Theoretical study of the permeation of water through TiO2 nanotubes

2024年6月大学英语六级考试真题和答案（第1套）Part I Writing (30 minutes)Directions: For this part, you are allowed 30 minutes to write an essay that begins with the sentence “There is a growing awareness of the importance of digital literacy and skills in today’s world.” You can make comments, cite examples or use your personal experiences to develop your essay. You should write at least 150 words but no more than 200 words. You should copy the sentence given in quotes at the beginning of your essay.Part Ⅱ Listening Comprehension (30 minutes)Section ADirections:In this section, you will hear two long conversations. At the end of each conversation, you will hear four 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). Then mark the corresponding letter on Answer Sheet 1 with a single line through the centre.Questions 1 to 4 are based on the conversation you have just heard.1. A) Reply to the man’s last proposal within a short time.B) Sign the agreement if one small change is made to it.C) Make a sponsorship deal for her client at the meeting.D) Give the man some good news regarding the contract.2. A) They are becoming impatient.B) They are afraid time is running out.C) They are used to making alterations.D) They are concerned about the details.3. A) To prevent geographical discrimination.B) To tap the food and beverage market.C) To avoid any conflict of interest.D) To reduce unfair competition.4. A) It is a potential market for food and beverage.B) It is very attractive for real estate developers.C) It is a negligible market for his company.D) It is very different from other markets.Questions 5 to 8 are based on the conversation you have just heard.5. A) They are thrilled by a rare astronomic phenomenon.B) They are celebrating a big event on mountain tops.C) They are enthusiastic about big science-related stories.D) They are joined by astronomers all across North America.6. A) It will be the most formidable of its kind in over a century.B) It will come closest to Earth in more than one hundred years.C) It will eclipse many other such events in human history.D) It will be seen most clearly from Denver’s mountain tops.7. A) A blur.B) Stars.C) The edge of our galaxy.D) An ordinary flying object.8. A) Use professional equipment.B) Climb to the nearby heights.C) Fix their eyes due north.D) Make use of phone apps.Section BDirections: In this section, you will hear two passages. At the end of each passage, you will hear three or four questions. Both the passage 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). Then mark the corresponding letter on Answer Sheet 1 with a single line through the centre.Questions 9 to 11 are based on the passage you have just heard.9. A) Whether consumers should be warned against ultra-processed foods.B) Whether there is sufficient scientific consensus on dietary guidelines.C) Whether guidelines can form the basis for nutrition advice to consumers.D) Whether food scientists will agree on the concept of ultra-processed foods.10. A) By the labor cost for the final products.B) By the degree of industrial processing.C) By the extent of chemical alteration.D) By the convention of classification.11. A) Increased consumers’ expenses.B) Greater risk of chronic diseases.C) People’s misunderstanding of nutrition.D) Children’s dislike for unprocessed foods.Questions 12 to 15 are based on the passage you have just heard.12. A) They begin to think of the benefits of constraints.B) They try to seek solutions from creative people.C) They try hard to maximize their mental energy.D) They begin to see the world in a different way.13. A) It is characteristic of all creative people.B) It is essential to pushing society forward.C) It is a creative person’s response to limitation.D) It is an impetus to socio-economic development.14. A) Scarcity or abundance of resources has little impact on people’s creativity.B) Innovative people are not constrained in connecting unrelated concepts.C) People have no incentive to use available resources in new ways.D) Creative people tend to consume more available resources.15. A) It is key to a company’s survival.B) It shapes and focuses problems.C) It is essential to meeting challenges.D) It thrives best when constrained.Section CDirections: In this section, you will hear three recordings of lectures or talks followed by three or four questions. The recordings will be played only once. After you hear a question, you must choose the best answer from the four choices marked A), B), C) and D). Then mark the corresponding letter on Answer Sheet 1 with a single line through the centre.Questions 16 to 18 are based on the recording you have just heard.16. A) Because they are learned.B) Because they come naturally.C) Because they have to be properly personalized.D) Because there can be more effective strategies.17. A) The extent of difference and of similarity between the two sides.B) The knowledge of the specific expectation the other side holds.C) The importance of one’s goals and of the relationship.D) The approaches one adopts to conflict management.18. A) The fox.B) The owl.C) The shark.D) The turtle.Questions 19 to 21 are based on the recording you have just heard.19. A) Help save species from extinction and boost human health.B) Understand how plants and animals perished over the past.C) Help gather information publicly available to researchers.D) Find out the cause of extinction of Britain’s 66,000 species.20. A) It was once dominated by dinosaurs.B) It has entered the sixth mass extinction.C) Its prospects depend on future human behaviour.D) Its climate change is aggravated by humans.21. A) It dwarfs all other efforts to conserve, protect and restorebiodiversity on earth.B) It is costly to get started and requires the joint efforts of thousands of scientists.C) It can help to bring back the large numbers of plants and animals that have gone extinct.D) It is the most exciting, most relevant, most timely and most internationally inspirational.Questions 22 to 25 are based on the recording you have just heard.22. A) Cultural identity.B) Social evolution.C) The Copernican revolution.D) Human individuality.23. A) It is a delusion to be disposed of.B) It is prevalent even among academics.C) It is a myth spread by John Donne’s poem.D) It is rooted in the mindset of the 17th century.24. A) He believes in Copernican philosophical doctrines about the universe.B) He has gained ample scientific evidence at the University of Reading.C) He has found that our inner self and material self are interconnected.D) He contends most of our body cells can only live a few days or weeks.25. A) By coming to see how disruptive such problems have got to be.B) By realising that we all can do our own bit in such endeavours.C) By becoming aware that we are part of a bigger world.D) By making joint efforts resolutely and persistently.Part III Reading Comprehension (40 minutes)Section ADirections: In this section, there is a passage with ten blanks. You are required to select one word for each blank from a list of choices given in a word bank following the passage. Read the passage through carefully before making your choices. Each choice in the bank is identified by a letter. Please mark the corresponding letter for each item on Answer Sheet 2with a single line through the centre. You may not use any of the words in the bank more than once.It’s quite remarkable how different genres of music can spark unique feelings, emotions, and memories. Studies have shown that music can reduce stress and anxiety before surgeries and we are all attracted toward our own unique life soundtrack.If you’re looking to____26____stress, you might want to give classical music a try.The sounds of classical music produce a calming effectletting____27____pleasure-inducing dopamine (多巴胺) in the brain thathelps control attention, learning and emotional responses.It can also turn down the body’s stress response, resulting in an overall happier mood. It turns out a pleasant mood can lead to____28____in a person’s thinking.Although there are many great____29____of classical music like Bach, Beethoven and Handel, none of these artists’ music seems to have the same health effects as Mozart’s does. According to researchers, listening to Mozart can increase brain wave activity and improve____30____function. Another study found that the distinctive features of Mozart’s music trigger parts of the brain that are responsible for high-level mental functions. Even maternity____31____use Mozart to help newborn babies adapt to life outside of the mother’s belly.It has been found that listening to classical music____32____reduces a person’s blood pressure. Researchers believe that the calming sounds of classical music may help your heart____33____from stress. Classical music can also be a great tool to help people who have trouble sleeping. One study found that students who had trouble sleeping slept better while they were listening to classical music.Whether classical music is something that you listen to on a regular basis or not, it wouldn’t____34____to take time out of your day to listen to music that you find____35____. You will be surprised at how good it makes you feel and the potentially positive change in your health.A) alleviateB) clarityC) cognitiveD) composersE) hurtF) inhibitingG) interrogationH) intrinsicallyI) looseJ) majesticK) mandatoryL) recoverM) significantlyN) soothingO) wardsSection BDirections: In this section, you are going to read a passage with ten statements attached to it. Each statement contains information given in one of the paragraphs. Identify the paragraph from which the information is derived. You may choose a paragraph more than once. Each paragraph is marked with a letter. Answer the questions by marking the correspondingletter on Answer Sheet 2.The Curious Case of the Tree That Owns ItselfA) In the city of Athens, Georgia, there exists a rather curious local landmark—a large white oak that is almost universally stated to own itself. Because of this, it is considered one of the most famous trees in the world. So how did this tree come to own itself and the land around it?B) Sometime in the 19th century a Georgian called Colonel William Jackson reportedly took a liking to the said tree and endeavored to protect it from any danger. As to why he loved it so, the earliest documented account of this story is an anonymously written front page article in the Athens Weekly Banner published on August 12, 1890. It states, “Col. Jackson had watched the tree grow from his childhood, and grew to love it almost as he would a human. Its luxuriant leaves and sturdy limbs had often protected him from the heavy rains,and out of its highest branches he had many a time gotten the eggs of the feathered singers.He watched its growth, and when reaching a ripe old age he saw the tree standing in its magnificent proportions, he was pained to think that after his death it would fall into the hands of those who might destroy it.”C) Towards this end, Jackson transferred by means of a deed ownership of the tree and a little land around it to the tree itself. The deed read, “W. H. Jackson for and in consideration of the great affection which he bears the said tree, and his great desire to see it protected has conveyed unto the said oak tree entire possession of itself and of all land within eight feet of it on all sides.”D) In time, the tree came to be something of a tourist attraction, known as The Tree That Owns Itself. However, in the early 20th century, the tree started showing signs of its slow death,with little that could be done about it. Father time comes for us all eventually, even our often long lived, tall and leafy fellow custodians (看管者) of Earth. Finally, on October 9,1942, the over 30 meter tall and 200-400 year old tree fell, rumor has it, as a result of a severe windstorm and/or via having previously died and its roots rotted.E) About four years later, members of the Junior Ladies Garden Club (who’d tended to the tree before its unfortunate death) tracked down a small tree grown from a nut taken from the original tree. And so it was that on October 9, 1946, under the direction of Professor Roy Bowden of the College of Agriculture at the University of Georgia, this little tree was transplanted to the location of its ancestor. A couple of months later, an official ceremony was held featuring none other than the Mayor of Athens, Robert L McWhorter, to commemorate the occasion.F) This new tree became known as The Son of the Tree That Owns Itself and it was assumed that, as the original tree’s heir, it naturally inherited the land it stood on. Of course, there are many dozens of othertrees known to exist descending from the original, as people taking a nut from it to grow elsewhere was a certainty. Thatsaid, to date, none of the original tree’s other children have petitioned the courts for their share of the land, so it seems all good. In any event, The Son of the Tree That Owns Itself still stands today, though often referred to simply as The Tree That Owns Itself.G) This all brings us around to whether Jackson ever actually gave legal ownership of the tree to itself in the first place and whether such a deed is legally binding.H) Well, to begin with, it turns out Jackson only spent about three years of his life in Athens,starting at the age of 43 from 1829 to 1832, sort of dismissing the idea that he loved the tree from spending time under it as a child and watching it grow, and then worrying about what would happen to it after he died. Further, an extensive search of land ownership records in Athens does not seem to indicate Jackson ever owned the land the tree sits on.I) He did live on a lot of land directly next to it for those three years, but whether he owned that land or not isn’t clear. Whatever the case, in 1832 a four acre parcel, which included the land the tree was on and the neighboring land Jackson lived on, among others, was sold to University professor Malthus A Ward. In the transaction, Ward was required to pay Jackson a sum of $1,200 (about $31,000 today), either for the property itself or simply in compensation for improvements Jackson had made on the lot. In the end, whether he ever owned the neighboring lot or was simply allowed to use it while he allegedly worked at the University, he definitely never owned the lot the tree grew on, which is the most important bit for the topic at hand.J) After Professor Ward purchased the land, Jackson and his family purchased a 655 acre parcel a few miles away and moved there. Ten years later, in 1844, Jackson seemed to have come into financial difficulties and had his little plantation seized by the Clarke County Sheriff’s office and auctioned off to settle the mortgage. Thus, had he owned some land in Athens itself, including the land the tree sat on, presumably he would have sold it to raise funds or otherwise had it taken as well.K) And whatever the case there, Jackson would have known property taxes needed to be paid on the deeded land for the tree to be truly secure in its future. Yet no account or record indicates any trust or the like was set up to facilitate this.L) On top of all this, there is no hard evidence such a deed ever existed, despite the fact that deed records in Athens go back many decades before Jackson’s death in 1876 and that it was supposed to have existed in 1890 in the archives according to the original anonymous news reporter who claims to have seen it.M) As you might imagine from all of this, few give credit to this sideof the story. So how did all of this come about then?N) It is speculated to have been invented by the imagination of the said anonymous author at the Athens Weekly Banner in the aforementioned 1890 front page article titled “Deeded to Itself”, which by the way contained several elements that are much more easily proved to be false. As to why the author would do this, it’s speculated perhaps it was a 19th century version of a click-bait thought exercise on whether it would be legal for someone to deed such a non-conscious living thing to itself or not.O) Whatever the case, the next known instance of the Tree That Owns Itself being mentioned wasn’t until 1901 in the Centennial Edition of that same paper, the Athens Weekly Banner. This featured another account very clearly just copying the original article published about a decade before, only slightly reworded. The next account was in 1906, again in the Athens Weekly Banner, again very clearly copying the original account, only slightly reworded, the 19th century equivalent of re-posts when the audience has forgotten about the original.36. Jackson was said to have transferred his ownership of the oak tree to itself in order to protect it from being destroyed.37. No proof has been found from an extensive search that Jackson had ever owned the land where the oak tree grew.38. When it was raining heavily, Jackson often took shelter under a big tree that is said to own itself.39. There is no evidence that Jackson had made arrangements to pay property taxes for the land on which the oak tree sat.40. Professor Ward paid Jackson over one thousand dollars when purchasinga piece of land from him.41. It is said the tree that owned itself fell in a heavy windstorm.42. The story of the oak tree is suspected to have been invented as a thought exercise.43. Jackson’s little plantation was auctioned off to settle his debt in the mid-19th century.44. An official ceremony was held to celebrate the transplanting of a small tree to where its ancestor had stood.45. The story of the Tree That Owns Itself appeared in the local paper several times, with slight alterations in wording.Section CDirections:There are 2 passages in this section. Each passage is followed by some questions or unfinished statements. For each of them there are four choices marked A), B), C) and D). You should decide on the best choice and mark the corresponding letter on Answer Sheet 2 with a single line through the centre.Passage OneQuestions 46 to 50 are based on the following passage.It is irrefutable that employees know the difference between right and wrong. So why don’t more employees intervene when they see someone exhibiting at-risk behavior in the workplace?There are a number of factors that influence whether people intervene. First, they need to be able to see a risky situation beginning to unfold. Second, the company’s culture needs to make them feel safe to speak up. And third, they need to have the communication skills to say something effectively.This is not strictly a workplace problem; it’s a growing problem off the job too. Every day people witness things on the street and choose to stand idly by. This is known as the bystander effect—the more people who witness an event, the less likely anyone in that group is to help the victim. The psychology behind this is called diffusion of responsibility. Basically, the larger the crowd, the more people assume that someone else will take care of it—meaning no one effectively intervenes or acts in a moment of need.This crowd mentality is strong enough for people to evade their known responsibilities. But it’s not only frontline workers who don’t make safety interventions in the workplace. There are also instances where supervisors do not intervene either.When a group of employees sees unsafe behavior not being addressed at a leadership level it creates the precedent that this is how these situations should be addressed, thus defining the safety culture for everyone.Despite the fact that workers are encouraged to intervene when they observe unsafe operations, this happens less than half of the time. Fear is the ultimate factor in not intervening. There is a fear of penalty, a fear that they’ll have to do more work if they intervene. Unsuccessful attempts in the past are another strong contributing factor to why people don’t intervene—they tend to prefer to defer that action to someone else for all future situations.On many worksites, competent workers must be appointed. Part of their job is to intervene when workers perform a task without the proper equipment or if the conditions are unsafe. Competent workers are also required to stop work from continuing when there’s a danger.Supervisors also play a critical role. Even if a competent person isn’t required, supervisors need a broad set of skills to not only identify and alleviate workplace hazards but also build a safety climate within their team that supports intervening and open communication among them.Beyond competent workers and supervisors, it’s important to educate everyone within the organization that they are obliged to intervene ifthey witness a possible unsafe act, whether you’re a designated competent person, a supervisor or a frontline worker.46. What is one of the factors contributing to failure of intervention in face of risky behavior in the workplace?A) Slack supervision style.B) Unfavorable workplace culture.C) Unforeseeable risk.D) Blocked communication.47. What does the author mean by “diffusion of responsibility” (Line 4, Para. 3)?A) The more people are around, the more they need to worry about their personal safety.B) The more people who witness an event, the less likely anyone will venture to participate.C) The more people idling around on the street, the more likely they need taking care of.D) The more people are around, the less chance someone will step forward to intervene.48. What happens when unsafe behavior at the workplace is not addressed by the leaders?A) No one will intervene when they see similar behaviors.B) Everyone will see it as the easiest way to deal with crisis.C) Workers have to take extra caution executing their duties.D) Workers are left to take care of the emergency themselves.49. What is the ultimate reason workers won’t act when they see unsafe operations?A) Preference of deferring the action to others.B) Anticipation of leadership intervention.C) Fear of being isolated by coworkers.D) Fear of having to do more work.50. What is critical to ensuring workplace safety?A) Workers be trained to operate their equipment properly.B) Workers exhibiting at-risk behavior be strictly disciplined.C) Supervisors create a safety environment for timely intervention.D) Supervisors conduct effective communication with frontline workers.Passage TwoQuestions 51 to 55 are based on the following passage.The term “environmentalist” can mean different things. It used to refer to people trying to protect wildlife and natural ecosystems. In the 21st century, the term has evolved to capture the need to combat human-made climate change.The distinction between these two strands of environmentalism is thecause of a split within the scientific community about nuclear energy.On one side are purists who believe nuclear power isn’t worth the risk and the exclusive solution to the climate crisis is renewable energy. The opposing side agrees that renewables are crucial, but says society needs an amount of power available to meet consumers’ basic demands when the sun isn’t shining and the wind isn’t blowing. Nuclear energy, being far cleaner than oil, gas and coal, is a natural option, especially where hydroelectric capacity is limited.Leon Clarke, who helped author reports for the UN’s Intergovernmental Panel on Climate Change, isn’t an uncritical supporter of nuclear energy, but says it’s a valuable option to have if we’re serious about reaching carbon neutrality.“Core to all of this is the degree to which you think we can actually meet climate goals with 100% renewables,” he said. “If you don’t believe we can do it, and you care about the climate, you are forced to think about something like nuclear.”The achievability of universal 100% renewability is similarly contentious. Cities such as Burlington, Vermont, have been “100% renewable” for years. But these cities often have small populations, occasionally still rely on fossil fuel energy and have significant renewable resources at their immediate disposal. Meanwhile, countries that manage to run off renewables typically do so thanks to extraordinary hydroelectric capabilities.Germany stands as the best case study for a large, industrialized country pushing into green energy. Chancellor Angela Merkel in 2011 announced Energiewende, an energy transition that would phase out nuclear and coal while phasing in renewables. Wind and solar power generation has increased over 400% since 2010, and renewables provided 46% of the country’s electricity in 2019.But progress has halted in recent years. The instability of renewables doesn’t just mean energy is often not produced at night, but also that solar and wind can overwhelm the grid during the day, forcing utilities to pay customers to use their electricity. Lagging grid infrastructure struggles to transport this overabundance of green energy from Germany’s north to its industrial south, meaning many factories still run on coal and gas. The political limit has also been reached in some places, with citizens meeting the construction of new wind turbines with loud protests.The result is that Germany’s greenhouse gas emissions have fallen by around 11.5% since 2010—slower than the EU average of 13.5%.51. What accounts for the divide within the scientific community about nuclear energy?A) Attention to combating human-made climate change.B) Emphasis on protecting wildlife and natural ecosystems.C) Evolution of the term ‘green energy’ over the last century.D) Adherence to different interpretations of environmentalism.52. What is the solution to energy shortage proposed by purists’opponents?A) Relying on renewables firmly and exclusively.B) Using fossil fuel and green energy alternately.C) Opting for nuclear energy when necessary.D) Limiting people’s non-basic consumption.53. What point does the author want to make with cities like Burlington as an example?A) It is controversial whether the goal of the whole world’s exclusive dependence on renewables is attainable.B) It is contentious whether cities with large populations have renewable resources at their immediate disposal.C) It is arguable whether cities that manage to run off renewables have sustainable hydroelectric capabilities.D) It is debatable whether traditional fossil fuel energy can be done away with entirely throughout the world.54. What do we learn about Germany regarding renewable energy?A) It has increased its wind and solar power generation four times over the last two decades.B) It represents a good example of a major industrialized country promoting green energy.C) It relies on renewable energy to generate more than half of its electricity.D) It has succeeded in reaching the goal of energy transition set by Merkel.55. What may be one of the reasons for Germany’s progress having halted in recent years?A) Its grid infrastructure’s capacity has fallen behind its development of green energy.B) Its overabundance of green energy has forced power plants to suspend operation during daytime.C) Its industrial south is used to running factories on conventional energy supplies.D) Its renewable energy supplies are unstable both at night and during the day.Part IV Translation (30 minutes)Directions: For this part, you are allowed 30 minutes to translate a passage from Chinese into English. You should write your answer on Answer Sheet 2.中国的传统婚礼习俗历史悠久，从周朝开始就逐渐形成了一套完整的婚礼仪式，有些一直沿用至今。

伟大的头脑是天生的还是后天养成的英文文章1. Introduction1.1 Statement of the issue1.2 Explanation of the significance of the issue1.3 Thesis statement2. Nature vs. Nurture Debate: The Origins of Great Minds2.1 Brief history of the nature vs. nurture debate2.2 Definitions of 'nature' and 'nurture' in the context of intelligence and greatness2.3 Overview of key arguments and evidence on both sides3. The Case for Nature: Innate Intelligence and Genetic Factors 3.1 The role of genetics in determining intelligence3.2 Evidence from twin and adoption studies3.3 Examples of prodigies and geniuses with exceptional abilities from a young age3.4 Scientific research on the heritability of intelligence4. The Case for Nurture: Environmental and Experiential Factors 4.1 The influence of education and upbringing on cognitive development4.2 Socioeconomic and cultural factors that contribute to or hinder intellectual growth4.3 Case studies of individuals who have ovee challenging circumstances to achieve greatness4.4 The impact of early childhood experiences on br本人n development5. The Interactionist Perspective: The Complex Interplay of Nature and Nurture5.1 The theory of gene-environment interaction5.2 The importance of both genetic predispositions and environmental influences5.3 Examples of research demonstrating thebined effects of nature and nurture on cognitive abilities5.4 Implications for education, parenting, and social policy6. Conclusion6.1 Restatement of the thesis6.2 Summary of key points6.3 Discussion of the implications for understanding and fostering greatness in individuals6.4 Suggestions for future research on the origins of great mindsGreat Minds: Nature or Nurture?The question of whether great minds are born or made has long been a topic of debate in the fields of psychology, genetics, and education. This essay will explore theplex interplay of genetic and environmental factors in the development of exceptional cognitive abilities, with a focus on the implications for understanding and fostering greatness in individuals.The nature vs. nurture debate has a rich and contentious history, with proponents on both sides presentingpelling arguments for their respective positions. On one hand, advocates of the'nature' side argue that intelligence is largely determined by genetic factors. Studies of twins and adopted siblings have provided convincing evidence for the heritability of intelligence, suggesting that genetic predispositions play a significant role in the development of cognitive abilities. Furthermore, the existence of prodigies and geniuses who exhibit exceptional talents from a young age supports the notion of innate intelligence.Conversely, proponents of the 'nurture' perspective emphasizethe crucial influence of environmental and experiential factors on cognitive development. Numerous studies have demonstrated the impact of education, socioeconomic status, and family upbringing on intellectual growth. Moreover, inspiring stories of individuals who have ovee adversity and achieved greatness through hard work and perseverance serve as powerful testaments to the potential for nurturing extraordinary minds.In reality, the origins of great minds are likely to be the result of aplex interplay of nature and nurture. The theory of gene-environment interaction suggests that genetic predispositions interact with environmental influences to shape cognitive abilities. This interactionist perspective underscores the importance of recognizing the multifaceted nature of human intelligence and the need to consider both genetic and environmental factors in understanding and fostering greatness.In conclusion, the origins of great minds are best understood as the result of the intricate interplay of nature and nurture. Genetics may provide a foundational predisposition for cognitive abilities, but environmental and experiential factors also play a crucial role in shaping and fostering greatness inindividuals. Recognizing the dynamic interplay of genetic and environmental influences has important implications for education, parenting, and social policy, as well as for the ongoing pursuit of knowledge in the field of psychology. Continued research into the origins of great minds will further our understanding of theplex factors that contribute to exceptional cognitive abilities and guide efforts to nurture and support the development of greatness in individuals.。

托福（阅读）历年真题试卷汇编4(题后含答案及解析) 题型有： 3. Reading ComprehensionSections Three：Reading ComprehensionEarly Theories of Continental DriftP1: The idea that the geography of Earth was different in the past than it is today is not new. As far back as 1620, Francis Bacon spotted that the west coast of Africa and the east coast of South America looked as if they would fit together, like pieces of a jigsaw puzzle. Between then and 1912, other people identified further similarities between other continental coastlines. But because much of the early support for mobilism was based on far-flung intercontinental similarities, geologists tended to be skeptical of the fieldwork of others.P2: During the late nineteenth century, Austrian geologist Eduard Suess proposed the name “Gondwanaland”in his book The Face of the Earth (1885) and gave far greater emphasis to the evolutionary nature of the earth and he noted the similarities among the Late Paleozoic plant fossils of India, Australia, South Africa, and South America. Based upon glossopteris fern fossils in such regions, he explained that the three land masses were once connected in a supercontinent which he names Gondwanaland, and that the ocean flooded the spaces currently between those lands. Thus, in his view, the similarities of fossils on these continents could be accounted for by postulating the concept of a land bridge that existed once but subsided later.P3: Later, a number of refinements to Suess’s theory were made. The American geologist Frank Taylor published a pamphlet in 1910 presenting his concept of “horizontal displacement”. He explained the formation of mountain ranges as a result of the lateral movements of continents. With the earth’s capture of the moon, the gravitational forces between them generated a pull towards lower latitudes where they thickened and formed folded mountain belts especially in middle latitudes. Although we now know that Taylor’s explanation of continental drift is erroneous, one of his most significant contributions was his suggestion that the Mid-Atlantic Ridge—an underwater mountain range discovered by the 1872-1876 British HMS Challenger expeditions—might mark the site at which an ancient continent broke apart, forming the present-day Atlantic Ocean.P4: However, it is Alfred Wegener, a German meteorologist, who is generally credited with developing the hypothesis of continental drift. In his monumental book, The Origin of Continents and Oceans (1915), Wegener theorized that a single supercontinent he called “Pangaea”existed sometime between 350 million to 225 million years ago. Wegner portrayed his grand concept of continental movement in a series of maps showing the breakup of Pangaea and the movement of various continents to their present-day locations. What evidence did Wegener use to support his hypothesis of continental drift? First, Wegener noted that there is geographical similarity along both the coasts of the Atlantic Ocean. The opposing coasts of the Atlantic can be fitted together in the same way as two cut off pieces of wood can be refitted. Furthermore, mountain ranges and glacial deposits seem tomatch up in such a way that suggests continents could have once been a single landmass. Finally, many of the same fossils and vegetative remains are found today on widely separated continents, indicating that the continents must have been in proximity at one time. During his days, Wegener was regarded as an advocate rather than as an impartial scientific observer, appearing to ignore vast evidence unfavorable to his ideas and distort other evidence to bring it into harmony with the theory.P5: After Wegener’s death, a South African geologist Alexander Du Toit continued to assemble fossil evidence for Pangaea. He noted that fossils of the now extinct reptile “Mesosaurus”occur in rocks of the same age in both Brazil and South Africa. Because the physiology of freshwater and marine animals is completely different, it is hard to imagine how a freshwater reptile could have swum across the Atlantic Ocean and then found a freshwater environment nearly identical to its former habitat. Moreover, if Mesosaurus could have swum across the ocean, its fossil remains should be widely dispersed. It is more logical to assume that Mesosaurus lived in lakes in what were once adjacent areas of South America and Africa when it was united into a single continent. Notwithstanding all of the empirical evidence in favor of continental drift theory presented here, most geologists at the time refused to entertain the idea.P6 :The debate over continental drift has the same role and stature in the history of the earth sciences as the debate over Darwinian evolution in the history of life sciences and the debates over relativity and quantum theory in the history of physics. In the largest sense, the history of earth science, the history of biology, and the history of physics in the 20th century are all histories of the consolidation of opinion and the formation of broad consensus—that these theories were the best way to organize and advance these sciences.P5: After Wegener’s death, a South African geologist Alexander Du Toit continued to assemble fossil evidence for Pangaea.■He noted that fossils of the now extinct reptile “Mesosaurus” occur in rocks of the same age in both Brazil and South Africa.■Because the physiology of freshwater and marine animals is completely different, it is hard to imagine how a freshwater reptile could have swum across the Atlantic Ocean and then found a freshwater environment nearly identical to its former habitat. ■Moreover, if Mesosaurus could have swum across the ocean, its fossil remains should be widely dispersed. It is more logical to assume that Mesosaurus lived in lakes in what were once adjacent areas of South America and Africa when it was united into a single continent. ■Notwithstanding all of the empirical evidence in favor of continental drift theory presented here, most geologists at the time refused to entertain the idea.1．According to paragraph 2, Eduard Suess believed that similarities of plant and animal fossils on the southern continents were due toA．living in the southern climateB．crossing the land bridgesC．fossilization in the coal layersD．movements of the supercontinent正确答案：B解析：【事实信息题】题目问动植物化石的相似性是因为什么，文中提到“the similarities of fossils on these continents could be accounted for by”所以原因是之前有陆桥后来消失了，答案是B。

41. The most thrilling explanation is, unfortunately,a little defective. Some economists argue that powerful structural changes in the world have upended the old economic models that were based upon the historical link between growth and inflation. [参考译⽂]很不幸，这最令⼈震惊的解释有⼀点缺陷。

⼀些经济学家认为世界经济结构的强有⼒的变化已经结束了那个以经济增长和通货膨胀的历史关联为基础的旧的经济模式。

42. The Aswan Dam, for example, stopped the Nile flooding but deprivedEgypt of the fertile silt that floods left-all in return for a giant reservoir of disease which is now so full of silt that it barely generates electricity.[参考译⽂]例如，阿斯旺⼤坝使得尼罗河不再洪⽔泛滥，但是它也夺去了埃及以前所享有的洪⽔留下的肥沃淤泥--这些换宋的就是这么个疾病滋⽣的⽔库，现在这个⽔库积满了淤泥，⼏乎不能发电了。

43. New ways of organizing the workplace--all that re-engineering anddownsizing--are only one contribution to the overall productivity of an economy, which is driven by many other factors such as joint investment in equipment and machinery, new technology, and investment in education and training,[参考译⽂]企业重组的新⽅法--所有那些重新设计、缩⼩规模的做法--只是对⼀个经济的整体⽣产⼒做出了⼀⽅⾯的贡献。

第40卷第3期Vol.40㊀No.3重庆工商大学学报(自然科学版)J Chongqing Technol &Business Univ(Nat Sci Ed)2023年6月Jun.2023C4旋转对称光子晶体平板中的对称保护连续谱束缚态张铭洋重庆工商大学数学与统计学院,重庆400067摘㊀要:在光子晶体平板中,连续谱束缚态关于C2和C6旋转对称的依赖性已经在数值上进行了广泛研究,但是缺少严格的理论分析过程,此外还缺少对C4旋转对称的研究,鉴于此,构建了系统分析连续谱束缚态关于所有旋转对称的依赖性的理论,并且重点研究了C4旋转对称的情况;首先,通过分析具有旋转对称的结构中麦克斯韦方程组特征解的性质,将连续谱束缚态的存在性问题转变为旋转矩阵的特征值是否与一个简单代数方程的解相同的问题;其次,给出了C4旋转对称的结构中连续谱束缚态存在时所对应的条件;然后,证明了破坏C4旋转对称保持C2旋转对称时,连续谱束缚态依然存在;最后,利用有限元软件FreeFEM 进行了大量的数值验证;上述理论可适用于所有旋转对称的情况,深入揭示了旋转对称对连续谱束缚态存在的重要性,深入揭示了高阶旋转对称性与低阶旋转对称性之间的依赖关系,为连续谱束缚态的实际应用提供了理论指导㊂关键词:光子晶体;旋转对称;连续谱束缚态中图分类号:O436㊀㊀文献标识码:A㊀㊀doi:10.16055/j.issn.1672-058X.2023.0003.09㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀收稿日期:2022-05-13㊀修回日期:2022-06-20㊀文章编号:1672-058X(2023)03-0064-07基金项目:重庆市自然科学基金面上项目(CSTC2019JCYJ -MSXMX0717).作者简介:张铭洋(1997 ),女,重庆忠县人,硕士研究生,从事光子晶体㊁麦克斯韦方程组数值计算研究.引用格式:张铭洋.C4旋转对称光子晶体平板中的对称保护连续谱束缚态[J].重庆工商大学学报(自然科学版),2023,40(3):64 70.ZHANG Mingyang.Symmetry-protected bound states in the continuum in C4rotationally symmetric photonic crystal plates J .Journal of Chongqing Technology and Business University Natural Science Edition 2023 40 3 64 70.Symmetry-protected Bound States in the Continuum in C 4Rotationally Symmetric Photonic Crystal Plates ZHANG MingyangSchool of Mathematics and Statistics Chongqing Technology and Business University Chongqing 400067 ChinaAbstract The dependence of bound states in the continuum BICs on C2and C6rotational symmetry in photonic crystalplates has been extensively studied numerically.However a rigorous theoretical analysis process is lacking and there is alack of studies on C4rotational symmetries.In view of this a theory of systematic analysis of the dependence of BICs on all rotational symmetries was constructed and the case of C4rotational symmetry was mainly studied.Firstly by analyzingthe characteristic solutions of Maxwell s equations with rotationally symmetric structure the problem of the existence of BICs was transformed into the question of whether the eigenvalue of the rotation matrix was the same as the solution of asimple algebraic equation.Secondly the conditions for the existence of BICs in C4rotationally symmetric structures weregiven.Then it was proved that the BICs still existed when C4rotation symmetry was destroyed and C2rotatory symmetrywas maintained.Finally the finite element software FreeFEM was used to do a lot of numerical verifications.The abovetheory can be applied to all cases of rotational symmetries revealing the importance of rotational symmetry for the existenceof BICs.The dependence between high-order and low-order rotational symmetries was revealed providing theoretical guidance for applying BICs.Keywords photonic crystal rotational symmetry bound states in the continuum第3期张铭洋,等:C4旋转对称光子晶体平板中的对称保护连续谱束缚态1㊀引㊀言光学连续谱束缚态(bound states in the continuum, BIC)是指位于连续谱中的导模,其不能与辐射场耦合,没有能量辐射,被完美地束缚在结构中[1-3]㊂数学上,光学连续谱束缚态是指开结构中麦克斯韦方程组的一类频率位于连续谱内的平方可积特征解㊂通常,导模(即平方可积特征解)的特征频率位于连续谱外㊂1929年冯诺依曼等[4]从数学模型上发现在一些特殊的结构中存在特征频率位于连续谱内的导模㊂直到1985年文献[5]才构造出具有连续谱束缚态的真实物理系统㊂2008年,文献[3]研究了光子晶体结构中的连续谱束缚态㊂此后,连续谱束缚态受到广泛关注,与之有关的研究快速发展㊂目前,连续谱束缚态的概念和研究已推广到水波㊁声波等其他波动现象[1]㊂连续谱束缚态可看成为品质因子为无穷大的共振,只存在于若干离散的频率上㊂连续谱束缚态由共振模式所包围㊂通过扰动波矢,可在连续谱束缚态附近找到任意大小品质因子的共振模式[6]㊂此性质使得连续谱束缚态在光学㊁光子学等领域都拥有广阔的应用前景㊂目前,连续谱束缚态已在波导㊁光栅㊁光子晶体及超材料等结构中被广泛研究[7],光子晶体中的连续谱束缚态现在已经被用于传感器,激光器和滤波器的设计当中[8-10]㊂通常,共振模式的品质因子与波矢之差的平方成反比㊂文献[6]证明了存在特殊的连续谱束缚态使得附近共振模式的品质因子与波矢之差的四次方和六次方成反比,并给出了两类特殊连续谱束缚态的条件㊂从实际应用角度来讲,在这些特殊的连续谱束缚态附近更容易构造出高品质因子的共振模式㊂连续谱束缚态可以大致分为两类:对称保护的连续谱束缚态[11-15]和非对称保护的连续谱束缚态[2,16-19]㊂对称保护的连续谱束缚态(Symmetry Protected Bound states in the continuum,SPBIC)的机理是:在对称结构中,布洛赫模的对称性与结构中辐射场的对称性不相容,从而与辐射场不耦合,变成一个连续谱束缚态[3]㊂而非对称保护连续谱束缚态的存在机理是:共振模式的辐射场之间发生干涉相消现象,造成没有辐射,成为连续谱束缚态[3]㊂连续谱束缚态的存在性与结构的对称性具有密切联系㊂早期研究结果都是在对称结构中研究连续谱束缚态,学术界一度认为连续谱束缚态只存在于对称结构中㊂目前数学上还没有非对称保护连续谱束缚态的存在性理论㊂非对称保护连续谱束缚态关于结构对称性的依赖关系非常复杂㊂文献[20-23]从数值和实验上演示了破坏二维结构的C2旋转对称性后,连续谱束缚态演化为共振模式㊂这间接说明了结构的对称性对非对称保护连续谱束缚态的存在性具有重要影响㊂但是破坏对称性后连续谱束缚态是否一定会演化为共振模式并没有明确的结论㊂最近,文献[24-26]证明了只要引入足够多的结构扰动参数,连续谱束缚态可以连续存在于非对称的结构中,且对于不同类型的连续谱束缚态,所需要引入的最小参数的数量是不同的㊂上述结论表明,非对称保护连续束缚态可以存在于非对称结构中,只要结构的自由参数足够多㊂对称保护连续谱束缚态只存在于对称结构中㊂光子晶体平板可具有四类旋转对称性:C2㊁C3㊁C4和C6旋转对称性,即分别旋转180㊁120㊁90和60度后结构不变㊂文献[11-12]首先从数学理论上证明了在具有C2旋转对称的二维介质结构中对称保护连续谱束缚态的存在性,在非对称结构中一定不存在对称保护连续谱束缚态㊂研究对称保护连续谱束缚态对上述四种不同类型对称性的连续依赖性具有十分重要的意义㊂文献[21]研究了C6旋转对称性对具有拓扑电荷为q=-2的对称保护连续谱束缚态存在性的影响㊂通过数值计算发现破坏C6对称保持C2对称,对称保护连续谱束缚态依然存在,但是变成拓扑电荷q=-1;破坏C6对称保持C3对称,对称连续谱束缚态演化为共振模式,而且会产生两个非对称保护连续谱束缚态㊂上述研究结果给出了一种产生非对称保护连续谱束缚态的方法㊂目前,对于拓扑电荷为q=-1或q=1的对称保护连续谱束缚态关于结构对称性的依赖关系没有进行系统讨论,缺乏严格系统的依赖性理论㊂研究C4旋转对称结构中对称保护连续谱束缚态关于对称性的依赖关系㊂建立了严格数学理论证明破坏C4对称保持C2对称,对称保护连续谱依然存在㊂并利用有限元软件FreeFEM进行数值验证㊂相比于以前的研究,研究既有严格的数学理论,又有数值验证㊂研究成果具有一般性,可推广到分析对称保护连续谱束缚态关于C6对称性的依赖性,有利于深入理解连续谱束缚态关于对称性的依赖关系,为其实际应用提供理论指导㊂2㊀连续谱束缚态考虑一个在x与y方向为周期,在z方向上厚度有限的光子晶体平板㊂光子晶体平板通过在平板上构造正方形空气柱晶格所构成㊂设平板厚度为2D,晶格常数为L,平板的介电常数为ε1,空气的介电常数为ε0= 1㊂记整个结构的介电常数为ε(r),其中r=(x,y,z),则z>D时有ε(r)=1,且ε(r)满足ε(r)=ε(x+mL,y+nL,z)(1)其中,m与n为任意整数㊂设光子晶体平板是无磁性㊁各向同性的,由麦克斯56重庆工商大学学报(自然科学版)第40卷韦方程组可知,具有时间依赖e -iwt 的时谐波的电场E 满足如下的控制方程:∇ˑ∇ˑE -k 2εE =0∇㊃(εE )=0其中k =ωc为真空中的波速,ω为角频率,c 为真空中的光速㊂光子晶体平板中的布洛赫模(即麦克斯韦方程组的特征解)可写成:E (r )=Φ(r )e ik ㊃r其中,k =(α,β,0)为布洛赫波矢,实数α与β分别为x 与y 方向的布洛赫波速,Φ(r )满足周期条件式(1)㊂由于在z >D 时,结构是均匀的,由傅立叶展开式与分离变量法可知,满足向外辐射条件的布洛赫模可以展开为[18]E (r )=ð+ɕm ,n =-ɕd ʃm ,neik ʃm ,n㊃r ʃz >D (2)其中,常数向量d ʃm ,n满足d ʃm ,n ㊃k ʃm ,n =0,k ʃm ,n=(αm ,βn ,ʃγm ,n ),αm =α+2m πL ,βn =β+2n πL,γm ,n =k 2-α2m -β2n ㊂若结构是无耗散的,即ε(r )为非负实函数,则布洛赫模可以分为三类:导模㊁共振以及连续谱束缚态㊂(1)若k 为实数,则布洛赫模为导模㊂可以证明k 为实数等价于E (r )满足lim z ңɕE (r )=0,即没有能量辐射,没有能量损失㊂当波速和布洛赫波矢满足条件0<k <α2+β2时,导模关于α与β连续存在㊂在上述条件下,γm ,n 的虚部都大于0,所以展开式(2)中每一个平面波都在无穷远出衰退到0,即E (r )自动满足lim z ңɕE (r )=0,从而是一个导模㊂(2)若k 为复数,则布洛赫模为共振模式㊂共振模式的波速k 满足[Re(k )]2-[lm(k )]2>α2+β2㊂由于共振的波速k 为复数且满足向外辐射条件,共振满足条件lim z ңɕE (r )=ɕ,即在空间上是无限增大,但随时间指数衰退㊂共振波速k (或共振频率ω)的虚部小于零,即lm(k )<0,它表示共振随着时间衰退的速度㊂共振的品质因子Q 定义为Q =-12Re(k )lm(k ),表示共振模式的振幅衰退到原来的e -1时所需要的振荡周期㊂共振模式关于α与β也是连续存在的㊂(3)若k 为实数且满足k >α2+β2,则布洛赫模是一个连续谱束缚态㊂连续谱束缚态可以看成是一个Q 因子为无穷大的共振,只在离散的(α,β)点上存在,在连续谱束缚态的附近,通过调整α与β可以获得任意大小Q 因子的共振㊂由于k 为实数等价于条件limz ңɕE (r )=0,在展开式(2)中,若lm(γm ,n )ȡ0,则e ikm ,n㊃r可向z ңɕ辐射能量,(m ,n )是对应一个开放的辐射通道㊂若lm(γm ,n )<0,则eik ʃn ,m ㊃r在z ңɕ时衰退到零,对应一个关闭的辐射通道㊂若Re(k )>α2+β2,则至少有lm(γ0,0)ȡ0,即(0,0)处辐射通道是开放的㊂记Z 0表示所有开放的辐射通道,即Z 0=(m ,n )lm(γm ,n )ȡ0{},则条件lim z ңɕE (r )=0等价于d ʃm ,n=0,∀(m ,n )ɪZ 0(3)式(3)是布洛赫模的一个附加条件,在一般情况下,连续谱束缚态不容易存在㊂3　对称保护连续谱束缚态当光子晶体平板具有旋转对称性时,可能存在对称保护连续谱束缚态㊂下面给出具有C n 旋转对称的光子晶体平板中对称保护连续谱束缚态的定义,并分析其关于对称性的依赖关系㊂利用旋转对称性下布洛赫模的性质,将连续谱束缚态的存在性问题转变为旋转矩阵的特征值是否与一个简单代数方程的解相同的问题;其次,给出了C4旋转对称的结构中连续谱束缚态存在时所对应的条件;然后,证明了破坏C4旋转对称保持C2旋转对称时,连续谱束缚态依然存在;具有C n 旋转对称性结构的介电函数ε(r )满足条件ε(r )=ε(T -1r )其中,T =cos φ-sin φ0sin φcos φ0001éëêêêùûúúú表示旋转矩阵,φ=2πn ,只考虑n =2与n =4的情况㊂其理论可推广到n =3与n =6的情况㊂C n 旋转对称光子晶体平板中的布洛赫模具有以下性质[24]:若E (r )=Φ(r )e ik ㊃r 是一个对应于波矢k =(α,β,0)和波速k 的布洛赫模,则TE (T -1r )是一个对应于波矢Tk 和波速k 的布洛赫模㊂特别地,取α=β=0,即k =(0,0,0),有Tk =k ,此时E (r )与TE (T -1r )是对应同一个波矢与波速的两个布洛赫模㊂若特征值问题是非退化的,则E (r )与TE (T -1r )线性相关,即存在常数τ使得:TE (T -1r )=τE (r )(4)由于任何结构旋转n 次2πn角度(即2π)后都不变,有T n=I ,其中I 表示单位算子㊂所以有τn=1,即τ=e i 2πn j,j =0,1,2, ,n -1㊂更具体地,当n =2时,τ=ʃ1;当n =4时,τ=ʃ1,ʃi ㊂注意到τ的取值对应于C n 点群的不可约表示的特征㊂设布洛赫模的波矢为k =(0,0,0)且频率满足0<Re(k )ε0<2πL,即只有(0,0)处辐射通道是开放的,则当n =2时,对应于τ=1以及当n =4时,对66第3期张铭洋,等:C4旋转对称光子晶体平板中的对称保护连续谱束缚态应于τ=ʃ1的布洛赫模一定是连续谱束缚态,称为对称保护连续谱束缚态㊂在上述条件下,布洛赫模是一个连续谱束缚态的充要条件是d ʃ0,0=0㊂下面证明当n =2时,τ=1以及当n =4时,τ=ʃ1,有d ʃ0,0=0㊂将展开式(2)代入条件式(4)得:Td ʃ0,0=τd ʃ0,0(5)令,d ʃ0,0=d ʃx ,d ʃy ,d ʃz []T ,T ^=cos φ-sin φsin φcos φéëêêùûúú,d ʃʅ=d ʃx ,d ʃy []T 表示d ʃ0,0的x 与y 分量所构成的向量㊂由d ʃ0,0㊃k ʃ0,0=0且α=β=0,知d ʃz k =0,即d ʃz =0㊂所以d ʃ0,0=0等价于d ʃʅ=0㊂由式(5)得:T ^d ʃʅ=τd ʃʅ若τ不是T ^的特征值,则必有d ʃʅ=0㊂当n =2时,T ^=-100-1éëêêùûúú,T ^只有一个特征值-1㊂所以当τ=1时,有d ʃʅ=0㊂当n =4时,T ^=0-110éëêêùûúú,此时T ^的特征值为ʃi ㊂所以当τ=ʃ1时有d ʃʅ=0㊂由上面的证明过程可知,条件d ʃ0,0=0是由C n 对称性所保证的㊂在具有C4旋转对称的结构中,对称保护连续谱束缚态对应的τ=1或-1㊂注意到无论是τ=1还是-1,都有τ2=1,即这些连续谱束缚态也同时由C2旋转对称所保护㊂有以下结论:具有C4旋转对称结构中的对称保护连续谱束缚态都是由C2旋转对称所保护的,即破坏C4旋转对称,保持C2旋转对称,这些连续谱束缚态依然存在㊂4㊀拓扑电荷连续谱束缚态对应于动量空间中辐射场的漩涡,因此其存在性与拓扑性质有关㊂前面提到了布洛赫模是一个连续谱束缚态的充要条件是d ʃ0,0=0,通过d ʃ0,0的x 与y 分量可以计算得到辐射场的极化角㊂极化椭圆的长轴与y 轴的夹角称之为极化角度,记为θ㊂θ可以看成是α与β的函数,即θ=θ(α,β)㊂在αβ平面上,任意给定一条曲线Γ,让(α,β)沿着Γ绕一圈重新定义θ,使其为连续函数㊂拓扑电荷的定义为q =12πɥΓd θ=12πɥΓ∇θ㊃ n d s拓扑电荷q 表示αβ上的一点绕Γ走一圈后,极化角度旋转了q 圈,q 是一个整数㊂若Γ所围区域内无圆极化与连续谱束缚态,则q =0;若Γ所围区域内有且仅有一个连续谱束缚态则q =ʃ1,ʃ2,ʃ3, ;若Γ所围区域内只有一个圆极化,则q =ʃ12㊂需要注意的是圆极化和连续谱束缚态是αβ平面中的一个极化奇点㊂5㊀数值实验由于辐射边界条件下的特征值问题定义在无穷区间上,无法用数值方法来计算㊂所以在实际计算连续谱束缚态的时候,可以用完美匹配层的方法来将无穷区域截断为有限区域㊂用完美匹配层截断后的特征值问题是原特征值问题的一个近似,它们之间的误差关于完美匹配层的参数σ∗㊁H 2-H 1(即完美匹配层的厚度)指数衰退到零㊂所以只需要选择合适的σ∗与H 2-H 1,便可以得到足够精确的特征解,即可以计算得到连续谱束缚态的频率㊂相对于拟周期边界条件,在有限元方法中周期边界条件更容易实现㊂用有限元方法求解偏微分方程最重要的是弄清楚解空间和变分形式㊂在用有限元求解时,变分问题被近似为下列代数方程的特征值问题:A Φ=k 2B Φ其中,A 与B 为矩阵㊂考虑如图1所示的具有正方形晶格空气柱的光子晶体平板,其俯视图如图2所示㊂平板是由空气所包围的㊂平板的厚度为2D =0.5L ,介电常数为ε1=4,空气中的介电常数ε0=1㊂空气柱体横截面参数分别为w =0.2L ,a =w2㊂若h 1=h 2,则结构具有C4旋转对称性㊂若h 1ʂh 2,则结构只有C2旋转对称性㊂为了验证前面得到的理论,用完美匹配层[27]的方法将无穷区域上的特征值问题转化为有限区域上的特征值问题,并用有限元[28]的方法求解㊂数值计算时,需要用完美匹配层方法将z 方向截断为-H 2,H 2[],如图3所示㊂取完美匹配层的厚度为H 2-H 1=L ,σ∗=18ˑm +1β0(H 2-H 1)[29],β0=k 20ε0-α2-γ2,m =3㊂其中H 1-D 表示完美匹配层的远近㊂采用基于FreeFEM 软件的有限元方法来数值求解特征值问题,以计算对称保护连续谱束缚态的频率㊂计算时在平板的每个边界的离散点个数取N =10,PML 层的离散点个数也取N =10㊂考虑如此复杂结构的原因是为了避免其他对称性(例如镜面反射对称)对结果的影响㊂ε=ε0ε=ε0z =D z =-Dzxy图1㊀光子晶体平板结构图Fig.1㊀Structure diagram of photonic crystal plates76重庆工商大学学报(自然科学版)第40卷h 2h 1h 1h 2L Lαωωxy图2㊀光子晶体平板结构的俯视图Fig.2㊀Top view of the photonic crystal flat plate structurez=H2z=H1z=D z=-Dz=-H1z=-H2ε=εε=εzxy 图3㊀PML截断后的计算区域Fig.3㊀Computation region after PML truncation 若取h1=h2=0.15L,这时结构具有C4旋转对称性㊂通过数值计算,可以找到5个TM-Like模式下(即E z是z变量的奇函数)的对称保护连续谱束缚态,其频率如表1的第2列所示㊂图4(a) 图8(a)分别是SPBIC1-SPBIC5在具有C4旋转对称的结构中log10Q 关于α与β的值㊂可以通过观察得到当(α,β)ң(0, 0)时,Q的值趋近于无穷大㊂图4(c) 图8(c)分别是SPBIC1到SPBIC5在C4旋转对称结构中的磁场z分量H z在z=0时的场图㊂从下面的场图可以观察得到, SPBIC1与SPBIC5对应于τ=1,其他3个对称保护连续谱束缚态对应于τ=-1㊂表1的最后一列表示为对称保护连续谱束缚态的拓扑电荷㊂若保持h1=0.15L,令h2=0.1L,参数扰动后结构的C4旋转对称性被破坏,但保持了C2旋转对称性㊂通过数值计算表明,SPBIC1-SPBIC5在扰动后的结构中依然存在,其频率如表1的第三列所示,可以发现两种结构下连续谱束缚态的频率近乎相等㊂图4(b) 图8(b)分别是SPBIC1到SPBIC5在具有C2旋转对称的结构中log10Q关于α与β的值㊂可以通过观察得到当(α,β)ң(0,0)时,Q的值趋近于无穷大,并且可以发现两种结构下,log10Q关于α与β的值很相近㊂图4(d) 图8(d)分别代表的是扰动后SPBIC1-SPBIC5在C2旋转对称结构中的磁场z分量H z在z=0时的场图㊂从下面的场图可以观察得到,SPBIC1与SPBIC5仍然对应于τ=1,其他3个对称保护连续谱束缚态也依旧对应于τ=-1㊂通过对比,可以发现两种结构下的场图几乎一模一样,并且可以发现结构扰动不改变对称保护连续谱束缚态的拓扑电荷㊂表1㊀C4与C2旋转对称结构中对称保护连续谱束缚态的频率ωL2πc的值Table1㊀Value ofωL2πc the frequency of symmetrically protected bound states in the continuum in the rotationallysymmetric structure of C4and C2C4C2q SPBIC10.61590.6101+1 SPBIC20.63670.6282-1 SPBIC30.85690.8485-1 SPBIC40.93830.9338-1 SPBIC50.95140.9492+10.05-0.05-0.0500.05βL/(2π)Q f a c t o rαL/(2π)8640.05-0.05-0.0500.05αL/(2π)864βL/(2π)Q f a c t o r㊀㊀(a)(b )0.5-0.5-0.500.5y/LR e/H zx/L0.5-0.5-0.500.5R e/H zx/L㊀㊀(c)(d)图4㊀SPBIC1的Q因子图和场图Fig.4㊀Q factor diagram and field diagram of SPBIC10.05-0.05-0.0500.05βL/(2π)Q f a c t o rαL/(2π)8765430.05-0.05-0.0500.05αL/(2π)βL/(2π)Q f a c t o r876543㊀㊀(a)(b)86第3期张铭洋,等:C4旋转对称光子晶体平板中的对称保护连续谱束缚态0.50-0.5-0.50.5y /LR e /H zx /L0.5-0.5-0.50.5y /LR e /H zx /L㊀㊀(c )(d )图5㊀SPBIC2的Q 因子图和场图Fig.5㊀Q factor diagram and field diagram of SPBIC20.010-0.01-0.010.01βL /(2π)Q f a c t o rαL /(2π)8765430.010-0.01-0.0100.01αL /(2π)βL /(2π)Q f a c t o r76543㊀㊀(a )(b )0.5-0.5-0.50.5y /LR e /H zx /L0.5-0.5-0.50.5y /LR e /H zx /L㊀㊀(c )(d )图6㊀SPBIC3的Q 因子图和场图Fig.6㊀Q factor diagram and field diagram of SPBIC30.010-0.01-0.010.01βL /(2π)Q f a c t o rαL /(2π)76540.010-0.01-0.0100.01αL /(2π)βL /(2π)Q f a c t o r7654㊀㊀(a )(b )0.5-0.5-0.50.5y /LR e /H zx /L0.5-0.5-0.50.5y /LR e /H zx /L㊀㊀(c )(d )图7㊀SPBIC4的Q 因子图和场图Fig.7㊀Q factor diagram and field diagram of SPBIC40.020-0.02-0.0200.02βL /(2π)Q f a c t o rαL /(2π)8640.020-0.02-0.0200.02αL /(2π)βL /(2π)Q f a c t o r6543㊀㊀(a )(b )0.5-0.5-0.50.5y /LR e /H zx /L0.5-0.5-0.500.5y /LR e /H zx /L㊀㊀(c )(d )图8㊀SPBIC5的Q 因子图和场图Fig.8㊀Q factor diagram and field diagram of SPBIC5经过数值计算,从扰动前后不同结构下对称保护连续谱束缚态的频率以及对比分析它们的Q 因子图和场图可以观察得到具有C4旋转对称结构的光子晶体平板中的对称保护连续谱束缚态都是由C2旋转对称性所保护的㊂即若破坏C4旋转对称但保持C2旋转对称,原有的对称保护连续谱束缚态依然存在㊂进一步反映了C4旋转对称与C2旋转对称之间的依赖关系㊂6㊀结束语构建了系统分析连续谱束缚态关于旋转对称性的依赖理论,并且重点研究了C4旋转对称的情况,分别从理论和数值两个方面证明了具有C4旋转对称光子晶体平板中的对称保护连续谱束缚态都是由C2旋转对称性所保护的㊂即破坏C4旋转对称但是保持C2旋转对称性,原对称保护连续谱束缚态依然存在㊂虽然只考虑了C4旋转对称光子晶体平板中的对称保护连续谱束缚态,但提出的理论和数值分析方法都可以用于研究具有C6旋转对称的光子晶体平板,不过由于此结构同时具有C2与C3旋转对称性,对称保护连续谱束缚态与对称性的依赖关系可能会更加复杂㊂提出的理论分析方法也可以适用于所有旋转对称的情况㊂由于是从麦克斯韦方程组出发,没有引入模型近似,并且分析过程根据严格㊂研究结果有利于深入理解对称保护连续谱束缚态的性质,为其理论分析和实际应用提供指导㊂96重庆工商大学学报(自然科学版)第40卷参考文献References1 ㊀HSU C W ZHEN B STONE A D et al.Bound states in thecontinuum J .Nature Reviews Materials 2016 1 9 1 13.2 ㊀HSU C W ZHEN B LEE J et al.Observation of trappedlight within the radiation continuum J .Nature 2013 4997457 188 191.3 ㊀MARINICA D C BORISOV A G SHABANOV S V.Boundstates in the continuum in photonics J .Physical Review Letters 2008 100 18 1 4.4 ㊀NEUMANN J WIGNER E P.Über 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Study on the properties of percolativeconductorsIntroductionPercolation theory provides a useful framework for understanding the behavior of materials that conduct electricity. A percolative conductor is a material that has a random network of conductive pathways, allowing for the flow of current. In this article, we will explore the properties of percolative conductors and the factors that affect their behavior.ConductivityThe primary property of percolative conductors is conductivity, which is a measure of the ability of a material to conduct electricity. The conductivity of a percolative conductor is determined by the density and connectivity of the network of conductive pathways. As the density of the network increases, so does the conductivity of the material.Another important factor that affects conductivity is the shape of the conductive pathways. The more tortuous the pathways, the lower the conductivity of the material. This is because the electrons have a longer path to travel, and thus encounter more resistance.ResistanceThe resistance of a material is the inverse of its conductivity and is a measure of the opposition to the flow of current. The resistance of a percolative conductor is determined by the density and connectivity of the network of conductive pathways as well as the shape of the pathways.In general, as the density of the network of conductive pathways increases, the resistance of the material decreases. Similarly, as the connectivity of the network increases, the resistance decreases. However, the shape of the pathways can have asignificant impact on the resistance of the material, with more tortuous pathways leading to higher resistance.Percolation ThresholdThe percolation threshold is the point at which the network of conductive pathways becomes connected, allowing for the flow of current. Below this threshold, the material is insulating, and above it, the material is conducting.The percolation threshold is determined by a number of factors, including the size and shape of the conductive particles, the nature of the insulating material, and the concentration of the conductive particles. In general, smaller particles lead to lower percolation thresholds, and more spherical particles lead to higher percolation thresholds.Temperature DependenceOne of the key factors that affect the conductivity of percolative conductors is temperature. In general, as the temperature increases, the conductivity of the material increases. This is because the higher temperature allows for more energy to be available for the flow of current.However, there are limits to this effect, and at some point, the temperature will become too high, and the conductivity will decrease due to other factors such as thermal expansion and changes in the material structure.ApplicationsPercolative conductors have a wide range of applications, including in the electronic and optoelectronic industries. They are used in the production of printed circuit boards, organic solar cells, and transparent conductive coatings.ConclusionThe study of percolative conductors provides a useful framework for understanding the behavior of materials that conduct electricity. The conductivity, resistance, percolation threshold, and temperature dependence are all important properties that affectthe behavior of these materials. By understanding these properties, researchers can develop new and improved materials for a wide range of applications.。