Effect of Short Range Correlation

Effect of Short Range Correlation The effect of short range correlation on the nuclear charge density distribution, elastic and inelastic electron scattering coulomb form factors of 16O nucleus Abdullah S. Mdekil Abstract The effect of the short range correlation on the charge density disribution, elastic electron scattering form factors and inelastic Coulomb form factors is studied for the two excited states (6.92 and 11.52 MeV) in is analyzed. This effect (which depends on the correlation parameter) is inserted into the ground state charge density distribution through the Jastrow type correlation function. The single particle harmonic oscillator wave function is used with an oscillator size parameter The parameters and are considered as free parameters, adjusted for each excited state separately so as to reproduce the experimental root mean square charge radius of In inelastic coulomb (longitidinal) form factors of 16O, two different models are employed for . In the first model (model A), is considered as a closed shell nucleus. Here, the model space in does not contribute to the transition charge density, because there are no protons outside the closed shell nucleus . In the second model (mo del B), the nucleus of is assumed as a core of with 2 protons and 2 neutrons move in and model space. It is found that the introduction of the effect of short range correlations is necessary for obtaining a remarkable modification in the calculated inelastic Coulomb form factors and considered as an essential for explanation the data amazingly throughout the whole range of considered momentum transfer. Keywords: charge density distribution, elastic charge form factors, inelastic longitudinal form factors, short range correlation. 1-Introduction Electron scattering provides more accurate information about the nuclear structure for example size and charge distribution. It provides important knowledge about the electromagnetic currents inside the nuclei. Electron scattering have been provided a good test for such evaluation since it is sensitive to the spatial dependence on the charge and current densities [1, 2, 3]. Depending on the electron scattering, one can distinguish two types of scattering: in the first type, the nucleus is left in its ground state, that is called elastic electron scattering while in the second type, the nucleus is left on its different excited states, this is called inelastic electron scattering [4, 5]. In the studies of Massen et al. [6-8], the factor cluster expansion of Clark and co-workers [9-11] was utilized to derive an explicit form of the elastic charge form factor, truncated at the two-body term. This form, which is a sum of one- and two-body terms, depends on the harmonic oscillator parameter and the correlation parameter through a Jastrow-type correlation function [12]. This form is employed for the evaluation of the elastic charge form factors of closed shell nuclei and in an approximate technique (that is, for the expansion of the two-body terms in powers of the correlation parameter, only the leading terms had been kept) for the open and shell nuclei. Subsequently, Massen and Moustakidis [13] performed a systematic study of the effect of the SRC on and shell nuclei with entirely avoiding the approximation made in their earlier works outlined in [6-8] for the open shell nuclei. Explicit forms of elastic charge form factors and densities were found utilizing the fac tor cluster expansion of Clark and co-workers and Jastrow correlation functions which introduce the SRC. These forms depends on the single particle wave functions and not on the wave functions of the relative motion of two nucleons as was the case of our previous works [14-20] and other works [6,21,22]. It is important to point out that all the above studies were concerned with the analysis of the effect of the SRC on the elastic electron scattering charge form factors of nuclei. There has been no detailed investigation for the effect of the SRC on the inelastic electron scattering form factors of nuclei. We thus, in the present work, perform calculations with inclusion this effect on the inelastic Coulomb form factors for closed shell nucleus. As a test case, the is considered in this study. To study the effect of SRC (which depends on the correlation parameter on the inelastic electron scattering charge form factors of considered nucleus, we insert this effect on the ground state charge density distribution through the Jastrow type correlation function [12]. The single particle harmonic oscillator wave function is used in the present calculations with an oscillator size parameter The effect of SRC on the inelastic Coloumb form factors for the two excited states (6.92 and 11.52 MeV) in is analyzed. 2. Theory Inelastic electron scattering longitudinal (Coulomb) form factor involves angular momentum and momentum transfer and is given by [23] (1) where and are the initial and final nuclear states (described by the shell model states of spin and isospin ), is the longitudinal electron scattering operator, is the center of mass correction (which removes the spurious states arising from the motion of the center of mass when shell model wave function is used), is the nucleon finite size correction and assumed to be the same for protons and neutrons, A is the nuclear mass number, is the atomic number and is the harmonic oscillator size parameter. The form factor of eq.(1) is expressed via the matrix elements reduced in both angular momentum and isospin [24] (2) where in eq. (2), the bracket ( ) is the three- symbol, where and are restricted by the following selection rule: (3) and is given by The reduced matrix elements in spin and isospin space of the longitudinal operator between the final and initial many particles states of the system including configuration mixing are given in terms of the one-body density matrix (OBDM) elements times the single particle matrix elements of the longitudinal operator [25] (4) where and label single particle states (isospin included) for the shell model space. The in eq. (4) is calculated in terms of the isospin-reduced matrix elements as [26] (5) where is the isospin operator of the single particle. (6) The model space matrix element, in eq. (6), is given by (7) where is the spherical Bessel function and is the model space transition charge density, expressed as the sum of the product of the times the single particle matrix elements, given by [26]. (8) Here, is the radial part of the harmonic oscillator wave function and is the spherical harmonic wave function. The core-polarization matrix element, in eq. (6), is given by (9) where is the core-polarization transition charge density which depends on the model used for core polarization. To take the core-polarization effects into consideration, the model space transition charge density is added to the core-polarization transition charge density that describes the collective modes of nuclei. The total transition charge density becomes (10) According to the collective modes of nuclei, the core polarization transition charge density is assumed to have the form of Tassie shape [27] (11) where is the proportionality constant given by [14] (12) which can be determind by adusting the reduced transition probability to the experimental value, and is the ground state charge density distribution of considered nuclei. For the ground state charge densities of closed shell nuclei may be related to the ground state point nucleon densities by [28, 29] (13) in unit of electronic charge per unit volume (e.fm-3). An expression of the correlated density (where the effect of the SRCs is included), consists of one- and two-body terms, is given by [13] (14) where is the normalization factor and is the one body density operator given by (15) The correlated density of eq. (14), which is truncated at the two-body term and originated by the factor cluster expansion of Clark and co-workers [10-12], depends on the correlation parameter through the Jastrow-type correlation (16) where is a state-independent correlation function, which has the following properties: for large values of and for It is so clear that the effect of SRCs, inserted by the function becomes large for small values of SRC parameter and vice versa. The one-body term, in eq. (14), is well known and given by (17) where is the occupation probability of the state and is the radial part of the single particle harmonic oscillator wave function. The two-body term, in eq. (14), is given by [13] (18) where (19) The form of the two-body term is then originated by expanding the factor in the spherical harmonics and expressed as [13] (20) where (21) and is the Clebsch-Gordan coefficients. It is important to point out that the expressions of eqs. (17) And (20) are originated for closed shell nuclei with where the occupation probability is 0 or 1. To extend the calculations for isotopes of closed shell nuclei, the correlated charge densities of these isotopes are characterized by the same expressions of eqs. (17) and (20) (this is because all isotopic chain nuclei have the same atomic number but this time different values for the parameters and are utilized. The mean square charge radii of nuclei are defined by (22) where the normalzation of the charge density distribution is given by (23) 3-Results and discussion The ground state CDD is calculated by eq.(13) together with eqs. (14), (17) and (20). The calculated CDD without (with) the effect of the SRC [i.e., when the correlation parameter is obtained by adjusting only the parameter (the two parameters and ) so as to reproduce the experimental root mean square (rms) charge radii of nuclei under study. The elastic electron scattering charge form factors which is simply the Fourier transform of the ground state CDD. In Fig. 1, we compare the calculated CDD [Fig. 1(a)] and elastic charge form factors [Fig. 1(b)] of with those of experimental data (the open circles). In Fig. 1, we compare the calculated CDD [Fig. 1 (a)] and elastic charge form factors [Fig. 1 (b)] of with those of experimental data (the open circles). The dashed curves are the calculated results without the inclusion of the effect of the SRC obtained with and fm. The solid curves are the calculated results with including the effect of the SRC obtained with fm-2 and fm. It is important to point out that the parameters and employed in the calculations of the dashed and solid curves are chosen so as to reproduce the experimental rms charge radius of Fig. 1 (a) illustrates that the calculated CDD of the dashed curve (without the effect of the SRC) is in such a good agreement with that of the experimental data, and the solid curve (with the effect of the SRC) is not in such a good agreement with that of the experimental data, e specially in the central region ( fm) of the distributions. The inclusion of SRC has the feature of reducing the central region of the distribution as seen in the solid curve of this figure. Inspection to the Fig. 1 (b) gives an indication that the solid curve is better describing the experimental data than that of the dashed curve, particularly in the region of momentum transfer fm-1. The rms charge radius calculated with the above values of and is 2.621 fm, which is less than the experimental value by 0.097fm, which corresponds to a decrease of nearly 3.6 % of the experimental value. Fig. 1. The calculated CDD and elastic charge form factors are compared with those of experimental data. The dashed curve corresponds to the values for the parameters and fm, the solid curve corresponds to the values for the parameters fm-2 and fm while the open circles and the triangles in Figs. 1 (a) and 1 (b) are the experimental data taken from [30] and [31], respectively. The effect of the SRC on the inelastic Coulomb form factors is studied for the two excited states (6.92 and 11.52 MeV) in. Core polarization effects are taken into consideration by means of the Tassie model [eq. (11)], where this model depends on the ground state charge density distribution. The proportionality constant [eq. (12)] is estimated by adjusting the reduced transition probability to the experimental value. The effect of the SRC is incorporated into the ground state charge density distribution through the Jastrow type correlation function [12]. The single particle harmonic oscillator wave function is employed with an oscillator size parameter The charge density distribution calculated without the effect of the SRC depends only on one free parameter (namely the parameter), where is chosen in such away so as to reproduce the experimental rms charge radii of considered nuclei. The charge density distribution calculated with the effect of the SRC depends on two free parameters (namely the harmonic oscillator size parameter and the correlation parameter), where these parameters are adjusted for each excited state separately so as to reproduce the experimental rms charge radii of considered nuclei. Two different models are employed for. In the first model (model A), is considered as a closed shell nucleus. In this model, the proton occupation probabilities in are assumed to be and Here, the model space in does not contribute to the transition charge density [i.e. ], because there are no protons outside the closed shell nucleus . Accordingly, the Coloumb form factors of come entirely from the core polarization transition charge density. In the second model (model B), the nucleus of is assumed as a core of with 2 protons and 2 neutrons move in and model space. In this model, the proton occupation probabilities in are assumed to be and Here, the total transition charge density [eq. (10)] comes from both the model space and core polarization transition charge densities. The OBDM elements of are generated, via the shell model code OXBASH [32], using the REWIL [33] as a realistic effective interaction in the isospin formalism for 4 particles move in the and model spac e with a core. In Table 1, the experimental excitation energies (MeV), experimental reduced transition probabilities (fm) and the chosen values for the parameters and for each excited state (used in the calculations of model A and B) in and are displayed. The root mean square (rms) charge radius calculated in both models with the effect of SRC is also displayed in this table and compared with that of experimental result. It is evident from this table that the values of the parameter employed for calculations with the effect of SRC are smaller than that of without SRC ( fm) . This is attributed to the fact that the introduction of SRC leads to enlarge the relative distance of the nucleons (i.e., the size of the nucleus) whereas the parameter (which is proportional to the radius of the nucleus) should become smaller so as to reproduce the experimental rms charge radius of the considered nuclei. Inelastic Coloumb form factors for different transitions in are displayed in Figs. 1 and 2. The calculated inelastic form factors obtained with model A are shown in the upper panel [Figs. 1(a)-2(a)] of the above figures whereas those obtained with model B are shown in the lower panel [Figs. 1(b)- 2(b)] of the above figures. It is obvious that all transitions considered in, presented in the above figures, are of an isoscalar character. Besides, the parity of them does not change. Here, the calculated inelastic form factors are plotted versus the momentum transfer and compared with those of experimental data. The dashed and solid curves are the calculated inelastic Coloumb form factors without and with the inclusion of the effect of the SRC, respectively. The open symbols are those of experimental data taken from [34, 35]. Table1. The experimental excitation energies and reduced transition probabilities, the chosen values for and as well as the rms charge radius calculated with the effect of the SRC of 16O. (fm) Model B Model A fm2L) (MeV) State (fm) (fm-2) (fm) (fm) (fm-2) (fm) [30] 2.704

The Parmenidean Paradox Of Motion Essay -- essays research papers

Philosophical thought begins with the Milesians, where intellectual curiosity propelled thinkers like Anaximander and Heraclitus to attempt to explain the phenomena of the universe by means of specific physical elements. During the 6th century BC, Eleatics, like Parmenides and Zeno, had rejected physical phenomena and propounded metaphysical paradoxes that cut at the roots of belief in the very existence of the natural world. Parmenides uproots the theories of his predecessors by bearing to light the logical possibilities of any philosophical inquiry. He argues that that the only things about which we can inquire about must exist, else our search is fruitless. Through deductive reasoning, Parmenides proves that if something exists, then it cannot come to be or perish, change or move, nor be the subject to any imperfection. His proteges were left with an enormous problem: how could one reconcile Parmenides’ rejection of change with the possibility of giving a rational account of the changing world of sense experience? By accepting only certain parts of his doctrine of being, his successors ultimately fail in their attempts to explain the changing universe in light of the Parmenidean paradox. How does Parmenides draw the conclusion that if something is, then it is unchanging? A more formal examination of his arguments regarding subjects of inquiry shows how he comes to the conclusion that all is one. The only ways of inquiry there are for thinking: the one, that it is and that it is not possible for it not to be, is the path of Persuasion (for it attends upon the Truth), the other, that it is not and that it is necessary for it not to be, this I point out to you to be a path completely unlearnable, for neither may you know that which is not (for it is not to be accomplished) nor may you declare it (Curd fr.2 ll.3-8, pg.45). Parmenides’ subject of inquiry, as show in the fragment, either you must assume that your subject is or it is not. Careful consideration of the statement ‘is not’ shows that it is impossible to point out what does not exist, because it has no attributes or true predicate. Parmenides concludes that if something does not exist, then its non-existence cannot allow for it to come into being or perishing, because if it comes to be, then formally, it previously did not exist. Since we cannot know anything about things that do not exist, coming... ...rmenidean doctrine that substances are uncreated and eternal; however, by positing that there are four creative and two controlling substances, he dubiously maintains that combination and separation, through their endless cycles bring about a whole. If Empedocles were to follow the Parmenidean notion of being absolutely, then his separation and combination would never take place, because each element would be continuously attracted and negated, so that no combination could ever take place.   Ã‚  Ã‚  Ã‚  Ã‚  The Pluralists want to reconcile everything that they perceive through their senses with the Parmenidean idea of an uncreated, eternal, unchanging whole. The problem of such a task lies in the fact that Parmenides’ notion of being goes against everything that our sense experience tells us. With our eyes we see motion and change every day, be it our own self-motion or that of others around us. Furthermore, we experience coming-into-being and perishing through the cycle of birth and death. The Pluralists would had made better progress in extrapolating their own ideas if they would have either sided completely with Parmenides or taken means to discredit his work.

Pros and Cons of exercising Essay

Question 3: Do you agree that exercising is the only way to keep fit and healthy? When it comes to the the phrase ‘a healthy lifestyle’, many people often think of the gym as the only way to keep your body in shape. It seems difficult for them to achieve, and is often seen very negatively. However, exercising is not the only way. In fact, there are many ways to keep fit and healthy, such as having a regular diet, having enough rest, and keeping a positive mindset. Having a regular diet is one of the simplest yet hardest way to keep fit. It requires one to adjust the way they eat, and be cautious of their food intake. However, it may not seem as difficult as it may seem to be. For example, fish and chips seems very unhealthy, but there are ways to make it healthier. Baking it instead of deep-frying it, and just adding a few vegetables to go along with your meal would be an easy way to â€Å"neutralize† your unhealthy food. Make a few simple changes to your diet, and you would soon feel much healthier. To make sure your body has abundant energy to do your daily activities, having enough rest is very important. While you rest, your brain stays busy, and prepares you for the next day. Minimal deprivation of sleep takes a toll on your mood, energy, and ability to handle stress. Adults should sleep betweeen 6-7 hours each day. However, if one is not getting average sleep time, it is best to catch up on weekends where there is no work. Sleep should not be deprived of as it can affect your mental state, energy level and ability to focus. Lastly, having a postitive mindset can help boost one’s morale or motivation spirit. When one is postive about his or her life, he or she would be able to solve obstacles they face with ease as they are able to find ways to solve their problems rather than just dwelling over it. This helps keep both the mind and brain healthy. Rather than thinking that life is unfair, one should appreciate the many things that one has.

What Are Unethical About Stem Cell Research - 956 Words

Introduction In the early 20th century a Russian American named Alexander A. Maximow, established the theory that every cell comes from a precursor cell. Maximow was deeply involved in histology, the study of plant and animal tissue. Being the first to show that blood cells come from a common precursor cell, he is given the most credit in discovering what is known as stem cells (source #7). Stem cells have the ability to change into other cells such as blood, bone, tissue, and muscle cells. Researchers are trying to better understand these foundation cells to create cures and treatments for diseases and injuries. The report will be based on two articles. The next point will evaluate each source. Afterwards, the following information will†¦show more content†¦Funding for the Research in 2005 the Public Health Committee of the General Assembly passed a bill that permits the harvesting of human embryotic stem cells; obtained from the embryo. The bill also allows the harvesting of adult stem cells; found in bone marrow or umbilical cords. President Bush put a limitation on the funds (315.6 million 2005) for creating embryotic stem cells prior to the approval. About 4.4 billion dollars was funded towards stem cell research from California’s taxpayers fund approval; Wisconsin’s commitment towards biotech research and stem cell projects, and New Jersey. (Silverman, pg. CT1) Pros and Cons Silverman introduces The State Representative Lawrence Miller, he gave his testimony about adult stem cells. Miller is a cancer survivor who had adult stem stem cells implanted in him. He said in the article that he was hesitant in sending the bill because of research controversy. He resolve and said, â€Å"why aren’t we doing this† referring to the research. Although Miller rejoiced over the approval, some religious groups denounced the bill because of ethics. Others in the medical community oppose the bill as well. The article zoned in on Dr. Mathews-Roth of Harvard. She gave information about embryonic stem cells that might have discouraged supporters of the bill. She said, â€Å"Embryonic stem cells can form tumors when implanted and run a higher risk of being rejected than adult stem cells when transplanted.† (Silverman, Pg. CT1) Ethical