INVESTMENT AND PORTFOLIO MANAGEMENT Assignment Example | Topics and Well Written Essays - 2000 words

INVESTMENT AND PORTFOLIO MANAGEMENT - Assignment Example The investor who prefers to bank his funds to generate a fixed ‘certain’ interest at the end of a term is the classic case of risk-averse individual while a casino gambler who bets against high ‘uncertain’ odds is at the other end of the spectrum (Pietersz, 2009). In scenario whereby an individual investment is assured of a  £500 return, in the uncertain situation, a bet is considered that with a toss of a penny, the individual can get  £1,000 or naught, while in the certain situation the individual will definitely receive the  £500. Although both situations have a guaranteed return of  £500, the uncertain situation has a 50 percent chance of garnering  £1,000 or nothing. Therefore, three possible scenarios emerge: Risk aversion is therefore a characteristic case of martingale effect whereby the most likely scenario is the investor risk-taker only gaining the original amount (Yates, 2009). In modern portfolio theory, risk aversion is calculated as the added subsidiary return an investor needs to admit supplementary risk, which is calculated through the standard deviation of the ROI or the square root of its variance (Baker, 2001). Modern portfolio theory established mean-variance efficient portfolios in a fixed time horizon that ignored future market movements hence not applicable to multi-period investment horizon. Sharpe (1964), Lintner (1965) and Mossin (1966) separately have been ascribed to establishing the Capital Asset Pricing Model (CAPM) model that was developed from Markowitzs (1959) exposition of the Modern Portfolio Theory (MPT) particularly the mean-variance model. The fundamental theory of the CAPM indicates that there is a linear link involving systematic risk, as measured by beta, and projected share returns (Brewton, 2009). The CAPM model endeavours to illustrate the linkage by applying beta to describe the differences involving the likely proceeds from shares and share portfolios (Laubscher, 2002, p.

History of the English Language Essay Example | Topics and Well Written Essays - 500 words

History of the English Language - Essay Example did not take place overnight since the language developed over hundreds of years with influences from many different languages therefore it has interesting origins. The English language began to form with the arrival of three Germanic tribes who invaded Britain in the fourth century. The three tribes were the Angles, the Saxons and the Jutes and came from the northern areas of Germany. The Celtic language speakers of Britain were repelled by the invaders to areas of Wales and Scotland (Wells, 1982). At the same time, the language of the Angles tribe became the predominant language of the region from which the words England and English are taken (English Club, 2006). The language developed into what we know as ‘Old English’. This language would be difficult if not impossible for us to understand since there would be many more modifications to the language before it would develop into modern English. Even so, many of the words we use in English today have their roots in Old English. For instance words like Strong and Water come from Old English even though they are in common use today (Wells, 1982). In 1066, the same tribes were conquered by William the Conqueror who came from the area of Normandy which is in modern day France. He and his forces spoke a version of French which was used in the royal court while the lower classes spoke English with a mixture of French words. This version of English came to be known as Middle English and was used by Chaucer to write his Canterbury tales. It is difficult for modern speakers of English to understand or read this language but it does show how French words came to be accepted in English use (English Club, 2006). Modern English as we see it today has two distinct periods of development, the first is the early period which lasted from 1500-1800 where changes were made in pronunciation, character use and many more words were added to English. With the coming of the Renaissance, words from classical languages found

Historical events of the US Essay Example | Topics and Well Written Essays - 1250 words

Historical events of the US - Essay Example The conventional Protestant spiritual groups set up religious bases in Asia and Africa, as well as over 450 missions in China by the year 1900.The 20th century started with the US entangled in a severe and bloody war. However, that war is no longer remembered, a war that cost millions of dollars, thousands of lives and displacement of millions. ThePhilippine-American War that was fought between1899 to 1902 resulted in approximately 265,000 human deaths. Thewarassistedthe US establish itself as a military powerhouse in the Pacific regions. While the European powers were engaging themselves in the First World War (WWI), the then president of the US Wilson Woodrow was apprehensive that Germany might take up Haiti and destabilize the pathway leading to the Panama Canal. In order to assert and protect its interest, Wilson sent a few hundred US Marines and Navy men to Haiti. In 1898 and 1899, The US took over Hawaii and obtained the Philippines, Puerto Rico, and some parts of the Pacific I slands. Theexpansionraisedthesignificantquery of if their recently annexed states would be given the American citizens civil rights. HavingwonboththeSpanish-American War and Philippines War symbolized both an addition to earlier expansionist desires and a quick departure from suppositions that had steered American foreign strategy before. Fortheveryfirsttime, the US made significant strategic dedications in the Far East; acquiring regions that had never been planned for statehood, and dedicated itself to law enforcement actions.

INVESTMENT AND PORTFOLIO MANAGEMENT Assignment Example | Topics and Well Written Essays - 2000 words

INVESTMENT AND PORTFOLIO MANAGEMENT - Assignment Example The investor who prefers to bank his funds to generate a fixed ‘certain’ interest at the end of a term is the classic case of risk-averse individual while a casino gambler who bets against high ‘uncertain’ odds is at the other end of the spectrum (Pietersz, 2009). In scenario whereby an individual investment is assured of a  £500 return, in the uncertain situation, a bet is considered that with a toss of a penny, the individual can get  £1,000 or naught, while in the certain situation the individual will definitely receive the  £500. Although both situations have a guaranteed return of  £500, the uncertain situation has a 50 percent chance of garnering  £1,000 or nothing. Therefore, three possible scenarios emerge: Risk aversion is therefore a characteristic case of martingale effect whereby the most likely scenario is the investor risk-taker only gaining the original amount (Yates, 2009). In modern portfolio theory, risk aversion is calculated as the added subsidiary return an investor needs to admit supplementary risk, which is calculated through the standard deviation of the ROI or the square root of its variance (Baker, 2001). Modern portfolio theory established mean-variance efficient portfolios in a fixed time horizon that ignored future market movements hence not applicable to multi-period investment horizon. Sharpe (1964), Lintner (1965) and Mossin (1966) separately have been ascribed to establishing the Capital Asset Pricing Model (CAPM) model that was developed from Markowitzs (1959) exposition of the Modern Portfolio Theory (MPT) particularly the mean-variance model. The fundamental theory of the CAPM indicates that there is a linear link involving systematic risk, as measured by beta, and projected share returns (Brewton, 2009). The CAPM model endeavours to illustrate the linkage by applying beta to describe the differences involving the likely proceeds from shares and share portfolios (Laubscher, 2002, p.

Historical events of the US Essay Example | Topics and Well Written Essays - 1250 words

Historical events of the US - Essay Example The conventional Protestant spiritual groups set up religious bases in Asia and Africa, as well as over 450 missions in China by the year 1900.The 20th century started with the US entangled in a severe and bloody war. However, that war is no longer remembered, a war that cost millions of dollars, thousands of lives and displacement of millions. ThePhilippine-American War that was fought between1899 to 1902 resulted in approximately 265,000 human deaths. Thewarassistedthe US establish itself as a military powerhouse in the Pacific regions. While the European powers were engaging themselves in the First World War (WWI), the then president of the US Wilson Woodrow was apprehensive that Germany might take up Haiti and destabilize the pathway leading to the Panama Canal. In order to assert and protect its interest, Wilson sent a few hundred US Marines and Navy men to Haiti. In 1898 and 1899, The US took over Hawaii and obtained the Philippines, Puerto Rico, and some parts of the Pacific I slands. Theexpansionraisedthesignificantquery of if their recently annexed states would be given the American citizens civil rights. HavingwonboththeSpanish-American War and Philippines War symbolized both an addition to earlier expansionist desires and a quick departure from suppositions that had steered American foreign strategy before. Fortheveryfirsttime, the US made significant strategic dedications in the Far East; acquiring regions that had never been planned for statehood, and dedicated itself to law enforcement actions.

State of the U.S. economy for the first half of 2008 Essay Example for Free

State of the U.S. economy for the first half of 2008 Essay In the first half of 2008, four things are pummeling the typical American: the housing bust, the credit crunch, higher fuel and food costs and, most recently, a weakening labor market. The unemployment rate rose to 5. 1% in March, while the private sector lost jobs for the fourth month in a row. Consumer confidence is in a 26-year low. Feeling poorer and with fewer people prepared to lend money, consumers are cutting back. Seeing that consumer spending accounts for 70% of American demand, makes it more difficult for the economy especially when it is coupled with a collapse in the once mighty construction industry. The International Monetary Fund (IMF) now officially predicts an American recession in 2008. (The Economist). For the first time, Federal Reserve Chairman Ben Bernanke conceded that the U. S. economy may slip into recession but said growth should pick up later this year as the impact of interest rate cuts and other emergency steps take root. Bernanke told a congressional panel that the economy appeared to be growing, but warned it could shrink in the first half of 2008 as housing and financial markets remain distressed despite dramatic Fed interest rate cuts and emergency lending (Reuters). Volatility and crisis were the watchwords of the U. S. economy in 2007 buffeted by rising oil prices and the subprime mortgage crisis. Their combined impact stretched into 2008 especially in the first half, slowing economic growth. The impact of these factors was felt in the broader economy undermining both business investment and consumer confidence. The U. S. Chamber of Commerce appears to be more optimistic predicting modest growth averaging about 2% in the first half of the year, an improvement over a weak fourth quarter of 2007. Growth should increase to 2. 5% in the second half, as the housing sector bottoms out. Job growth will continue, albeit at a relatively slow pace-with the unemployment rate rising slightly above 5%. Corporate profits should improve as 2008 progresses and business investment will pick up modestly by the middle of the year. Export growth will continue to brighten the overall economic picture. Inflation will be in the moderate range, making further interest rate reductions less likely as the year progresses (US Chamber of Commerce). As 2008 progressed, many Americans experienced a growing despair as they watched their largest asset- the family home- decline in value. The United States is experiencing its worst housing recession in more than 15 years. Underscoring the breadth of the real estate recession, sales of existing homes fell in 45 states and Washington D. C. and prices dropped in more than half the metro areas it tracks according to the National Association of Realtors. In fact, homes are selling at a price 24% less than year ago as foreclosures continue to increase dramatically at 57% in Southern California. The slide in sales are predicted to persist and prices will likely fall throughout 2008, according to a majority of economists surveyed last month by USA TODAY (Knox). The housing downturn is spreading more broadly through the economy. Employers are shedding jobs, consumer confidence and spending have been shaken, and lenders have pulled back. If not for stronger demand for U. S. goods brought on by a weaker dollar, the economy would be in worse shape. To cushion the effect of the on-going crisis, the Federal Reserve has slashed interest rates, promised more cuts if the economy stays weak and perhaps most importantly sharply reduced the odds of financial-market catastrophe by extending its safety net to investment banks. The Federal Reserve has lowered benchmark interest rates by three percentage points to 2. 25 percent since mid-September to help put a floor under an economy hit hard by a housing slump and credit market turmoil. Bernanke said those rate cuts and other emergency measures to thaw frozen credit markets should promote growth over time. Also, Bernanke staunchly defended the Federal Reserves decision last month to broker JPMorgan Chases (JPM) takeover of investment bank Bear Stearns, (BSC) including approval of a loan backed by $30 billion of Bear Stearns assets. According to Bernanke, â€Å"A Bear Stearns default could have sparked a chaotic unwinding affecting the overall economy. Given exceptional pressures on the global economy and financial system, the damage caused by a potential Bear Stearns default could have been severe and extremely difficult to contain† (Kirchhoff). In retrospect, if the Federal Reserve did not intervene bankruptcy was inevitable which might have caused the U. S. and global markets to collapse. Even though the economy is slowing, inflation, boosted by skyrocketing energy and food prices remains a concern and constraint for the central bank. Another factor would be the declining value of the dollar; if the dollar buys less, inflation rises. The Federal Reserve data show that the U. S. dollar has declined about 10 percent over the past year against a trade-weighted basket of currencies from major U. S. trading partners. Moreover, the dollar sank to new lows against the euro in the days following March 4, 2008 after a series of dour reports on the U. S. economy and expectations that the Federal Reserve will continue slashing interest rates. Indeed, with the falling of the dollar, prices of all the things bought are rising. The Producer Price Index for one month was up 1. 1% in March with the price of wheat and rice up at least 100 percent than last year. The price of oil reached $114 a barrel and gasoline is also at a new high of $3. 38 per gallon. Purchases of crude oil jumped, reflecting increases in the number of barrels bought and a record price surpassing $100 a barrel on the New York Mercantile Exchange early this year. The U. S. is the worlds biggest consumer of crude oil and higher fuel costs are making imports more expensive (Bloomberg). America is in fact the worst food and energy inflation in two decades and consumers take a direct hit. The pace of job losses in America has been relatively mild compared with previous downturns. However, data showed the economy during February shed the biggest amount of jobs in nearly five years. Because of the high fuel prices, budget airlines are bleeding and are filing bankruptcies resulting to further job losses. Frontier airline was the fourth airlines to do so in just two weeks. There is also slow spending in malls everywhere as increasing number of consumers struggling to make ends meet causing retailers such as Sharper image, Levitz Furniture and stores like Ann Taylor, Zales Jeweler and Footlocker to close. General Electric (GE), second largest company on Earth, reported a 6% loss in income for the first quarter. It was a surprise result that rattled Wall Street with the stocks of GE experiencing its worst one-day loss since the 1987 crash. GE had a lot of trouble selling commercial real state and people are not buying appliances. Economic figures indicate a weakening economy. The Energy Information Administration (EIA) forecast that U. S. real gross domestic product would contract from $11. 577 trillion in the fourth quarter of 2007 to $11. 563 trillion in the first quarter of 2008, then decline to $11. 542 trillion in the second quarter. U. S. Stock prices continue to slip, with the blue chip Dow Jones industrial average closing down at the 12,000-level. Prices for U. S. government bonds also fell, as did the value of the dollar. As waves of bad news began to wash in- foreclosures, tumbling dollar, falling retail sales and more recently investment bank rescues- exporters were the only thing keeping the national nose and lips above the recessionary waters. The decline in the value of the dollar relative to other currencies, notably the euro, has helped make U. S. goods less expensive for overseas buyers and therefore more attractive. This fuelled a healthy global demand for U. S. products and services resulting to earnings from exports rising to a new record (Chandra). So what can be done in order to address the crisis? The government should introduce a fiscal stimulus, which would help create new jobs in 2008, and induce consumer spending through tax rebates in order to keep the economy moving. To try and keep the subprime-mortgage crisis from escalating, the Federal Reserve should propose changes for loans including requirements that creditors should strictly verify a prospective borrower’s income and assets before providing actual loans. The government must offer mortgage advice to homebuyers who may not have understood the risks. Often borrowers did not even realize that their monthly payment would rise if interest rates went up. Subprime borrowers on adjustable interest rates, whose mortgages make up just 7% of the total, accounted for more than 40% of the foreclosures begun in the fourth quarter of last year (The Economist). And if the subprime mortgage crisis gets worse, the government should consider a possible bailout for the housing market. For a long-term solution, a new version of a deliberate program to build up the middle class must be instituted similar in nature with postwar programs like the G. I. Bill, interstate highway system and other measures with job-creating investments in biomedical research, alternative energy, roads, railroads and education (McIntyre). The G. I. Bill, created after World War II, was one of America’s most successful investments. According to the 1988 report for Congresss Subcommittee on Education and Health of the Joint Economic Committee by 1952, the US government had spent $14 billion (1952 dollars) on educational and job training benefits for 7. 8 million veterans. Of these funds, $7 billion was spent on college and graduate school for 2. 2 million G. I. ’s. ? The first benefit from this investment was increased growth in the economy. The report calculated that about 40 percent of those who took advantage of the G. I. Bill would not otherwise have been able to attend college. The extra output those people created in the economy amounted to $35. 6 billion (1952 dollars after factoring out inflation) over the next 35 years. America is in the bleak of a recession and a sound economic plan should be in place to cushion its effect. References Chandra, Shobhana. â€Å"U. S. January Trade Deficit Rises 0. 6%; Exports Gain (Update6)†. Bloomberg. 29 April 2008. 29 April 2008. http://www. bloomberg. com/apps/news? pid=20601087refer=homesid=aynvcWrnI 8w. Doggett, Tom. â€Å"UPDATE 2-U. S. economy to contract in 1st half of 2008 –EIA†. Reuters. 11 March 2008. 2 May 2008. http://www. reuters. com/article/oilRpt/idUSN1148909720080311. Kirchhoff, Sue. â€Å"Fed chief Bernanke defends Bear Stearns deal†. USA Today. 4 April 2008. 2 May 2008. http://www. usatoday. com/money/economy/2008-04-02-bernanke economy_N. htm . Knox, Noelle. â€Å"Falling home sales problem spreads to 45 States†. USA Today. 14 Feb 2008. 2 May 2008. http://www. usatoday. com/money/economy/housing/2008-02-14 housing-q4 nar_N. htm. McIntyre, Jamie â€Å"Tenth anniversary of the Gulf War: A look back† CNN. com In-depth specials Gulf War. 16 Jan. 2001. 24 April 2008. http://archives. cnn. com/2001/US/01/16/gulf. anniversary/index. html. MoneyNews. â€Å"Treasury: Economy May Improve in 2nd Half 2008†. MoneyNews . 24 April 2008. 2 May 2008. http://moneynews. newsmax. com/money/archives/articles/2008/4/24/085350. cfm. Subcommittee on Education and Health of the Joint Economic Committee (1988). A Cost Benefit Analysis of Government Investment in Post-Secondary Education Under the World War II GI Bill. The Economist. â€Å"Getting it right on the money†. The Economist. 3 April 2008. 8 April 2008. http://www. economist. com/displaystory. cfm? story_id=10958702. US Chamber of Commerce. â€Å"The Economy in 2008†. US Chamber of Commerce. 2 May 2008.

Calculations of Rare Earth (Y, La and Ce) Diffusivities

Calculations of Rare Earth (Y, La and Ce) Diffusivities First-principles calculations of rare earth (Y, La and Ce) diffusivities in bcc Fe Xueyun Gaoa,b,[*], Huiping Rena, Chunlong Lia,c, Haiyan Wanga, Yunping Jia, Huijie Tan a ABSTRACT: The impurity diffusivities of rare earth elements, Y, La and Ce, in bcc Fe have been investigated by the first-principles calculations within nine-frequency model. The microscopic parameters in the pre-factor and activation energies have been calculated. For the three elements, the first nearest-neighbor solute-vacancy interactions are all attractive, in which Y and La solute atoms more favorably bond to the vacancy. The solute-vacancy binding energy can be explained in terms of the combination of the distortion binding energy and the electronic binding energy, and the decomposition results of the total solute-vacancy binding energy suggest that the strain-relief effect accounts for larger portion of the binding energy for Y and La than that for Ce. The diffusion coefficients of Y are one order of magnitude larger than that of La, and predicted to be comparable to that of Fe self-diffusion. Compared with Y and La, Ce shows large migration energy and small solute-vacancy att ractive interaction, which accounts for the lowest diffusivity of this element. Keywords: Diffusion; Rare earth; Bcc Fe; First-principles calculations 1. Introduction In the past years, the addition of rare earth (RE) elements has been regarded promising in steels. A series of beneficial research for the development of rare earth addition have been focused on the purification and modification of inclusion, since RE elements are characterized by significant negative free energy changes for compound formations [1-3]. RE doping also improve the high-temperature oxidation resistance and the corrosion resistance of steels due to the reactive-element effect (REE) [4,5].In addition, the solidification, phase transformations, recrystallization behavior, of steel can be improved by adding RE [6, 7]. Knowledge of the above mentioned mechanism is essential to understand the influence of RE additions on the physical, chemical and various properties of steels. In spite of the progress so far in RE application, it is apparent that many questions still remain rather controversial. A thorough theoretical study on the diffusivities of RE elements in Fe-based alloy is still lacking, which is essential for understanding the effects of RE on the structure and properties of steel, and is also helpful for designing and preparing RE doped steels [8]. For the three commonly used RE elements, Y, La and Ce, to our knowledge, only the diffusion coefficient of Y has been reported [9]. The diffusion of substitutional-type solute is mainly controlled by vacancy mechanism. In this case, the interaction of solute atom with vacancy plays significant roles in understanding of the diffusion properties of solutes. To understand the microstructure evolution in bcc Fe alloy, DFT method has been applied in calculations of the binding energies of solute atom with vacancy [10]. Based on the nine-frequency model of Le Claire, Ding and Huang et al. [11,12] developed a computational framework to calculate the solute diffusion coefficients in bcc Fe, which presents an effective method in investigation of the diffusion properties in bcc Fe alloy. The purpose of this work is to investigate the impurity diffusivities of rare earth elements, Y, La and Ce, in bcc Fe by the first-principles calculations within nine-frequency model and the semi-empirical corrections for magnetization[11,13], calculate the associated solute-vacancy binding energies, migration energies, pre-factors and activation energies for these impurity diffusivities, as well as the self-diffusion coefficient of Fe, thus discuss the related factors of the diffusion coefficients. 2. Methodology The temperature dependence of diffusion coefficient D is expressed in the Arrhenius form D=D0exp(-Q/kBT), where D0 and Q are the pre-factor and activation energy, respectively. Below the Curie temperature, the self-diffusion and solute diffusion coefficients in bcc Fe deviate downward from the Arrhenius type relationship extrapolated from the paramagnetic state [14]. These deviations are attributed to the change of magnetization which affects the diffusion activation energy. To investigate the effect of magnetic disorder on the diffusion activation energy of spin-polarized metals, Ding et al. [15] reported a first-principles approach based on the spin-wave DFT method for studying the self-diffusion of bcc Fe and fcc Co, and the calculated values of ÃŽÂ ± agree well with the experimental data. More recently, by combination of the first-principle calculations and Heisenberg Monte Carlo simulations, Sandberg et al. [16] presented a quasi-empirical model to study the magnetic contribu tion to the self-diffusion activation energy of bcc Fe. Murali et al. [17] conducted a systematic study of the effects of phonon and electron excitations on the free formation energy of vacancy, the solute-vacancy binding energy, and the vacancy migration energy in bcc Fe. The authors then calculated the Fe self-diffusion coefficient based on the computed free energies, by employing the semi-empirical model presented in Ref. [18]. The diffusion associated data yielded by these methods are in good agreement with experiments. We employ the semi-empirical model to describe the dependence of the diffusion activation energy on the magnetization in the ferromagnetic state [18]: QF(T)=QP[1+ÃŽÂ ±s(T)2] (1) where QP is the activation energy in the paramagnetic state; s(T) is the ratio of the magnetization of bcc Fe at a certain temperature T to that at 0K, and has been experimentally measured [19,20]; the constant ÃŽÂ ± quantifies the extent of the influence of magnetic on activation energy, the measured value of ÃŽÂ ± for Fe self-diffusion is 0.156 [11].In the case of the solute species investigated in this context that have no measured ÃŽÂ ± values available, the values can be estimated from a linear correlation with the induced changes in local magnetization on Fe atoms in the first and second neighbor shells of a solute atom [21]. The first-principles calculations give direct access to the magnitude of the activation energies for self- and solute diffusion in the fully ordered ferromagnetic state (T=0K). Given the values of ÃŽÂ ± and s(T), we can compute QP through the relation , and QF(T) from Eq. (1). We based the diffusion calculations on the assumption that the mechanism of diffusion is primarily monovacancy mediated. The self-diffusion and solute diffusion coefficients on a bcc lattice can be expressed as following [11,22]: Dself=a2f0Cvw0 (2) Dsolute=a2f2Cvw2(-ΆGb/kBT) (3) where a is the bcc lattice constant, f0=0.727 is the self-diffusion correlation factor, f2 is the correlation factor for solute diffusion which depends on the relative vacancy jump frequencies around the solute atom. Cv denotes the equilibrium vacancy concentration, can be written as Cv=exp(ΆSf /kB)exp(-ΆHf /kBT), where ΆSf and ΆHf are the vacancy formation enthalpy and entropy, respectively, the harmonic approximation makes these two become temperature-independent constants. and kB is Boltzmanns constant. w0 and w2 are the vacancy hopping frequencies for Fe and solute atoms to exchange with a nearest-neighbor vacancy, respectively. Based on transition state theory (TST), the vacancy hopping frequency w is written as , where and are the phonon frequencies in the initial state and transition state, and the product in the denominator ignores the unstable mode; ΆHmig is the migration energy, gives the energy difference for the diffusin g atom located at its initial equilibrium lattice position and the saddle-point position. The solute-vacancy binding free energy ΆGb can be expressed as ΆGb =ΆHbTΆSb, where ΆHb and ΆSb are the binding enthalpy and entropy, respectively. The correlation factor f2 can be calculated using the nine-frequency model developed by Le Claire [13] which involves different jump frequencies of vacancies to their first neighbor position in the presence of the solute atoms, as illustrated in Fig. 1. In this model, the interaction of solute-vacancy is assumed up to second neighbor distance. The nine frequencies shown in Fig. 1 illustrate all of the distinct vacancy jumps in a bcc system with a dilute solute concentration, including the host Fe atom jump w0 without impurity. The detailed calculation procedures could be found in Ref. [11]. Fig. 1. Schematic illustration of the nine-frequency model for the bcc Fe crystalline with a solute atom. The arrows denote the jump directions of the vacancy. The numbers in the circle represent the neighboring site of the solute atom. For convenience, we can represent the self- and solute diffusion equations (Eqs.(2) and (3)) in Arrhenius form to obtain the pre-factor and activation energy of diffusion. By combining the above Eqs., the diffusion coefficient for Fe self-diffusion and solute diffusion can be expressed as: (4) For self-diffusion, the pre-factor is, and the activation energy given as . Also, the solute diffusion coefficient can be expressed in an Arrhenius form with the pre-factor is, and. The first-principles calculations presented here are carried out using the Vienna Ab initio Simulation Package(VASP) with the projector augmented wave(PAW) method and the generalized gradient approximation of Perdew-Burke-Ernzerhof functional(GGA-PBE) [23]. All calculations were performed in spin polarized. The computations performed within a 4à ¯Ã¢â‚¬Å¡Ã‚ ´4à ¯Ã¢â‚¬Å¡Ã‚ ´4 supercell including 128 atoms. The binding, vacancy formation and migration energies were calculated with 300eV plane-wave cutoff and 12à ¯Ã¢â‚¬Å¡Ã‚ ´12à ¯Ã¢â‚¬Å¡Ã‚ ´12 k-point meshes. The residual atomic forces in the relaxed configurations were lower than 0.01eV/Ã…. The transition states with the saddle point along the minimum energy diffusion path for vacancy migration were determined using nudged elastic band (NEB) method [24] as implemented in VASP. We adopt the harmonic approximation (HA) to consider the contribution of normal phonon frequencies to free energy. The normal phonon frequencies were calcu lated using the direct force-constant approach as implemented in the Alloy Theoretic Automated Toolkit (ATAT) [25] package. Similar cutoff energy, k-point mesh size and supercell size used for the total energies were used for the vibrational calculations. 3. Result and discussion Table 1 illustrates our calculated energies for vacancy formation, migration and binding, as well as the constant ÃŽÂ ± for solute species, the associated paramagnetic activation energies and fully ordered ferromagnetic activation energies for both self- and solute-diffusion. For pure bcc Fe, the vacancy formation energy and migration energy obtained here are consistent with the reported range of values, ΆHf=2.16-2.23 eV and ΆHmig=0.55-0.64 eV [11,26,27]. For Y impurity in bcc Fe, the calculated vacancy binding energy in full ordered ferromagnetic state also compare well with the previous first principles work [28], in which ΆHb=-0.73 eV. It can be seen that Y and La have smaller activation energy than that for Fe self-diffusion, while Ce is predicted to have a lager value of activation energy than that for Fe self-diffusion, in both the ordered ferromagnetic and paramagnetic state. Table 1 Vacancy formation energy ΆHf , solute-vacancy binding energy ΆHb, migration energy ΆHmig, the ferromagnetic activation energy and the paramagnetic activation energy QP; the variable dependence parameter of activation energy on magnetization ÃŽÂ ±. Fe Y La Ce ΆHf(eV) 2.31 à ¢Ã¢â€š ¬Ã¢â‚¬â„¢ à ¢Ã¢â€š ¬Ã¢â‚¬â„¢ à ¢Ã¢â€š ¬Ã¢â‚¬â„¢ ΆHb (eV) à ¢Ã¢â€š ¬Ã¢â‚¬â„¢ -0.69 -0.66 -0.43 ΆHmig (eV) 0.54 0.09 0.17 1.09 ÃŽÂ ± 0.156 0.088 0.038 0.125 (eV) 2.85 1.71 1.82 2.97 QP (eV) 2.47 1.57 1.75 2.64 Solute-vacancy binding energy plays a crucial role in understanding solute diffusion kinetics. Table 2 presents the binding energies of Y, La and Ce atoms with vacancy in their 1nn, 2nn and 3nn coordinate shells. From Table 2 it can be seen that referring to the first nearest-neighbor solute-vacancy pairs, the binding energies are all negative, which implies the solute-vacancy pairs are favorable. Specifically, Y and La impurities are computed to have higher values of solute-vacancy binding energies -0.69 eV and -0.66 eV in 1nn configuration, respectively, while that for Ce is -0.43 eV. Correspondingly, we found that Y, La and Ce atoms relax towards the 1nn vacancy by 22.3%, 19.6% and 12.2% of the initial 1nn distance (2.488 Ã…) after the structure optimization. The interactions of the solute-vacancy pair at the 2nn shells tend to be smaller in magnitude than that of 1nn, and that of Ce-vacancy predicted to be repulsive. The interactions of the 3nn solute-vacancy are relatively we ak, indicating that the interactions of the solute-vacancy are local. According to Le Clair model [13], in the situation that the interactions of the first and second nearest solute-vacancy neighbors are appreciable, the nine different jump frequencies should be considered. To obtain information on the origin of these attractive behaviors, we decompose the total binding energy into the distortion binding energyand the electronic binding energy as [29] . The distortion binding energy can be obtained by the distortion reducing of the bcc Fe matrix when a solute atom and a vacancy combine to form a solute-vacancy pair, and can be expressed as: (5) where and can be calculated as follows: after the supercell containing a solute-vacancy pair (or a substitutional atom) has been fully relaxed, the solute-vacancy pair (or the substitutional atom) is removed from the system, then the total energy can be calculated. denotes the total energy of the pure bcc Fe supercell, and denotes the total energy of the supercell containing a vacancy. Then can be calculated from . The calculated solute-vacancy binding energies of 1nn, 2nn and 3nn are shown in Table 2, along with the energy decomposition for 1nn solute-vacancy binding. The distortion energies (-0.31 to -0.65 eV) for all solute elements(Y, La and Ce) are negative, and much bigger than their corresponding electronic binding energies (-0.04 to -0.12 eV). This implies that the distortion energy accounts for a major part of the total solute-vacancy binding energy, i.e. the strain relief effect contributes significantly to the interaction between the impurity atom and the vacancy, esp ecially for the solute Y and La, which accounted for 94.2% and 97.0% of the total binding energy, respectively. Furthermore, there is a strong correlation between the binding energy and the distance of the solute-vacancy, and the lattice relaxation around the vacancy is local. For the case of Ce-vacancy , specifically, we found that Ce atom relax away from the 2nn vacancy by 4.3% of the initial 2nn distance, which leads to the positive binding energy. Table 2 Decomposition of the total solute-vacancy binding energy into distortion binding energy and electronic binding energy. Units are eV. Solute element Y La Ce ΆHb (1nn) -0.69 -0.66 -0.43 (1nn) -0.65 -0.64 -0.31 (1nn) -0.04 -0.02 -0.12 ΆHb (2nn) -0.16 -0.21 0.10 ΆHb (3nn) -0.06 0.09 -0.05 The calculated migration energies of the different vacancy jumps corresponding to the paths in Fig. 1 are listed in Table 3. The migration energies of w2 jump for Y and La are lower than that of w0 jump for host Fe atom (0.54 eV), while the migration barrier of Ce in bcc Fe is higher than that of Fe self-diffusion. The migration barrier of w2 jump for Y is 0.09 eV, comparable to the reported value of 0.03 eV and 0.02 eV [9, 30]. The results indicate that there is a correlation between the binding energy of solute-vacancy and the migration energy, i.e. the strong attraction of solute-vacancy in 1nn configuration gives rise the low migration energy of the corresponding vacancy jump. For the three solute atoms, because of the strong attraction of 1nn solute-vacancy, the migration barriers of which the 1nn vacancy jump away from the solute atom, i.e. w3, w3, and w3, are higher than that of the opposite jumps, i.e. w4, w4, and w4, as well as that of Fe self-diffusion in pure bcc Fe. And t he same tendency can be found in the results of jump w5 and w6. Table 3 Migration energies for different jumps in the presence of Y, La and Ce in bcc Fe matrix. Units are eV. Jump Y La Ce w2 0.09 0.17 1.08 w3 1.81 1.84 1.55 w4 0.91 0.99 0.92 w3 0.93 1.23 1.07 w4 0.04 0.03 0.08 w3 0.86 0.92 0.87 w4 0.12 0.05 0.11 w5 0.94 0.98 0.89 w6 0.69 0.67 0.82 The correlation factor f2 is related to the probability of the reverse jump of a solute atom to its previous position [31]. Table 4 lists the calculated values of correlation factors for Y, La and Ce at representative temperatures of 850, 1000 and 1150K. The correlation factor of Y is 3.3ÃÆ'-10-5 at 1000K, close to the value of 1ÃÆ'-106 obtained by Murali [9]. For the three elements, the correlation factors of Ce have the highest values, and the correlation factors of La are one order of magnitude lower than that of Y. Therefore, Ce atom is the most difficult to return back to its original position in the temperature range of our investigation. Including the smallest binding energy, highest migration energy and correlation factor, provides an explanation for the low diffusivity of Ce atom. Table 4 Correlation factors (f2) for Y, La and Ce solute-diffusion at representative temperatures of 850, 1000 and 1150K. T(K) Y La Ce f2 f2/ f0 f2 f2/ f0 f2 f2/ f0 850 6.4ÃÆ'-106 1.111 2.9ÃÆ'-107 1.264 0.379 1.373 1000 3.3ÃÆ'-10-5 1.070 2.4ÃÆ'-106 1.223 0.381 1.370 1150 1.2ÃÆ'-104 1.034 1.4ÃÆ'-10-5 1.188 0.383 1.367 Table 5 lists the calculated diffusion activation energies and pre-factors for Fe self-diffusion and Y, La and Ce impurity diffusion. For pure bcc Fe, we find our calculated results are in good agreement with the published values. For Y impurity in bcc Fe, the calculated activation energy in full ordered ferromagnetic state is lower than the previous first principles work, and the pre-factor is as much as two orders of magnitude lager than the reported value. The experimental or calculated diffusion coefficients of La and Ce are not available to the best of our knowledge. For the case of experimental investigation, due to the very small solubilities of La and Ce in iron, the measured data may be affected by segregation of solutes, grain boundary, other impurities and the method of detection. Besides, the theory calculations, e.g. molecular dynamics (MD), first-principles etc. have not been applied widely in the study of RE contained steel yet, so the fundamental data of RE elements i n iron, such as the potential functions of Fe-La and Fe-Ce, is lacking. Table 5 Activation energies in the fully ordered ferromagnetic state () and paramagnetic state (QP), along with diffusion pre-factors for Fe self-diffusion and impurity diffusion of Y, La and Ce in bcc Fe. Reference (kJ mol-1) QP(kJ mol-1) D0(m2/s) Fe Present work 275.3 238.1 2.99ÃÆ'-10-5 Huang et al. [11] 277 239 6.7ÃÆ'-10-5 Nitta et al. [32] 289.7 ±5.1 250.6 ±3.8 2.76ÃÆ'-10-4 Seeger[33] 280.7 242.8 6.0ÃÆ'-10-4 Y Present work 165.9 159.9 1.09ÃÆ'-109 Murali et al. [9] 218.1 à ¢Ã¢â€š ¬Ã¢â‚¬â„¢ 8.0ÃÆ'-107 La Present work 175.6 169.2 2.88ÃÆ'-1010 Ce Present work 286.3 275.8 7.66ÃÆ'-106 Fig. 2 presents a direct comparison between the calculated and published temperature dependent diffusion coefficients for Fe self-diffusion and Y solute diffusion. For Fe self-diffusion, the calculated values are in good agreement with Huang et al. [11] and Nitta et al. [29], but smaller than the measured d