Dispersion Properties of the Propagation of Linear Waves

Dispersion Properties of the Propagation of Linear Waves ABSTRACT In electron-positron plasmas some of the plasma modes are decoupled due to the equal charge to mass ratio of both species. The dispersion properties of the propagation of linear waves in degenerate electron–positron magnetoplasma are investigated. By using the quantum hydrodynamic equations with magnetic fields of the Wigner–Maxwell system, we have obtained a set of new dispersion relations in which ions’ motions are not considered. The general dielectric tensor is derived using the electron and positron densities and its momentum response to the quantum effects due to Bohm potential and the statistical effect of Femi temperature. It has been demonstrated the importance of magnetic field and its role with the quantum effects in these plasmas which support the propagation of electromagnetic linear waves. Besides, the dispersion relations in case of parallel and perpendicular modes are investigated for different positron-electron density ratios. Keywords: Quantum Plasma; Dispersion relation ; Electron –Positron 1- INTRODUCTION Electron-positron (e-p) plasmas are found in the early universe, in astrophysical objects (e.g., pulsars, super nova remnants, and active galactic nuclei, in ÃŽ ³ -ray bursts, and at the center of the Milky Way galaxy [1]. In such physical systems, the e-p pairs can be created by collisions between particles that are accelerated by electromagnetic and electrostatic waves and/or by gravitational forces. Intense laser-plasma interaction experiments have reported the production of MeV electrons and conclusive evidence of positron production via electron collisions. Positrons have also been created in post disruption plasmas in large tokamaks through collisions between MeV electrons and thermal particles. The progress in the production of positron plasmas of the past two decades makes it possible to consider laboratory experiments on e-p plasmas [2]. The earlier theoretical studies on linear waves in electron–positron plasmas have largely focused on the relativistic regime relevant to astrophysical contexts [3]. This is largely due to the fact that the production of these electron–positron pairs requires high-energy processes. In laboratory plasmas non-relativistic electron–positron plasmas can be created by using two different schemes. In one scheme, a relativistic electron beam when impinges on high Z-target produces positrons in abundance. The relativistic pair of electrons and positrons is then trapped in a magnetic mirror and cools down rapidly by radiation, thus producing non-relativistic pair plasmas. In another scheme positrons can be accumulated from a radioactive source. Such non-relativistic electron–positron plasmas have been produced in the laboratory by many researchers. This has given an impetus to many theoretical works on non-relativistic electron–positron plasmas. Stewart and Laing [4] studied the dispersion properties of linear waves in equal-mass plasmas and found that due to the special symmetry of such plasmas, well known phenomena such as Faraday rotation and whistler wave modes disappear. Iwamoto [5] studied the collective modes in non-relativistic electron–positron plasmas using the kinetic approach. He found that the dispersion relations for longitudinal modes in electron–positron plasma for both unmagnetized and magnetized electron–positron plasmas were similar to the modes in one-component electron or electron–ion plasmas. The transverse modes for the unmagnetized case were also found to be similar. However, the transverse modes in the presence of a magnetic field were found to be different from those in electron–ion plasmas. Studies of wave propagation in electron–positron plasmas contin ue to highlight the role played by the equal mass of electrons and positrons. For example, the low frequency ion acoustic wave, a feature of electron–ion plasmas due to significantly different masses of electrons and ions, has no counterpart in electron–positron plasma. Shukla et al [6] derived a new dispersion relation for low-frequency electrostatic waves in strongly magnetized non-uniform electron–positron plasma. They showed that the dispersion relation admits a new purely growing instability in the presence of equilibrium density and magnetic field inhomogeneties. Linear electrostatic waves in a magnetized four-component, two-temperature electron–positron plasma are investigated by Lazarus et al in Ref. [7]. They have derived a linear dispersion relation for electrostatic waves for the model and analyzed for different wave modes. Dispersion characteristics of these modes at different propagation angles are studied numerically. In this work, The dispersion properties of the propagation of linear waves in degenerate electron–positron magnetoplasma are investigated. By using the quantum hydrodynamic equations with magnetic fields of the Wigner–Maxwell system, we have obtained a set of new dispersion relations in which ions’ motions are not considered. The general dielectric tensor is derived using the electron and positron densities and its momentum response to the quantum effects due to Bohm potential and the statistical effect of Femi temperature. 2- MODELING EQUATIONS We consider quantum plasma composed of electrons and positrons whose background stationary ions. The plasma is immersed in an external magnetic field . The quasi-neutrality condition reads as . From model, the dynamics of these particles are governed by the following continuity equation and the momentum equation: (1) (2) Here and are the number density, the velocity and the mass of particle respectively () and is the plank constant divided by. Let electrons and positrons obey the following pressure law: Where, is the Fermi thermal speed, is the particle Fermi temperature, is the Boltzmann’s constant and is the equilibrium particle number density. We have included both the quantum statistical effects through Fermi temperature and the quantum diffraction in the –dependent. If we set equal to zero and equal the temperature of electrons and positrons, we obtain the classical hydrodynamic equation. Assuming that the plasma is isothermal, the Fermi speeds for different particles may be equal. Using the perturbation technique, assume the quantity representing (n, u, B, E) has the following form where is the unperturbed value and is a small perturbation . Assuming the equilibrium electric field is zero and linearizing the continuity and the momentum equations, we have: (3) (4) Multiplying equation (4) by and Simplifying, we can obtain the following equation: (5) where, , , and Assuming, , then the three components of the fluid velocity can be written as: (6a) (6b) (6c) Where, and The current density and the dielectric permeability of the medium are given: (7) (8) where is the unit tensor. So, we can obtain the dielectric tensor as follows: (9) Where, Then, according to equations (8), (9) The propagation of different electromagnetic linear waves in quantum plasma can be obtained from the following general dispersion relation: (10) Where, is the plasma frequency and . 3- DISCUSSION In this section, we focus our attention on the discussion of some different modes in two cases that the wave vector parallel and perpendicular to the magnetic field . (3.I) Parallel modes So, this case leads to, with . Therefore the general dispersion relation (10) becomes: (11) This gives two dispersion relations. The first one () investigates the dispersion of electrostatic quantum waves included the quantum effects as follows (12) By neglecting the quantum effects, equation (11) describes the following well-known classical modes The second dispersion equation gives: (13) Equation (13) is similar to the dispersion of left and right waves (L- and R- modes). Owing to the symmetry between the positively and negatively charged particles, the dispersion relation for the right circularly polarized wave is identical to the left circularly polarized wave. It has been noted that no quantum effects on these modes. For unmagnetized plasma , the dispersion relation becomes: (14) (3.II) Perpendicular mode In this case, we have So, the general dispersion relation (10) becomes: (15) Where it has the following new elements , , , , , , , In the case of unmagnetized plasma , we have the following two dispersion equations: (16) and (17) The equation (16) is the well known dispersion relation which investigates the propagation of electromagnetic waves in classical unmagnetized plasma.The damping is absent because the phase velocity of the wave obtained from this equation is always greater than the velocity of light, so that no particles can be resonant with the wave. This results is analogous to the one-component electron plasma [5]. While the other relation (17) indicates the dispersion of the waves in electron-positron plasma under the quantum effects. 4- NUMERICAL ANALYSIS AND RESULTS In this section, we are going to investigate the above dispersion relations numerically. Introducing the normalized quantities , , , , and the plasmonic coupling () which describes the ratio of plasmonic energy density to the electron Fermi energy density, we rewrite some of the dispersion relations in both of parallel and perpendicular modes. (4.I) Parallel modes In the first, equation (12), () becomes: (18) Where, . The dispersion relation (17) has two positive solutions, Fig 1, for positron electron density ration with and .One of solutions of the dispersion equation (19) can be investigated in Fig. (2) to study the parallel modes for different density ratios with in quantum plasma . The solution of the normalized dispersion equation (17) has been also displayed in 3D figure (3) for quantum unmagnetized plasma . It is clear from the previous figures that the dispersion relations depend strongly on the density ratio of positron to electron. As the positron density is increased to equal to the electron density, the phase velocity has been increased. In the beginning, with very small positron density the wave frequency equals the electron plasma frequency and decreased with positron density increased. Besides, in the Fig. (4), the dispersion relation of parallel modes is shown for different quantum ratios , in the case of positron-electron density ratio and equal velocities of them . It is clear that the phase velocity of the mode is increased with the increases of plasmonic coupling ratio. (4.II) Perpendicular mode In the case of perpendicular modes, equation (15) can be normalized and solved numerically (here, ). Figure (5) displays the dispersion curves of electromagnetic modes under the effect of different density ratios in classical plasma. Also, the other equation (16) can be solve numerically to give two real solutions. One of them is the same solution approximately of equation (15) (which is clear in Figure (6). The other solution of dispersion equation (16) is displayed in figure (7). It is clear in the figures that the dispersion curves at depend essentially on the positron-electron density ratio . As the positron density increases to equal electron density, the wave frequency is increased to be bigger than the plasma frequency. On the dispersion curves (figures (5) and (6)), it has been noted the phase velocity of modes (+ve slope of the curves) decreases as density ratio increases. But, on the figure (7), the phase velocities of these modes (-ve slope) are the same with changes of the density ratio. They tend to zero with large wave number which means that these modes cannot propagate in plasmas. Figure (8) investigates the dispersion relations of the electromagnetic waves in electron-positron plasma under the quantum effects. It is clear that, in the case of classical plasma, the wave frequency decreases as wave number increases (the phase velocity is negative). But, in the case of quantum plasma (for small ratio ), the wave frequency deceases as wave number increases (the phase velocity is negative). Then, the phase velocity and group velocity tends to zero at definite wave number () depends on the quantum ratio (). For high quantum ratio, the phase velocity starts to be +ve and increases again. 5-CONCLOUSION In this work, The dispersion properties of the propagation of linear waves in degenerate electron–positron magnetoplasma are investigated by using the quantum hydrodynamic equations with magnetic fields of the Wigner–Maxwell system. The general dielectric tensor is derived using the electron and positron densities and its momentum response to the quantum effects due to Bohm potential and the statistical effect of Femi temperature. We have obtained a set of new dispersion relations in two cases that the wave vector parallel or perpendicular to the magnetic field to investigate the linear propagation of different electromagnetic waves. It is clear that the quantum effects increase or decrease the phase velocity of the modes depends on the external magnetic field. Besides, it has shown that the dispersion curves at depend essentially on the positron-electron density ratio such as the positron density is increased to equal electron density, the wave frequency of the modes is increased.. Fig.(1). The dispersion relation (5.19) has two positive solutions for positron electron density ration with and Fig. (2) The dispersion relations of the modes for different density positron-electron ratios with and Fig. (3). The dispersion relations of the parallel modes along density ratioaxis with and Fig.(4). The dispersion relations of different modes for different quantum effects with positron-electron density ratio and velocity ratio .. , Fig. (5.5). The dispersion relations of electromagnetic modes for different ratios in classical plasma. Fig.(6). The dispersion solutions of the equations (5.17) and (5.18) for different density ratios . Fig. (7). The other dispersion solutions of the equation (18) for different density ratios . Fig.(8). 3D plotting for dispersion relation for perpendicular modes in quantum unmagnetized plasma along quantum ratio axis with

Space – Should the costing be spent?

I have chosen to write a report entitled, â€Å"Space – Can the expense be justified?†. I chose this as I had a keen interest in the matter; once we had discussed this as a class. Another reason in which why I have selected this topic is because as the world of media is advancing, this subject has been under debate for many years now, but with no final conclusion. Humans are said to have landed on the moon in 1969, however again through media many accusations have been made against this. This proves media manipulates decisions made; therefore the public should be more aware of the situation. Sources of information For this assignment, my main source of information will be the BBC news archive, due to the fact this is a government based company. Therefore the information will be very much factual and reliable, rather than opinion based data from other sources of media. Further on, I will use the British National Space Centre (BNSC) as this is Britain's main space exploration organisation. Following on, I will be using â€Å"Encarta Encyclopedia† which is a screen-based archive of various facts and figures including extended explanations of various topics. This proves I will be using a vast range of sources which I have selected, however most are screen-based as these are constantly being updated throughout key advancements. Space The universe contains everything – all of space and time and all the matter and energy within it. The universe is unknowably vast, and ever since it formed, it has been expanding, carrying some of its most distant regions forever beyond the naked eye. The universe contains everything from the smallest atom to the largest galaxy cluster and yet it seems they all have the same laws. [1] For many years, space was out of reach for humans on our planet. People thought day and night on how to make space exploration a reality. Everyone at some time in their life asks questions similar to: â€Å"Where does space begin? Where does it end? What is the difference between space and the universe?† Space is built of so little matter that we cannot consider it as empty. However on Earth, there is matter everywhere, in the form of liquid/solid/ and gas. In space there is no night or day, this is due to the fact our atmosphere scatters the sun's rays, ultimately giving the blue colour in our sky. [2] A crucial property of the universe is that it is expanding. [3] It must be growing, because distant galaxies are quickly withdrawing from Earth. Assuming that the universe has always been growing, it once must have had to be smaller and denser; this is the face which strongly supports the Big Bang theory. [4] Benefits of space exploration Space exploration has many benefits for us, which could eventually save mankind from extinction if the time came. The following examples are a handful of positive effects if we did invest more into space exploration. Population increase/Colonization Since history has been written, our population has grown rapidly with 6.5 billion people today. It is said that the world's population has quadrupled in the last one hundred years. If the population carries on dramatically increasing we will have many problems including housing issues, for the number of people. By the year 2050, from predictions, it shows that there will be eventually ten billion people living on Earth, this is a growth around 75 million people per year. [5] The graph on the left shows the predicted population grown for the next 50 years, as well as the â€Å"population boom† which has occurred earlier. The key shows different areas of the world and the population increase there. Asia seems to be increasing the most; this clearly is a developing region of the world. On the other hand a developed region such as Northern America seems to have a steady increase. For this issue to be rectified, space exploration can be extremely helpful. If we can locate a suitable area beyond the Earth then we can â€Å"Colonize†. We have determined that many materials can be available in space, however human space flight advancements and engineering is vital for this to occur. The moon seems to be a viable location for us to â€Å"Colonize†, due to the fact, it is extremely close to us compared to other planets therefore easy to transport goods and supplies. The only issue is that there seems to be low amounts of Hydrogen and Carbon. The low gravity is also a major concern. Lack of resources As above, if the population carries on increasing as predicted, Earth's resources will eventually start to run out. This will have a dramatic effect on the habitants on Earth, as many necessities we take for granted will be depleted. These items include: * Clean water * Natural resources such as Fossil fuels Experts say that seas will become emptied of fish while forests – which absorb Carbon Dioxide emissions – are completely destroyed and fresh water supplies become scarce and polluted. For example since 1970 forests have been reduced by twelve percent. This proves that if space exploration improves and is funded, we can get numerous materials from out of space. For example, on the Moon there seems to be a great deal of silicon and metals such as iron, aluminium and titanium. [6] Counter-benefits of space exploration On the other hand of this debate, there are many disadvantages against investing in space exploration. This topic has been under debate for many years now, many people feel that the money is better spent elsewhere, ultimately spent on this planet prior to exploring others. NHS and Healthcare The amount of money spent on space exploration may have better uses, as the NHS are currently in debt, this should be rectified. Spending for example à ¯Ã‚ ¿Ã‚ ½150m on a certain probe could instead cut the NHS's debt by a fifth. Also, there are many people who are fighting disease, however some which aren't currently curable, as the research hasn't been paid for; there is always scientist's salaries to fund. Helping cure disease ultimately seems imperative. Space exploration can't bring people's health therefore as healthcare is vital, exploration should be halted until many basic lives saving problems are treated for. The diagram on the right shows how much per year the NHS is spending, this is also projected for up till 2008.[7] They seem to be spending currently around 90 million per year. This proves to me that developing a space instrument which tends to cost 100 million plus is not needed, as this can be fund the NHS for the year (the entire nation's treatment). Human space flight Unlike robotic space exploration, human space flight costs much more however with little benefit and outcome. Additionally, sending a robotic device to space can increase the scientific knowledge attained. Robots are clearly more efficient than humans, the speed and technology is much faster. The majority of astronauts sent to space also unfortunately pass away due to the tasks being extremely difficult regardless of how much training they have completed. Conclusion After completing this report, I have clearly shown the benefits and counter-benefits of space exploration, and wherever it should be funded. I believe my main benefit is how we can search for resources elsewhere, therefore not having to worry greatly on the amount we are using, (within reason). We evidently have two sides of the debate, which can become extremely ethical; â€Å"For, or against, science†. However this would be political of me to discuss. I have used evidence from both sides of the argument, the scientific aspect as well as the ethical reasons, which have been shown mainly within the health care section. It is also debateable that why isn't the UK investing in human space flight, even though the UK holds the world's fourth largest economy however they have no presence in manned space flight, or any interest in such activities. On the other hand they have invested largely into unmanned space flight, which have been a helping hand to improve the quality of life. From this report, evidence shows that the positives unmistakably outweigh the negatives for the title â€Å"Space – Can the expense be justified?†. Personally I am supporting space exploration due to the fact I am intrigued into the question many people ask, â€Å"What is out there?† I also believe with the advancements of the modern world, during my life time I will be able to maybe benefit from space exploration in one way or another.

Literal/ Golden/ Mischief Rules Essay

The literal rule is the primary rule which takes precedence over the others. Words and phrases should be construed by the courts in their ordinary sense, and the ordinary rules of grammar and punctuation should be applied. If, applying this rule, a clear meaning appears, then this must be applied, and the courts will not inquire whether what the statute says represents the intention of the legislature: ‘The intention of Parliament is not to be judged by what is in its mind, but by the expression of that mind in the statute itself’. The literal rule is strongly criticised by many lawyers. It has been said to be ‘†¦.a rule against using intelligence in understanding language. Anyone who in ordinary life interpreted words literally, being indifferent to what the speaker or writer meant, would be regarded as a pedant, a mischief-maker or an idiot’. Such criticism, it is submitted, is misguided. For example, the Hotel Proprietors Act 1956 provides that in certain circumstances an hotel proprietor is liable for loss of or damage to guests’ property, but that this liability does not usually extend to guests’ motor vehicles or property left ‘therein’. The question arises – is the hotel proprietor liable for property left on, rather than in, a vehicle, for example, on a roof rack. On a literal interpretation, the hotel proprietor is liable, because if Parliament had intended to exclude property left on a vehicle, the Act would have said ‘therein or thereon’. The ‘common-sense’ school would say that it is ridiculous to make a distinction between property left in or on a vehicle. That may be so in the admittedly trivial example given, but if this line of argument is accepted, it means that the courts would have power to rewrite Acts of Parliament, which many people would consider to be highly dangerous, particularly where it takes the form of assuming that Parliament ‘intended’ something, when in truth it is more than likely that Parliament never gave that matter a moments’ thought. It is better that the courts interpret statutes strictly, and if this leads to unsatisfactory or inequitable results, then Parliament should pass amending legislation to indicate clearly what its intention was. The full force of the literal rule was demonstrated in the case of Whitely v, Chappell (1869). The defendant had voted in the name of a person who had died, but was found not guilty of the offence of personating ‘any person entitled to vote’: a dead person is not entitled to vote. Golden Rule Where the meaning of words in a statute, if strictly applied, would lead to an absurdity, the golden rule is that the courts are entitled to assume that Parliament did not intend such absurdity, and they will construe the Act to give it the meaning which Parliament intended. So, for example, the Offences Against the Person Act 1861 provided that ‘whosoever being married shall marry another person during the life of the former husband or wife’ is guilty of bigamy. Re Sigsworth 1935 provided that the Defendant was not entitled to inherit because it would be manifestly repugnant to allow a murderer to reap the benefit of his crime even if the Defendant is the only inheritor. Interpreted literally, this definition is absurd on two counts. First, the phrase ‘shall marry another person’ is meaningless in the context, as the essence of bigamy is that a married person cannot marry again while his first marriage subsists. Secondly, the reference to a ‘former’ husband or wife is quite inappropriate. The word ‘former’ suggests that the original marriage no longer exists, but if that were the case the person marrying again would not be guilty of bigamy. Despite the slipshod draftsmanship of the Act, however, the intention was clear, and the courts have interpreted the relevant section as meaning that a person who purports to marry another while his or wife or husband is still alive is guilty of bigamy. Mischief Rule When it is not clear whether an act falls within what is prohibited by a particular piece of legislation, the judges can apply the mischief rule. This means that the courts can take into account the reasons why the legislation was passed; what ‘mischief’ the legislation was designed to cure, and whether the act in question fell within the ‘mischief’. For example, the Street Offences Act 1959 made it an offence for a prostitute to solicit men ‘in a street or public place’. In Smith v. Hughes the question was whether a woman who had tapped on a balcony and hissed at men passing by was guilty of an offence under the Act. Parker, L.C.J., found her guilty: ‘I approach the matter by considering what is the mischief aimed at by this Act. Everybody (sic) knows that this was an Act intended to clean up the streets, to enable people to walk along the streets without being molested or solicited by common prostitutes. Viewed in that way, it can matter little  whether the prostitute is soliciting while in the street or standing in a doorway or on a balcony’. In the case mentioned, it was comparatively easy to apply the mischief rule as the circumstances which caused the passing of the Act were well known. The rule does, however, have limitations as it is by no means always easy to discover the ‘mischief’ at which particular Act was aimed. The rules of interpretation discussed above do not apply to the interpretation of EEC legislation. The European Communities Act 1972 provides that questions of interpretation of EEC law must be decided in accordance with the principles laid down by any relevant decision of the European Court. Therefore, although EEC legislation has the force of law in England and thus becomes part of English law, the courts cannot interpret it by the methods which they apply to the main body of English law. In interpreting statutes, the courts make certain presumptions: (a) that the statute is not intended to have retrospective effect; (b) that it applies only to the United Kingdom; (c) that it is not intended to interfere with existing vested rights; (d) that the property of any person will not be confiscated without compensation; (e) that there is no intention to interfere with existing contractual rights; (f) that there is no intention to interfere with personal liberty; (g) that any person to whom judicial or quasi-judicial power is given will exercise such power in accordance with the rules of natural justice; (h) that the statute is not intended to derogate from the requirements of international law. Any of these presumptions may be overruled by the precise words of the statute. Private Acts (but not public Acts) always have a preamble which sets out the objects of the legislation. Preambles can on occasion be of considerable assistance to the courts in interpreting the Acts.

Abraham s Theory Of Behaviorism - 1573 Words

Abraham Harold Maslow was born on April 1, 1908 in Brooklyn, New York. He was the first born to his parents, Samuel and Rose Maslow. He was a lonely and unhappy Jewish boy who spent most of his time in the library and among books as a means of comfort and refuge. However, in 1925 at the age of 17 he enrolled at the City College of New York. In 1926, he registered for evening classes at the Brooklyn Law School, then transferred to Cornell University in Ithaca, New York in 1927. In 1928, he transferred to the University of Wisconsin and earned his Bachelor, Maters and Doctorate within the years 1930-1934. Shortly after Maslow married his longtime sweetheart and first cousin Bertha Goodman, he had discovered J.B. Watson and his theories on behaviorism which sparked an interest in him. However, the birth of his two daughters Anna and Ellen made him forget behaviorism. After working for 18 months at the Columbia University he met with well-known learning theorist Edward Thorndike who sparked an interest in him that he decided to research on the relationship between dominance and sexuality in humans. During the period 1937-1951, Abraham taught at Brooklyn College and there continued with his research on human sexuality. He continued seeking to understand humans, more so, Max Wertheimer and Ruth Benedick, who had great influence on him. These influential assisted him in formulating an interest in self-actualizing people. In 1951 Abraham moved to Brandeis University and served asShow MoreRelatedEvolution Of Educational Theories : Learning And Student Centered Learning848 Words   |  4 Pa gesEvolution of Educational Theories Looking into most classrooms an educator can be seen implementing Abraham Maslow, B.F. Skinner, or Maria Montessori’s motivational techniques. 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