiluz, "Electronic stopping power of aluminum crystal", Phys. Rev. B 58, 10307 - 10314 (1998).
DOI: 10.1103/PhysRevB.58.10307.
[Lindhard1954] J. Lindhard, "On the properties of a gas of charged particles", Mat. Fys. Medd. Dan. Vid. Selsk. 28, 8 (1954).

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