The motion of an ion in an electron gas
<p>We give a semi-classical treatment of the problem of electrical conduction around a charged impurity ion in a metal. As the electrons flow past the ion they are scattered by its screened Coulomb potential and so introduce a velocity dependent distribution of charge. This charge distributio...
| Main Authors: | , |
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| Format: | Thesis |
| Language: | English |
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1970
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| _version_ | 1797066369794048000 |
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| author | Aish, M Aish, M. R. |
| author_facet | Aish, M Aish, M. R. |
| author_sort | Aish, M |
| collection | OXFORD |
| description | <p>We give a semi-classical treatment of the problem of electrical conduction around a charged impurity ion in a metal. As the electrons flow past the ion they are scattered by its screened Coulomb potential and so introduce a velocity dependent distribution of charge. This charge distribution leads, through the Poisson equation, to a modification of the potential surrounding the ion through which the electrons are initially scattered; we look for a self-consistent or simultaneous solution to the Boltzmann and Poisson equations. This final velocity dependent charge distribution induces an electric field at the ion, so exerting a force on that ion. We identify two effects: (i) a screening force acting so as to reduce the effect of the external field on the ion, (ii) a drift force resulting from the direct collisions between electron, and ion.</p> <p>The process is described by a Boltzmann Transport equation which we solve at large distances from the ion. We derive the Green's Function for the equivalent integral equation, and then solve that equation by means of a Fourier Transform. In this technique we do not assume as in previous approaches that the ion is stationary or that its charge is small. The evaluation of the Green's Function does however involve the construction of an orbit model in which all scattered electrons pass through the ion itself and we do not expect this model to be of any use inside the screening radius.</p> <p>We turn to a method of expansion by spherical harmonics, suggested by Das in his thesis, in the hope that it may be of use at short range. In understanding a paradox of the method we lose a criterion for convergence and our confidence in the method. To do this we consider the electron mean free path to be infinite in this short range region. In this limit a quantum mechanical calculation of the self-consistent field gives a result identical to that from a classical weak charge calculation due to Peierls, in the same limit, with the significant addition of a cut-off in momentum space at twice the Fermi-momentum. It is this cut-off, for which Peierls had argued on physical grounds, that eliminates a divergence at the origin. We use it to calculate the force.</p> <p>We find that we may add Peierls' result where the Green's Function method is too insensitive. There are then three extra terms in the force, two due to drift and one to screening. However, the term we interpret as a screening correction may for large enough charge on the ion become greater than the direct field force. This may be due to an error in our interpretation as screening, or in the calculation itself. If it is the latter we propose a further calculation to check. In as far as the force on the ion derived in this thesis may be larger than the direct field force there is no disagreement with experimental data. Until we can understand the screening problem the result is not suitable for a detailed comparison with experiment.</p> |
| first_indexed | 2024-03-06T21:41:03Z |
| format | Thesis |
| id | oxford-uuid:47f1365c-c642-4be7-9149-34d3605dd36f |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:41:03Z |
| publishDate | 1970 |
| record_format | dspace |
| spelling | oxford-uuid:47f1365c-c642-4be7-9149-34d3605dd36f2022-03-26T15:22:54ZThe motion of an ion in an electron gasThesishttp://purl.org/coar/resource_type/c_db06uuid:47f1365c-c642-4be7-9149-34d3605dd36fEnglishPolonsky Theses Digitisation Project1970Aish, MAish, M. R.<p>We give a semi-classical treatment of the problem of electrical conduction around a charged impurity ion in a metal. As the electrons flow past the ion they are scattered by its screened Coulomb potential and so introduce a velocity dependent distribution of charge. This charge distribution leads, through the Poisson equation, to a modification of the potential surrounding the ion through which the electrons are initially scattered; we look for a self-consistent or simultaneous solution to the Boltzmann and Poisson equations. This final velocity dependent charge distribution induces an electric field at the ion, so exerting a force on that ion. We identify two effects: (i) a screening force acting so as to reduce the effect of the external field on the ion, (ii) a drift force resulting from the direct collisions between electron, and ion.</p> <p>The process is described by a Boltzmann Transport equation which we solve at large distances from the ion. We derive the Green's Function for the equivalent integral equation, and then solve that equation by means of a Fourier Transform. In this technique we do not assume as in previous approaches that the ion is stationary or that its charge is small. The evaluation of the Green's Function does however involve the construction of an orbit model in which all scattered electrons pass through the ion itself and we do not expect this model to be of any use inside the screening radius.</p> <p>We turn to a method of expansion by spherical harmonics, suggested by Das in his thesis, in the hope that it may be of use at short range. In understanding a paradox of the method we lose a criterion for convergence and our confidence in the method. To do this we consider the electron mean free path to be infinite in this short range region. In this limit a quantum mechanical calculation of the self-consistent field gives a result identical to that from a classical weak charge calculation due to Peierls, in the same limit, with the significant addition of a cut-off in momentum space at twice the Fermi-momentum. It is this cut-off, for which Peierls had argued on physical grounds, that eliminates a divergence at the origin. We use it to calculate the force.</p> <p>We find that we may add Peierls' result where the Green's Function method is too insensitive. There are then three extra terms in the force, two due to drift and one to screening. However, the term we interpret as a screening correction may for large enough charge on the ion become greater than the direct field force. This may be due to an error in our interpretation as screening, or in the calculation itself. If it is the latter we propose a further calculation to check. In as far as the force on the ion derived in this thesis may be larger than the direct field force there is no disagreement with experimental data. Until we can understand the screening problem the result is not suitable for a detailed comparison with experiment.</p> |
| spellingShingle | Aish, M Aish, M. R. The motion of an ion in an electron gas |
| title | The motion of an ion in an electron gas |
| title_full | The motion of an ion in an electron gas |
| title_fullStr | The motion of an ion in an electron gas |
| title_full_unstemmed | The motion of an ion in an electron gas |
| title_short | The motion of an ion in an electron gas |
| title_sort | motion of an ion in an electron gas |
| work_keys_str_mv | AT aishm themotionofanioninanelectrongas AT aishmr themotionofanioninanelectrongas AT aishm motionofanioninanelectrongas AT aishmr motionofanioninanelectrongas |