Stability of the vortex lattice in D-wave superconductors

2025年全国大学英语CET六级考试模拟试卷及解答参考一、写作（15分）Task 1: Writing (30 minutes)Part AWrite an email to your friend about a recent movie you watched. In your email, you should:1.Briefly introduce the movie and its main theme.2.Share your personal feelings about the movie.3.Recommend the movie to your friend, explaining why you think they would enjoy it.You should write about 100 words on the ANSWER SHEET 2.Do not sign your own name at the end of the letter. Use “Li Ming” in stead. Do not write the address.Example:Dear [Friend’s Name],I hope this email finds you well. I wanted to share with you a movie I recently watched that I thought you might find interesting.The movie I’m talking about is “Inception,” directed by Chris topher Nolan.It revolves around the concept of dream manipulation and the layers of reality. The story follows Dom Cobb, a skilled thief who specializes in extracting secrets from within the subconscious during the dream state.I was deeply impressed by t he movie’s intricate plot and the exceptional performances of the cast. The visual effects were breathtaking, and the soundtrack was perfectly matched to the action sequences. The movie made me think a lot about the nature of reality and the power of dreams.I highly recommend “Inception” to you. I believe it will be a captivating experience, especially if you enjoy films that challenge your perceptions and make you think.Looking forward to your thoughts on this movie.Best regards,Li MingAnalysis:This example follows the structure required for Part A of the writing task. It starts with a friendly greeting and a brief introduction to the subject of the email, which is the movie “Inception.”The writer then shares their personal feelings about the movie, highlighting the plot, the cast’s performances, the visual effects, and the soundtrack. This personal touch helps to engage the reader and provide a more authentic recommendation.Finally, the writer makes a clear recommendation, explaining that theybelieve the movie would be enjoyable for their friend based on itsthought-provoking nature and entertainment value. The email concludes with a friendly sign-off, maintaining a warm and inviting tone.二、听力理解-长对话（选择题，共8分）第一题听力原文：M: Hi, Lisa. How was your trip to Beijing last weekend?W: Oh, it was amazing! I’ve always wanted to visit the Forbidden City. The architecture was so impressive.M: I’m glad you enjoyed it. By the way, did you manage to visit the Great Wall?W: Yes, I did. It was a long journey, but it was worth it. The Wall was even more magnificent in person.M: Did you have any problems with transportation?W: Well, the subway system was very convenient, but some of the bus routes were confusing. I ended up getting lost a couple of times.M: That’s a common problem. It’s always a good idea to download a map or use a GPS app.W: Definitely. I also found the people in Beijing to be very friendly and helpful. They spoke English well, too.M: That’s great to hear. I’m thinking of visiting Beijing next month. Arethere any other places you would recommend?W: Oh, definitely! I would suggest visiting the Summer Palace and the Temple of Heaven. They are both beautiful and culturally significant.M: Thanks for the ti ps, Lisa. I can’t wait to see these places myself.W: You’re welcome. Have a great trip!选择题：1、Why did Lisa visit Beijing?A. To visit the Great Wall.B. To see her friends.C. To experience the local culture.D. To study Chinese history.2、How did Lisa feel about the Forbidden City?A. It was boring.B. It was too crowded.C. It was impressive.D. It was not as beautiful as she expected.3、What was the biggest challenge Lisa faced during her trip?A. Finding accommodation.B. Getting lost.C. Eating healthy food.D. Visiting all the tourist spots.4、What other places does Lisa recommend visiting in Beijing?A. The Summer Palace and the Temple of Heaven.B. The Great Wall and the Forbidden City.C. The National Museum and the CCTV Tower.D. The Wangfujing Street and the Silk Market.答案：1、C2、C3、B4、A第二题Part Two: Listening ComprehensionSection C: Long ConversationsIn this section, you will hear one long conversation. At the end of the conversation, you will hear some questions. Both the conversation and the questions will be spoken only once. After you hear a question, you must choose the best answer from the four choices marked A), B), C), and D).1.What is the main topic of the conversation?A) The importance of cultural exchange.B) The challenges of teaching English abroad.C) The experiences of a language teacher in China.D) The impact of language barriers on communication.2.Why does the speaker mention studying Chinese?A) To show his respect for Chinese culture.B) To express h is gratitude for the Chinese students’ hospitality.C) To emphasize the importance of language learning.D) To explain his reasons for choosing to teach English in China.3.According to the speaker, what is one of the difficulties he faced in teaching English?A) The students’ lack of motivation.B) The limited resources available.C) The cultural differences between Chinese and Western students.D) The high expectations from the school administration.4.How does the speaker plan to overcome the language barrier in his future work?A) By learning more Chinese.B) By using visual aids and non-verbal communication.C) By collaborating with local language experts.D) By relying on his previous teaching experience.Answers:1.C2.C3.C4.B三、听力理解-听力篇章（选择题，共7分）第一题Passage:A new study has found that the way we speak can affect our relationships and even our physical health. Researchers at the University of California, Los Angeles, have been investigating the connection between language and well-being for several years. They have discovered that positive language can lead to better health outcomes, while negative language can have the opposite effect.The study involved 300 participants who were monitored for a period of one year. The participants were asked to keep a daily diary of their interactions with others, including both positive and negative comments. The researchers found that those who used more positive language reported fewer physical symptoms and a greater sense of well-being.Dr. Emily Thompson, the l ead researcher, explained, “We were surprised to see the impact that language can have on our health. It’s not just about what we say, but also how we say it. A gentle tone and supportive language can make a significant difference.”Here are some examples of positive and negative language:Positive Language: “I appreciate your help with the project.”Negative Language: “You always mess up the project.”The researchers also looked at the effects of language on relationships. They found that couples who used more positive language were more likely toreport a satisfying relationship, while those who used negative language were more likely to experience relationship stress.Questions:1、What is the main focus of the study conducted by the University of California, Los Angeles?A) The impact of diet on physical health.B) The connection between language and well-being.C) The effects of exercise on mental health.D) The role of social media in relationships.2、Which of the following is a positive example of language from the passage?A) “You always mess up the project.”B) “I can’t believe you did that again.”C) “I appreciate your help with the project.”D) “This is a waste of time.”3、According to the study, what is the likely outcome for couples who use negative language in their relationships?A) They will have a more satisfying relationship.B) They will experience fewer physical symptoms.C) They will report a greater sense of well-being.D) They will likely experience relationship stress.Answers:1、B2、C3、D第二题Passage OneIn the United States, there is a long-standing debate over the best way to educate children. One of the most controversial issues is the debate between traditional public schools and charter schools.Traditional public schools are operated by government and are funded by tax dollars. They are subject to strict regulations and are required to follow a standardized curriculum. Teachers in traditional public schools are typically unionized and receive benefits and pensions.On the other hand, charter schools are publicly funded but operate independently of local school districts. They are free to set their own curriculum and teaching methods. Charter schools often have a longer school day and a more rigorous academic program. They are also subject to performance-based evaluations, which can lead to their closure if they do not meet certain standards.Proponents of charter schools argue that they provide more choices for parents and that they can offer a more personalized education for students. They also claim that charter schools are more accountable because they are subject to more direct oversight and can be closed if they fail to meet their goals.Opponents of charter schools argue that they take resources away fromtraditional public schools and that they do not provide a level playing field for all students. They also claim that charter schools can be more selective in their admissions process, which may lead to a lack of diversity in the student body.Questions:1、What is a key difference between traditional public schools and charter schools?A) Funding sourceB) CurriculumC) Teacher unionsD) Academic rigor2、According to the passage, what is a potential advantage of charter schools?A) They are subject to fewer regulations.B) They offer more choices for parents.C) They are more likely to receive government funding.D) They typically have a shorter school day.3、What is a common concern expressed by opponents of charter schools?A) They are less accountable for their performance.B) They may lead to a lack of diversity in the student body.C) They are more expensive for local taxpayers.D) They do not follow a standardized curriculum.Answers:1、B) Curriculum2、B) They offer more choices for parents.3、B) They may lead to a lack of diversity in the student body.四、听力理解-新闻报道（选择题，共20分）第一题News ReportA: Good morning, everyone. Welcome to today’s news broadcast. Here is the latest news.News Anchor: This morning, the World Health Organization (WHO) announced that the number of confirmed cases of a new strain of the H1N1 flu virus has reached 10,000 worldwide. The WHO has declared the outbreak a public health emergency of international concern. Health officials are urging countries to take immediate measures to contain the spread of the virus.Q1: What is the main topic of the news report?A) The announcement of a new strain of the H1N1 flu virus.B) The declaration of a public health emergency.C) The measures taken to contain the spread of the virus.D) The number of confirmed cases of the new strain.Answer: BQ2: According to the news report, who declared the outbreak a public health emergency?A) The World Health Organization (WHO)B) The Centers for Disease Control and Prevention (CDC)C) The European Union (EU)D) The United Nations (UN)Answer: AQ3: What is the main purpose of the health officials’ urging?A) To increase awareness about the flu virus.B) To encourage people to get vaccinated.C) To take immediate measures to contain the spread of the virus.D) To provide financial assistance to affected countries.Answer: C第二题News Report 1:[Background music fades in]Narrator: “This morning’s top news includes a major announcement from the Ministry of Education regarding the upcoming changes to the College English Test Band Six (CET-6). Here’s our correspondent, Li Hua, with more details.”Li Hua: “Good morning, everyone. The Ministry of Education has just announced that starting from next year, the CET-6 will undergo significant modifications. The most notable change is the inclusion of a new speaking section, which will be mandatory for all test-takers. This decision comes in response to the increasing demand for English proficiency in various fields. Let’s goto the Education Depar tment for more information.”[Background music fades out]Questions:1、What is the main topic of this news report?A) The cancellation of the CET-6 exam.B) The addition of a new speaking section to the CET-6.C) The difficulty level of the CET-6 increasing.D) The results of the CET-6 exam.2、Why has the Ministry of Education decided to include a new speaking section in the CET-6?A) To reduce the number of test-takers.B) To make the exam more difficult.C) To meet the demand for English proficiency.D) To replace the written test with an oral test.3、What will be the impact of this change on students preparing for the CET-6?A) They will need to focus more on writing skills.B) They will have to learn a new type of test format.C) They will no longer need to take the exam.D) They will be able to choose between written and oral tests.Answers:1、B2、C3、B第三题You will hear a news report. For each question, choose the best answer from the four choices given.Listen to the news report and answer the following questions:1、A) The number of tourists visiting the city has doubled.B) The city’s tourism revenue has increased significantly.C) The new airport has attracted many international tourists.D) The city’s infrastructure is not ready for the influx of tou rists.2、A) The government plans to invest heavily in transportation.B) Local businesses are benefiting from the tourism boom.C) The city is experiencing traffic congestion and overcrowding.D) The city is working on expanding its hotel capacity.3、A) Th e city’s mayor has expressed concern about the impact on local culture.B) The tourism industry is collaborating with local communities to preserve traditions.C) There are concerns about the negative environmental effects of tourism.D) The city is implementing strict regulations to control tourist behavior.Answers:1.B) The city’s tourism revenue has increased significantly.2.C) The city is experiencing traffic congestion and overcrowding.3.B) The tourism industry is collaborating with local communities to preserve traditions.五、阅读理解-词汇理解（填空题，共5分）第一题Read the following passage and then complete the sentences by choosing the most suitable words or phrases from the list below. Each word or phrase may be used once, more than once, or not at all.Passage:In the past few decades, the internet has revolutionized the way we communicate and access information. With just a few clicks, we can now connect with people from all over the world, share our thoughts and experiences, and even conduct business transactions. This rapid advancement in technology has not only brought convenience to our lives but has also raised several challenges and concerns.1、_________ (1) the internet has made it easier for us to stay connected with friends and family, it has also led to a decrease in face-to-face interactions.2、The increasing reliance on digital devices has raised concerns about the impact on our physical and mental health.3、Despite the many benefits, there are also significant_________(2) associated with the internet, such as privacy breaches and cybersecuritythreats.4、To mitigate these risks, it is crucial for individuals and organizations to adopt robust security measures.5、In the future, we need to strike a balance between embracing technological advancements and maintaining a healthy lifestyle.List of Words and Phrases:a) convenienceb) challengesc) privacy breachesd) physicale) significantf) mentalg) privacyh) embracei) reliancej) face-to-face1、_________ (1)2、_________ (2)第二题Reading PassagesPassage OneMany people believe that a person’s personality is established at birthand remains unchanged throughout life. This view is supported by the idea that personality is determined by genetic factors. However, recent studies have shown that personality can be influenced by a variety of environmental factors as well.The word “personality” can be defined as the unique set of characteristics that distinguish one individual from another. It includes traits such as extroversion, neuroticism, and agreeableness. These traits are often measured using psychological tests.According to the passage, what is the main idea about personality?A. Personality is solely determined by genetic factors.B. Personality remains unchanged throughout life.C. Personality is influenced by both genetic and environmental factors.D. Personality is determined by a combination of psychological tests.Vocabulary Understanding1、The unique set of characteristics that distinguish one individual from another is referred to as ________.A. personalityB. genetic factorsC. environmental factorsD. psychological tests2、The view that personality is established at birth and remains unchanged throughout life is ________.A. supportedB. challengedC. irrelevantD. misunderstood3、According to the passage, traits such as extroversion, neuroticism, and agreeableness are part of ________.A. genetic factorsB. environmental factorsC. personalityD. psychological tests4、The passage suggests that personality can be influenced by ________.A. genetic factorsB. environmental factorsC. both genetic and environmental factorsD. neither genetic nor environmental factors5、The word “personality” is best defined as ________.A. the unique set of characteristics that distinguish one individual from anotherB. the genetic factors that determine personalityC. the environmental factors that influence personalityD. the psychological tests used to measure personalityAnswers:1、A2、A3、C4、C5、A六、阅读理解-长篇阅读（选择题，共10分）First QuestionPassage:In the digital age, technology has transformed almost every aspect of our lives, including education. One significant impact technology has had on learning is through online platforms that offer a wide variety of courses and educational materials to anyone with internet access. This democratization of knowledge means that individuals no longer need to rely solely on traditional educational institutions for learning. However, while online learning provides unprecedented access to information, it also poses challenges such as ensuring the quality of the content and maintaining student engagement without the structure of a classroom setting. As educators continue to adapt to these changes, it’s clear that technology will play an increasingly important role in s haping the future of education.1、According to the passage, what is one major advantage of online learning?A) It guarantees higher academic achievements.B) It makes educational resources more accessible.C) It eliminates the need for traditional learning methods entirely.D) It ensures that all students remain engaged with the material.2、What challenge does online learning present according to the text?A) It makes it difficult to assess the quality of educational content.B) It increases the reliance on traditional educational institutions.C) It decreases the amount of available educational material.D) It simplifies the process of student engagement.3、The term “democratization of knowledge” in this context refers to:A) The ability of people to vote on educational policies.B) The equal distribution of printed books among citizens.C) The process by which governments control online information.D) The widespread availability of educational resources via the internet.4、How do educators respond to the changes brought about by technology in education?A) By rejecting technological advancements in favor of conventional methods.B) By adapting their teaching practices to incorporate new technologies.C) By insisting that online learning should replace traditional classrooms.D) By ignoring the potential benefits of online learning platforms.5、Based on the passage, which statement best reflects the future outlook for education?A) Traditional educational institutions will become obsolete.B) Technology will have a diminishing role in the education sector.C) Online learning will complement but not completely replace traditional education.D) Students will no longer require any form of structured learning environment.Answers:1.B2.A3.D4.B5.CThis is a fictional example designed for illustrative purposes. In actual CET exams, the passages and questions would vary widely in topic and complexity.第二题Reading PassagesPassage OneGlobal warming is one of the most pressing environmental issues facing the world today. It refers to the long-term increase in Earth’s average surface temperature, primarily due to human activities, particularly the emission of greenhouse gases. The consequences of global warming are far-reaching, affecting ecosystems, weather patterns, sea levels, and human health.The Intergovernmental Panel on Climate Change (IPCC) has warned that if global warming continues at its current rate, we can expect more extreme weather events, such as hurricanes, droughts, and floods. Additionally, rising sealevels could displace millions of people, leading to social and economic instability.Several measures have been proposed to mitigate the effects of global warming. These include reducing greenhouse gas emissions, transitioning to renewable energy sources, and implementing sustainable agricultural practices. However, despite the urgency of the situation, progress has been slow, and many countries have failed to meet their commitments under the Paris Agreement.Questions:1、What is the primary cause of global warming according to the passage?A、Natural climate changesB、Human activitiesC、Ecosystem changesD、Increased carbon dioxide levels in the atmosphere2、Which of the following is NOT mentioned as a consequence of global warming?A、Extreme weather eventsB、Rising sea levelsC、Improved crop yieldsD、Increased global biodiversity3、What is the IPCC’s main concern regarding the current rate of global warming?A、It is causing a decrease in Earth’s average surface temperatu re.B、It is leading to more extreme weather events.C、It is causing the Earth’s magnetic field to weaken.D、It is causing the ozone layer to thin.4、What are some of the proposed measures to mitigate the effects of global warming?A、Reducing greenhouse gas emissions, transitioning to renewable energy sources, and implementing sustainable agricultural practices.B、Building more coal-fired power plants and expanding deforestation.C、Increasing the use of fossil fuels and reducing the number of trees.D、Ignoring the issue and hoping it will resolve itself.5、Why has progress in addressing global warming been slow, according to the passage?A、Because it is a complex issue that requires international cooperation.B、Because people are not concerned about the consequences of global warming.C、Because scientists do not have enough information about the issue.D、Because the Paris Agreement has not been effective.Answers:1、B2、C3、B4、A5、A七、阅读理解-仔细阅读（选择题，共20分）First QuestionPassage:In the age of rapid technological advancement, the role of universities has shifted beyond traditional academic pursuits to include fostering innovation and entrepreneurship among students. One such initiative taken by many institutions is the integration of technology incubators on campus. These incubators serve as platforms where students can turn their innovative ideas into tangible products, thereby bridging the gap between theory and practice. Moreover, universities are increasingly collaborating with industry leaders to provide practical training opportunities that prepare students for the challenges of the modern workforce. Critics argue, however, that this shift might come at the cost of undermining the foundational academic disciplines that have historically formed the core of higher education.Questions:1、What is one key purpose of integrating technology incubators in universities according to the passage?A) To reduce the cost of university education.B) To bridge the gap between theory and practice.C) To compete with other universities.D) To focus solely on theoretical knowledge.Answer: B) To bridge the gap between theory and practice.2、According to the text, how are universities preparing students for the modern workforce?A) By isolating them from industry professionals.B) By providing practical training through collaboration with industry leaders.C) By discouraging entrepreneurship.D) By focusing only on historical academic disciplines.Answer: B) By providing practical training through collaboration with industry leaders.3、What concern do critics raise about the new initiatives in universities?A) They believe it will enhance foundational academic disciplines.B) They fear it could undermine the core of higher education.C) They think it will make universities less competitive.D) They are worried about the overemphasis on practical skills.Answer: B) They fear it could undermine the core of higher education.4、Which of the following best describes the role of universities in the current era as depicted in the passage?A) Institutions that strictly adhere to traditional teaching methods.B) Centers that foster innovation and entrepreneurship among students.C) Organizations that discourage partnerships with industries.D) Places that prevent students from engaging with real-world challenges.Answer: B) Centers that foster innovation and entrepreneurship among students.5、How does the passage suggest that technology incubators benefit students?A) By ensuring they only focus on theoretical studies.B) By giving them a platform to turn ideas into products.C) By limiting their exposure to practical experiences.D) By encouraging them to avoid modern workforce challenges.Answer: B) By giving them a platform to turn ideas into products.This set of questions aims to test comprehension skills including inference, detail recognition, and understanding the main idea of the given passage. Remember, this is a mock example and should be used for illustrative purposes only.Second QuestionReading Passage:The Future of Renewable Energy SourcesIn recent years, there has been a growing interest in renewable energy sources due to their potential to reduce dependency on fossil fuels and mitigate the effects of climate change. Solar power, wind energy, and hydropower have all seen significant advancements in technology and cost-efficiency. However, challenges remain in terms of storage and distribution of these energy sources. For solar energy to become a viable primary energy source worldwide, it must overcome the limitations posed by weather conditions and geographical location. Wind energy faces similar challenges, particularly in areas with low wind speeds. Hydropower, while more consistent than both solar and wind energies, is limited。

超导-常规介质界面的可调表面等离激元韦德泉;张兴坊【摘要】研究了环境温度对超导材料和常规介质分界面处的表面等离激元色散曲线和特征长度的影响.结果表明:表面等离激元沿界面传播波矢的最大值和传播距离对温度变化极为敏感,均随环境温度的升高而迅速减小,但表面等离激元在超导材料中的穿透深度却随温度升高而增大.同时,表面等离激元波长与其在常规介质中的穿透深度还与频率有关,处于低频段时,两者都随着温度的升高而减小,而处于高频段时,两者却随着温度升高而增大.利用超导二流体模型的电子超导态和正常态转换机制对该现象进行了理论解释.【期刊名称】《枣庄学院学报》【年(卷),期】2017(034)002【总页数】5页(P24-28)【关键词】超导;表面等离激元;色散曲线;温度;二流体【作者】韦德泉;张兴坊【作者单位】枣庄学院光电工程学院,山东枣庄277160;枣庄市太赫兹工程技术研究中心,山东枣庄277160【正文语种】中文【中图分类】O441.6表面等离激元(Surface Plasmon Polartions, SPPs)具有亚波长尺度导波、光场局域增强性质，在新型光源、超高分辨率成像、光子回路、生化传感等领域具有广泛的应用前景[1-3].研究表明，SPPs存在于介电常数或磁导率符号相反的两种介质分界面处，是一种非辐射电磁模式，可通过TM或TE波激发，其性质与介质几何结构、物理参量和周围环境等因素有关[4].目前，SPPs的研究已从可见光波段拓展至THz、微波等波段[5,6]，分界面处的介质材料也从最常见的光滑金属或褶皱结构金属扩展至半导体[7]、左手材料[8]和各种人工结构晶体[9]等，且通过对材料施加光照、温度、电磁场等外界因素还可实现对SPPs的调制[10,11].但上述材料不同程度的存在着欧姆损耗，使SPPs的潜在应用受到限制.近年来，超导材料的临界温度不断被突破，多种高温超导材料被研制出来，采用低损耗的高温超导材料实现对SPPs的调控引起了人们兴趣[12-14].超导材料的光学性质受外界环境温度影响，进而影响着其表面SPPs的性质.考虑由两种半无限大、各向同性介质组成的界面，介质1为常规材料，其介电常数和磁导率分别为ε1和μ1，介质2为超导材料，其介电常数和磁导率分别为ε2和μ2.为简单计算，假设μ1=μ2=1，则在超导体和常规介质分界面处可能存在着由TM波激发的SPPs，其传播常数为β=ω/c，传播距离为LSPPs=1/，SPPs波长为λSPPs=2π/，在两种介质内的穿透深度分别为δi=1/，其中(i=1,2)为SPPs 在两种介质内垂直于界面方向的波矢[15].超导材料的介电常数ε2是一个与入射频率ω和环境温度有关的量，由超导二流体模型可得[16]其中，ωsp和ωnp分别为超导态电子和正常态电子的等离子体频率，其与超导体内的电子密度有关，εc为超导的介电常数，γ为正常态电子的阻尼系数.而ωsp和ωnp又可表示为[16]，其中，T和Tc分别为超导材料的环境温度和临界温度，λL(0)表示超导材料在T=0K时的伦敦深度，c为真空中的光速.本文中选用的超导材料为铌，其参数为Tc=9.2K，λL(0)=83nm，εc设为1，γ为1014 Hz.设常规介质材料为空气，相对介电常数ε1=1.图1给出了当环境温度分别为1 K、6 K和9 K时，在光滑超导材料铌表面产生的SPPs的色散曲线，图中纵坐标归一化频率中的ωP为当环境温度为0 K时的超导材料的等离子体频率.由图可见，温度对色散曲线的影响与频率有关，当频率较低时，色散曲线几乎不受温度影响，当频率增大到ω/ωP～0.68时，三条色散曲线开始分离，温度越高分离越快，且对应的波矢相对较小.当频率继续增大时，SPPs波矢将快速减小直至接近于零，温度越高对应的波矢反而相对越大，当ω/ωP>1时，色散曲线与温度无明显的关系.超导材料铌表面产生的SPPs的色散曲线与贵金属表面SPPs的色散曲线类似，当不考虑材料的阻尼作用时，两者均在频率ωP/～ωP范围内存在一个禁带，在禁带内SPPs不存在.当频率高于禁带频率上限ωP时，存在着可在金属或超导体内传播的辐射模式，而低于禁带频率下限ωP/时，才存在沿界面传播的局域SPPs[15].但是，如果考虑材料的阻尼作用时，在禁带频率内将存在波矢随频率增大而减少的SPPs模式.对于超导体来说，当温度较低(T=1 K)时，此时超导体内的电子大多处于超导状态而不产生阻尼，少量正常态电子引起的阻尼可以忽略不计，因此产生了较明显的禁带.随着温度的升高，超导体内的电子逐渐由超导态转变为正常态，产生的阻尼将增大，引起超导体材料的介电常数虚部变大，从而在禁带频域内产生可传播的SPPs，虚部越大，SPPs波矢也越大.同时，超导体具有的介电常数虚部将导致传播波矢最大值为一有限值，并且，随着温度的升高，超导体的介电常数虚部虽然增大，但其实部变化却较小，使得SPPs波矢最大值变小.图2给出了铌表面SPPs的传播距离与温度的关系曲线.由图可见，随着温度的减小，SPPs的传播距离将迅速增大.对频率为4μm的SPPs，当温度为9K时传播距离仅为约160μm，随着温度减小到6K，传播距离将增大到约1mm，增大了约6倍，当温度继续减小到1K时，此时的传播距离将达到约1m，增大了约6000倍，且波长越大传播距离也越大.这是因为，超导体内的超导态电子数量与温度四次方成反比关系，温度越低超导体内存在的超导态电子也越多，从而引起超导体的介电常数虚部随温度降低而迅速减小，进而导致SPPs在界面的传播损耗减小，传播距离增大.当温度低到一定程度时，超导体的介电常数虚部将远小于其实部，此时的传播距离表达式可近似表示为LSPPs=cRe(ε2)/ωlm(ε2)[17]，其与温度的关系近似成四次方反比关系.因此，要有效的增大SPPs的传播距离，需要尽可能的降低超导材料周围的环境温度.当环境温度分别为1 K、6 K和9 K时，SPPs波长随温度的变化关系如图3所示.由图可见，在计算频率范围内，三条SPPs波长随着温度的升高几乎无明显变化.但是，当将计算频率范围缩小后发现，在频率较大时，温度越高SPPs波长越大，而且温度为1 K和6 K时的SPPs波长在较大频域范围内差别较小，很难将其区分(如插图(a)可见).当频率减小到一定程度时，温度低(1 K)时的SPPs波长将比温度高时的SPPs波长大(如插图(b)可见).显然，当频率靠近禁带频率下限时，由于SPPs传播波矢最大值随温度升高而减小，将引起有效折射率cRe(β)/ω减小，SPPs波长变大.当频率较低时，此时超导材料介电常数的实部和虚部受环境温度影响较大，实部和虚部值均随着温度的升高而增大，导致有效折射率变大，SPPs波长变小，如插图(b)中所示T=1 K时的SPPs波长最大.当频率变得更低时，如在30μm时，此时将可以看到6 K时的SPPs将比9 K时的波长大(图中未显示). SPPs在超导材料和空气中的穿透深度随温度的变化曲线如图4(a)和(b)所示.由图4(a)可见，当处于较高频域(0.75-2 μm)时，SPPs在温度较低的超导材料内的穿透深度对温度变化不敏感，但当温度接近临界温度时，穿透深度随温度升高而增大.当频率超过2μm时，穿透深度随温度的升高而明显增大，在频率为10μm时，穿透深度将由T=1 K时的83 nm增大到T=9 K时的87 nm，约有几个纳米厚度的变化.可以这样解释，当温度较低时，处于低频域的超导体内的超导态电子占大多数，此时超导体几乎完全处于超导状态，外界电磁波穿透极限为伦敦深度，即83nm，但随着温度的升高，超导态电子逐渐转换为正常态电子，超导体也由超导态向正常态转变，电子碰撞几率增大，阻尼明显，伦敦穿透深度增大.而处于高频段接近禁带频率下限时，SPPs在界面的模场受到压缩，局域性最好，穿透深度较小，温度对其影响不易分辨.由图4(b)可见，温度对SPPs在空气中的穿透深度影响与频段有关.在低频段时，穿透深度随着温度升高而减小，但在高频段时，穿透深度随着温度的升高而增大(插图中可见).这种变化与超导体内电子状态对介电常数的贡献相联系，当处于低频段时，超导材料的介电常数实部和虚部值均随着温度升高而增大，当处于高频段时，随着温度的升高，超导实部值与温度变化关系不大，而虚部则成四次方增大，导致穿透深度在不同频段随温度变化规律不同.值得说明的是，尽管温度变化影响SPPs在两种介质内的穿透深度，但影响程度很小.而且，SPPs在空气中的穿透深度随着频率减小而快速增大，如在频率为6μm时就达到十微米量级，不利于微纳光子器件在微米波段和更长波段的集成，为控制SPPs在介质中的穿透深度，应找到合适的手段和方法将超导材料的介电常数零点移动到所需频段.基于超导二流体模型，理论研究了在超导材料铌表面存在的SPPs色散曲线和特征长度随温度的变化关系.结果表明，SPPs的传播波矢最大值和传播距离对温度变化极为敏感，均随着温度的升高而快速减小，而SPPs在超导材料中的穿透深度则随温度升高缓慢增大，增大程度较小仅有几个纳米的变化.同时，SPPs波长及其在常规介质中的穿透深度还与频率有关，处于低频段时，两者都随着温度的升高而减小，而处于高频段时，两者却随着温度升高而增大，且只有当温度接近超导材料的临界温度时，变化才变的较为明显.这是由超导体内超导态电子和正常态电子与温度的变化关系决定的，其导致超导材料介电常数在低频段和高频段随温度的变化规律不同.该结果为设计长传输距离的波导器件提供一定的理论依据.【相关文献】[1]Zayats A V, Smolyaninov I I, Maradudin A A. Nano-optics of surface plasmon polaritons[J]. Physics Reports, 2005, 408(3): 131-314.[2] Zhang X, Liu Z. Superlenses to overcome the diffraction limit[J]. Nature Materials, 2008, 7(6): 435-441.[3]Cai W, Vasudev A P, Brongersma M L. Electrically controlled nonlinear generation of light with plasmonics[J]. Science, 2011, 333(6050): 1720-1723.[4]Ruppin R. Surface polaritons of a left-handed material slab[J]. Journal of Physics: Condensed Matter, 2001, 13(9): 1811-1819.[5]Hendry E, Garcia-Vidal F J, Martin-Moreno L, et al.. Optical control over surface-plasmon-polariton-assisted THz transmission through a slit aperture[J]. Physical Review Letters, 2008, 100(12): 123901.[6]Rance H J, Hooper I R, Hibbins A P, et al.. Structurally dictated anisotropic “designer surface plasmons”[J]. Applied Physics Letters, 2011, 99(18): 181107.[7]Strelniker Y M, Bergman D J. Transmittance and transparency of subwavelength-perforated conducting films in the presence of a magnetic field[J]. 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Surface plasmons on smooth surfaces[M]. Springer Berlin Heidelberg, 1988, 5-7.[16]Li Chunzao, Liu Shaobin, Kong Xiangkun, et al.. Tunable photonic bandgap in a one-dimensional superconducting-dielectric superlattice[J]. Applied Optics, 2011, 50(16): 2370-2375.[17]Barnes W L. Surface plasmon-polariton length scales: a route to sub-wavelength optics[J]. Journal of Optics A: Pure and Applied Optics, 2006, 8(4): 87-93.。

托福阅读第三篇tpo75R-3原文+译文+题目+答案+背景知识原文 (1)译文 (4)题目 (7)答案 (13)背景知识 (14)原文Seismic Waves①Seismic waves-energy waves produced by earthquakes-permit scientists to determine the location,thickness,and properties of Earth's internal zones.They are generated when rock masses are suddenly disturbed,such as when they break or rupture.Vibrations spread out in all directions from the source of the disturbance, traveling at different speeds through parts of Earth's crust and interior that differ in chemical composition and physical properties.The principal categories of these waves are primary,secondary,and surface. All three types of waves are recorded on an instrument called a seismograph.②Primary waves,or P-waves,are the speediest of the three kinds of waves and therefore the first to arrive at a seismograph station after there has been an earthquake.They travel through the upper crust of Earth at speeds of4to5kilometers per second,but near the base of the crust they speed along at6or7kilometers per second.In these primary waves,pulses of energy are transmitted as a succession of compressions and expansions that parallel the direction of propagation of the wave itself.Thus,a given segment of rock set in motion during an earthquake is driven into its neighbor and bounces back.The neighbor strikes the next particle and rebounds and subsequent particles continue the motion.Vibrational energy is an accordion-like push-pull movement that can be transmitted through solids,liquids and gases.Of course,the speed of Pwave transmission will differ in materials of different density and elastic properties.③Secondary waves,or S-waves,travel1to2kilometers per second slower than do P-waves.Unlike the movement of P-waves,rock vibration in secondary waves is at right angles to the direction of propagation of the energy.This type of wave is easily demonstrated by tying a length of rope to a hook and then shaking the free end.A series of undulations will develop in the rope and move toward the hook-thatis,in the direction of propagation.Any given particle along the rope, however,will move up and down in a direction perpendicular to the direction of propagation.It is because of their more complex motion that S-waves travel more slowly than Pwaves.They are the second group of oscillations to arrive at a seismograph station.Unlike Pwaves, secondary waves will not pass through liquids or gases.④Both P-and S-waves are sometimes also termed body waves because they are able to penetrate deep into the interior or body of our planet.Body waves travel faster in rocks of greater elasticity,and their speeds therefore increase steadily as they move downward into more elastic zones of Earth's interior and then decrease as they begin to make their ascent toward Earth's surface.The change in velocity that occurs as body waves invade rocks of different elasticity results in a bending or refraction of the wave.The many small refractions cause the body waves to assume a curved travel path through Earth.⑤Not only are body waves subjected to refraction,but they may also be partially reflected off the surface of a dense rock layer in much the same way as light is reflected off a polished surface.Many factorsinfluence the behavior of body waves.An increase in the temperature of rocks through which body waves are traveling will cause a decrease in velocity,whereas an increase in confining pressure will cause a corresponding increase in wave velocity.In a fluid where no rigidity exists,S-waves cannot propagate and P-waves are markedly slowed.⑥Surface waves are large-motion waves that travel through the outer crust of Earth.Their pattern of movement resembles that of waves caused when a pebble is tossed into the center of a pond.They develop whenever P-or S-waves disturb the surface of Earth as they emerge from the interior.Surface waves are the last to arrive at a seismograph station.They are usually the primary cause of the destruction that can result from earthquakes affecting densely populated areas.This destruction results because surface waves are channeled through the thin outer region of Earth,and their energy is less rapidly scattered into the large volumes of rock traversed by body waves.译文地震波①地震波是由地震产生的能量波，它们使科学家能够确定地球内部区域的位置、厚度和性质。

a r X i v :c o n d -m a t /9906144v211J un1999Bose-Einstein condensates with vortices in rotating trapsY.Castin 1and R.Dum 1,21Laboratoire Kastler Brossel ∗,´Ecole normale sup´e rieure,24rue Lhomond,F-75231Paris Cedex 05,France2Institut d’optique,BP 147,F-91403Orsay Cedex,France (16March 1999)We investigate minimal energy solutions with vortices for an interacting Bose-Einstein condensate in a rotating trap.The atoms are strongly confined along the axis of rotation z ,leading to an effective 2D situation in the x −y plane.We first use a simple numerical algorithm converging to local minima of energy.Inspired by the numerical results we present a variational Ansatz in the regime where the interaction energy per particle is stronger than the quantum of vibration in the harmonic trap in the x −y plane,the so-called Thomas-Fermi regime.This Ansatz allows an easy calculation of the energy of the vortices as function of the rotation frequency of the trap;it gives a physical understanding of the stabilisation of vortices by rotation of the trap and of the spatial arrangement of vortex cores.We also present analytical results concerning the possibility of detecting vortices by a time-of-flight measurement or by interference effects.In the final section we give numerical results for a 3D configuration.I.INTRODUCTION After the achievement of Bose-Einstein condensates in trapped atomic gases [1]many properties of these systems have been studied experimentally and theoretically [2].However a striking feature of superfluid helium,quantized vortices [3],[4],has not yet been observed in trapped atomic gases.There is an abundant literature on vortices in helium II,an overview is given in [4].The atomic gases have interesting properties which justify efforts to generate vortices in these systems:the core size of the vortices is adjustable,as in contrast to helium the strength of the interaction can be adjusted through the density;the number of vortices in atomic gases can be in principle well controlled;for a small number of particles in the gas metastability of the vortices can be studied,that is one can watch spontaneous transitions between configurations with different number of vortices.Several ways to create vortices in atomic gases have been suggested.A method inspired from liquid helium consists in rotating the trap confining the atoms [5];at a large enough rotation frequency it becomes energetically favorable at low temperatures to produce vortices;two different paths could be in principle followed:(1)producing first a condensate then rotating the trap,or (2)cooling the gas directly in a rotating trap.It has been recently proposed in [6]to use quantum topological effects to obtain a vortex.Other methods that do not rely on thermal equilibrium have been suggested [7],[8].Here we study theoretically the minimal energy configurations of vortices in a rotating trap [9].The model is defined in section II;in sections III to VI we assume a strong confinement of the atoms along the rotation axis z so that we face an effective 2D problem in the transverse plane x −y .We present numerical results for solutions with vortices that are local minima of the Gross-Pitaevskii energy functional (section III).These solutions contain only vortices with a charge ±1,the vortices with a charge larger than or equal to 2are thermodynamically unstable (section IV).We discuss possibilities to get experimental evidence of vortices in atomic gases in section V.Finally,we concentrate on the regime where the interaction energy is much larger than the trap frequenciesωx,y ,the so-called Thomas-Fermi limit [2].This is complementary to the work of [10].We obtain inthis “strong interacting”regime analytical predictions based on a variational Ansatz that reproducesatisfactorily the numerical results (section VI).In section VII we present results for vortices in 3D,that is in a trap with a weak confinement along the rotation axis.II.MODEL CONSIDERED IN THIS PAPERThe atoms are trapped in a potential rotating at angular velocity Ω.In the laboratory frame theHamiltonian of the gas is therefore time dependent.To eliminate this time dependence we introducea rotating frame at the angular velocity Ωso that the trapping potential becomes time independent;this change of frame is achieved by the single-atom unitary transform:U(t)=e i Ω· Lt/¯h(1) where L is the angular momentum operator of a single atom.As the unitary transform is timedependent the Hamiltonian in the rotating frame contains an extra inertial term,given for eachatom byi¯h U†(t)dφ|φ +1φ|φ 2.(3)In this energy functional H0contains the kinetic energy and the trapping potential energy of theparticles:H0=−¯h22mω2αr2α.(5)Furthermore in all but in section VII we will assume that the trapping potential is much strongeralong the z axis than along the x,y axis,with an oscillation frequency much larger than the typicalinteraction energy Ng3D|φ|2per particle.This situation,although not realized experimentally yet,is not out of reach,in particular when one uses optical traps rather than magnetic traps[11].Inthis strong confining regime the motion of the particles along z is frozen in the ground state of thestrong harmonic potential:φ(x,y,z)≃ψ(x,y) mωz2¯hωz:E[ψ,ψ∗]= d2 rψ∗( r)[H⊥−ΩL z]ψ( r)2Ng|ψ|42m ∆x,y+12π¯h 1/2.(9)Most of the results of the paper are dealing with the2D energy functional;a numerical result for a local minimum of the full3D energy functional will be given in the section VII.We concentrate on the so-called Thomas-Fermi regime,where the interaction energy per particle is much larger than ¯hωx,y.The opposite regime has already been studied in[10].III.LOCAL MINIMA OF ENERGY WITH VORTICES In this section we briefly discuss the general problem of minimizing energy functionals of the type Eq.(7).We present the numerical algorithm that we have used and we give numerical results for the2D problem.A.A numerical algorithm tofind local minimaThe algorithm in our numerical calculations is commonly used in the literature to minize energy functionals E[ψ,ψ∗]of the form Eq.(7).The intuitive idea is to start from a randomψand move it opposite to the local gradient of E[ψ,ψ∗]that is along the local downhill slope of the energy. Numerically this is implemented by an evolution ofψparametrized by afictitious timeτ:−dδψ∗[ψ,ψ∗].(10)Assuming aψnormalized to unity we get the following equation of motion forψ:−ddτE[ψ,ψ∗]=−2 d2 r δE2mΩ2r2cannot exceed the trapping potential.Therefore E has to convergeto afinite value forτ→∞.Asymptotically dE/dτ=0andψsatisfiesδEωx=ω/(1+ǫ)(14)ωy=ω(1+ǫ).(15) In Fig.1we show different local minima configurations obtained forǫ=0.3and a rotation frequencyΩ=0.2ω;each configuration has been obtained for different random initialψ’s.The holes observedin the spatial density correspond to the vortex cores.We have always found that the phase ofψchanges by2πaround a vortex core;we have not found vortices with a charge±q,where the integerq is strictly larger than one;this fact will be explained in the next section.Furthermore the senseof circulation is the same for all vortices.To quantify the effect of the non-axisymmetry of the trap we have plotted in Fig.2the dependenceof energy of different vortex configurations onωx/ωy for afixedω;we measure the energies fromE iso,the energy of the zero-vortex solution in the axisymmetric caseǫ=0.The zero-vortex solutionexhibits a significant variation of energy withǫ;for a non-zeroǫthe wavefunctionψdevelops a phase proportional toΩfor weakΩ’s,which accounts for the energy change as explained in section VI B.The solutions with vortices experience quasi the same energy shift as function ofǫ.As only theenergy difference between the various local minima matters we will from now on only consider the axisymmetric caseǫ=0to identify the solution with the absolute minimal energy.Note that the solutionsψwith several vortices obtained in the limiting caseǫ=0are not eigen-vectors of L z;this reflects a general property of non-linear equations such as the Gross-Pitaevskiiequations to have symmetry broken solutions;it is explained in[10]how to reconcile this symmetrybreaking with the fact that eigenvectors of the full N-atom Hamiltonian are of well defined angular momentum.IV.STABILITY PROPERTIES OF VORTICESIn this section we recall that a(normalized)wavefunctionψsuch that E[ψ,ψ∗]has a local min-imum inψ,describes a condensate having all the desired properties of stability,that is dynamicaland thermodynamical stability.We then show that a vortex centered at r= 0with an angular mo-mentum strictly larger than¯h is not a local minimum of energy and is therefore thermodynamicallyunstable.A.Stability properties of local minimaLet us express the fact thatψcorresponds to a local minimum of the energy.Afirst condition isthat the energy functional is stationary forψ,that isψsolves the Gross-Pitaevskii equation Eq.(13).To get the second condition,we consider a small variation ofψ,ψ→ψ+δψ(16) preserving the normalization of the condensate wavefunction to unity:||ψ+δψ||2−||ψ||2=0= ψ|δψ + δψ|ψ + δψ|δψ .(17) We expand the energy functional E[ψ,ψ∗]in powers ofδψ,neglecting terms of orderδψ3or higher.Using Eq.(17)and Eq.(13)wefind that terms linear inδψvanish so that1δE=1.Dynamical stabilityConsiderfirst the problem of so-called“dynamical stability”:to be a physically acceptable con-densate wavefunction,ψhas to be a stable solution of the time dependent Gross-Pitaevskii equationi¯h∂tψ=H GPψ(21) otherwise any small perturbation ofψ,e.g.the effect of quantumfluctuations or experimental noise,may lead to an evolution ofψfar from its initial value.To determine the evolution of a smalldeviationδψas in Eq.(16)we linearize Eq.(21):i¯h∂t |δψ |δψ∗ =L |δψ |δψ∗ (22) where the operator L is related to L c byL c= 100−1 L.(23)Asψis time independent,so is L and dynamical stability is equivalent to the requirement that theeigenvalues of L have all a negative or vanishing imaginary part.As we now show the positivity ofL c leads to a purely real spectrum for L.Consider an eigenvector(u,v)of L with the eigenvalueε.Contracting Eq.(23)between the ket(|u ,|v )and the bra( u|, v|)we getε[ u|u − v|v ]=( u|, v|)L c |u |v .(24)Note that the matrix element of L c is real positive as L c is a positive hermitian operator.We nowface two possible cases for the real quantity u|u − v|v :• u|u − v|v =0.In this case L c has a vanishing expectation value in(|u ,|v );as L c is positive(|u ,|v )has to be an eigenvector of L c with the eigenvalue zero;from Eq.(23)and the fact that 100−1 is invertible wefind that(|u ,|v ) is also an eigenvalue of L with the eigenvalue0,so thatε=0is a real number.• u|u − v|v >0:we getεas the ratio of two real numbers,so thatεis real.2.Thermodynamical stabilityA second criterion of stability is the so-called“thermodynamical”stability.For zero temperature,this condition can be formulated in the Bogoliubov approach[2],where the particles out of thecondensate,which always exist because of the interactions,are described by a set of uncoupledharmonic oscillators with frequenciesεsign[ u|u − v|v ]/¯h,where(u,v)is an eigenvector of L withthe eigenvalueε.In order for a thermal equilibrium to exist for these oscillators,their frequenciesshould be strictly positive,which is the case here in virtue of Eq.(24)[13].If a mode with a negativefrequency were present thermalization by collisions would transfer particles from the condensateψto this mode,leading to a possible evolution of the system far from the initial stateψ[14].What happens for solutionsψof the Gross-Pitaevskii equations that are not local minima of energy?The operator L c has at least an eigenvector with a strictly negative eigenvalue.In thiscase one cannot have thermodynamical stability,that is one cannot haveε[ u|u − v|v ]>0forall modes[13].From the non-positivity of L c one cannot however distinguish between a simplethermodynamically instability or a more dramatic dynamical instability.B.Why not a vortex of angular momentum larger than¯h?For simplicity we consider only a single vortex in the center of an axi-symmetric trap.We show that vortices with a change of phase of2qπare not local minima of energy,that is are(at leastthermodynamically)unstable.We have found numerically a solution of the Gross-Pitaevskii equationEq.(13)by an evolution in complex time,starting from a wavefunctionψwith an angular momentumq¯h along z,as already done in[15];our solution of the Gross-Pitaevskii equation with imposedsymmetry is a local mimimum of energy in the subspace of functions with angular momentum q¯halong z,but not necessarily a local minimum in the whole functional space,as we will see for|q|>1.In the Thomas-Fermi regimeµ≫¯hωwefind that the solutions can be well reproduced bya variational Ansatz of the formψ(x,y)=e iqθ[tanhκq r]|q| ˜µ−1Ng 1/2(25) whereθis the polar angle in the x−y plane and where˜µ,the chemical potential in the lab frame˜µ=µ+q¯hΩ(26) does not depend onΩ.In this Ansatz the vortex core is accounted for by tanh|q|,a function thatvanishes as r|q|in zero as it should,and the condensate density outside the core coincides withthe Thomas-Fermi approximation commonly used for the zero-vortex solution[2].We calculate themean energy Eq.(7)of the variational Ansatz and we minimize it with respect to the variationalparameterκq;we getκq= ˜µmq2 +∞0du u tanh2|q|(u)−1 2(28) is a number(c1=0.7687,c2=0.5349,...).In order for the vortex of charge q to be a local minimum of energy,the operator L c of Eq.(19)has to be positive.This implies that the operator on thefirst line,first column of L c,the so-calledHartree-Fock Hamiltonian,be positive:H HF=H⊥+2Ng|ψ|2−˜µ+q¯hΩ−ΩL z≥0.(29) To show that this is not the case it is sufficient tofind a wavefunction f(x,y)leading to a negative expectation value for H HF.As the potential appearing in H HF has a dip at r=0we have taken fof a form localized around r=0:1f(x,y)=¯h2 1/2r (30) whereγis adjusted to minimize the expectation value.For e.g.q=2we takeγ=1leading tof|H HF|fsection the trap is axi-symmetry with a time dependent frequencyω(t).We consider the evolutionin the laboratory frame,as the detection is performed in this frame:i¯h∂tψlab= −¯h22mω2(t)r2+Ng|ψlab|2 ψlab.(32)As shown in[19,20]the effect of the time dependence ofω(t)can be absorbed by a scaling and gaugetransform of the wavefunction:1ψlab( r,t)=−ω2(t)λ(34)λ3with initial conditionsλ(0)=1,˙λ(0)=0;if the trap in the x−y plane is abruptly switched offatt=0+the scaling parameter is given byλ(t)==dτ(36)λ2(t)wefind that˜ψsolves the same equation asψlab with a constant trap frequency equal toω(0):i¯h∂τ˜ψ= −¯h22mω2(0)r2+Ng|˜ψ|2 ˜ψ.(37)Asψlab rotates in the trap at the frequencyΩin the lab frame,so does˜ψin terms of the renormalizedtimeτ.In the limit of t→∞,τtends to afinite valueτmax,so that˜ψis rotated by afinite angleduring the ballistic expansion:Ωτmax=Ω ∞0dt2ΩVI.INTUITIVE V ARIATIONAL CALCULATIONTo get a better understanding of the numerical results we now proceed to an intuitive Ansatzfor the wavefunction with several vortices.It coincides very well with the numerical results andallows an easy construction of the minimal energy configurations with vortices.It gives a physical understanding of the stability conditions and of the structure of the solutions:a set of n vortices isequivalent to a gas of interacting particles in presence of an external potential adjusted by the rota-tion frequency of the trap.We restrict to the case of an axi-symmetric trap,a good approximationfor weak(<10%)non-axisymmetries(see section III B).A.Ansatz for the densityTo construct the Ansatz we splitψin a modulus and a phase:ψ(x,y)=|ψ|e iS.(40) In the Thomas-Fermi regime,the modulus in presence of n vortices appears as a slowly varyingenvelope given by the Thomas-Fermi approximation used in the0-vortex case:ψslow= µ−1Ng 1/2(41)with narrow holes digged by the vortices with charge q=±1,represented by tanh functions ofadjustable widths and with zeros at adjustable positions:|ψ|=ψslow×nk=1tanh[κk| r− αk R|].(42)The positions of the vortex cores αk are expressed in units of the Thomas-Fermi radius R of the condensate:R= mω2.(43)From section IV B we expect as typical values for the inverse width of the vortex coresκk≃(mµ/¯h2)1/2.The chemical potential is not an independent variable but is expressed as a functionof the other parameters from the normalization condition ψ|ψ =1;neglecting overlap integralsbetween the holes we getµ=µ0 1+2n k=1(1−α2k)ln2(κR)4) (44)where1/(κR)4∼(¯hω/µ)4≪1and whereµ0is the Thomas-Fermi approximation for the condensatechemical potential without vortices:µ0= mω2Ngcentered on the vortex core and S 0is the single-valued part of the phase.The function S 0can in principle be determined from the modulus of ψfrom the continuity equation:div[|ψ|2 v ]=0.(47)The local velocity field v is related to the phase S byv =¯h ¯h [x∂y −y∂x ]|ψ|2=0.(49)This can be turned into an equation for the single-valued part S 0of the phase;because the density |ψ|2in a trap vanishes at the border of the condensate S 0is uniquely determined (up to a constant)by the resulting equation (see Appendix);this is to be contrasted to the case of superfluid helium in a container,where the flux,not the density,vanishes at the border,which requires a boundary condition on the gradient of the phase.Eq.(49)can be solved for a non-axisymmetric trap in the absence of vortices.The solution is given byS (x,y )=−m Ωω2x +ω2yxy (50)which leads to a change in the energy per particleδE =−1(ω2x +ω2y )ω2x ω2y(51)where µT F is the Thomas-Fermi approximation for the chemical potential for Ω=0,µT F =(mωx ωy Ng/π)1/2[23].As can be seen in Fig.2this prediction is in good agreement with our numerical results.In presence of vortices the equation for S is more difficult to solve analytically.From now on we consider the case of an axi-symmetric trap,as the energy ordering of the vortices solutions is not affected for weak (<10%)non-axisymmetries (see section III B).For a single vortex at the center of the trap one can see that S 0=0solves Eq.(49).From the spatial dependence of the phase obtained numerically (section III B)for a displaced vortex or several vortices we have identified the following heuristic Ansatz,obtained in setting ωx =ωy =ωin Eq.(50):S 0(x,y )≡0(52)that we will use in the remaining part of the section.C.Further approximations for the mean energyIn the calculation of the mean energy,we make some further approximations in the spirit of the Ansatz Eq.(42).The reader not interested by these more technical considerations can proceed to the next subsection.The kinetic energy involves an integral of the gradient squared of the wavefunction:| ∇ψ|2=|ψ|2 ( ∇ln |ψ|)2+( ∇S )2 .(53)For the gradient of the modulus of ψwe neglect the variation of the slow envelope ψslow :∇ln |ψ|≃nk =1κktanh ′of ψinvolves holes with a density varying as 1−tanh 2=sech 2.In the following we keep the sech 2for the vortex k only if it is multiplied by ( ∇θk )2,a quantity diverging in the center of the core;the other terms lead toconverging integrals smaller by a factor (µ/¯h ω)2,which is the inverse surface of a vortex core ( d 3 r |ψslow |2sech 2κr ∝1/(κR )2).This finally leads toE kin ≃¯h 22mω2r 2+13µ0+nk =1W ( αk ,κk )+1µ012ln(1−α2)−q ¯h Ω (1−α2)2 +µ0(κR )2(60)where C =0.495063.The lines inEq.(60)correspond successively to E kin ,E rot and E pot .This can be seen as an effective potential for the vortices.One can check that the part of W independent of Ωexpells the vortex core from the trap center,whereas the part proportional to Ωprovides a confinement of the vortex core (see the following subsection).The vortex interaction potential is given by12m q αq β d 2 r |ψslow |2 ∇θ αR · ∇θ βR .(61)This interaction term is equivalent to the one found in the homogeneous case and describes a repulsive interaction for vortices turning in the same direction (q αq β>0)and is attractive for vortices with opposite charges [9].An attractive interaction will lead to the coalescence and consequently annihilation of vortices with opposite charges.Therefore we find in stationary systems always vortices with equal charges.As the interaction potential V ( α, β)does not depend on the parameters κwe can optimize sepa-rately the self-energy part with respect to κand find(κR)2=ξ2(1−α2) µ03(4ln2−1) 1/2≃1.08707.By rewriting the above equation as¯h2κ22ξ2 µ−1µ0 13+lnνµ0¯hω2(1−α2) (64) whereν=0.49312.E.Case of a single vortex:critical frequenciesIn Fig.4we have plotted the self-energy of a vortex as a function of the displacement of the corefrom the trap center,for different values of the rotation frequencyΩ.The analytical predictioncoincides very well with the numerical value[24].ForΩ=0the position of the vortex at the trap center gives an energy maximum.ForΩ>0therotation of the trap provides an effective confinement of the vortex core at the center of the trap forpositive charges q(see the term proportional toΩin Eq.(64));from now one we therefore take allthe charges q k to be equal to+1.For a large enoughΩwe reach a situation where a vortex at thetrap center corresponds to a local energy minimum,by further increasingΩthe vortex state at thetrap center becomes a global minimum with energy less than the condensate without vortex.The above suggests that we have to distinguish two critical rotation frequencies:Thefirst onedefines the frequencyΩstab above which the vortex is a local minimum of energy.Above the frequencyΩc the single vortex solution has an energy lower than the condensate without vortex.We calculateΩstab from the condition d2W/dα2=0atα=0andΩc from the condition W=0atα=0:Ωc=¯hω2¯hω (65)Ωstab=¯hω2¯hω (66)where C′=e(2ln2+1)/3+1/2ν≃1.8011.As we are in the regimeµ0≫¯hωΩc is approximately twice Ωstab[25].Our prediction forΩc scale as(logµ0)/µ0as in[15],with a coefficient C′leading to better agreement with the numerics.F.Case of several vorticesBy integrating Eq.(61)we get an explicit form for the vortex interaction potential for vortices with equal charges:V( α, β)=(¯hω)2(1− α· β)+1| α− β|4 (67)At short distances between the two vortex cores the logarithmic term in the above expression dominates,leading to a repulsive potential∼−2(1− α· β)log| α− β|(¯hω)2/µ0.In Fig.5we plot the interaction energy between a vortex at the center of the trap and one of equal charge displaced by αR;the interaction is purely repulsive.A conclusion which essentially holds as well for arbitrary vortex positions.In Fig.6we show the total(interaction+self-energy)for two vortices symmetrically displaced from the trap center,as function of the displacement;the analytical prediction coincides again very well with the numerical results[24].To obtain the equilibrium distance between the two vortex cores one minimizes the total energy overαin Fig.6.To get the minimal energy configurations as function of the rotation frequency of the trap,we minimize our analytical prediction for the energy over the positions of the n=1,2,...vortex cores.The result is shown in Fig.7.Each curve corresponds to afixed value of n;it starts atΩ=Ωstab(n)(forΩ<Ωstab(n)there is no local minima of energy with n vortices);it becomes the global energyminimum forΩ=Ωc(n).We have plotted these two critical frequencies as function of n in Fig.8.We have also given numerical results(circles)in Fig.7.Even if there is good agreement between analytical and numerical results,we still need a numerical calculation to check the stability of thesolutions;our simple analytical Ansatz is indeed not sufficient to predict the destabilization of agiven vortex configuration at highΩ,a phenomenon studied with a numerical calculation of theBogoliubov spectrum for a single vortex in[26].For afixed value of the number of vortices n there may exist local minima of energy,in addition to the global minimum plotted in Fig.7,a situation known from superfluid helium[4].E.g.for n=6(see Fig.9)the global minimum of energy is given by a configuration with six vortex cores on acircle;there exists also a local minimum of energy with one vortex core at the center of the trap andfive vortex cores on a circle.The energy difference per particle between the two configurations isvery small,δE≃0.002¯hωfor the parameters of thefigure and probably beyond the accuracy of ourvariational Ansatz.For relatively large rotation frequenciesΩone canfind local minima of energyconfigurations with many vortices(see[4]for superfluid helium);we plot two configurations with18vortices in Fig.10,with an energy differenceδE=0.0034¯hω.In estimating the physical relevance of these energy differences one should keep in mind that NδE matters,rather thanδE,where N is the number of particles in the condensate: e.g.at afinite temperature T the ground energy configuration is statistically favored as compared to themetastable one when NδE≫k B T.VII.VORTICES IN A3D CONFIGURATIONWe have extended the numerical calculation to the case of a3D cigar-shaped trap,that is witha confinement weaker along the rotation axis than in the x−y plane.Even in this case rotationof the trap can stabilize the vortex.We show in Fig.11density cuts of a solution with5vortices;the vortex cores are almost straight lines in the considered Thomas-Fermi regime,except at vicinityof the borders of the condensate.As in section VI the core diameter is determined by the localchemical potential in the gas.This suggests that our2D Ansatz(section VI)can be generalized to3D situations,with αk and κk depending on z.VIII.CONCLUSION AND PERSPECTIVESWe have presented in this paper an efficient numerical algorithm and a heuristic variational Ansatz to determine the local minima energy configurations for a Bose-Einstein condensate stronglyconfined along z and subject to a rotating harmonic trap in the x−y plane.Our results can be used as afirst step towardsfinite temperature calculations.Interesting prob-lems are e.g.the critical temperature for the vortex formation and the Magnus forces induced bythe non-condensed particles on the vortex core[27].Acknowledgement:We acknowledge useful discussions with Sandro Stringari and Dan Rokhsar.We thank J.Dalibard for useful comments on the manuscript.We thank the ITP at Santa Barbarafor its hospitality and the NSF for support under grant No.PHY94-07194.This work was partiallysupported by the TMR Network“Coherent Matter Wave Interactions”,FMRX-CT96-0002.∗L.K.B.is an unit´e de recherche de l’Ecole Normale Sup´e rieure et de l’Universit´e Pierre et Marie Curie,associ´e e au CNRS.APPENDIX A:UNIQUENESS OF THE PHASE FROM THE CONTINUITY EQUATION IN A TRAPConsider two solutions S1and S2of the continuity equation:mΩdiv[|ψ|2 ∇S]=。

a r X i v :a s t r o -p h /0607328v 1 14 J u l 2006Mon.Not.R.Astron.Soc.000,1–34(2006)Printed 5February 2008(MN L A T E X style file v2.2)Global structures in a composite system of twoscale-free discs with a coplanar magnetic fieldYu-Qing Lou 1,2,3⋆and Xue-Ning Bai 11Physics Department and Tsinghua Centre for Astrophysics (THCA),Tsinghua University,Beijing 100084,China;2Departmentof Astronomy and Astrophysics,The University of Chicago,5640South Ellis Avenue,Chicago,IL 60637,USA;3National Astronomical Observatories,Chinese Academy of Science,A20,Datun Road,Beijing 100012,China.Accepted .Received ;in original formABSTRACTWe investigate a theoretical magnetohydrodynamic (MHD)disc problem involv-ing a composite disc system of gravitationally coupled stellar and gaseous discs with acoplanar magnetic field in the presence of an axisymmetric dark matter halo.The two discs are expediently approximated as razor-thin,with a barotropic equation of state,a power-law surface mass density,a ring-like magnetic field,and a power-law rota-tion curve in radius r .By imposing the scale-free condition,we construct analytically stationary global MHD perturbation configurations for both aligned and logarithmic spiral patterns using our composite MHD disc model.MHD perturbation configura-tions in a composite system of partial discs in the presence of an axisymmetric dark matter halo are also considered.Our study generalizes the previous analyses of Lou &Shen and Shen &Lou on the unmagnetized composite system of two gravitation-ally coupled isothermal and scale-free discs,of Lou and Shen et al.on the cases of a single coplanarly magnetized isothermal and scale-free disc,and of Lou &Zou on magnetized two coupled singular isothermal discs.We derive analytically the station-ary MHD dispersion relations for both aligned and unaligned perturbation structures and analyze the corresponding phase relationships between surface mass densities and the magnetic fipared with earlier results,we obtain three solution branches corresponding to super fast MHD density waves (sFMDWs),fast MHD density waves (FMDWs)and slow MHD density waves (SMDWs),respectively.We examine the m =0cases for both aligned and unaligned MHD perturbations.By evaluating the unaligned m =0case,we determine the marginal stability curves where the two unstable regimes corresponding to Jeans collapse instability and ring fragmentation instability are identified.We find that the aligned m =0case is simply the limit of the unaligned m =0case with the radial wavenumber ξ→0(i.e.,the breath-ing mode)which does not merely represent a rescaling of the equilibrium state.We further show that a composite system of partial discs behaves much differently from a composite system of full discs in certain aspects.We provide numerical examples by varying dimensionless parameters β(rotation velocity index),η(ratio of effective sound speed of the two discs),δ(ratio of surface mass density of the two discs),q (a measure of coplanar magnetic field strength),F (gravity potential ratio),ξ(radial wavenumber).Our formalism provides a useful theoretical framework in the study of stationary global perturbation configurations for MHD disc galaxies with bars,spirals and barred spirals.Key words:MHD waves—ISM:magnetic fields —galaxies:kinematics and dynam-ics —galaxies:spiral —star:formation —galaxies:structure.⋆E-mail:xiaobai2Y.-Q.Lou,X.N.Baivarious complementary aspects.Lin&Shu(1964,1966)and their co-workers pioneered the classic density wave theory and achieved a great success in understanding the dynami-cal nature of spiral galaxies(Lin1987;Binney&Tremaine 1987;Bertin&Lin1996).The basic idea of analyzing such a large-scale density wave problem is to treat coplanar perturbations in a background axisymmetric rotating disc; this procedure has been proven to be powerful in prob-ing the galactic dynamics to various extents.In such a model development,perturbations(either linear or non-linear)are introduced onto a background equilibrium to form local or large-scale structures and to perform stabil-ity analysis under various situations.Broadly speaking,ax-isymmetric and non-axisymmetric perturbations may lead to aligned configurations while non-axisymmetric perturba-tions can naturally produce spiral wave patterns.Theoreti-cally,a disc system may be treated as razor-thin for simplic-ity(Syer&Tremaine1996;Shen&Lou2004b;Lou&Zou 2004;Shen et al.2005),with a mass densityρ=Σ(r,θ)δ(z) under the situation where the thickness of a galactic disc is sufficiently small compared with its radial size scale;in this manner,the model problem reduces to a two-dimensional one.In theoretical model investigations,two classes of disc models are frequently encountered.One class is the so-called singular isothermal discs(SIDs),which bears aflat rotation curve and a constant‘temperature’with a diverg-ing surface mass density towards the centre[e.g.,Shu et al. (2000)].Since the early study by Mestel(1963)more than four decades ago,this idealized theoretical SID model has attracted considerable interest among astrophysicists in various contexts of disc dynamics[e.g.,Zang(1976); Toomre(1977);Lemos et al.(1991);Goodman&Evans (1999);Shu et al.(2000);Lou(2002);Lou&Fan(2002); Lou&Shen(2003);Lou&Zou(2004);Lou&Wu(2005); Lou&Zou(2006)].The other class is the so called scale-free discs[e.g.Syer&Tremaine(1996);Shen&Lou(2004b); Shen et al.(2005);Wu&Lou(2006)]and is the main focus of this paper.Being of a more general form for differentially rotating discs,a scale-free disc has a rotation curve in the form of v∝r−β(the case ofβ=0corresponds to aflat rotation curve)and a barotropic equation of state in the form ofΠ=KΣn whereΠis the vertically integrated two-dimensional pressure.There is no characteristic spatial scale and all quantities in the disc system vary as powers of ra-dius r.In our analysis,the rotation velocity indexβsatisfies −1/4<β<1/2for warm discs.Furthermore,it is possible to construct stationary(i.e.ω=0)density wave patterns in scale-free discs(Syer&Tremaine1996;Shen&Lou2004b; Shen et al.2005;Wu&Lou2006),clearly indicating that the pattern speed of a density wave is in the opposite sense of the disc rotation speed.Because of the self-gravity,one important technical as-pect in dealing with thin disc galaxies involves the Poisson integral,relating the mass density distribution and grav-itational potential.In the early development of the den-sity wave theory,this Poisson integral is evaluated by an asymptotic analysis valid in the large wavenumber regime of the Wentzel-Kramers-Brillouin-Jeffreys(WKBJ)approx-imation.Based on this,the pursuit of analytical solu-tions leads to a further exploration of the problem.Sev-eral potential-density pairs are generalized in Chapter2of Binney&Tremaine(1987).One special utility is the logarithmic spiral potential-density pairfirst derived by Kalnajs(1971).This is a powerful tool in analyzing non-axisymmetric perturbations in an axisymmetric background disc.Furthermore,Qian(1992)made use of the techniques of Lynden-Bell(1989)and found a larger family of potential-density pairs in terms of the generalized hypergeometric ing these results for scale-free discs under our consideration,the Poisson integral is solved analytically for both aligned and spiral perturbations(Shen&Lou2004b; Shen et al.2005;Wu&Lou2006).Magneticfield is an important ingredient in various as-trophysical disc systems.In the interstellar medium(ISM), the partially or fully ionized gases make the cosmic space an ideal place for applying the magnetohydrodynamics(MHD) equations.In many astrophysical situations involving large-scale dynamics,the magneticfield can be effectively con-sidered as completely frozen into the gas.While in some cases the magnetic force is relatively weak and has lit-tle impact on the dynamical process.There do exist cer-tain situations when the magneticfield plays a crucial role in both dynamics and diagnostics,such as in spi-ral galaxies and accretion processes[e.g.Balbus&Hawley (1998);Balbus(2003);Fan&Lou(1996,1997);Shu et al. (2000);Lou&Fan(1998a);Lou(2002);Lou&Fan(2003); Lou&Zou(2004,2006);Shen et al.(2005);Lou&Wu (2005)].In terms of MHD model development,we need to prescribe the magneticfield geometry.Shu&Li(1997)in-troduced a model in which a disc is‘isopedically’magnetized such that the mass-to-flux ratio remains spatially uniform and the effect of magneticfield is subsumed into two pa-rameters(Shu&Li1997;Shu et al.2000;Lou&Wu2005; Wu&Lou2006).In fact,Lou&Wu(2005)have shown ex-plicitly that a constant mass-to-flux ratio is a natural conse-quence of the frozen-in condition from the ideal MHD equa-tions.In parallel,the coplanar magneticfield also serves as an interesting model(Lou2002;Lou&Zou2004;Shen et al. 2005)with the magneticfield being azimuthally embedded into the disc system.In particular,with the inclusion of an azimuthal magneticfield,one can construct the so-called fast and slow MHD density waves(FMDWs and SMDWs) (Fan&Lou1996,1997;Lou&Fan1998a;Lou2002)and the interlaced optical and magnetic spiral arms in the nearby spiral galaxy NGC6946are sensibly explained along this line.The stability of such MHD discs is yet another lively debated issue.In the WKBJ or tight-winding regime,the well-known Q parameter criterion(Safronov1960;Toomre 1964)was suggested to determine the galactic local ax-isymmetric stability in the absence of magneticfield. Meanwhile,there have been numerous studies concern-ing the global stability of the disc problem[e.g.,Zang (1976);Toomre(1977);Lemos et al.(1991);Evans&Read (1998a,b);Goodman&Evans(1999);Shu et al.(2000)].For example,stationary perturbations of the zero-frequency neu-tral modes are emphasized as the marginal instability modes in scale-free discs.In this context,Syer&Tremaine(1996) made a breakthrough in obtaining semi-analytic solutions for stationary perturbation configurations in a class of scale-free discs.Shu et al.(2000)analyzed the stability of isope-dically magnetized SIDs and derived stationary perturba-tion solutions.They interpreted these aligned and unalignedCoupled Magnetized Scale-Free Discs3configurations as onsets of bar-like instabilities.Lou(2002) performed a coplanar MHD perturbation analysis on az-imuthally magnetized SIDs from a perspective of stationary FMDWs and SMDWs.Our analysis on two-dimensional coplanar MHD pertur-bations has avoided at least two major issues.If perturbation velocity and magneticfield components perpendicular to the disc plane are allowed,it is then possible to describe Alfv´e nic fluctuations and model disc warping process.If one further takes into account of vertical variations across the disc(i.e., the disc thickness is not negligible),magneto-rotational in-stabilities can develop(e.g.,Balbus2003).These two as-pects are important and should bear physical consequences in modelling disc galaxies.In a typical disc galaxy system,the basic component involves stars,gases,dusts,cosmic rays,and a massive dark matter halo(Lou&Fan1998a,2003).In terms of theoreti-cal analysis,it would be a great challenge to include all these factors into one single model consideration.While limited in certain aspects,it remains sufficiently challenging and inter-esting to consider a composite system consisting of a copla-narly magnetized gas disc,a stellar disc as well as an axisym-metric background of a massive dark matter halo.A seminal analysis concerning a composite disc system dates back to Lin&Shu(1966,1968)who combined a stellar distribution function and a gasfluid description to derive a local disper-sion relation for galactic spiral density waves in the WKBJ approximation.Since then,there have been extensive the-oretical studies on perturbation configurations and stabil-ity problems of the composite disc system.Jog&Solomon (1984a,b)investigated the growth of local axisymmetric per-turbations in a composite stellar and gaseous disc system. Bertin&Romeo(1988)considered the spiral modes con-taining gas in a two-fluid model.Vandervoort(1991a,b) studied the effect of interstellar gas on oscillations and the stability of spheroidal galaxies.Romeo(1992)considered the stability of two-componentfluid discs withfinite thickness. Two-fluid approach was adopted into modal analysis mor-phologies of spiral galaxies by Lowe et al.(1994),support-ing the notion that spiral structures are long-lasting and slowly evolving.Elmegreen(1995)and Jog(1996)suggested an effective Q ef f(or Q s−g)parameter criterion(Safronov 1960;Toomre1964)for local axisymmetric two-fluid insta-bilities of a disc galaxy.Lou&Fan(1998b)explored basic properties of open and tight spiral density-waves modes in a two-fluid model to describe a composite system of coupled stellar and gaseous discs.Recently,Lou&Shen(2003)stud-ied a composite SID system to derive stationary global per-turbation configurations and further explored the axisym-metric instability properties(Shen&Lou2003)where they proposed a fairly straightforward D criterion for the ax-isymmetric instability problem for a composite SID system. Shen&Lou(2004b)extended these analysis to a compos-ite system of two scale-free discs and carried out analytical analysis on both aligned and logarithmic spiral perturba-tion configurations.By adding a coplanar magneticfield to the background composite SIDs,Lou&Zou(2004)obtained MHD perturbation configurations and further studied the axisymmetric instability problem(Lou&Zou2006).The main objective of this paper is to construct global scale-free stationary configurations in a two-fluid gaseous and stellar disc system with an embedded coplanar mag-neticfield in the gas disc.Meanwhile,an axisymmetric in-stability analysis is also performed.There are several new features compared with previous works(Lou&Zou2004; Shen&Lou2004b;Shen et al.2005)that may provide cer-tain new clues to understand large-scale structures of disc galaxies.This paper is structured as follows.In Section2,we present the theoretical formalism of the problem;both the stationary equilibrium state and the linearized MHD pertur-bation equations are summarized.In Section3,we perform numerical calculations for the aligned perturbation configu-rations.Both the dispersion relation and phase relationship between density and the magneticfield perturbations are deduced and evaluated.In Section4,we apply the same procedure to the analysis of global logarithmic spiral con-figurations.The m=0marginal stability is also discussed. Finally,we summarize and discuss our results in Section5. Several technical details are included in Appendices A,B, and C.2FLUID-MAGNETOFLUID DISCSWe adopt thefluid-magnetofluid formalism in this paper to construct large-scale stationary aligned and unaligned copla-nar MHD disturbances in a background MHD rotational equilibrium of axisymmetry[Lou&Shen(2003);Lou&Zou (2004);Shen&Lou(2004b);Lou&Wu(2005);Lou&Zou (2006);Wu&Lou(2006)].All the background physical quantities are assumed to be axisymmetric and to scale as power laws in radius r.Specifically,the rotation curves bear an index of−β(viz.,v∝r−β)and the vertically integrated mass density has the formΣ0∝r−αwith−αbeing another index.Physically,the magnetofluid formalism is directly ap-plicable to the magnetized gas disc,while thefluid formal-ism is only an approximation when applied to the stellar disc,where a distribution function approach would give a more comprehensive description.For our purpose of mod-elling large-scale stationary MHD perturbation structures and for mathematical simplicity,as well as the similarity between the two sorts of descriptions(Shen&Lou2004b), it suffices to work with thefluid-magnetofluid formalism.In this section,we present basic MHD equations of the fluid-magnetofluid description for a composite system con-sisting of a scale-free stellar disc and a coplanarly magne-tized gas disc.In our approach,the two gravitationally cou-pled discs are treated using the razor-thin approximation (i.e.,we use vertically integratedfluid-magnetofluid equa-tions and neglect vertical derivatives of physical variables along z)and the two-dimensional barotropic equation of state to construct global MHD perturbation structures.A coplanar magneticfield is involved in the dynamics of the thin gas disc following the basic MHD equations.The back-ground state of axisymmetric rotational equilibrium isfirst derived.We then superpose coplanar MHD perturbations onto the equilibrium state and obtain linearized equations for MHD perturbations in the composite disc system.2.1Ideal Nonlinear MHD EquationsBy our conventions,we use either subscript or superscript‘s’and‘g’to denote physical variables in association with the stellar disc and the magnetized gas disc,respectively.For4Y.-Q.Lou,X.N.Bailarge-scale MHD perturbations of our interest at this stage, all diffusive effects such as viscosity,resistivity,thermal con-duction,and radiative losses etc.are ignored for simplicity. In cylindrical coordinates(r,θ,z)and coincident with the z=0plane,we readily write down the basic nonlinear ideal MHD equations for the composite disc system,namely∂Σgr ∂r2∂∂t +u g∂u gr2∂u gr3=−1∂r−∂φΣg d zBθ∂r−∂B r∂t +u g∂j gr2∂j gΣg∂Πg∂θ+14π ∂(rBθ)∂θ(3)for the magnetized gas disc,and∂Σsr ∂r2∂∂t +u s∂u sr2∂u sr3=−1∂r−∂φ∂t +u s∂j sr2∂j sΣs∂Πs∂θ(6)for the stellar disc in the‘fluid’approximation,respectively. Here,Σdenotes the surface mass density,u is the radial component of the bulkflow velocity,j≡rv is the spe-cific angular momentum in the vertical z-direction and v is the azimuthal bulkflow velocity,Πis the vertically inte-grated two-dimensional pressure,B r and Bθare the radial and azimuthal components of the coplanar magneticfield B≡(B r,Bθ,0),andφis the total gravitational potential. Thisφcan be expressed in terms of the Poisson integral Fφ(r,θ)= dϕ ∞0−GΣ(r′,ϕ,t)r′d r′∂r +∂Bθ∂t=1∂θ(u g Bθ−v g B r),(9)∂Bθ∂r(u g Bθ−v g B r).(10)Among(8)−(10),only two of them are independent.Thebarotropic equation of state for the scale-free discs isΠi=K iΣn i,(11)where K i>0is a constant proportional coefficient andn i>0is the barotropic index with subscript i denotingeither g or s for the two discs(we use this convention forsimplicity).An isothermal equation of state has n i=1.Forthe stellar disc and the magnetized gas disc,K i are allowedto be different,but we need to require n g=n s=n in orderto meet the scale-free requirement(see the next subsection).It follows that the sound speed a i(in the stellar disc the ve-locity dispersion mimics the sound speed)for either disc isreadily defined bya2i≡dΠi04πΣg0 B20dz(14)varying with r in general,andv s02=−αa2s+rdφ0/dr(15)in the stellar disc.Poisson integral(7)yieldsFφ0=−2πGrΣ0P0(α),(16)where the numerical factor P0(α)is explicitly defined byP0(α)≡Γ(−α/2+1)Γ(α/2−1/2)Coupled Magnetized Scale-Free Discs5with Γ(···)being the standard gamma function.Expression (17)can alsobeincludedin a more general form of P m (β)defined later in equation (55).Inordertosatisfy the scale-free condition,radial force balances (13)and (15)should hold for all radii,leading to the following simple relation among the four indices 2β=α(n −1)=2γ−α=α−1.(18)This in turn immediately gives the explicit expressions ofindices α,γand n in terms of β,namely α=1+2β,γ=(1+4β)(1+2β).(19)Since n >0for warm discs,we have β>−1/4.Further-more,Poisson integral (7)converges when 1<α<2which corresponds to 0<β<1/2.In addition,for a finite total gravitational force a larger β-range of −1/2<β<1/2is allowed.Therefore,the physical range for βis constrained by −1/4<β<1/2(Syer &Tremaine 1996;Shen &Lou 2004b;Lou &Zou 2004;Shen et al.2005).For simplicity,we introduce the following parameters A i ,D i ,and q defined by a 2i=A 2ir β,C 2A≡q 2a 2g=q 2A 2gr=A i D ird (r 2Ωi )4β+2q2=A 2s (D 2s +1)=2πG 2βP 0Σ0r 1+2β/F .(22)We now introduce two dimensionless parameters to com-pare the properties of the two scale-free discs.The first oneis simply the surface mass density ratio δ≡Σg 0/Σs0,and the second one is the square of the ratio of effective sound speedsη≡a 2g /a 2s =A 2g /A 2s .For disc galaxies,the ratio δcan be either greater (i.e.,younger disc galaxies)or less (i.e.,older disc galaxies)than 1,depending on the type and evolution stage of a disc galaxy.Meanwhile,the ratio ηcan be gen-erally taken as less than 1because the sound speed in themagnetized gas disc is typically less than the stellar velocity dispersion (regarded as an effective sound speed).With these notations,we now have from condition (22)Σg 0=F A 2g [D 2g +1+(4β−1)q 2/(4β+2)]δ2πG (2βP 0)(1+δ)r (1+2β).(23)Note that expressions (22)and (23)reduce to expression (14)of Shen &Lou (2004b)when the magnetic field van-ishes (i.e.,q =0)and are also in accordance with Shen et al.(2005)when a single magnetized scale-free gas disc is con-sidered.Since the magnetic field effect is represented by the q parameter multiplied by a factor (1−γ)or (4β−1)[see eq.(13)or eq.(22)],we know that in the special case of β=1/4,the Lorentz force vanishes due to the cancella-tion between the magnetic pressure and tension forces in the background equilibrium (Shen et al.2005).Moreover,when −1/4<β<1/4,the net Lorentz force arising from the azimuthal magnetic field points radially inward,while for 1/4<β<1/2,the net Lorentz force points radially outward (Shen et al.2005).Another point should also be noted here.From equa-tion (22),there exists another physical requirement for the rotational Mach number D g ,i.e.,it should be large enoughto warrant a positive D 2s .This requirement for D 2g is simplyD 2s=ηD 2g +1+4β−1∂t+1∂r(r Σg 0u g1)+Ωg∂Σg 1r 2∂j g1∂t+Ωg∂u g 1r=−∂Σg 0+φ1+(1−4β)C 2A Σg 1Σg 0d zb θ∂r −14πr∂(rb θ)∂θ,(26)6Y.-Q.Lou,X.N.Bai∂j g12Ωg u g1+Ωg∂j g1∂θ a2gΣg1Σg0 d zb r∂r(27)in the magnetized gas disc and∂Σs1r ∂∂θ+Σs0∂θ=0,∂u s1∂θ−2Ωsj s1∂r a2sΣs1∂t +rκ2s∂θ=−∂Σs0+φ1(28)in the stellar disc,respectively.All dependent variables are taken to thefirst-order smallness with nonlinear terms ig-nored.The perturbed Poisson integral appears asφ1= dϕ ∞0−G(Σg1+Σs1)r′d r′∂r +∂bθ∂t =1∂θ(u g1B0−rΩg b r),∂bθ∂r (u g1B0−rΩg b r)(30)from the divergence-free condition and the magnetic induc-tion equation.2.3.1Non-Axisymmetric Cases of m 1As the background equilibrium state is stationary and ax-isymmetric,these perturbed physical variables can be de-composed in terms of Fourier harmonics with the periodic dependence exp(iωt−i mθ)whereωis the angular frequency and m is an integer to characterize azimuthal variations. More specifically,we writeΣi1=S i(r)exp(iωt−i mθ),u i1=U i(r)exp(iωt−i mθ), j i1=J i(r)exp(iωt−i mθ),φ1=V(r)exp(iωt−i mθ),b r=R(r)exp(iωt−i mθ),bθ=Z(r)exp(iωt−i mθ),(31) where we use the italic superscript i to indicate associations with the two discs and the roman i for the imaginary unit. We also define V i(i=s,g)for the corresponding gravi-tational potentials associated with the stellar and gaseous discs such that V(r)≡V s(r)+V g(r).By substituting ex-pressions(31)into equations(25)−(30),we readily attaini(ω−mΩg)S g+1dr(rΣg0U g)−i mΣg0J gr=−dΨg2Σg0r−14πr d(rZ)Σg0d zZ dr,(33)i(ω−mΩg)J g+rκ2gΣg0 d zR dr(34)for the magnetized gas disc,andi(ω−mΩs)S s+1dr(rΣs0U s)−i mΣs0J sr=−dΨs2ΩsU s=i mΨs(35)for the stellar disc,respectively,where we defineΨi≡a2i S i/Σi0+V.(36)The perturbed Poisson integral now becomesV(r)= dϕ ∞0−G(S g+S s)cos(mϕ)r′dr′dr−i mZ=0,i(ω−mΩg)R+i mB0dr(rΩg R−B0U g).(38)A combination of equations(32)−(35)and(37)−(38)consti-tutes a complete description of coplanar MHD perturbationsin a composite disc system.From equation(38),we obtainZ=−idr,R=−mB0U gr=−dΨg2Σg0r−C2A d22r d r2 i U g2ΩgU g=i mΨg−mC2A(1−4β)U gCoupled Magnetized Scale-Free Discs7 with a coplanar magneticfield.For m 1,we setω=0inMHD perturbation equations to deducemΩg S g+1dr(rΣg0i U g)+mΣg0J gr=dΨg2Σg0r−C2Adr2+1−12βdr+2β(1+4β)−m2Ωg,(43)mΩg J g+rκ2g2ri U gdr B0i U g rΩg,(45) for the magneticfield perturbation.For coplanar perturba-tions in the stellar disc,we derivemΩs S s+1dr(rΣs0i U s)+mΣs0J sr=dΨs2Ωsi U s=−mΨs.(46)As part of the derivation,we deduce from the last two equa-tions of(46)for the expressions of U s and J s asU s=−i mΩsr+dr =−1r+κ2sdr Ψs.(47)A substitution of equation(47)into equation(46)gives a single equation relating S s andΨs for coplanar perturba-tions in the stellar disc,namelymΩs S s+1dr mΩs rΣs0r+d m2Ω2s−κ2s m2Ωs2rΩs dr dr =−ωdΨg2Σg0r−C2A d22r d r2 i U g,(50)ωJ g=rκ2gdr(B0i U g)(52) for the magnetized gas disc.For coplanar perturbations in the stellar disc,we have in parallelωS s=1dr(rΣs0i U s),ωi U s−2ΩsJ sdr,ωJ s=rκ2s8Y.-Q.Lou,X.N.Bairespectively.The numerical factor P m (β)is defined explic-itly by P m (β)≡Γ(m/2−β+1/2)Γ(m/2+β)ri U g+m Σg 02m Ωg r 2i U g+2Ωg2r Σg 0S g+4πGβP m S s,(57)m Ωg J g +(1−β)r Ωg −C 2A (1−4β)r Σg 0S g +2πGmr P m S s ,(58)Z =−1m Ωg r,R =B 0U g(m 2+4β−4β2)−a 2s +2πGr Σs 0P mS s+2πGr Σs 0P m Sg =0(60)for the stellar disc.Notethat equations (56)−(59)arethe same as equation (43)of Shen et al.(2005)for a sin-gle magnetized scale-free disc by setting S s =0and equa-tion (60)is exactly the same as the first one of equation (35)in Shen &Lou (2004b).A combination of equations (56)−(58)and (60)then gives a complete solution for the stationary dispersion relation in the gravitational coupled MHD discs that are scale free.Because these equations are linear and homogeneous,to obtain non-trivial solutions for (S g ,S s ,U g ,J g ),the determinant of the coefficients of equa-tions (56)−(58)and (60)should vanish.This actually gives rise to the stationary MHD dispersion relation.Meanwhile,in order to get a physical sense of the dispersion relation,we solve the above MHD equations directly.A combination of equations (56)and (58)produces re-lations of i U g and J g in terms of S g and S s ,namely i U g =m Ωg rΣg 0{[−1+3β(ΘA +ΞA )]S g+3βΘA S s},(61)where for notational simplicity,we define ΘA ≡(2πGr P m Σg 0)/∆A ,ΞA ≡(−a 2g +Ω2g r 2)/∆A ,∆A ≡(1+2β)Ω2g r 2+(2β−1/2)C 2A .(62)Here,we use the subscript A to indicate the ‘aligned case’.In particular,ΘA and ΞA are two dimensionless constant parameters.A substitution of equation (61)into equation (57)and a further combination with equation (60)lead to[K A (ΘA +ΞA )+L A −2πG Σg 0r (2βP m )]Sg+[K A ΘA −2πG Σg 0r (2βP m )]S s=0,2πG Σs 0r P m Sg+[M A +2πG Σs 0r P m ]Ss=0,(63)where for notational simplicity we defineK A ≡(m 2+6β)Ω2g r 2−[m 2−(1−4β)2/2]C 2A ,L A ≡(1/2−2β)C 2A −2Ω2g r 2+2βa 2g ,M A ≡(m 2+2β−2)Ω2s r 2/(m 2+4β−4β2)−a 2s .(64)We now obtain the stationary MHD dispersion relation bycalculating the coefficient determinant of equation (63).Af-ter a proper rearrangement,we obtain ∆AK A,(65)where the left-hand side consists of two factors.One can show that the left bracket is exactly the dispersion relation for a single coplanar magnetized scale-free disc discussed by Shen et al.(2005)and the second parentheses denotes the dispersion relation for a single hydrodynamic scale-free disc.The right-hand side denotes the effect of gravitational coupling between perturbations in the two scale-free discs.By setting β=0for an isothermal composite disc system,equation (65)then reduces to dispersion relation (62)of Lou &Zou (2004)where MHD perturbations in isothermal fluid-magnetofluid discs are investigated.These calculations are straightforward but tedious and we turn to the following subsections for further analyses.3.2The Aligned m =0CaseAs already discussed earlier,the m =0case is special andshould be treated in the procedure of setting m =0and then taking the limit of ω→0.By letting ω→0in equa-tions (49)−(53),we find U =0and R =0similar to the earlier work [e.g.,Shu et al.(2000);Lou &Shen (2003);Shen &Lou (2004b);Lou &Zou (2004);Shen et al.(2005)].By further requiring the scale-free condition,other physical quantities should be in the forms of Z ∝r −γand J ∝r 1−βwhich are exactly the same as those of the background equi-librium.Since these perturbations are axisymmetric,it turns out that such perturbations simply represent a sort of rescal-ing of the axisymmetric background equilibrium.Neverthe-less,by carefully taking limits in calculations,we can also find a stationary ‘dispersion relation’.We leave this anal-ysis to Appendix A for a further discussion and focus our attention on m 1cases in the next subsection.。

AVL-AthenaVortexLattice-MITAVL 3.30 User Primer last update 18 Aug 10 Mark Drela, MIT Aero & AstroHarold Youngren, Aerocraft, Inc.History=======AVL (Athena Vortex Lattice) 1.0 was originally written by Harold Youngren circa 1988 for the MIT Athena TODOR aero software collection. The code was based on classic work by Lamar (NASA codes), E. Lan and L. Miranda (VORLAX) and a host of other investigators. Numerous modifications have since been added by Mark Drela and Harold Youngren, to the point where only stubborn traces of the original Athena code remain.General Description===================AVL 3.xx now has a large number of features intended for rapidaircraft configuration analysis. The major features are as follows:Aerodynamic componentsLifting surfacesSlender bodiesConfiguration descriptionKeyword-driven geometry input fileDefined sections with linear interpolationSection propertiescamberline is NACA xxxx, or from airfoil filecontrol deflectionsparabolic profile drag polar, Re-scalingScaling, translation, rotation of entire surface or body Duplication of entire surface or bodySingularitiesHorseshoe vortices (surfaces)Source+doublet lines (bodies)Finite-core optionDiscretizationUniformSineCosineBlendControl deflectionsVia normal-vector tiltingLeading edge flapsTrailing edge flapsHinge lines independent of discretizationGeneral freestream descriptionalpha,beta flow anglesp,q,r aircraft rotation componentsSubsonic Prandtl-Glauert compressibility treatmentSurfaces can be defined to "see" only perturbation velocities(not freestream) to allow simulation ofground effectwind tunnel wall interferenceinfluence of other nearby aircraftAerodynamic outputsDirect forces and momentsTrefftz-planeDerivatives of forces and moments, w.r.t freestream, rotation, controls In body or stability axes Trim calculationOperating variablesalpha,betap,q,rcontrol deflectionsConstraintsdirect constraints on variablesindirect constraints via specified CL, momentsMultiple trim run cases can be definedSaving of trim run case setups for later recallOptional mass definition file (only for trim setup, eigenmode analysis) User-chosen units Itemized component location, mass, inertiasTrim setup of constraintslevel or banked horizontal flightsteady pitch rate (looping) flightEigenmode analysisRigid-body analysis with quasi-steady aero modelDisplay of eigenvalue root progression with a parameterDisplay of eigenmode motion in real timeOutput of dynamic system matricesVortex-Lattice Modeling Principles==================================Like any computational method, AVL has limitations on what it can do. These must be kept in mind in any given application. Configurations--------------A vortex-lattice model like AVL is best suited for aerodynamic configurations which consist mainly of thin lifting surfaces at small angles of attackand sideslip. These surfaces and their trailing wakes are representedas single-layer vortex sheets, discretized into horseshoe vortex filaments, whose trailing legs are assumed to be parallel to the x-axis. AVL provides the capability to also model slender bodies such as fuselages and nacellesvia source+doublet filaments. The resulting force and moment predictionsare consistent with slender-body theory, but the experience with this modelis relatively limited, and hence modeling of bodies should be done with caution. If a fuselage is expected to have little influence on the aerodynamic loads, it's simplest to just leave it out of the AVL model. However, the two wings should be connected by a fictitious wing portionwhich spans the omitted fuselage.Unsteady flow-------------AVL assumes quasi-steady flow, meaning that unsteady vorticity sheddingis neglected. More precisely, it assumes the limit of small reduced frequency, which means that any oscillatory motion (e.g. in pitch) must be slow enoughso that the period of oscillation is much longer than the time it takesthe flow to traverse an airfoil chord. This is true for virtually any expected flight maneuver. Also, the roll, pitch, and yaw rates usedin the computations must be slow enough so that the resulting relativeflow angles are small. This can be judged by the dimensionlessrotation rate parameters, which should fall within the followingpractical limits.-0.10 < pb/2V < 0.10-0.03 < qc/2V < 0.03-0.25 < rb/2V < 0.25These limits represent extremely violent aircraft motion, and are unlikely to exceeded in any typical flight situation, except possibly duringlow-airspeed aerobatic maneuvers. In any case, if any of theseparameters falls outside of these limits, the results should beinterpreted with caution.Compressibility---------------Compressibility is treated in AVL using the classical Prandtl-Glauert (PG) transformation, which converts the PG equation to the Laplace equation, which can then be solved by the basic incompressible method. Thisis equivalent to the compressible continuity equation, with the assumptions of irrotationality and linearization about the freestream. The forcesare computed by applying the Kutta-Joukowsky relation to each vortex,this remaining valid for compressible flow.The linearization assumes small perturbations (thin surfaces) and is not completely valid when velocity perturbations from the free-stream become large. The relative importance of compressible effects can be judged bythe PG factor 1/B = 1/sqrt(1 - M^2), where "M" is the freestream Mach number. A few values are given in the table, which shows the expected range of validity.M 1/B--- -----0.0 1.000 |0.1 1.005 |0.2 1.021 |0.3 1.048 |- PG expected valid0.4 1.091 |0.5 1.155 |0.6 1.250 |0.7 1.400 PG suspect (transonic flow likely)0.8 1.667 PG unreliable (transonic flow certain)0.9 2.294 PG hopelessFor swept-wing configurations, the validity of the PG modelis best judged using the wing-perpendicular Mach numberMperp = M cos(sweep)Since Mperp < M, swept-wing cases can be modeled up to higherM values than unswept cases. For example, a 45 degree swept wing operating at freestream M = 0.8 hasMperp = 0.8 * cos(45) = 0.566which is still within the expected range of PG validityin the above table. So reasonable results can be expectedfrom AVL for this case.When doing velocity parameter sweeps at the lowest Mach numbers,say below M = 0.2, it is best to simply hold M = 0. This willgreatly speed up the calculations, since changing the Mach numberrequires recomputation and re-factorization of the VL influence matrix,which consumes most of the computational effort. If the Mach numberis held fixed, this computation needs to be done only once.Input Files===========AVL works with three input files, all in plain text format. Ideallythese all have a common arbitrary prefix "xxx", and the following extensions:xxx.avl required main input file defining the configuration geometry xxx.mass optional file giving masses and inertias, and dimensional units xxx.run optional file defining the parameter for some number of run casesThe user provides files xxx.avl and xxx.mass, which are typically created using any text editor. Sample files are provided for use as templates.The xxx.run file is written by AVL itself with a user command.It can be manually edited, although this is not really necessarysince it is more convenient to edit the contents in AVL and thenwrite out the file again.Geometry Input File -- xxx.avl==============================This file describes the vortex lattice geometry and aerodynamic section properties. Sample input files are in the runs/ subdirectory.Coordinate system-----------------The geometry is described in the following Cartesian system:X downstreamY out the right wingZ upThe freestream must be at a reasonably small angle to the X axis (alpha and beta must be small), since the trailing vorticity is oriented parallel to the X axis. The length unit used inthis file is referred to as "Lunit". This is arbitrary,but must be the same throughout this file.File format-----------Header data- - - - - -The input file begins with the following information in the first 5 non-blank, non-comment lines:Abc... | case title# | comment line begins with "#" or "!"0.0 | Mach1 0 0.0 | iYsym iZsym Zsym4.0 0.4 0.1 | Sref Cref Bref0.1 0.0 0.0 | Xref Yref Zref0.020 | CDp (optional)Mach = default freestream Mach number for Prandtl-Glauert correctioniYsym = 1 case is symmetric about Y=0 , (X-Z plane is a solid wall)= -1 case is antisymmetric about Y=0, (X-Z plane is at const. Cp)= 0 no Y-symmetry is assumediZsym = 1 case is symmetric about Z=Zsym , (X-Y plane is a solid wall)= -1 case is antisymmetric about Z=Zsym, (X-Y plane is at const. Cp)= 0 no Z-symmetry is assumed (Zsym ignored)Sref = reference area used to define all coefficients (CL, CD, Cm, etc)Cref = reference chord used to define pitching moment (Cm)Bref = reference span used to define roll,yaw moments (Cl,Cn)X,Y,Zref = default location about which moments and rotation rates are defined (if doing trim calculations, XYZref must be the CG location,which can be imposed with the MSET command described later)CDp = default profile drag coefficient added to geometry, applied at XYZref(assumed zero if this line is absent, for previous-version compatibility)The default Mach, XYZref, and CDp values are superseded by the valuesin the .run file (described later), if it is present. They can alsobe changed at runtime.Only the half (non-image) geometry must be input if symmetry is specified.Ground effect is simulated with iZsym = 1, and Zsym = location of ground.Forces are not calculated on the image/anti-image surfaces.Sref and Bref are assumed to correspond to the total geometry.In practice there is little reason to run Y-symmetric image cases,unless one is desperate for CPU savings.Surface and Body data- - - - - - - - - - -The remainder of the file consists of a set of keywords and associated data. Each keyword expects a certain number of lines of data to immediately follow it, the exception being inline-coordinate keyword AIRFOIL which is followedby an arbitrary number of coordinate data lines. The keywords must also be nested properly in the hierarchy shown below. Only the first four characters of each keyword are actually significant, the rest are just a mnemonic.SURFACECOMPONENT (or INDEX)YDUPLICATESCALETRANSLATEANGLENOWAKENOALBENOLOADSECTIONSECTIONNACASECTIONAIRFOIL CLAFCDCLSECTIONAFILE CONTROL CONTROLBODYYDUPLICATE SCALETRANSLATE BFILESURFACEYDUPLICATESECTIONSECTIONSURFACE..etc.The COMPONENT (or INDEX), YDUPLICATE, SCALE, TRANSLATE, and ANGLE keywords can all be used together. If more than one of these appears fora surface, the last one will be used and the previous ones ignored.At least two SECTION keywords must be used for each surface.The NACA, AIRFOIL, AFILE, keywords are alternatives.If more than one of these appears after a SECTION keyword,the last one will be used and the previous ones ignored. i.e.SECTIONNACAAFILEis equivalent toSECTIONAFILEMultiple CONTROL keywords can appear after a SECTION keyword and dataSurface-definition keywords and data formats- - - - - - - - - - - - - - - - - - - - - - -*****SURFACE | (keyword)Main Wing | surface name string12 1.0 20 -1.5 | Nchord Cspace [ Nspan Sspace ]The SURFACE keyword declares that a surface is being defined untilthe next SURFACE or BODY keyword, or the end of file is reached.A surface does not really have any significance to the underlyingAVL vortex lattice solver, which only recognizes the overallcollection of all the individual horseshoe vortices. SURFACEis provided only as a configuration-defining device, and alsoas a means of defining individual surface forces. This isnecessary for structural load calculations, for example.Nchord = number of chordwise horseshoe vortices placed on the surfaceCspace = chordwise vortex spacing parameter (described later)Nspan = number of spanwise horseshoe vortices placed on the surface [optional] Sspace = spanwise vortex spacing parameter (described later) [optional]If Nspan and Sspace are omitted (i.e. only Nchord and Cspace are present on line), then the Nspan and Sspace parameters will be expected for each section interval, as described later.*****COMPONENT | (keyword) or INDEX3 | LcompThis optional keywords COMPONENT (or INDEX for backward compatibility)allows multiple input SURFACEs to be grouped together into a compositevirtual surface, by assigning each of the constituent surfaces the sameLcomp value. Application examples are:- A wing component made up of a wing SURFACE and a winglet SURFACE- A T-tail component made up of horizontal and vertical tail SURFACEs.A common Lcomp value instructs AVL to _not_ use a finite-core modelfor the influence of a horseshoe vortex and a control point which lieson the same component, as this would seriously corrupt the calculation.If each COMPONENT is specified via only a single SURFACE block,then the COMPONENT (or INDEX) declaration is unnecessary.*****YDUPLICATE | (keyword)The YDUPLICATE keyword is a convenient shorthand device for creating another surface which is a geometric mirror image of the onebeing defined. The duplicated surface is _not_ assumed to bean aerodynamic image or anti-image, but is truly independent.A typical application would be for cases which have geometricsymmetry, but not aerodynamic symmetry, such as a wing in yaw.Defining the right wing together with YDUPLICATE will convenientlycreate the entire wing.The YDUPLICATE keyword can _only_ be used if iYsym = 0 is specified. Otherwise, the duplicated real surface will be identical to theimplied aerodynamic image surface, and velocities will be computeddirectly on the line-vortex segments of the images. This willalmost certainly produce an arithmetic fault.The duplicated surface gets the same Lcomp value as the parent surface,so they are considered to be the same COMPONENT. There is no significanteffect on the results if they are in reality two physically-separate surfaces.Ydupl = Y position of X-Z plane about which the current surface isreflected to make the duplicate geometric-image surface.*****SCALE | (keyword)1.0 1.0 0.8 | Xscale Yscale ZscaleThe SCALE allows convenient rescaling for the entire surface.The scaling is applied before the TRANSLATE operation described below.Xscale,Yscale,Zscale = scaling factors applied to all x,y,z coordinates(chords are also scaled by Xscale)*****TRANSLATE | (keyword)10.0 0.0 0.5 | dX dY dZThe TRANSLATE keyword allows convenient relocation of the entiresurface without the need to change the Xle,Yle,Zle locationsfor all the defining sections. A body can be translated withoutthe need to modify the body shape coordinates.dX,dY,dZ = offset added on to all X,Y,Z values in this surface.*****ANGLE | (keyword)The ANGLE keyword allows convenient changing of the incidence angleof the entire surface without the need to change the Ainc valuesfor all the defining sections. The rotation is performed aboutthe spanwise axis projected onto the y-z plane.dAinc = offset added on to the Ainc values for all the defining sections in this surface *****NOWAKE | (keyword)The NOWAKE keyword specifies that this surface is to NOT shed a wake,so that its strips will not have their Kutta conditions imposed.Such a surface will have a near-zero net lift, but it will stillgenerate a nonzero moment.。

Modeling of morphology evolution in the injection moldingprocess of thermoplastic polymersR.Pantani,I.Coccorullo,V.Speranza,G.Titomanlio* Department of Chemical and Food Engineering,University of Salerno,via Ponte don Melillo,I-84084Fisciano(Salerno),Italy Received13May2005;received in revised form30August2005;accepted12September2005AbstractA thorough analysis of the effect of operative conditions of injection molding process on the morphology distribution inside the obtained moldings is performed,with particular reference to semi-crystalline polymers.The paper is divided into two parts:in the first part,the state of the art on the subject is outlined and discussed;in the second part,an example of the characterization required for a satisfactorily understanding and description of the phenomena is presented,starting from material characterization,passing through the monitoring of the process cycle and arriving to a deep analysis of morphology distribution inside the moldings.In particular,fully characterized injection molding tests are presented using an isotactic polypropylene,previously carefully characterized as far as most of properties of interest.The effects of both injectionflow rate and mold temperature are analyzed.The resulting moldings morphology(in terms of distribution of crystallinity degree,molecular orientation and crystals structure and dimensions)are analyzed by adopting different experimental techniques(optical,electronic and atomic force microscopy,IR and WAXS analysis).Final morphological characteristics of the samples are compared with the predictions of a simulation code developed at University of Salerno for the simulation of the injection molding process.q2005Elsevier Ltd.All rights reserved.Keywords:Injection molding;Crystallization kinetics;Morphology;Modeling;Isotactic polypropyleneContents1.Introduction (1186)1.1.Morphology distribution in injection molded iPP parts:state of the art (1189)1.1.1.Modeling of the injection molding process (1190)1.1.2.Modeling of the crystallization kinetics (1190)1.1.3.Modeling of the morphology evolution (1191)1.1.4.Modeling of the effect of crystallinity on rheology (1192)1.1.5.Modeling of the molecular orientation (1193)1.1.6.Modeling of theflow-induced crystallization (1195)ments on the state of the art (1197)2.Material and characterization (1198)2.1.PVT description (1198)*Corresponding author.Tel.:C39089964152;fax:C39089964057.E-mail address:gtitomanlio@unisa.it(G.Titomanlio).2.2.Quiescent crystallization kinetics (1198)2.3.Viscosity (1199)2.4.Viscoelastic behavior (1200)3.Injection molding tests and analysis of the moldings (1200)3.1.Injection molding tests and sample preparation (1200)3.2.Microscopy (1202)3.2.1.Optical microscopy (1202)3.2.2.SEM and AFM analysis (1202)3.3.Distribution of crystallinity (1202)3.3.1.IR analysis (1202)3.3.2.X-ray analysis (1203)3.4.Distribution of molecular orientation (1203)4.Analysis of experimental results (1203)4.1.Injection molding tests (1203)4.2.Morphology distribution along thickness direction (1204)4.2.1.Optical microscopy (1204)4.2.2.SEM and AFM analysis (1204)4.3.Morphology distribution alongflow direction (1208)4.4.Distribution of crystallinity (1210)4.4.1.Distribution of crystallinity along thickness direction (1210)4.4.2.Crystallinity distribution alongflow direction (1212)4.5.Distribution of molecular orientation (1212)4.5.1.Orientation along thickness direction (1212)4.5.2.Orientation alongflow direction (1213)4.5.3.Direction of orientation (1214)5.Simulation (1214)5.1.Pressure curves (1215)5.2.Morphology distribution (1215)5.3.Molecular orientation (1216)5.3.1.Molecular orientation distribution along thickness direction (1216)5.3.2.Molecular orientation distribution alongflow direction (1216)5.3.3.Direction of orientation (1217)5.4.Crystallinity distribution (1217)6.Conclusions (1217)References (1219)1.IntroductionInjection molding is one of the most widely employed methods for manufacturing polymeric products.Three main steps are recognized in the molding:filling,packing/holding and cooling.During thefilling stage,a hot polymer melt rapidlyfills a cold mold reproducing a cavity of the desired product shape. During the packing/holding stage,the pressure is raised and extra material is forced into the mold to compensate for the effects that both temperature decrease and crystallinity development determine on density during solidification.The cooling stage starts at the solidification of a thin section at cavity entrance (gate),starting from that instant no more material can enter or exit from the mold impression and holding pressure can be released.When the solid layer on the mold surface reaches a thickness sufficient to assure required rigidity,the product is ejected from the mold.Due to the thermomechanical history experienced by the polymer during processing,macromolecules in injection-molded objects present a local order.This order is referred to as‘morphology’which literally means‘the study of the form’where form stands for the shape and arrangement of parts of the object.When referred to polymers,the word morphology is adopted to indicate:–crystallinity,which is the relative volume occupied by each of the crystalline phases,including mesophases;–dimensions,shape,distribution and orientation of the crystallites;–orientation of amorphous phase.R.Pantani et al./Prog.Polym.Sci.30(2005)1185–1222 1186R.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221187Apart from the scientific interest in understandingthe mechanisms leading to different order levels inside a polymer,the great technological importance of morphology relies on the fact that polymer character-istics (above all mechanical,but also optical,electrical,transport and chemical)are to a great extent affected by morphology.For instance,crystallinity has a pro-nounced effect on the mechanical properties of the bulk material since crystals are generally stiffer than amorphous material,and also orientation induces anisotropy and other changes in mechanical properties.In this work,a thorough analysis of the effect of injection molding operative conditions on morphology distribution in moldings with particular reference to crystalline materials is performed.The aim of the paper is twofold:first,to outline the state of the art on the subject;second,to present an example of the characterization required for asatisfactorilyR.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221188understanding and description of the phenomena, starting from material description,passing through the monitoring of the process cycle and arriving to a deep analysis of morphology distribution inside the mold-ings.To these purposes,fully characterized injection molding tests were performed using an isotactic polypropylene,previously carefully characterized as far as most of properties of interest,in particular quiescent nucleation density,spherulitic growth rate and rheological properties(viscosity and relaxation time)were determined.The resulting moldings mor-phology(in terms of distribution of crystallinity degree, molecular orientation and crystals structure and dimensions)was analyzed by adopting different experimental techniques(optical,electronic and atomic force microscopy,IR and WAXS analysis).Final morphological characteristics of the samples were compared with the predictions of a simulation code developed at University of Salerno for the simulation of the injection molding process.The effects of both injectionflow rate and mold temperature were analyzed.1.1.Morphology distribution in injection molded iPP parts:state of the artFrom many experimental observations,it is shown that a highly oriented lamellar crystallite microstructure, usually referred to as‘skin layer’forms close to the surface of injection molded articles of semi-crystalline polymers.Far from the wall,the melt is allowed to crystallize three dimensionally to form spherulitic structures.Relative dimensions and morphology of both skin and core layers are dependent on local thermo-mechanical history,which is characterized on the surface by high stress levels,decreasing to very small values toward the core region.As a result,the skin and the core reveal distinct characteristics across the thickness and also along theflow path[1].Structural and morphological characterization of the injection molded polypropylene has attracted the interest of researchers in the past three decades.In the early seventies,Kantz et al.[2]studied the morphology of injection molded iPP tensile bars by using optical microscopy and X-ray diffraction.The microscopic results revealed the presence of three distinct crystalline zones on the cross-section:a highly oriented non-spherulitic skin;a shear zone with molecular chains oriented essentially parallel to the injection direction;a spherulitic core with essentially no preferred orientation.The X-ray diffraction studies indicated that the skin layer contains biaxially oriented crystallites due to the biaxial extensionalflow at theflow front.A similar multilayered morphology was also reported by Menges et al.[3].Later on,Fujiyama et al.[4] investigated the skin–core morphology of injection molded iPP samples using X-ray Small and Wide Angle Scattering techniques,and suggested that the shear region contains shish–kebab structures.The same shish–kebab structure was observed by Wenig and Herzog in the shear region of their molded samples[5].A similar investigation was conducted by Titomanlio and co-workers[6],who analyzed the morphology distribution in injection moldings of iPP. They observed a skin–core morphology distribution with an isotropic spherulitic core,a skin layer characterized by afine crystalline structure and an intermediate layer appearing as a dark band in crossed polarized light,this layer being characterized by high crystallinity.Kalay and Bevis[7]pointed out that,although iPP crystallizes essentially in the a-form,a small amount of b-form can be found in the skin layer and in the shear region.The amount of b-form was found to increase by effect of high shear rates[8].A wide analysis on the effect of processing conditions on the morphology of injection molded iPP was conducted by Viana et al.[9]and,more recently, by Mendoza et al.[10].In particular,Mendoza et al. report that the highest level of crystallinity orientation is found inside the shear zone and that a high level of orientation was also found in the skin layer,with an orientation angle tilted toward the core.It is rather difficult to theoretically establish the relationship between the observed microstructure and processing conditions.Indeed,a model of the injection molding process able to predict morphology distribution in thefinal samples is not yet available,even if it would be of enormous strategic importance.This is mainly because a complete understanding of crystallization kinetics in processing conditions(high cooling rates and pressures,strong and complexflowfields)has not yet been reached.In this section,the most relevant aspects for process modeling and morphology development are identified. In particular,a successful path leading to a reliable description of morphology evolution during polymer processing should necessarily pass through:–a good description of morphology evolution under quiescent conditions(accounting all competing crystallization processes),including the range of cooling rates characteristic of processing operations (from1to10008C/s);R.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221189–a description capturing the main features of melt morphology(orientation and stretch)evolution under processing conditions;–a good coupling of the two(quiescent crystallization and orientation)in order to capture the effect of crystallinity on viscosity and the effect offlow on crystallization kinetics.The points listed above outline the strategy to be followed in order to achieve the basic understanding for a satisfactory description of morphology evolution during all polymer processing operations.In the following,the state of art for each of those points will be analyzed in a dedicated section.1.1.1.Modeling of the injection molding processThefirst step in the prediction of the morphology distribution within injection moldings is obviously the thermo-mechanical simulation of the process.Much of the efforts in the past were focused on the prediction of pressure and temperature evolution during the process and on the prediction of the melt front advancement [11–15].The simulation of injection molding involves the simultaneous solution of the mass,energy and momentum balance equations.Thefluid is non-New-tonian(and viscoelastic)with all parameters dependent upon temperature,pressure,crystallinity,which are all function of pressibility cannot be neglected as theflow during the packing/holding step is determined by density changes due to temperature, pressure and crystallinity evolution.Indeed,apart from some attempts to introduce a full 3D approach[16–19],the analysis is currently still often restricted to the Hele–Shaw(or thinfilm) approximation,which is warranted by the fact that most injection molded parts have the characteristic of being thin.Furthermore,it is recognized that the viscoelastic behavior of the polymer only marginally influences theflow kinematics[20–22]thus the melt is normally considered as a non-Newtonian viscousfluid for the description of pressure and velocity gradients evolution.Some examples of adopting a viscoelastic constitutive equation in the momentum balance equations are found in the literature[23],but the improvements in accuracy do not justify a considerable extension of computational effort.It has to be mentioned that the analysis of some features of kinematics and temperature gradients affecting the description of morphology need a more accurate description with respect to the analysis of pressure distributions.Some aspects of the process which were often neglected and may have a critical importance are the description of the heat transfer at polymer–mold interface[24–26]and of the effect of mold deformation[24,27,28].Another aspect of particular interest to the develop-ment of morphology is the fountainflow[29–32], which is often neglected being restricted to a rather small region at theflow front and close to the mold walls.1.1.2.Modeling of the crystallization kineticsIt is obvious that the description of crystallization kinetics is necessary if thefinal morphology of the molded object wants to be described.Also,the development of a crystalline degree during the process influences the evolution of all material properties like density and,above all,viscosity(see below).Further-more,crystallization kinetics enters explicitly in the generation term of the energy balance,through the latent heat of crystallization[26,33].It is therefore clear that the crystallinity degree is not only a result of simulation but also(and above all)a phenomenon to be kept into account in each step of process modeling.In spite of its dramatic influence on the process,the efforts to simulate the injection molding of semi-crystalline polymers are crude in most of the commercial software for processing simulation and rather scarce in the fleur and Kamal[34],Papatanasiu[35], Titomanlio et al.[15],Han and Wang[36],Ito et al.[37],Manzione[38],Guo and Isayev[26],and Hieber [25]adopted the following equation(Kolmogoroff–Avrami–Evans,KAE)to predict the development of crystallinityd xd tZð1K xÞd d cd t(1)where x is the relative degree of crystallization;d c is the undisturbed volume fraction of the crystals(if no impingement would occur).A significant improvement in the prediction of crystallinity development was introduced by Titoman-lio and co-workers[39]who kept into account the possibility of the formation of different crystalline phases.This was done by assuming a parallel of several non-interacting kinetic processes competing for the available amorphous volume.The evolution of each phase can thus be described byd x id tZð1K xÞd d c id t(2)where the subscript i stands for a particular phase,x i is the relative degree of crystallization,x ZPix i and d c iR.Pantani et al./Prog.Polym.Sci.30(2005)1185–1222 1190is the expectancy of volume fraction of each phase if no impingement would occur.Eq.(2)assumes that,for each phase,the probability of the fraction increase of a single crystalline phase is simply the product of the rate of growth of the corresponding undisturbed volume fraction and of the amount of available amorphous fraction.By summing up the phase evolution equations of all phases(Eq.(2))over the index i,and solving the resulting differential equation,one simply obtainsxðtÞZ1K exp½K d cðtÞ (3)where d c Z Pid c i and Eq.(1)is recovered.It was shown by Coccorullo et al.[40]with reference to an iPP,that the description of the kinetic competition between phases is crucial to a reliable prediction of solidified structures:indeed,it is not possible to describe iPP crystallization kinetics in the range of cooling rates of interest for processing(i.e.up to several hundreds of8C/s)if the mesomorphic phase is neglected:in the cooling rate range10–1008C/s, spherulite crystals in the a-phase are overcome by the formation of the mesophase.Furthermore,it has been found that in some conditions(mainly at pressures higher than100MPa,and low cooling rates),the g-phase can also form[41].In spite of this,the presence of different crystalline phases is usually neglected in the literature,essentially because the range of cooling rates investigated for characterization falls in the DSC range (well lower than typical cooling rates of interest for the process)and only one crystalline phase is formed for iPP at low cooling rates.It has to be noticed that for iPP,which presents a T g well lower than ambient temperature,high values of crystallinity degree are always found in solids which passed through ambient temperature,and the cooling rate can only determine which crystalline phase forms, roughly a-phase at low cooling rates(below about 508C/s)and mesomorphic phase at higher cooling rates.The most widespread approach to the description of kinetic constant is the isokinetic approach introduced by Nakamura et al.According to this model,d c in Eq.(1)is calculated asd cðtÞZ ln2ðt0KðTðsÞÞd s2 435n(4)where K is the kinetic constant and n is the so-called Avrami index.When introduced as in Eq.(4),the reciprocal of the kinetic constant is a characteristic time for crystallization,namely the crystallization half-time, t05.If a polymer is cooled through the crystallization temperature,crystallization takes place at the tempera-ture at which crystallization half-time is of the order of characteristic cooling time t q defined ast q Z D T=q(5) where q is the cooling rate and D T is a temperature interval over which the crystallization kinetic constant changes of at least one order of magnitude.The temperature dependence of the kinetic constant is modeled using some analytical function which,in the simplest approach,is described by a Gaussian shaped curve:KðTÞZ K0exp K4ln2ðT K T maxÞ2D2(6)The following Hoffman–Lauritzen expression[42] is also commonly adopted:K½TðtÞ Z K0exp KUÃR$ðTðtÞK T NÞ!exp KKÃ$ðTðtÞC T mÞ2TðtÞ2$ðT m K TðtÞÞð7ÞBoth equations describe a bell shaped curve with a maximum which for Eq.(6)is located at T Z T max and for Eq.(7)lies at a temperature between T m(the melting temperature)and T N(which is classically assumed to be 308C below the glass transition temperature).Accord-ing to Eq.(7),the kinetic constant is exactly zero at T Z T m and at T Z T N,whereas Eq.(6)describes a reduction of several orders of magnitude when the temperature departs from T max of a value higher than2D.It is worth mentioning that only three parameters are needed for Eq.(6),whereas Eq.(7)needs the definition offive parameters.Some authors[43,44]couple the above equations with the so-called‘induction time’,which can be defined as the time the crystallization process starts, when the temperature is below the equilibrium melting temperature.It is normally described as[45]Dt indDtZðT0m K TÞat m(8)where t m,T0m and a are material constants.It should be mentioned that it has been found[46,47]that there is no need to explicitly incorporate an induction time when the modeling is based upon the KAE equation(Eq.(1)).1.1.3.Modeling of the morphology evolutionDespite of the fact that the approaches based on Eq.(4)do represent a significant step toward the descriptionR.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221191of morphology,it has often been pointed out in the literature that the isokinetic approach on which Nakamura’s equation (Eq.(4))is based does not describe details of structure formation [48].For instance,the well-known experience that,with many polymers,the number of spherulites in the final solid sample increases strongly with increasing cooling rate,is indeed not taken into account by this approach.Furthermore,Eq.(4)describes an increase of crystal-linity (at constant temperature)depending only on the current value of crystallinity degree itself,whereas it is expected that the crystallization rate should depend also on the number of crystalline entities present in the material.These limits are overcome by considering the crystallization phenomenon as the consequence of nucleation and growth.Kolmogoroff’s model [49],which describes crystallinity evolution accounting of the number of nuclei per unit volume and spherulitic growth rate can then be applied.In this case,d c in Eq.(1)is described asd ðt ÞZ C m ðt 0d N ðs Þd s$ðt sG ðu Þd u 2435nd s (9)where C m is a shape factor (C 3Z 4/3p ,for spherical growth),G (T (t ))is the linear growth rate,and N (T (t ))is the nucleation density.The following Hoffman–Lauritzen expression is normally adopted for the growth rateG ½T ðt Þ Z G 0exp KUR $ðT ðt ÞK T N Þ!exp K K g $ðT ðt ÞC T m Þ2T ðt Þ2$ðT m K T ðt ÞÞð10ÞEqs.(7)and (10)have the same form,however the values of the constants are different.The nucleation mechanism can be either homo-geneous or heterogeneous.In the case of heterogeneous nucleation,two equations are reported in the literature,both describing the nucleation density as a function of temperature [37,50]:N ðT ðt ÞÞZ N 0exp ½j $ðT m K T ðt ÞÞ (11)N ðT ðt ÞÞZ N 0exp K 3$T mT ðt ÞðT m K T ðt ÞÞ(12)In the case of homogeneous nucleation,the nucleation rate rather than the nucleation density is function of temperature,and a Hoffman–Lauritzen expression isadoptedd N ðT ðt ÞÞd t Z N 0exp K C 1ðT ðt ÞK T N Þ!exp KC 2$ðT ðt ÞC T m ÞT ðt Þ$ðT m K T ðt ÞÞð13ÞConcentration of nucleating particles is usually quite significant in commercial polymers,and thus hetero-geneous nucleation becomes the dominant mechanism.When Kolmogoroff’s approach is followed,the number N a of active nuclei at the end of the crystal-lization process can be calculated as [48]N a ;final Zðt final 0d N ½T ðs Þd sð1K x ðs ÞÞd s (14)and the average dimension of crystalline structures can be attained by geometrical considerations.Pantani et al.[51]and Zuidema et al.[22]exploited this method to describe the distribution of crystallinity and the final average radius of the spherulites in injection moldings of polypropylene;in particular,they adopted the following equationR Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3x a ;final 4p N a ;final 3s (15)A different approach is also present in the literature,somehow halfway between Nakamura’s and Kolmo-goroff’s models:the growth rate (G )and the kinetic constant (K )are described independently,and the number of active nuclei (and consequently the average dimensions of crystalline entities)can be obtained by coupling Eqs.(4)and (9)asN a ðT ÞZ 3ln 24p K ðT ÞG ðT Þ 3(16)where heterogeneous nucleation and spherical growth is assumed (Avrami’s index Z 3).Guo et al.[43]adopted this approach to describe the dimensions of spherulites in injection moldings of polypropylene.1.1.4.Modeling of the effect of crystallinity on rheology As mentioned above,crystallization has a dramatic influence on material viscosity.This phenomenon must obviously be taken into account and,indeed,the solidification of a semi-crystalline material is essen-tially caused by crystallization rather than by tempera-ture in normal processing conditions.Despite of the importance of the subject,the relevant literature on the effect of crystallinity on viscosity isR.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221192rather scarce.This might be due to the difficulties in measuring simultaneously rheological properties and crystallinity evolution during the same tests.Apart from some attempts to obtain simultaneous measure-ments of crystallinity and viscosity by special setups [52,53],more often viscosity and crystallinity are measured during separate tests having the same thermal history,thus greatly simplifying the experimental approach.Nevertheless,very few works can be retrieved in the literature in which(shear or complex) viscosity can be somehow linked to a crystallinity development.This is the case of Winter and co-workers [54],Vleeshouwers and Meijer[55](crystallinity evolution can be drawn from Swartjes[56]),Boutahar et al.[57],Titomanlio et al.[15],Han and Wang[36], Floudas et al.[58],Wassner and Maier[59],Pantani et al.[60],Pogodina et al.[61],Acierno and Grizzuti[62].All the authors essentially agree that melt viscosity experiences an abrupt increase when crystallinity degree reaches a certain‘critical’value,x c[15]. However,little agreement is found in the literature on the value of this critical crystallinity degree:assuming that x c is reached when the viscosity increases of one order of magnitude with respect to the molten state,it is found in the literature that,for iPP,x c ranges from a value of a few percent[15,62,60,58]up to values of20–30%[58,61]or even higher than40%[59,54,57].Some studies are also reported on the secondary effects of relevant variables such as temperature or shear rate(or frequency)on the dependence of crystallinity on viscosity.As for the effect of temperature,Titomanlio[15]found for an iPP that the increase of viscosity for the same crystallinity degree was higher at lower temperatures,whereas Winter[63] reports the opposite trend for a thermoplastic elasto-meric polypropylene.As for the effect of shear rate,a general agreement is found in the literature that the increase of viscosity for the same crystallinity degree is lower at higher deformation rates[62,61,57].Essentially,the equations adopted to describe the effect of crystallinity on viscosity of polymers can be grouped into two main categories:–equations based on suspensions theories(for a review,see[64]or[65]);–empirical equations.Some of the equations adopted in the literature with regard to polymer processing are summarized in Table1.Apart from Eq.(17)adopted by Katayama and Yoon [66],all equations predict a sharp increase of viscosity on increasing crystallinity,sometimes reaching infinite (Eqs.(18)and(21)).All authors consider that the relevant variable is the volume occupied by crystalline entities(i.e.x),even if the dimensions of the crystals should reasonably have an effect.1.1.5.Modeling of the molecular orientationOne of the most challenging problems to present day polymer science regards the reliable prediction of molecular orientation during transformation processes. Indeed,although pressure and velocity distribution during injection molding can be satisfactorily described by viscous models,details of the viscoelastic nature of the polymer need to be accounted for in the descriptionTable1List of the most used equations to describe the effect of crystallinity on viscosityEquation Author Derivation Parameters h=h0Z1C a0x(17)Katayama[66]Suspensions a Z99h=h0Z1=ðx K x cÞa0(18)Ziabicki[67]Empirical x c Z0.1h=h0Z1C a1expðK a2=x a3Þ(19)Titomanlio[15],also adopted byGuo[68]and Hieber[25]Empiricalh=h0Z expða1x a2Þ(20)Shimizu[69],also adopted byZuidema[22]and Hieber[25]Empiricalh=h0Z1Cðx=a1Þa2=ð1Kðx=a1Þa2Þ(21)Tanner[70]Empirical,basedon suspensionsa1Z0.44for compact crystallitesa1Z0.68for spherical crystallitesh=h0Z expða1x C a2x2Þ(22)Han[36]Empiricalh=h0Z1C a1x C a2x2(23)Tanner[71]Empirical a1Z0.54,a2Z4,x!0.4h=h0Zð1K x=a0ÞK2(24)Metzner[65],also adopted byTanner[70]Suspensions a Z0.68for smooth spheresR.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221193。

塑室堕窒堕盔奎堂堡主堂焦笙苎——摘要本课题对置鲨笪型条件下的星尘星煎闷堕握选进行了较为详细的分析和研究：首先对尾流的形成和消散机理以及尾流对飞行安全的影响做出了详细的分析，然后就目前实行的几种不同的最小尾流间隔标准进行了比较和分析，提出了相关的几条准则和一种改进的机型尾流分类标准；在这之后，采用理论分析和统计数据分析相结合的方法，建立了尾流危险遭遇基本模型和尾涡消散模型：并在此基础上对民用航空不同机型的尾流分类的合理性进行了计算和评估，为最小尾流间隔标准的确定和改进提供了初步的理论依据。

关键词：空中交通管制，飞行间隔，尾涡消散，飞行安全空中交通中的尾流安全间隔研究ＡｂｓｔｒａｃｔＷ酞ｅｖｏｒｔｅｘｓｅｐａｒａｔｉｏｎｓｔａｎｄａｒｄｓａｒｅｕｓｅｄｔｏｐｒｅｖｅｎｔｈａｚａｒｄｏｕｓｗａｋｅｖｏｒｔｅｘｅｎｃｏｕｎｔｅｒｓｗｈｉｃｈｍａｉｎｌｙｆｏｕｎｄｉｎｔｈｅｆｉｎａｌａｐｐｒｏａｃｈｃｏｕｒｓｅｗｉｔｈｉｎｔｈｅａｉｒｐｏｒｔｔｅｒｍｉｎａｔｉｏｎｚｏｎｅ．Ｆｉｒｓｔｌｙ，ｔｈｉｓｐａｐｅｒａｎａｌｙｓｉｓｔｈｅｍｅｃｈａｎｉｓｍｏｆｔｈｅｖｏｒｔｅｘｄｅｃａｙａｎｄｒｅｓｅａｒｃｈｔｈｅｉｎｆｌｕｅｎｃｅｏｎｔｈｅｓａｆｅｔｙｏｆｔｈｅｆｌｉｇｈｔ；ｔｈｅｎ，ｓｅｖｅｒａｌｄｉｆｆｅｒｅｎｔｒａｄａｒｗａｋｅｖｏｒｔｅｘｓｅｐａｒａｔｉｏｎｓｔａｎｄａｒｄｓａｒｅｃｏｍｐａｒｅｄａｎｄｓｏｍｅｂａｓｉｃｇｕｉｄｅｌｉｎｅｓａｒｅｄｅｖｅｌｏｐｅｄｔｏｅｓｔａｂｌｉｓｈｔｈｅｆｕｔｕｒｅｖｏｒｔｅｘｓｔａｎｄａｒｄｓ．Ｉｎａｄｄｉｔｉｏｎ，ｏｎｅｋｉｎｄｏｆａｄｖａｎｃｅｖｏｒｔｅｘｃｌａｓｓｉｆｉｃａｔｉｏｎｆｏｒｔｈｅｃｉｖｉｌａｉｒｐｌａｎｅｉｓｐｒｏｐｏｓｅｄ．Ｓｅｃｏｎｄｌｙ，ａｎｅｎｃｏｕｎｔｅｒｈａｚａｒｄｍｏｄｅｌｗｈｉｃｈｅｓｔａｂｌｉｓｈｅｄｆｏｒｍｅｆｏｌｌｏｗｉｎｇａｉｒｃｒａｆｔａｎｄａｓｉｍｐｌｅｖｏｒｔｅｘｄｅｃａｙｍｏｄｅｌｗｈｉｃｈｍｏｓｔｌｙｃｏｎｃｅｍｗｉｔｈｔｈｅｐｒｏｃｅｅｄｉｎｇｏｎｅａｒｅｄｅｖｅｌｏｐｅｄ；ａｎｄａｓａｆｅｓｅｐａｒａｔｉｏｎｍｏｄｅｌｃａｎｂｅｄｅｒｉｖｅｄｆｒｏｍｔｈｅｂｏｔｈ．Ｆｉｎａｌｌｙ，ｔｈｒｏｕｇｈｔｈｅｓｉｍｕｌａｔｉｏｎｃａｌｃｕｌａｔｉｏｎ，ｔｈｅｃｏｒｒｅｃｔｎｅｓｓｏｆｔｈｅｖｏｒｔｅｘｃｌａｓｓｉｆｉｃａｔｉｏｎｏｆｄｉｆｆｅｒｅｎｔｔｙｐｅｓｏｆａｉｍｒａｆｔｓａｒｅｔｅｓｔｅｄａｎｄｐｒｏｖｅｄ．Ｋｅｙｗｏｒｄｓ：ａｉｒｔｒａｆｆｉｃｃｏｎｔｒｏｌ，ｆｌｉｇｈｔｓｅｐａｒａｔｉｏｎ，ｗａｋｅｖｏｒｔｅｘｓｅｐａｒａｔｉｏｎ，ｖｏｒｔｅｘｄｅｃａｙ，ｆｌｉｇｈｔｓ嘶ＩＩ南京航空航天大学硕士学位论文第一章绪论１．１课题研究背景自从二十世纪初期美国的莱特兄弟发明飞机以来，航空运输以其他运输方式无可比拟的快速和安全迅速成为二十世纪中后期最为人们青睐的交通方式。

a r X i v :c o n d -m a t /9811106v 2 [c o n d -m a t .s u p r -c o n ] 18 J u n 1999Scaling of the Hall Resistivity in the Solid and Liquid Vortex Phases in TwinnedSingle Crystal YBa 2Cu 3O 7−δG.D’Anna,1V.Berseth,1L.Forr´o ,1A.Erb,2and E.Walker 21Institut de G´e nie Atomique,Ecole Polytechnique F´e d´e rale de Lausanne,CH-1015Lausanne,Switzerland2D´e partement de Physique de la Mati`e re Condens´e e,Universit´e de Gen`e ve,CH-1211Gen`e ve,Switzerland(February 1,2008)Longitudinal and Hall voltages are measured in a clean twinned Y Ba 2Cu 3O 7−δsingle crystal in the liquid and solid vortex phases.For magnetic fields tilted away from the c-axis more than about 2◦,a scaling law |ρxy |=Aρβxx with β≈1.4is observed,which is unaffected by the vortex-lattice melting transition.The vortex-solid Hall conductivity is non-linear and diverges to negative values at low temperature.When the magnetic field is aligned to the c-axis,the twin-boundary correlated disorder modifies the scaling law,and β≈2.The scaling law is unaffected by the Bose-glass transition.We discuss the scaling behaviour in terms of the dimension-dependent theory for percolation in metallic conductors.PACS numbers:74.25.Fy,74.60.Ge,74.72.BkA current flowing in a conductor exposed to a mag-netic field gives rise to a Hall voltage.The Hall effect has been a powerful probe of the mechanisms of charge transport in metals and semiconductors.Similarly,a Hall voltage is observed in superconductors in high magnetic fields and carrying large electric currents.The Hall ef-fect in this system is an intriguing phenomenon which has triggered a very large experimental and theoretical literature.Remarkable experimental facts include the ”Hall anomaly”,i.e.,the Hall effect sign reversal in the superconducting vortex state with respect to the normal state,as observed in various high-and low-temperature type-II superconductors [1],and the ”scaling law”,i.e.,the power-law dependence of the Hall resistivity with re-spect to the longitudinal resistivity [2].Many theoretical explanations have been proposed,most of them addressing the Hall anomaly which is be-lived to be a fundamental problem of vortex dynamics.These theories are developed either in terms of micro-scopic electronic processes [3][4][5][6][7],or includ-ing pinning [8],vortex-vortex interactions [9][10],time-dependent Ginzburg-Landau theories [11],phenomeno-logical models [12][13][14],or other ideas [15].The most frequently adopted approach is microscopic:it as-cribes the Hall effect in the vortex state to hydrodynamic and vortex-core forces which determine the single vor-tex trajectory (e.g.in ref.[4]).In the scenario the Hall sign reversal results from microscopic details of the Fermi surface.The situation remains,however,debated and a consensus is not achieved on fundamental points like the transverse force on a vortex moving in a superfluid [16][17],or on experimental problems like the doping depen-dence [18].Recently the vortex-lattice melting transition has been shown to influence the Hall conductivity [19],which has rised questions to which extent the microscopic approach of the Hall behavior in the vortex state is legitimate [20].We report here new measurements intended to study the scaling law as the system crosses the vortex-lattice melt-ing transition,or the Bose-glass transition when twin-boundary correlated disorder is relevant.Testing the scaling law in different vortex phases provides insights into the origin of the Hall effect and the mechanisms of magnetic flux transport in type-II superconductors.The experiments are performed in a very clean twinned Y Ba 2Cu 3O 7−δ(YBCO)single crystal in which the char-acteristic features associated to the vortex phase transi-tions are observed.The micro-twinned crystal has di-mensions 0.9×0.4mm 2in the a-b plane,and thickness 24µm in the c-direction.The major twin family is at 45◦from the long edge of the sample.Some untwinned do-mains and some twins at 90◦from the dominant family are also present.The sample displays a sharp resistive transition at about T c =93.5K .The longitudinal resis-tivity ρxx and Hall resistivity ρxy are measured simul-taneously by injecting an ac current (30Hz ),sometimes on the top of a dc current,along the longest dimension of the crystal,and by measuring the in-phase voltages parallel and perpendicular to the current.The experi-mental method is presented in detail in a previous work on the same sample [19].The Hall conductivity is ob-tained by σxy =ρxy / ρ2xx +ρ2xy ,and the Hall angle θH by tan θH =ρxy /ρxx .We begin by discussing the angular dependence and the effect of twin-boundaries in our sample.The Bose-glass theory [21]predicts that for magnetic fields well aligned to the twin-boundaries the vortex-solid phase is a smecticlike phase and the transition to the vortex-liquid is a Bose-glass transition.When the field is tilted away from the twin-boundaries the vortex-solid phase is a Bragg-glass [22]and the transition to the vortex-liquid is a vortex-lattice melting transition.We found experimentally evidence for this angular be-havior.Figure 1shows the longitudinal resistivity ρxx measured at 6T for zero dc current and a small ac cur-rent of j ac =1A/cm 2,and for different angles αbetween the applied magnetic field and the c-axis,as a function of the temperature.One can clearly see the effect oftwin-boundaries below T T B.The twin-boundary pin-ning reduces the longitudinal resistivity,as expected for correlated disorder[21].The inset of Fig.1shows the on-set temperature T onset ofρxx measured with a criterion of0.1µΩcm,as a function of the angle.For decreasing, large angles T onset(=T m)decreases according to a usual anisotropy law[23].For aboutα<2◦,the onset temper-ature T onset(=T BG)increases and reaches a maximum atα=0◦.This kind of behavior has been associated[24] to the change in the nature of the transition according to the Bose-glass theory and the crossover angle is about 2◦for twinned YBCO crystals similar to the one we use. Therefore,the onset in resistivity in Fig.1can be as-sociated to the vortex-lattice melting transition[25][26] forα>2◦,that we denote by T m,and to the Bose-glass transition[27]forα<2◦,that we denote by T BG. When the vortex-lattice response is probed by super-imposing the ac current on top of a large dc current,a longitudinal resistivity different from zero is observed be-low T m or T BG,as the vortex-solid is moving under the ef-fect of the large Lorentz force.We then also detect a Hall voltage,and obtain the Hall resistivity and the Hall con-ductivity in the vortex-solid phase[19].The inset of Fig. 2shows the Hall conductivityσxy as a function of the temperature at2T and atα=7◦andα=0◦,measured with large dc and ac current densities(j dc=150A/cm2, j ac=50A/cm2)so that the Hall signal is detected deep inside the vortex-solid.In the inset the small difference in temperature between the vortex-lattice melting tran-sition at T m forα=7◦and the Bose-glass transition at T BG forα=0◦is not visible.By reducing the temper-ature from the normal state the Hall conductivityσxy becomes negative below T c.In the vortex-liquid phase the Hall conductivities atα=0◦andα=7◦coincide down to about T T B.Below roughly T T B and forα=0◦we observe an approximately constant Hall conductivity until the large scattering of the data begins.For the an-gle tilted away from the c-axis,α=7◦,the Hall conduc-tivity decreases smoothly until the vortex-lattice melting transition.Below T m the Hall conductivity deviates from its behavior in the vortex-liquid phase and goes rapidly towards large negative values(see also the current de-pendence in Fig.3below).The Hall angle,not shown in Fig.2,tends to small values.We investigate now the scaling behavior between the Hall resistivity and the longitudinal resistivity,that is the existence of a scaling lawρxy∝ρβxx.The main panel of Fig.2shows the log-log plot of|ρxy|vsρxx forα=7◦andα=0◦at2T and large dc and ac current densities. The position of the vortex-lattice melting and Bose-glass temperatures are indicated.Thefit to a power-law de-pendence of a form|ρxy|=Aρβxx,gives forα=0◦the values A≈0.005andβ≈2.0,and forα=7◦it gives A≈0.02andβ≈1.4,as shown by the two straight dotted lines.A separatefit to the solid and liquid part gives the same result within the experimental error(we also obtainβ≈1.4in the whole range3◦to7◦).There is no change of the|ρxy|vsρxx dependence at the vortex-lattice melting transition or at the Bose-glass transition, suggesting that such a scaling law is effectively insensitive to the specific vortex phase.The Hall effect current dependence is shown in Fig.3 forα=3◦.The inset of Fig.3shows the Hall conduc-tivityσxy as a function of the magneticfield at89K and different dc currents.Atα=3◦the vortex phase tran-sition is the vortex-lattice melting at B m.The curves have larger noise over signal ratio than above.Neverthe-less the current dependence is clearly observable in the divergingσxy.Below the meltingfield B m the Hall con-ductivityσxy decreases faster,the smaller the dc current. Above B m in the vortex-liquid the Hall conductivity is linear.The current dependence of the scaling law is in-vestigated in the main panel of Fig.3,which shows a log-log plot of|ρxy|vsρxx,constructed from measure-ments as a function of the magneticfields at a constant temperature of89K and different current densities.The fit to a power-law dependence of a form|ρxy|=Aρβxx, with A andβfree parameters as above,gives the average values A≈0.012andβ≈1.4.There is no change of the scaling law with the current density,neither in liquid nor in the solid vortex phases.The data presented here prove that the general trend of the Hall conductivity is indeed captured by a very ro-bust scaling law|ρxy|=Aρβxx.The scaling law implies tanθH∝ρβ−1xxandσxy∝ρβ−2xx.Consistently,with an exponent less than two atα>2◦,as the longitudinal re-sistivity tends to zero,the Hall conductivity diverges,and the Hall angle is small.The strong non-linear dependence of the longitudinal resistivity in the vortex-solid phase is reflected in the Hall conductivity,which below the melt-ing transition diverges faster,the smaller the current den-sity.Forα=0◦and consistently with an exponentβ≈2 the Hall conductivity seems to be a constant below about T T B,the temperature of twin-boundary pinning onset.A Hall resistivity which vanishes as a power of the lon-gitudinal resistivity,ρxy∝ρβxx,has been observed by various authors[2][28][29],and predicted in different theoretical contexts.Dorsey et al.[13]developed a scal-ing theory near the vortex-glass transition with a power β<2for the three-dimensional regime.In the model the exponentβis universal,but the sign of the Hall effect is material specific and possibly related to microscopic pair-ing processes.Vinokur et al.[12]proposed that the scal-ing law withβ=2is a general feature of any vortex state with disorder-dominated dynamics,without the need to invoke the vortex-glass scaling.In this model the Hall conductivity is independent of disorder and is directly linked to the microscopic processes determining the sin-gle vortex equation of motion.Wang et al.[8]have pro-posed that the pinning affects the single vortex trajectory via the backflow current inside the normal vortex-core. This modifies the exponent in such a way thatβ=1.5for strong pinning andβ=2for weak pinning,and the Hall sign reversal is a pinning effect.Ao[9]derived the scaling law in the context of a vortex many-body linear theory,where the Hall voltage results from the motion of vortex-lattice defects(vacancies),withβ=2for pinning induced vacancies andβ=1for thermally(fluctuation) induced vacancies.A complete explanation of the puzzling vortex-Hall be-havior is not yet achieved.A consistent theory should explain,in addition to the Hall effect sign reversal,the robustness of the scalingρxy∝ρβxx reported here.We have found thatσxy becomes current dependent in the vortex-solid phase.This contradicts the idea that the Hall conductivity is independent of disorder[12].Pin-ning effects have to be considered.However,even if the temperature andfield dependence ofσxy change at vortex phase transitions(either vortex-lattice melting or Bose-glass transitions),the scaling law is found to remain inde-pendent of the specific vortex phase.This suggests that a comprehensive theory for the Hall effect far enough from the sign change does not require phase-dependent parameters.Provided it reproduces a general scaling law ρxy∝ρβxx and leads to the correct sign ofσxy,the Hall behavior is then completely determined by the longitu-dinal resistivity,which englobes many-body effects,e.g., collective pinning in the vortex-solid phase.A significant result of this paper is that the exponent βentering the scaling law is disorder-type dependent.In particularβ≈2.0for correlated planar disorder and β≈1.4for uncorrelated point disorder.This suggests an alternative explanation for the scaling behaviour,as pro-posed by Geshkenbein[30].If one views the vortex freez-ing as an inhomogeneous,non simultaneous process,with regions where vortices are pinned(thus with vanishing resistivity),and regions where they can still move(thus inducing a non zero electric resistivity),the vortex freez-ing behaviour in superconductors has strong analogies with the percolation transition in inhomogeneous conduc-tors.In the case of a mixed metallic/insulating system, the conductivity is governed by percolation processes and the longitudinal conducticity is expressed asσxx∝δp t, whereδp=p−p c is the difference between the conduct-ing metallic phase density p and the critical percolation threshold density p c.The critical exponent is t≈1.3 in two dimensions,and t≈1.6in three dimensions[31]. Similarly,the Hall number R H diverges as R H∝δp−g, where g=ν(d−2),[31]where d is the dimension.In two dimensions g=0and as consequence the Hall con-ductivityσxy≈HR Hσ2xx is exactly proportional toσ2xx (at constant magneticfield H).In three dimensions,g=ν≈0.9,[31]such thatσxy∝σ2−g/txx∝σ1.44xx .This immediatly leads to a percolation model for the vortex scaling behaviour,provided the vortex conductiv-ity is interpreted as the electric resistivity of the metal-lic/insulating system,since a high vortex mobility means large electric dissipation.With the identification that the conductivityσin the metallic/insulating system is the re-sistivityρin the vortex system,one obtains the scaling lawρxy∝ρ2−g/txx andβ=2−g/t for the vortex system. The percolation model is very appealing since it provides two universal exponents,i.e.,β=2andβ=1.44,which correspond to the most frequently reported experimental estimate.These exponents are determined by the dimen-sionality of the vortex system,that is,determined by the intrinsic anisotropy of the material,or by the vortex lo-calization along correlated defects,or by the geometry of the samples.For the magneticfield accurately aligned to the twin-boundaries,which localizes the vortices along the c-axis,the system is two dimensional andβ≈2.The same exponent is observed in two-dimensionalfilms[29]. When the magneticfield is”slightly”tilted away from the twin-boundaries(2to3degrees are enough),the vortices recover the third degree of freedom andβ≈1.4.This is also likely to happen when splayed defects are introduced by irradiation,bringing back the exponent form2to1.5.[28]The vortex-percolation model is indeed very interest-ing since it predicts a scaling law independent of the vor-tex phase,as observed here,and explains the robustness of the scaling since it is related only to the dimensionality of the vortex system.We are grateful to V.Geshkenbein and G.Blatter for many discussions.This work is supported by the Swiss National Science Foundation.[13]A.T.Dorsey and M.P.A.Fisher,Phys.Rev.Lett.68,694(1992).[14]R.Ikeda,J.Phys.Soc.Jpn.65,3998(1996).[15]For a more complete bibliography see,for example.in:E.H.Brandt,Rep.Prog.Phys.58,1465(1995).[16]D.J.Thouless et al.,Phys.Rev.Lett.76,3758(1996);M.R.Geller et al.,Phys.Rev.B57,R8119(1998);C.Wexler and D.J.Thouless,Phys.Rev.B58,R8897 (1998).[17]H.E.Hall and J.R.Hook,Phys.Rev.Lett.80,4356(1998);C.Wexler et al.,Phys.Rev.Lett.80,4357(1998);P.Ao,Phys.Rev.Lett.80,5025(1998);N.B.Kopnin and G.E.Volovik,Phys.Rev.Lett.80,5026(1998).[18]T.Nagaoka et al.,Phys.Rev.Lett.80,3594(1998).[19]G.D’Anna et al.,Phys.Rev.Lett.81,2530(1998).[20]P.Ao,Phys.Rev.Lett.82,2413(1999);R.Ikeda,Phys.Rev.Lett.82,3378(1999).[21]D.R.Nelson and V.M.Vinokur,Phys.Rev.Lett.68,2398(1992);.D.R.Nelson and V.M.Vinokur,Phys.Rev.B48,13060(1993).[22]See,for example,in:P.Le Doussal and T.Giamarchi,Phys.Rev.B57,11356(1998).[23]G.Blatter et al.,Rev.Mod.Phys.66,1125(1994).[24]E.Morr´e et al.,Phys.Lett.A233,130(1997);S.A.Grigera et al.,Phys.Rev.Lett.81,2348(1998). [25]H.Safar et al.,Phys.Rev.Lett.69,824(1992);W.K.Kwok et al.,Phys,Rev.Lett.69,3370(1992).[26]Some experimental discrepancies concerning the order ofthe melting transition remain to be elucidated,in partic-ular whether the transition remainsfirst-order down to lowfields,or is restricted to an intermediate section of the melting line(see,for example,in:M.Roulin et al., Phys.Rev.Lett.80,1722(1998)).[27]Because some parts of our sample seem untwinned,forα=0◦the transition is possibly not completely deter-mined by the correlated disorder.[28]R.C.Budhani et al.,Phys.Rev.Lett.71,621(1993);A.V.Samoilov et al.,Phys.Rev.Lett.74,2351(1995);W.N.Kang et al.,Phys.Rev.Lett.76,2993(1996). [29]P.J.M.W¨o ltgens et al.,Phys.Rev.Lett.71,3858(1993).[30]V.Geshkenbein,private communication.[31]B.I.Shklovskii,Sov.Phys.JETP45,152(1977);J.P.Straley,J.Phys.C13,4335(1980).Figure captionsFIG.1.The longitudinal resisitivityρxx in a twinned Y Ba2Cu3O7−δsingle crystal at6T as a function of the temperature,measured at low ac current density, j ac=1A/cm2,for different angles between thefield and the c-axis(-0.2◦,0◦,0.2◦,0.5◦,1◦,2◦,3◦,5◦,7◦,10◦, 12◦,20◦).Inset:the onset temperature T onst as a func-tion of the angle,at2T and6T.For aboutα>2◦the onset temperature follows the usual anisotropic law of the vortex-lattice melting temperature.For small angles the onset temperature increases as expected from the Bose-glass theory(see text for details).FIG.2.Log-log plot of|ρxy|versusρxx at2T and α=7◦andα=0◦.The linearfit according to a scalinglaw|ρxy|=Aρβxx givesβ≈1.4forα=7◦,andβ≈2.0 forα=0◦,as indicated by the two straigth dotted lines.The position of the vortex-lattice melting temperature, T m,and of the Bose-glass temperature,T BG,are indi-cated in the curves,as well as the temperature of twin-boundary pinning onset,T T B.Notice that the scaling law is unaffected by crossing the transitions.Inset:the Hall conductivity,σxy=ρxy/ ρ2xx+ρ2xy ,as a function of the temperature atα=7◦andα=0◦.The dotted vertical lines denote the transitions at T m,T BG and T T B.Thecurrent densities are j dc=150A/cm2,j ac=50A/cm2. FIG.3.Log-log plot of|ρxy|versusρxx for different current densities at89K andα=3◦.For each curve the current densities are indicated by(j dc,j ac),both in unit of A/cm2.The position of the vortex-lattice melting field is indicated.The linearfit according to a scaling law |ρxy|=Aρβxx gives the average values A≈0.012andβ≈1.4(see dotted line)and there is no current dependence of the parameters.Inset:Thefield dependence of the Hall conductivityσxy at89K,for various ac and dc current densities.The dotted vertical line denotes the vortex-lattice melting transition at B m.84ρx x (µΩ c m )878685848382T (K)-3-1-1。