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Return of the Microbes: How infections are once more taking over
Professor William Ayliffe FRCS PhD
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A number of infectious diseases cause blindness. After decades of improvement in treatment many were thought to be on the edge of extinction.
Whilst this was achieved with smallpox, the promise if eradication of trachoma and river blindness remains elusive.
Furthermore the increasing resistance if organisms to antibiotics is becoming and urgent challenge in all parts of the world. The resurgence if Tuberculosis is a particular problem.
How these organisms cause disease, blindness, how they are treated and how we may eradicate them concerns each and every one of us. The answer is becoming more difficult and more urgent to find.
Professor William Ayliffe is Emeritus Professor of Physic at Gresham College and a Consultant Ophthalmologist at the Lister Hospital in London. As well as being a practising clinician and teacher, he also continues to carry out clinical research into the prevention of blindness.
After taking a first in Immunology at Imperial College, Professor Ayliffe qualified in Medicine in St. Bartholomew's Hospital London. Specialising in inflammatory eye diseases and corneal and cataract surgery he held a research registrar post at Oxford, before training in clinical ophthalmology in Bristol, Manchester and Harvard USA. His PhD was on mechanisms of corneal transplant failure. He has worked in developing countries and also with ORBIS, the international flying eye hospital.
In addition to general ophthalmology, Professor Ayliffe has developed a local and tertiary referral service for cornea, uveitis and inflammatory eye disease. Professor Ayliffe is a winner of the prestigious Wix Prize for the History of Medicine and the Kabi-Pharmacia Prize for immunological mechanisms of corneal transplant rejection. He is a Reviewer for a number of professional journals including Eye, British Journal of Ophthalmology, Journal of Cataract and Refractive Surgery, he is an Examiner for the Royal College of Ophthalmologists and an Advisor to the UK Transplant Service. He has undertaken wide media and TV work for the national press, the BBC and Channel 4, has an extensive range of publications and has delivered prestigious lectures all over the world.
Professor Ayliffe took up his Gresham Professorship in 2009, the 200th anniversary of Louis Braille's birth. His lectures offered a cultural and scientific survey of the eye and vision, centring on the biological and cultural aspects of the human eye and vision, covering the overlapping medical, philosophical and humanitarian concerns of this area.
All of Professor Ayliffe's previous lectures may be accessed here.
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This is part of the series: Vision and the Eye
Immunology Medical science Vision and the eye
RETURN OF THE MICROBES:
HOW INFECTIONS ARE ONCE MORE TAKING OVER
PROFESSOR WILLIAM AYLIFFE
Good evening, ladies and gentlemen, and welcome to the first of the Physic lectures for 2014. Horrible day outside, so I really appreciate what effort you must have made to come in this evening, and I hope you find it is worthwhile.
I have got to do two things tonight: one is to control this screen independently of this screen, so bear with me for the first few slides until I get used to the new technique!
The microbe, of course, has been with us forever, and long before, for billions of years before we were an item, causes of much distress and death, and, throughout the years, we have had endemic, epidemic, minor outbreaks that have damaged human societies, in many cases irreparably, and sometimes destroyed them.
I have brought this up as the first painting to illustrate the relationship between pestilence, or infectious disease, between war, which it often accompanies, as I will show many times in the lecture, famine, which can lead to both of the above and be connected with pestilence following famine, as was found out with the typhus epidemic in Ireland after the Potato Famine, when it was commonly known as the Irish fever, and they brought it over to Liverpool and Manchester and decimated the populations around the ports. And then, finally, the outcome of all of this is…death.
Microbes have been with us ever since human beings have been around, but of course, they have been around long before that, and in fact, without microbes, the modern world, we would not recognise at all. There would be no animal life without microbes because there would be no oxygen. And for 3.5 billion years, until there was enough oxygen, all life on Earth was microscopic and single-cells. Gradually, with evolution, we developed the ability to have a nucleus and to separate different manufacturing components but still remain as single-cells, and eventually, the single-cells amalgamate and form multi-cellular animals, as I have previously shown in a lecture a year ago, and eventually, through evolution, to more complex life forms.
We can look at these various types of life fairly simplistically, from the smallest, which would be a virus – and it is argumented whether that's life or just a package of DNA, but it is certainly capable of reproducing itself. There are smaller items of reproducible material, such as certain prion proteins, but we are not going to be involved with those tonight. Bacteria, of course: archaea; chlamydia, which are intra-cellular and very small bacteria, and they are totally dependent on living in other cells, rather like viruses are dependent on living cells to reproduce themselves; there are protists, which are single-cell organisms; and then there are microfilaria, which technically do not come into the microbial world, but are often counted with it because the same people who deal with microbes deal with microfilaria, and they are the larval forms of various more complicated forms of life.
Once I have gone through some of those diseases, we will talk a little bit about drug resistance because it is very, very topical, and then come onto some 21st Century threats.
Just briefly, we were talking about archaea, and many of you may not have come across this term before, but it is a separate branch of life, and life is now being described in three different forms, based on the RNA, and there is: the bacteria; then there is the archaea, and they are more closely related to us than bacteria are, and they are more closely related to us than they are to bacteria. They are single-cells. Under the microscope, they look very similar.
In fact, we have them in our guts and they produce methane, which reminds me here of the Papal Belvedere which was drawn by Lucas Cranach, showing the Papal Bulls. The Papal Bulls were written in a room called the Belvedere in the Vatican, and the German peasants, on being presented one with its fire and brimstone, show it around with their fresh methane emittance, showing the Pope their beautiful view, which reflects that schism in religious belief in Europe that led to so much war, famine and pestilence, and the 30 Years War is probably one of the most horrible wars there has ever been.
The existence of microbes of course remains unimaginable, until Antonie van Leeuwenhoek picked his teeth and put it under his microscope and was impressed with what he saw, he immediately found two old men who had never cleaned their teeth and took some of their plaque, which he described as "a little white matter as thick as to a batter, then most always saw, with great wonder, that in that matter, there were many living animalcules," which he draws most beautifully here, including the little spiral movement it made, very quickly, in Organism B. Organism G is one of these spirochaetes, and then of course there is the long rod-shaped bacteria as well, and, recently, these have been put together with the modern microscopy saying what these many organisms might be. In fact, in the mouth, there may be 80 different types of bacteria between the plaque of the front teeth, and going further back, there are many anaerobes, which is very difficult for us to imagine, that having animals that cannot live in oxygen living in our mouth, which we imagine is full of oxygen. In fact, the more you go into the microbial world, the less you might wish to stay there.
Later in time, we were able through the multiple discoveries of many people studying these small organisms to identify them and classify them into two broad groups, according to Hans Gram's Stain, which is ones that stain with Gram's Stain and then ones that are bleached and do not keep the stain, and when they are counter-stained, they stain with this purplish-red, which enables us to find broadly two types of the bacterial microbial world, the gram-positive and the gram-negatives.
It is a moot point whether bacteria live on us, or whether we are just an organism that transports bacteria around, and it just depends on numbers. A human being, for example, has about a hundred trillion cells. There are ten to twenty times that number of bacteria that live on us. In the large intestine alone, we will have ten to the ten per centimetre squared, ten billion organisms on the lining for every centimetre. On the human skin, it is less densely populated, but there are 50 million organisms per centimetre squared, and what is amazing is how this is very different, this pattern of life forms, and it depends where you take the swab from. You can see, from the different colours, the different orders of bacteria that are present in different parts of the surface of the human body. So, if I take a scrap between your big toe and your next toe down, I am going to get a completely different set of organisms than if I take it from your nose.
You may be aware of MRSA and the swabbing of noses before you go into hospitals, and the swabbing of noses of surgeons and nurses to identify a particular organism called Staphylococcus aureus, and its favourite place to live is actually in the nose. It does live in some other places as well, but its favourite place is the nose.
Now, these bacteria that live on us or use us as a mobile transport medium are quite helpful. They secrete vitamins, for example – Vitamin K, B12 - and germ-free animals that are bred without any germs, via a Caesarean section, are deficient and need dietary supplementation to thrive. These bacteria prevent colonisation by nastier bugs, and for example, in a germ-free animal, you only need ten cells of salmonella to infect it, whereas, for us, we need a million cells. Intestinal bacteria produce many, many substances that prevent other bacteria taking over, and they inhibit or kill other bacterial growth and protect us from environmental water-borne diseases.
They also stimulate the development of tissues. We would not develop a normal bowel if we did not have bacteria living in it, particularly the immune system, and probably the systemic immune system's development is driven by our exposure to bacteria, and there is a story you may come across about immunisation of children may actually lead to dysregulation of the immune system as one of the explanations of why allergic diseases are much commoner nowadays than they ever used to be.
Also, they stimulate the production of natural antibodies. Now, as well as all these bacteria, there is a whole heap of viruses, mites, lice, protozoa and worms that also call the human being home, and often do us no harm at all. In fact, the mites on your eyelashes are part of the normal flora of the ocular surface, and occasionally can cause overgrowth and disease, called blepharitis.
I am going to take us back to the pre-antibiotic world, and this is the world of the early Industrial Revolution, in particular to the early-Victorian times, such as the old town of Manchester and "the frightful conditions in this hell on Earth, that everything here arouses horror and indignation". That was Friedrich Engels on "The Condition of the Working Classes".
The Poor Relief Act, the Elizabethan Poor Relief Act of 1601 that was based on two previous Acts in the previous two decades was no longer sufficient for looking after the amount of poor people in the country who were flooding into these nascent towns that were increasing in size exponentially. So, in 1834, the Poor Law was revised, and the new Poor Law came out. It was a pretty good law – it lasted from 1600 to 1834, and, as Robert in the front row here would tell you, that is pretty good for legislation, even nowadays. What they did, very similar to modern thoughts from certain governmental parties: "The provision of the poor relief must be so pleasant it should put off anybody but the most desperate seeking it." Now, this had some unintended consequences, some good and some bad, and we will come onto those later.
A decade later, Edwin Chadwick, also born in Manchester, and, interestingly, his mother died before he was even named, and this reflects this maternal and childhood mortality that is going to be a constant theme through this talk. He was a rather humourless secretary to the new Poor Law, and he reported on the sanitary conditions of the labouring population, and he concluded that urban poverty was not caused by individual immorality but by sickness caused by filth diseases, and he used quantitative methods to directly link poor living conditions and disease and life expectancy, as we can see in this table down below.
Much of poor relief of the country was spent on men who had died from infectious diseases, such as tuberculosis or from the variety of intestinal diseases that caused sporadic epidemics in these filthy, crowded and unhygienic places. In fact, what is striking, if you go to Manchester – I did a year of training there – this is one of the original slums, and these are the original privies that opened directly into the Irwell River. In fact, to rescue someone from this river became quite a sensational event, and if you survived it, as the rescuer, it happened that you might be made the Lord Mayor of Manchester and have a pub named after you, which is actually just a few blocks down the river from this remnant of this old early Victorian slum.
So, these people were being poisoned by their surroundings, by the sewage, the refuse, and it was thought that these poisonous smelly vapours, which they called miasmas, were the cause of this disease, and what we needed to do was to get in fresh water, clean out the smells, get rid of the rotting bodies and the sewage, and things would become better. It was not necessarily popular, and Chadwick had a particular way of saying things that did not enamour him to his colleagues. A letter to The Times says: "We would prefer to take our chance with cholera than be bullied into health. There is nothing a man hates so much as being cleansed against his will. It is a fact that many people have died from a good washing."
The 1848 Public Health Act comes in, which makes it the legal duty of the local authorities to improve the sanitary conditions of the towns, and if they were unable to do so, such as Hereford, they had to call in help to get rid of their slums, to make sure there was access to clean water, and in particular, to get rid of the medieval cesspit system that had been in use for a thousand years.
As well as this chronic and local contagion, there were also problems with infections and plagues in an epidemic fashion. Famine, war, social unrest, poverty, these things killed more soldiers than weapons did, and weakened populations on adulterated food were susceptible to these waves of epidemics. For example, in the measles and whooping cough – and by the way, that is how they spelt whooping cough in those days – in 1838 to 1840, 50,000 deaths in England and Wales alone. If this happened today, there would be a national outcry. In fact, there was a national outcry. People were beginning to realise that this was becoming unacceptable in a modern society. In the 1830s, the term "fever" though meant many different diseases: cholera, influenza, the new fever, which was typhus, which I mentioned before.
The worst outbreak of typhus occurs in London, and this caused the most deaths of all the fevers, in 1837-8. It had been known for a while, previously known as "jail fever" and used to cause outbreaks in Newgate Prison. But, after this epidemic, 16,000 cases a year in England… Unfortunately, it coincided with a smallpox outbreak that killed thousands, again mainly children. Scarlet Fever in 1840 – 20,000 deaths… It still caused deaths when I was a child. I grew up in the Tropics, and a few children died of Scarlet Fever each year. There are other implications with Scarlet Fever, which I will come onto, and many of us of our generation had elderly relatives who had leaky heart valves due to the problems with Streptococcal infections, which was one of the causes of the Scarlet Fever.
And Virchow, the great bacteriologist of his generation, was called out to an outbreak of typhus in Upper Silesia and he recorded that he felt this was "an artificial epidemic due to neglectful social and economic policies" and the medical recommendation was "full and unlimited democracy". "Medicine is a social science; politics is nothing else but medicine on a large scale."
The graph here shows, long before antibiotics were invented, how these infectious diseases were declining in their impact in causing death on societies, dramatic decreases. In fact, by the time we are developing very good vaccines, we have plateaued out and changed the causes of what people are dying of.
Plagues still occur, epidemics still occur, and they still kill people, but they are not killing people in the Western world like they used to. They are killing people in countries that have the same problems that Victorian England had.
Now, remember this thing of adulterated food. Just yesterday morning, on Radio 4, there was something about food adulteration, and, here, it shows an example. It was endemic in the early-Victorian period. "If you please, Sir, mother says can you let her have a quarter of a pound of your best tea to kill the rats and an ounce of chocolate that will get rid of the black beetles." That was in Punch in 1855.
Three years later, a Bradford sweet shop owner ordered Plaster of Paris, which is what he used to adulterate the sugar with, which was expensive, but a novice supplied arsenic instead and they killed twenty people and hundreds were seriously ill. There must have been many cases where adulterated food went undetected.
Which brings me on to death and diet in Victorian England. Because, in eighteenth century France, half the children did not reach the age of two, in nineteenth century England, a third of children died before one year, and most of these diseases were due to infectious disease. The English death rate in the 1740s was about 80 per 1,000. It had declined significantly by the 1840s, despite those dreadful conditions in the towns, and to put this in perspective, it is about the same as what we have in Sierra Leone today, which is the worst death rate in the world.
What did they die of? Well, infection, very big – TB, pneumonia, epidemics, poor sanitation… Accidents were a big cause of death as well, as we can see, and they come under the yellow, the other diseases, and these could be industrial accidents, trauma, burns, railway accidents, a lot of domestic accidents, the same as happens in India and Pakistan today, partly due to dress, with flowing clothes that can get trapped by fires, and partly due to the way they were cooking in cramped conditions on open fires. But the main thing here was infant and mother mortality, and also maternal haemorrhage. The second cause of these circulatory failures is a very tiny little pink blob at the bottom here, and most of this was due to the complications of infectious disease, such as rheumatic fever, causing leaky heart valves and heart failure. Angina does not even appear as a cause of death until 1857 and then it's recorded as "disease of old age". Cancer was rare.
Now, look at the modern day. Look where cancer is now. But, guess what? We still die! And what is interesting, if you take out the childhood mortality, the Victorian person between 1850 and 1880 lived slightly longer, if he was a male, than you do today. So, your life expectancy at five, in England, as a male, in 1870 was slightly longer than it is now, which is an extraordinary statistic, slightly shorter then if you were a female. So, the childhood deaths was the big thing to get rid of. But if you survived these childhood deaths, or you eliminate them, you are still going to die, but you are going to die of things like angina, you are going to die of cancer, you are going to die of other diseases, what we call degenerative diseases, which I have spoken about a lot. But this tells me there is something very fishy going on here because, if these are degenerative diseases, why did they not degenerate of them in Victorian England?
If we look at the diet, and this has been looked at in some great detail, it turns out their diet was actually much better than ours. They had about double the amount of calories that we do. They were thinner than we were and had a much better body mass index, with muscle to fat ratio, but the nutritional value of all of the Victorian vegetables and fruit were much higher than they are today. They also ate far more nutritional cuts of meat, particularly offal, rather than just stripping out the least nutritional parts of the animal. So, in 2011, the life expectancy in Blackpool was 73.8, according to the Office of National Statistics, and the average, even if you take it from age five, is 78. A Clayton & Rowbotham study actually shows that, in Victorian Britain, that was absolutely pretty much identical: 75 for a male, 73 for a female.
So, this diet, where did this come from? It may have come from Viscount Charles, known as Turnip Townsend, who, after a political argument, left the House of Lords. He was originally a Tory, became a Whig, fell out of course when you do things like that with lots of people, and eventually left Government forever to go and set up in the country, where he grew turnips and invented the rotational field system which produced agricultural production in this country immensely, allowing export of food for the first time, and also increasing the nutritional content of the food, as we still do today, using clover to increase the nitrogen in the soils.
All of these advantages were squandered by high sugar intake, margarine and tobacco in the late-1800s. In fact, by 1901 and later on, in the First World War, we had problems recruiting people of correct height and weight to join the Army. You may have noticed that today about the policemen: they do not seem to be – they are younger, but they do not seem to be as big as they were when I was a kid, and I suspect, even at my age, I might be able to outrun some of them as they stagger out of their rather large cars.
Now, I promised I would tell you a little bit about these infections. So, viruses, the small things, they might be small but they are deadly. We have several strains of virus: Influenza C affects humans and pigs – it is a relatively mild disease in humans; B, it is a little bit more severe and affects humans, and, recently, it has been discovered in seals; and Influenza A that infects birds and many different types of mammals, particularly pigs, and it can leak out of birds and pigs into humans. We classify this on H and N, which are the little studs on the outside of the flu virus. Now, the flu virus buds out, taking with it the surface of the host cell that it is in, which gives it a lipid coat, with these little studs of proteins which are these H and N antigens, and we can classify, such as H1, H2, H3, and it goes up to H18, which has just been found in a Peruvian fruit-bat, and then we can combine it with an N, and there are a bunch of different Ns.
The one that we are scared about is H5N1 because this is a flu that is very, very specific to birds, but it leaks out into humans who handle birds, particularly in the Far East, and what really scared people was it went from humans to humans. If this had caused a pandemic, it would have been as lethal as the Spanish flu pandemic, which was H1N1, a swine flu variant, in 1918.
Now, it first came to attention in Fort Riley in Kansas in a recruiting camp, and over five weeks, 46 of the recruits died, out of a thousand infected. A second wave hits the Army in France. The mildly ill of course stayed in the trenches. The sicker soldiers were hospitalised, and they came together, selecting these deadly strains, and it was a very high mortality, 10-20%. Normally, flu would cause a 0.1% mortality, so already this flagged things up, and as soon as you flag things up in wartime or in times of political turmoil, you censor it, so nobody knew this was happening. But Spain was not involved in the First World War, so it was not censored, so hence everybody in Spain was being recorded in the newspapers, this terrible epidemic, and it got known as Spanish flu. There was absolutely nothing Spanish about it whatsoever. It was a global pandemic. We think it spread from birds to pigs, and probably originated in some mammal in the 1880s, slowly evolving into an ancestor virus that diverged in 1913 into two groups, swine and humans, and coming into contact with the pigs, which were much more readily infected and always around Army camps as a food source, spread into humans, causing this global pandemic in 1918 that killed more American soldiers than the War did.
We also see viral infections in the eye, commonly. This time of year, we are seeing the pink eye, and this is what it causes. If you get these spots on the cornea, you cannot see. Interesting, not lethal, can be sight-threatening, temporarily, but there are two things: there is a lethal version – Adenovirus type fourteen can kill people after gastroenteritis; and secondly, there is a link to obesity with Adenovirus type 36, and it turns out that obese individuals in America have a higher incidence of being positive to Adeno 36 than thin people are. So, there is evolving stories about viruses and human health that are also linked to degenerative diseases and mortality. In fact, many cancers of course, we now know, are caused by viruses, such as cervical cancers, papilomas, some leukaemias, some lymphomas, and it may turn out there is a whole bunch more that we have not come across that are either directly viral-related or indirectly supported by viral problems.
Herpes, it is horrible down below, but it is beautiful in the eye. Look at these branching lesions that we see staining in green, hence its name, dendritic ulcer. It is a large virus. We have good treatments for it in the eye. There is one particular devastating complication that can occur, and this is genital herpes in the eye that has been acquired by the child during birth and then it comes out at the age of six or twelve, ripping through the brain and the eye, causing devastating disease and blindness if it is not recognised within days and treated.
Shingles is chicken pox virus. It is also a herpes virus. It is in the same family. It is not directly related. This is the classic chicken pox – of course, you really want to get this when you are a child. If you get this when you are an adult, it is very, very serious and often can be lethal if it causes you pneumonia. We get two types of viral disease in the eye from it. As well as the shingles in the eye here, we can get retinal involvement at the top, and in immune-suppressed people, a devastating retinal disease that causes a complete whiteout of the retina, within days and hours, and that is progressive outer retinal necrosis, or PORN in America, where they like to use acronyms that they think are funny.
Now, HIV brings us onto this because these are the people that we see the progressive outer retinal necrosis in, and the early cases were also associated with another viral disease here called cytomegalovirus, which caused this pizza-pie retina, which was the cottage cheese and tomato splodges – again, American acronym, food on the mind type of thing.
It is estimated there is about 35.3 million people who have HIV worldwide, and deaths running at about 1.6 million, and 6,300 new cases a day. Most of these are in developing economies. They can cause cancer, and here is a lesion on the eye which is actually a malignant tumour.
Bacteria were the next largest thing up, and we can actually see these under the microscope, whereas we cannot see viruses under the light microscope. This is what they look like, streptococci, where were originally called streptos because they were twisted like a chain, and they were first cultured a long time ago, back in 1883. The ones we are interested in medically are these streptococcus pyogenes, or Group A streptococci, which cause a bunch of different diseases, as you can see here, from the strep sore throat to fevers and rashes. They are divided into three groups, depending on how they haemolyse, that is on how they digest in lysed blood, and if we grow the colonies, you can see clear rings where they have lysed the blood into a clear yellowish ring, into a less clear green ring, which is one of the pigment products, and hardly lysing it at all, with the gamma haemolytic ones. The most important problem with strep is what it does afterwards, and what it does is lead to a bunch of different things that can cause problems with kidneys and they can cause problems with the heart, and these can be slowly progressive and horrible, and, as well as affecting quality of life, can also lead to death.
Now, some strep of course are good, and that is one of my favourite cheeses down there that relies on three different types of strep to make it, and two of them make the sugar that the third one makes into the bubbles, which is carbon dioxide bubbles, and Emmentaler cheese is there for your delight to show you that not all bacteria are bad. Even bad bacteria, on occasions, can be good, and if you are a bacteria of course, like a boy, you are always good – it just depends on which environment you are in and who is calling you good or bad.
This one was bad. This one was Puerperal fever. You may have heard about washing hands in hospitals reduces infections. You may have heard of somebody called Semmelweis, who is credited with having invented this simple thing. Unfortunately, he was not the guy who invented it. It was invented in the previous century by Alexander Gordon, an ex-naval surgeon, who reported on the epidemic of maternal deaths in Aberdeen. He did a table and he found out who was responsible for spreading these diseases, and guess what, it was the doctors and certain midwives who were passing it on, and then he realised that what you needed to do here was to wash your hands and fumigate your clothes, so no more going in with filthy old clothes. He reported this.
It was not received particularly too well, and in fact, there was a whole epidemic of building these things, which were maternal hospitals, in those days, and they were built to help the morals of the poor women, who otherwise would have to deliver at home and could not afford to deliver at home because it was expensive. Unfortunately, these things became a hotbed for spreading this horrible infection, and what Semmelweis found was that one of his hospitals had a very high incidence of mortality, between 5-30%, and only 4% in the other clinic, which did not train midwives and medical students. So, the midwives and the medical students were transmitting, particularly the medical students, from the autopsy wards, where they were doing autopsies, then they were coming straight out into the maternal wards and delivering the babies – they were passing on the streptococcal infection, and mothers were dying. In fact, there are stories of mothers begging on their knees and crying, saying "Please do not take me to Hospital A, take me to Hospital B or let me deliver my baby in the street," because they knew what was going around. He published this, indirectly – one of his juniors published it, in 1848, at the Royal Medical & Surgical Society, and it was published in the Lancet.
At that time, there was a lot of unrest in Vienna. His native Hungary was going into revolution and wanted to be released from direct control from Vienna, so he eventually leaves and goes to Hungary, thinking that he is going to face less prejudice there and be able to get a better position. He eventually does write "The Etiology of Childhood Fever", but he is incensed because he is ignored. He is overlooked for jobs. He is not helped by the fact he goes around calling his colleagues murderers, and eventually, all of this drives him mad and his wife reports him to his friend for behavioural change, where he has taken into asylum at the age of 47 and is beaten, and dies immediately after this beating by the guards. Interestingly, there is a Hungarian proverb and it says: "Tell the truth and people will bash in your head". Is that not extraordinary?
Staphylococci are the ones that we all need to know about because these are the ones that are allegedly infecting the NHS, but they are also out there in the community and in fact, some of the drug-resistant forms of Staphylococci in the community are more dangerous than some of those in the hospital. They are a bunch of grapes, they are single individual things, they cause a whole heap of nasty infections of the skin and sties and skin infections, as you can see from this poor character here, with a large puss lesion in his eyelid, and they become dangerous, these bugs, because they can form and manufacture a number of different things. Some of them stop the immune cells killing them. Some of them allow invasion because they cause digestion of proteins. Some of them cause puss formation. And some of them, if they get on your food, form a type of food poisoning that causes immediate vomiting from enterotoxins A and G. So, this is not, the Staphylococci, this is not the one where you have not cooked it and you have not cooked it properly. This is properly cooked, but it has been handled with Staphylococci that make these enterotoxins and not heated properly and they form in the fridge and in the kitchen, these toxins.
Talking of gut diseases, we come onto gram-negative bacteria. Salmonella. I have often been asked why it is called salmonella – is it in salmon? That is not the reason. It was discovered by somebody called Daniel Salmon, who was a nineteenth century veterinary pathologist. In fact, it is found widely in the environment. It is found in animals, both cold and warm-blooded, so, handle your turtle with care, and it is a major cause of food poisoning, and it is quite a nasty food poisoning. There is a subspecies of this, salmonella typhi, that causes a food poisoning that may be fatal.
In fact, Typhoid Mary, here, on the left, Mary Mallon, was an asymptomatic carrier. She had no problems herself, although she was carrying salmonella typhi, and she infected 50 people, and at least three of those people died. She was forcibly isolated by the public health authorities, and the second time, she never was allowed out of hospital and dies in hospital. It caused panic in New York at the time. She knew that she was carrying this and disappeared, but took no responsibility for it herself, feeling that this could not possibly be her, and it is a failure of food hygiene. This cartoons and drawings came round to teach food preparers how to do this healthily.
Talking of salmonella, we might remember Edwina, here, who is forced to resign in December 1988 because she said the following thing: "Most of the egg production in this country, sadly, is now affected with salmonella." Immediately, sales of eggs declined by 60% and four million hens are slaughtered over the next two weeks. Many, many farmers and egg producers are furious and so she has to stand down.
Well, let us have a look and see what was going on there. In 1997, there were 22,000 cases of food poisoning caused from poultry. By 2005, this had declined to 6,000, and by 2010, it is just down to a paltry 581, and in fact, now, salmonella in British eggs is one of the lowest in the European Community, much less than Germany and far less than Spain.
The US decided not to vaccinate their chickens, and, just in 2010, had to recall 550 million eggs from two farm producers in Iowa that had caused a nationwide outbreak of thousands of cases of salmonella poisoning, traced back to the eggs on these farms.
So, Edwina Currie, a controversial figure, had actually highlighted something that was important: too many people in this country were getting severe diarrhoea and time off work and severe complications because of the egg production. She was absolutely correct to have said that. I think, really, it is like I pointed out with Chadwick, is it now? It is how you tell them. This was probably not the way to do it, announcing it on mainstream TV, before discussing it with your senior political colleagues and your infectious disease experts.
There is another gram-negative bug called e-coli that you will have heard of. It resides in our gut and represents about 0.1% of our normal flora. Mostly, it is harmless, but if you infect it or give it an extra bit of DNA that can make toxins, particularly verotoxin, it becomes lethal, and this e-coli can kill people, as we know from the infections from the butchers' pies, what was it, two or three Christmases ago, as we know from epidemics, and it causes renal failure, haemolytic anaemia and very low platelet counts. So, there are about a thousand hospitalisations and 30 deaths each year from this bug in the United States. Unfortunately for us, it lives in the intestine of cattle, causing no problem at all, so it can easily get contaminated into milk that is not been pasteurised. It can get into the food source if you are making a hamburger and you do not clean the inside and make sure the inside of the hamburger is heated. So, if you are eating a rare hamburger in the United States, you are running a risk if the meat is infected with this particular bug. This is an issue, and you see warnings in every hamburger stall now, being America, that says you may suffer intestinal disease if you eat your hamburger meat rare.
Which brings me onto Cholera. Cholera is the most devastating of all of the infectious diseases in pandemics, sweeping over from Bengal. This is one of the first victims who died in the first epidemic in England, and this is her appearance immediately after death, but more striking, this is a portrait of someone in Venice, in the morning, and this is a portrait in the afternoon, and she was more than likely dead before bedtime. It was rapid.
When it struck in the second epidemic, Soho, 200 people died overnight, 600 died over the next ten days, and of course, this dramatic spike in death rates required some explanation, and the local GP, John Snow, eventually, by drawing this map, identifies the epicentre is due to this pump, from which he took the handle off. Now, the rest of London was supplied by piped water. This area was poor. But what he also identified was there were two companies in London: one of them was supplying from this pumping station here, the Southwark & Vauxhall Water Company, supplying untreated Thames water, which was causing cholera in the people they were supplying, in their customers; the other water company took it from above the tidal reaches of the Thames, where they did not get cholera. This cartoon shows, under a magnifying glass, what they were actually drinking.
There is another type of bacteria that is really important, both in warfare and also, because this bacteria, which actually lives in cells – this one cannot live free, it can only live by being passed from human to human, and it is called Egyptian ophthalmia. This was the reason that Moorfields Eye Hospital, Manchester Royal Eye Hospital, and various other eye hospitals were founded in the early-1800s. So, this bug is of particular importance.
It is also of importance because it caused the change in the outcome of several battles and in fact of the war in Egypt. As we know, Napoleon arrives in Egypt on July 21st and, after marching up from the coast, has the first Battle of Pyramids, where his large squares stand firm against the unexpected, unprovoked charge of the fearsome Mamluk cavalry. He then enters into Cairo and, immediately, his troops are all struck down with blindness, this horrible disease, and in fact, he has to set up a hospital specifically for them by the Pyramids.
Now, this ophthalmia had been known for a long time in Egypt. In the fourteenth century, an Egyptian oculist, he had noted that the inhabitants of Egypt were much more frequently attacked than people due to the abundance of dust. Other people, they called Egypt "the land of the blind". It was certainly a very, very important disease, and in fact, was described by European physicians of the time.
This is what it did. It caused an acute inflammation where the lining of your eye came out. When it goes back in, it is left so much scarring that your corneas go opaque and you cannot see anymore.
Several people wrote up whole dissertations on this, including Paul Gachet, who you see here. Now, he is more famous because he was the subject of Van Gogh's painting, and he was Van Gogh's last physician, and this was one of the most expensive paintings ever sold. But he is the guy who wrote the book on Egyptian ophthalmia, which unfortunately was never published and distributed widely.
Trachoma is tremendously important in the foundation of the American myth. These people are all Indians, on Indian Reservations. You can see, the father is blind, both the children are blind – this was devastating. Furthermore, the Americans thought that it was a better idea to put Native Americans into boarding schools rather than leave them out on the plains and occasionally go and shoot them and murder them in other ways, so they did the dutiful thing and moved them. And what happens in these institutes? They get TB and they also get trachoma, at massive rates. This is what they are trying to do: they are trying to convert what they euphemistically, in those days, called the Red Indian, from that product into this product, which is a good wholesome American, and that boarding school was the idea of it, and hundreds, thousands of these children were banged up.
There was one small advantage of this, was that, from the Proctor Foundation, one of the doctors there discovered that a newly marketed dyestuff, used as an antibiotic, called protosil, could cure this disease, and so he did.
But it was not just America. You go to Australia and here is Fred Hollows, who is probably the hero of any ophthalmologist you have ever seen, trained in Dunedin, goes to Moorfields, goes back to Australia, and is shocked by the poverty and disease that is present in the native Australians and sets up a number of schemes to actually reduce the impact of trachoma, this disease.
This is the grading, the simplified grading system we used in the field for doing the treatment. So, you have got a normal conjunctiva, with just grading under the top lid. You can see the early infectious, the lumpy, bumpy bit, the collections of the white cells, and then the scaring, and then, finally, the corneal scarring disease that is caused by the eye turning in and the sharp skin lid edge continuously scratching your eyes. It is like, they say, having eyelids lined by barbed wire. It is not only blinding, it is also painful.
And it turns out that what Hollow said was correct: it is transmitted by flies. Now, this was controversial, and still remains controversial, but you will notice that, way back in the Egyptian times, they knew that flies were a problem associated with this disease, and this has now become an important component of control of this disease to prevent the devastating blindness of here. The strategy is now, they use, is called the Safe Strategy, which is to provide surgery to correct these eyelid abnormalities, antibiotics to prevent the infection, controlling face washing, and environmental change, which is to control the flies.
This is the first bit, the surgery, to invert the lids, with a straightforward operation, done with a spoon and some large sutures, and I was amazed how very good the African surgeons are at doing this. This is an operation that I thought was ghastly and I would not personally do – no problem, see one, do one, teach one, and off we went, and they were just bashing these operations out at a fair rate of knots, ever hour. Face-washing is part of this, and environmental control.
We also have a version of chlamydia in England as well, which is venereal chlamydia, and this can be a significant problem. It tends not to be as blinding but can cause cicatrisation – that is scarring.
Fungal diseases can cause a bunch of not only eye diseases but also systemic things. These tend to occur in people who are immunio-suppressed, who are having transplants or other reasons, severe illnesses.
I wanted to move on to these protozoa because we have a former Professor of Physic present with us who worked for many years in the London School of Hygiene and Tropical Medicine. These are little single-cell tiny animals. They are the principal hunters and grazers of the microbial world, and we will be familiar with them because of malaria. This is Ross's page, from which he was so excited about in India when he jumped up from his lab and he had discovered the intermediate lifecycle of malaria which gave the clue as to where this was coming from – from the mosquito bites, and this was the blood from someone who had been infected by the mosquito, and what we are looking at is the middle stages there, when the, what are called, the sporozoites get into the host. Now, the main thing that happens is, these things have a lifecycle: they have bits when they are not active, when they can be a cyst, and that can either be outside the body, in the park; and then they have got an active bit, which is the trophozoite, which is their feeding bit, which can go on and cause disease, and also when they can go on and form the sexual aspects of their lifecycle to move on.
Malaria, of course, is a fabulously important disease, and it kills lots and lots of people. We used to have it in England. In the 1590s, John Norden, that famous cartographer who did the map of London, complains about the poor conditions in English marshlands and the fevers. But the biggest thing is here is that 660,000 people are killed by this disease, mostly African children. So, now what we are beginning to understand is we do still have the same as the Victorians had, the early-Victorians – it is just it is not killing us, it is killing Africans, mainly, and other people in other developing economies, and they have exactly the same problems that we had 200 years ago. We now know the answer to this, and it is not bumping in lots and lots of antibiotics; it is bumping in lots and lots of things that are going to improve their economy and improve their sanitation and their diet.
I would love to talk to you about Jesuit bark and quinine, which was brought over and became the mainstay of treatment, but newer drugs are coming from China, also derived from plants, particularly Artemisia.
The importance of the eye here is that cerebral malaria is what these children die of, and they come in with a coma. One third of comas in children are not caused by malaria, so you do not want to give them the malarial treatment. How can you identify the difference when you do not have things like CT scans and so on when you are in the bush? You take out your ophthalmoscope, and if you see this, it is malaria. This was an important, very straightforward, fantastic study done in the field, with minimal cost, and it is saving the lives of thousands and thousands of children.
You will have heard of this disease called toxoplasmosis because it is one of the commonest infections of protozoa there is in all humanity. In some parts of the world, 80%, 90% of humans are infected with this, and it does none of them any harm, unless it gets to the eye and reactivates and breaks out, causing this devastating inflammation, and in fact it becomes the commonest cause of posterior uveitis in the entire world, and one of the commonest causes of blindness due to infection.
Its native host is this, Ctenodactylus gundi, which is the comb-rat. Ctenodactylus means comb-like paws, so it has got paws that it can scrape itself with, which was used for Leishmania research in the Pasteur Lab in North Africa, which is where they discovered it. It lives in a cycle between cats, mainly, and cats can excrete it. It can be picked up by the humans. If that human is pregnant, she will pass it on to her foetus. The foetus will then get a congenital infection or may abort, if it is in the first trimester of pregnancy.
It lives in the tissue of animals, so if you eat steak tartar in certain countries where this is endemic, or undercooked meats, then you are highly likely to get infected. It probably will not do you any harm. And there are loads of Spaniards walking around with toxoplasmosis, who have had no illness whatsoever in their lives of any significance. But, it is this, when these cysts hatch out in the eye or the brain, that cause the problems, which neatly brings me on to worms, which are not really microbes, but there is one stage of their lifecycle that is, and that is their larvae, which we call microfilarae. Some of these, we pick up from cysts in the park, some of them, we pick up from being bitten by flies, and they have quite a complicated lifecycle, and some of them are quite beautiful things.
If you look here, here is a cluster of roundworms in the dog intestine. When these actually secrete their larvae, they migrate through the dog, and some of them are excreted as cysts into the park, where young footballers can pick them up. Now, they are not exactly going to eat the poo themselves – that is called coprophagia – but what they do is they eat or lick their hands, because they are boys and they do not wash their hands, after they have been playing football, and the ball has been in the dog poo. So, this is why they do not allow dogs to defecate in parks in civilised societies, and why you should be requested to pick up after your dog, because if you do not, these children have a risk of getting visceral larva migrans, and if that larvae migrates into the eye - and you can see the track in this child where it has gone through they eye, bump-bump-bump-bump, you can see, as it is wriggling along, causing local inflammation, and eventually, it ends up causing this vast inflammatory mass where it is walled off to stop it going further and causing further damage.
There are other worms that migrate through. This is the African eye worm – look at it, horrible thing! – loa loa, transmitted by something that is even more horrible, because you know when these flies bite you! This is a mango fly. It is really painful. They live in the forest but they bite in the savannah, so you have got no chance here – mosquito nets are not going to protect you, Deet is not going to protect you. You have got to walk around dressed up, with long trousers, long hands, gloves and hats, 24/7, or do not go on holiday to the areas striped in red, and you can go to other places instead. If you get this infection, it lives in the blood for a while and then migrates through. It does not cause much harm unless it gets into eye. It has been known about for a long time. On the African slaves on the French ships, we have a record of Dr Guillard removing one of these, as you can see here, from the eye in a slave, where it was seen wriggling around.
Now, there is another worm that is much more important because it causes more widespread blindness, called onchocerciasis or river blindness, transmitted by this fly, called Simulium damnosum, very appropriately named, and it is economically important because you can see the older people go blind. The younger ones are not blind yet, but they are destined to go blind, and whole villages and areas are depopulated, costing Africa millions and billions of pounds over decades, so something had to be done to prevent this load. What they did was to do the blackfly control programme, where they went around and sprayed to get rid of the blackfly early larval stage, before they hatched out, and to catch the early emerging blackfly, as you can see here. This did pretty well, but unfortunately, environmentalists do not like rivers being sprayed with pesticides, so this was officially closed. Secondly, this may not have been as benign and nice as you thought because this involved helicopter contracts with private American helicopter factories using pesticides from private American firms, all subsidised and paid for by the American Government, so it is one of those things where aid circles round a little bit and comes back in, but nevertheless was certainly effective.
Unfortunately, it was closed in 2002, so now we need a different way of controlling this, and this is with this ivermectin, which is the wonder drug, and in fact, if you can be sure you can distribute this drug properly, the company will give it to you for free – is that not amazing?! Now, you have got to treat the people for a long time, not very often, probably optimally about twice a year, but you have got to treat them for the lifetime of this worm, and this worm lives in you for fifteen to twenty years. So, got to case-source all these people and treat them because this tablet does not kill the worm, it kills the babies, the larvae, and they are being secreted in a cycle and they are in the tissues. This drug achieves pretty much 100% death of them.
These old pictures that we see of river blindness will hopefully be a thing of the past within one generation. Unfortunately, there is some resistance to ivermectin, and there are some problems with it.
Antimicrobials, I think we will probably deal with in more detail on another occasion, but I would like just to briefly go through, in the last few minutes remaining, to point out how people from history have used arsenic, which is very, very poisonous. It was known as the "poudre de succession" because Britannicus, the son of Emperor Claudius and the person who should have inherited the throne, was murdered by Locusta, who was hired by his stepbrother Nero. So, here is Nero practising out the poison on a slave, and Britannicus takes the poison by a devious route, because he had a wine-taster, but he had his wine diluted because it was too warm, with water, and the arsenic was in the water, which had not been tasted, and that is what killed him.
Now, organic arsenic was then invented because it was safer and was not as poisonous. It was still pretty poisonous, but salvarsan was called "the arsenic that saves". Here we have the inventor, Paul Ehrlich. There was a lot of controversy over this at the time, but it was the only treatment, and here are the advertisements: he took his syphilis shots every week until cured, which was a long time and very painful; he did not, so he ends up blind and debilitated. This is a recurring story that we are going to hear with other drugs, and particularly going on with sulphonamides.
Sulphonamides were a German invention, and they mimic something in the body that is needed for folates. They saved the life of Churchill – not penicillin. So, a German invention saved Churchill's life – is that not extraordinary? And many Americans as well because they are used on the battlefield, but more Americans are saved because they were used as prophylactic treatment for VD, and it says "Prophylaxis prevents venereal disease – so fool the Axis, use prophylaxis" and here are the Axis leaders coming up in this cartoon, and what you had to do was to use this prophylaxis if you were going to be infected.
Penicillin was the next antibiotic after sulphonamides to come along, and here is this lovely thing, "Thanks to penicillin, he will come home." But of course, penicillin was not used for this. Penicillin was far too precious to use on battlefield wounds. It was going to be used for treating this disease, which of course was the great pox, a night with Venus and a lifetime with mercury.
I am not going to read the slides to you, but if you want some more information, it is on the left-hand side, so I will just run through the story. The big problem that we had in the First World War was the VD rates of British soldiers were seven times that of the Germans, and many more times that of the French, both of whose armies had a system of organised regulated brothels, which the British refused to admit there was even a problem, and the morality of the high-ranking English generals felt that this actually was a Christian war. Now, in 1915, these Christian soldiers in one street were found, in one week, to visit 171,000 visits to the brothels, which were the unregulated ones. The regulated ones were the ones with these lamps outside. They were red for the general men and blue for the officers. They were too expensive for the English troops to afford and they were pretty much discouraged from going to them as well. Also, they were paid less, you see – they were paid three to five times less than the Australian, New Zealand and Canadian troops, who had a VD rate five times that of the British troops, which is extraordinary. So, you have got 20% of Canadian troops arriving on the battlefield unable to fight because they have gonorrhoea, and the treatment for gonorrhoea, as we saw, was those horrible injections that is going to go on and on and on for ages because prontosil and penicillin are a long time coming.
The French had a better system, setting up the Maison du Tolerance, which was organised, so the VD rate was low there because, if you were infected, or if you were a prostitute who was infected or had obvious symptons or obvious discharge, you were not allowed in. So, just very briefly, the story here was a General trying to help out this epidemic with the British troops, writes to his opposite number and says "I have opened an establishment just behind your front lines and I am going to offer this to the British and Commonwealth soldiers on Tuesdays and Thursdays," to which his opposite number wrote back and said, "This is completely unacceptable – we are British and we cannot possibly degenerate to this level," to which the reply came back, "Ah, of course, my General, I am terribly sorry, I meant to say it will be open to you on Tuesdays, Thursdays and Saturdays!"
With that, I will stop at this juncture, and, with my next lecture, which is going to be on new aspects of vision and vision-related things, I will deal with the emergency of antibiotic infections and new antibiotics at that time.
© Professor William Ayliffe, 2014
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text:
Дерягино — деревня в Вашкинском районе Вологодской области.

Входит в состав Киснемского сельского поселения, с точки зрения административно-территориального деления — в Киснемский сельсовет.

Расстояние по автодороге до районного центра Липина Бора — 46 км, до центра муниципального образования села Троицкое — 22 км. Ближайшие населённые пункты — Гаврилово, Лунево, Семенчево.

По переписи 2002 года население — 8 человек.

В Дерягино расположен памятник архитектуры жилой дом А. Н. Харламова.

Примечания

Ссылки
  в реестре населённых пунктов Вологодской области

Населённые пункты Вашкинского района

redpajama_set_name: RedPajamaWikipedia


text:
\section{Introduction}
\label{sec-introduction}

Ion trapping by radiofrequency (rf) electric fields was first demonstrated in 1954 by Paul in a trap with hyperbolic electrodes \cite{paul90}.  Attempts to maximize the field-free trapping volume led to the development of the linear Paul trap by Prestage in 1989 \cite{prestage89}.  Although less harmonic than hyperbolic traps, the field-free central axis of linear traps often make theirs the preferred geometry for experiments involving multiple laser-cooled ions, including precision spectroscopy \cite{rosenband05, koelemeij07}, quantum information processing \cite{haffner05, hume07}, and cavity quantum electrodynamics applications \cite{herskind09}.

When the thermal energy of a trapped ion cloud is cooled sufficiently below the repulsive Coulomb interaction energy, the ions settle into a Coulomb crystal with geometry defined by the trapping potential \cite{willitsch08, setsuo82}.  Crystallization of certain atomic ion species with closed-cycle electronic transitions can be accomplished directly by laser cooling \cite{drewsen98, removille09}; crystallization of other species (e.g., molecular ions) can be accomplished sympathetically by laser cooling of a co-trapped atomic species \cite{blythe05} or \emph{in situ} formation from a pre-cooled reactant \cite{molhave00}. Crystallization can be prevented by various heating mechanisms.  Two such heating mechanisms, \textit{micromotion heating} and \textit{nonlinear resonance heating}, become more problematic as a crystal grows, due to sampling of larger rf fields and of nonharmonic regions of the trapping potential, respectively. Thus, these heating mechanisms can limit the minimum temperature obtained for a given crystal size and also the maximum obtainable crystal size. Our work is motivated in part by recent difficulties growing large Coulomb crystals in a non-standard trap geometry, leading to speculation that axial rf fields near the endcaps (fringing fields) might limit crystal size \cite{kielpinski10}.

In this paper, we compare the suitability of four linear Paul trap designs for growth and study of large Coulomb crystals.  First, we compare the total micromotion (the unavoidable driven motion at the rf frequency) which results in non-resonant transfer of rf energy into thermal energy of the trapped sample (micromotion heating).  Second, we compare the component of micromotion along the trap axis (the axis of smallest micromotion), which induces first-order Doppler shifts in precision spectroscopy on large Coulomb crystals. Finally, we compare trap anharmonicity, which can lead to excitation of nonlinear heating resonances as a crystal grows and its normal mode frequencies are shifted onto a resonance. To characterize these anharmonicity-induced frequency shifts, we use a simple non-deforming crystal approximation, which allows a straightforward estimation of when crystal growth can lead to excitation of different nonlinear heating resonances.

\section{Trap Designs}
\label{sec-trapdesigns}

A linear quadrupole ion trap (Fig.~\ref{fig-trap}) is formed from four parallel cylindrical electrodes of radius $r_e$, held at a separation $r_0$, with an rf voltage signal applied to the electrodes.  The resulting electric field confines the ion motion radially; axial confinement (in the $z$-direction) is provided by endcap electrodes (not shown in Fig.~\ref{fig-trap}), to which a static voltage is applied.

Near the trap center, the electric potential is given by
\begin{eqnarray}
\label{eq-potlatcenter}
   \phi (x,y,z,t) &=& \frac{V_{rf}cos(\Omega_{rf}t)}{r_0^2}(x^2-y^2) + \nonumber \\
   && \frac{\kappa V_{ec}}{z_0^2}(\frac{2z^2-x^2-y^2}{2})
\end{eqnarray}
where $V_{rf}$ and $\Omega_{rf}$ are the voltage and frequency of the applied rf drive, $V_{ec}$ is the static voltage applied to the endcap electrodes, $z_0$ is half the distance between the endcap electrodes, and $\kappa$ is a geometric factor.  While this description is accurate near the trap center and along the $z$-axis, significant deviations can occur elsewhere in the trap interior.

\begin{figure}
\includegraphics[width=85mm]{fig1.jpg}
\caption{\label{fig-trap} End view of a linear quadrupole trap.  A sinusoidally varying voltage is applied to the rod electrodes, with one pair held $180^\circ$ out of phase with the other.}
\end{figure}

The radial motion of an ion in the potential of Eq.~\eqref{eq-potlatcenter} is described by the Mathieu differential equations, and the stability of this motion is expressed using Mathieu parameters which depend only on $V_{rf}$, $V_{ec}$, $\Omega_{rf}$, $\kappa$, and the charge-to-mass ratio $q/m$.  Solutions to the Mathieu equations contain regions of parameter space where ion motion is stable \cite{mclachlan47}, and trap parameters are chosen to operate in one such stable region.

The motion of a trapped ion can be described approximately as a superposition of micromotion, in which the ion oscillates at the rf frequency, with secular motion, in which the ion oscillates in a time-independent pseudopotential $\tilde{\phi}_{rf}(x,y,z)$ at slower secular frequencies $\omega_x$, $\omega_y$, and $\omega_z$.   An analytic relationship exists between the time-dependent rf potential and the time-independent pseudopotential \cite{dehmelt67}:
\begin{equation}
\tilde{\phi}_{rf}(x,y,z) =\frac{q}{4m\Omega_{rf}^2} |\nabla{}\phi_{rf}(x,y,z,t=0)|^2
\end{equation}
The endcap electrodes generate a potential $\phi_{ec}$, and the total potential governing secular motion is
\begin{equation}
\phi_{trap}=\tilde{\phi}_{rf}+\phi_{ec}.
\end{equation}

We compare four different linear trap implementations (Fig.~\ref{fig-trapgeoms}, with corresponding voltage given in Table~\ref{tbl-voltages}): a conventional in-line endcap design with two different voltage configurations (designs $A$ and $B$), a plate endcap design (design $C$, similar to a design analyzed in Ref. \cite{pedregosa10}), and an original rotated endcap design (design $D$).  In all trap designs, $z_0$ = 8.5 mm,  $r_0$ = 4 mm, and $r_e$ = 4.5 mm, with $r_0$ and $r_e$ chosen to agree with the optimal ratio $r_e$=1.14511$r_0$ for minimizing the leading-order contribution to anharmonicity in the rf potential \cite{reuben96}. Design $C$ uses thin circular plates of radius $2r_e+r_0$ as endcap electrodes, with a 2 mm radius hole left for axial optical access.  Design $D$ has four small cylindrical endcap electrodes of radius 1.75 mm, rotated 45$^{\circ}$ around the $z$-axis; the central axes of these endcap electrodes are on the same radius as the central axes of the rf electrodes. Our group has built a functional trap whose key geometric features are shared with design $D$, currently being used to trap $^{138}$Ba$^{+}$ ions.  All numerical simulations in this paper use values corresponding to $^{138}$Ba$^{+}$  ($q=1.602\times10^{-19}$ C, $m=2.292\times10^{-25}$ kg) and our operating rf frequency ($\Omega_{rf}=2\pi\times3.00$ MHz).

\begin{figure}
\includegraphics[width=85mm]{trap_geometries.jpg}
\caption{\label{fig-trapgeoms} Trap designs analyzed in this
paper.  Designs $A$ and $B$ share the same geometry, but differ in applied voltages.  In the offset view at right, designs $C$ and $D$ are shown with the nearest plate endcap pulled away from the rest of the trap to allow the trap interior to be seen. }
\end{figure}

\begin{table}
\caption{\label{tbl-voltages} Operating voltages applied in different trap designs. Here $V(t)=+V_{rf}cos(\Omega_{rf}t)$.}
\begin{ruledtabular}
\begin{tabular}{ c  c  c  c  c }
 & $V_1$ & $V_2$  & $V_3$ & $V_4$\\
 \hline
 $A$ & $V(t)$ & $-V(t)$ & $V_{ec}$ & $V_{ec}$ \\
 $B$ & $V(t)$ & $-V(t)$ & $V_{ec}+V(t)$ & $V_{ec}-V(t)$ \\
 $C$ & $V(t)$ & $-V(t)$ & $V_{ec}$ & $N/A$ \\
 $D$ & $V(t)$ & $-V(t)$ & $V_{ec}$ & $N/A$ \\
\end{tabular}
\end{ruledtabular}
\end{table}

\section{Computation of Trapping Potentials}
\label{sec-compmethod}

Our metrics of trap design comparison, presented in Sec.~\ref{sec-mmotion} and Sec.~\ref{sec-comfrequency}, are calculated from the trap potentials and their gradients.  The potentials $\phi_{ec}$ and $\phi_{rf}$ are found using the finite element method to solve the Laplace equation, with the trap geometries discretized by the meshing software Gmsh \cite{geuzaine09}.

To allow evaluation of the potentials at locations not on the mesh, the numerical solutions are fitted to an expansion in associated Legendre polynomials
\begin{eqnarray}
\label{alpseries}
\phi_{fit}(r, \theta, \phi) = \sum_{l=0}^{l_{max}} \ \sum_{m=-l_{max}}^{l_{max}} C_{lm}\ r^l\ P_{l}^m(cos~\theta) \nonumber \\
   \times
   \begin{cases}
      cos(m\phi) & \text{for $m \ge 0$} \\
      sin(m\phi) & \text{for $m < 0$}.
   \end{cases}
\end{eqnarray}
We exclude terms with odd values of $l$ and $m$ from our fitting due to symmetry.

An analytic form of the gradient of Eq.~\eqref{alpseries} is straightforward to find using
\begin{eqnarray}
\label{alpgrad}
\lefteqn{ \frac{\partial P_{l}^m(cos~\theta)}{\partial \theta} = }  \nonumber \\
&& \frac{l \: cos\, \theta \: P_{l}^m(cos~\theta) - (l+m) P_{-l}^m(cos~\theta))}{\sqrt{1-cos^2\theta}},
\end{eqnarray}
which can be derived from a more general identity given by \cite{abramowitz64}.  Fitting the Laplace solutions to the above expansion thus permits efficient evaluation of $\phi_{fit}$ or its gradient $\nabla \phi_{fit}$ at arbitrary location from the coefficients $\{C_{lm}\}$.  In this way, we determine $\tilde{\phi}_{rf}$ and $\phi_{ec}$ up to the scaling set by choosing $V_{rf}$ and $V_{ec}$.  Our choices of these voltages, shown in Table~\ref{tbl-kappas}, are made to equalize the single particle secular frequencies $\omega_z$ and $\omega_r$ across all designs at experimentally reasonable values $\omega_x=\omega_y=2\pi\times418$ kHz and $\omega_z=2\pi\times18.9$ kHz.

Single particle secular frequencies are calculated by evaluation of $\tilde{\phi}_{rf}$ and $\phi_{ec}$ along a radial or axial trace through the origin from $r$=$-\frac{r_0}{2}$ to $r$=$+\frac{r_0}{2}$ or $z$=$-\frac{z_0}{2}$ to $z$=$\frac{z_0}{2}$ respectively.  The resulting values fit a polynomial whose quadratic coefficient rapidly converges with finer discretization of the domain or increase of the polynomial fitting order.  From this coefficient $C_2$, we also find the values of $\kappa=\sqrt{2qC_2/m}$ in Table~\ref{tbl-kappas}.

\begin{table}
\caption{$V_{rf}$ and $V_{ec}$ used in comparisons and calculated value of $\kappa$ for each trap.}
\begin{ruledtabular}
\begin{tabular}{ c c c c }
 Trap Design & $V_{rf}$ (volts) & $V_{ec}$ (volts) & $\kappa$ \\
 \hline
 \ A & 800 & 5.0 & 0.15 \\
 \ B & 800 & 5.0 & 0.15 \\
 \ C & 800 & 2.7 & 0.27 \\
 \ D & 800 & 51 & 0.014 \\
\end{tabular}
\end{ruledtabular}
\label{tbl-kappas}
\end{table}

The effect of inaccuracy in the Laplace solutions is estimated by observing convergence of calculated values such as $\kappa$ with increasing meshing density, which is found to cause variation in all reported results of less than a few percent unless otherwise noted as we approach the highest mesh densities eventually used -- of order $10^6$ nodes for all designs.  We discuss the implications of the resulting uncertainty in Sec.~\ref{sec-discussion}.  Tuning of other parameters, such as the fitting order in Eq.~\eqref{alpseries}, is found to cause negligible variance by comparison.

\section{Micromotion}
\label{sec-mmotion}

Although radial and axial micromotion vanish, respectively, along the trap axis and at $z=0$, a large crystal necessarily extends beyond these regions. Micromotion heating results from the transfer of micromotion energy into secular energy, which leads to an elevated secular temperature. Although the scaling is non-trivial, simulations show that the micromotion heating rate strongly increases with micromotion amplitude and with secular temperature \cite{schiffer00, ryjkov05}. As a crystal grows, additional ions are held at locations with increasingly strong rf fields, causing them to experience increasingly large micromotion. The resulting heating limits the minimum obtainable temperature for a crystal of a given size, and eventually prevents further crystal growth.

The electric field created by the rf electrodes may be written as
\begin{equation}
\overrightarrow{\mathbf{E}}(x,y,z,t) = \overrightarrow{\mathbf{E}}_\mathbf{0}(x,y,z) cos(\Omega_{rf} t).
\end{equation}
For an ion held in place in a stationary crystal, in the limit of small micromotion amplitude $\overrightarrow{\mathbf{E}}_\mathbf{0}$ may be taken as constant as the ion moves, and the micromotion of the ion is harmonic with amplitude
\begin{equation}
A(x,y,z)=\frac{q}{m\Omega_{rf}^2}|\nabla \phi_{rf}(x,y,z,t=0)|
\end{equation}

\begin{figure}
\includegraphics[width=85mm]{total_mmotion_ampl_16-eps-converted-to.pdf}
\caption{\label{fig-totmmotion} (Color online.) Contour plots of total micromotion amplitude, labeled in microns. Designs $B$ and $D$ are indistinguishable at this scale.}
\end{figure}

With $V_{rf}$ as set in Table~\ref{tbl-kappas}, we find $\phi_{rf}(x,y,z,t=0)$ and calculate the resulting total micromotion amplitude $A(x,y,z)$.  The resulting micromotion contour is plotted in Fig~\ref{fig-totmmotion}.  All designs are comparable in the amplitude of total micromotion for any realistic trapping volume.  Interpretation of this result is discussed in Sec.~\ref{sec-discussion}.

We also compare the size of the $\hat{z}$ component of micromotion in each trap, which can limit the precision of spectroscopy.  Due to the smallness of these values (several orders of magnitude smaller than the total amplitude) numerical convergence comparable to Fig.~\ref{fig-totmmotion} is not observed.  Rather than plot contours, we compute the root mean square (RMS) average over a large trapping volume for each design in Table~\ref{tbl-axialmmotion}, producing a single quantity whose convergence is verified.  See Sec.~\ref{sec-discussion} for discussion.

\begin{table}
\caption{RMS average axial micromotion amplitudes over the cylindrical volume $r<500\mu$m, $|z|<z_0=8.5$mm.  Listed values for $A$ and $C$ change by no more than several percent when Laplace solutions are calculated over an order of magnitude in grid density.  Values for $B$ and $D$ change by up to tens of percent.  For averaging, the volume is discretized by a Cartesian grid with $h=20\mu$m, which is verified to be sufficiently dense.}
\begin{ruledtabular}
\begin{tabular}{ c c }
 Trap Design &  RMS Axial Micromotion Amplitude ($\mu$m)\\
 \hline
 \ A & 1.5 \\
 \ B & 0.04 \\
 \ C & 2.9 \\
 \ D & 0.08 \\
\end{tabular}
\end{ruledtabular}
\label{tbl-axialmmotion}
\end{table}

\section{Nonlinear Resonance Heating}
\label{sec-nlr}

Nonlinear resonance heating can occur if a resonance condition is satisfied between secular frequencies and the rf drive frequency.  For linear Paul traps, the condition is
\begin{equation}\label{eq-nlrcondition}
n_x\omega_x + n_y\omega_y + n_z\omega_z = \Omega_{rf}
\end{equation}
where $n_x$, $n_y$, and $n_z$ are integers \cite{drakoudis06}.  The corresponding condition for a single particle in a hyperbolic trap has been derived \cite{wang93}, with the resonances weakening with larger $n$.  This behavior has been confirmed in both hyperbolic traps \cite{alheit96} and linear traps \cite{drakoudis06}.  For sufficiently cold trapped samples, it is understood that the frequency of the center of mass (COM) modes, or other normal modes, should be used in Eq.~\eqref{eq-nlrcondition} rather than the single-particle frequencies \cite{wineland98}.  For an infinitely long linear Paul trap, or for a small crystal in a finite trap, where the axial rf drive vanishes, a non-deforming crystal is only expected to be heated on resonances with $n_z=0$.  However, a crystal which is large enough to sample axial fringing fields near the endcaps is expected to be excited also by resonances with $n_z > 0$.

Anharmonicity in the trapping potential causes the mode frequencies to shift as a crystal grows and samples less harmonic regions of the trap potential.  Eventually, the anharmonicity-shifted mode frequencies of a growing crystal will meet the resonance condition of Eq.~\eqref{eq-nlrcondition}, and some level of heating will occur, potentially halting further crystal growth. Even for a crystal of definite size, it is non-trivial to predict the precise response when it is swept through a given nonlinear heating resonance.  Predicting the heating response in the non-equilibrium scenario of a growing crystal, as anharmonic frequency shifts cause it to cross a resonance, is yet more complicated. Modeling of the heating rates and whether crossing a given subharmonic resonance will actually melt a crystal or prevent further growth is beyond the scope of this analysis.

Since it is difficult to determine whether crossing a heating resonance will limit crystal growth or cause other deleterious effects, in the current analysis, we take a conservative approach of attempting to design a trap in which a growing crystal will avoid low-order resonances altogether. We consider the anharmonicity-induced shifts of the COM frequencies for a non-deforming crystal; the non-deforming approximation is exact in the zero-temperature limit. In light of the resonant condition of Eq.~\eqref{eq-nlrcondition}, considering the COM frequency shifts provides a simple estimate of when trap anharmonicity could limit crystal growth.  Other modes of higher order than the COM mode can also be excited by nonlinear resonance heating, but the COM shifts are easily calculable and set the scale for anharmonicity-induced shifts in higher modes.

A previous analysis by the Marseille group (Pedregosa et. al. \cite{pedregosa10}) specified a single quantity describing the radial trap anharmonicity averaged over a trapped sample.  The Marseille approach has the utility of providing a single figure of merit by which to compare different trap designs; however, it does not consider axial anharmonicity (potentially important for long crystals) or provide an estimate of what level of anharmonicity is acceptable in order to avoid heating resonances.  The analysis we present here is similarly simple to calculate, and at the cost of not providing a single figure of merit, seeks to include axial anharmonicity and to provide a mechanism for determining whether a trap is sufficiently harmonic to safely work with crystals of some size.

\section{COM Frequency}
\label{sec-comfrequency}

For a zero-temperature crystal in a perfectly harmonic trapping potential, the COM oscillation frequency is the same as the secular frequency of a single particle. However, if the temperature of the crystal is nonzero or if the trapping potential is anharmonic, the crystal deforms during oscillation,  and the COM frequency and single particle frequency are no longer equal.  (Space charge shifts, which are well-understood in the limit of each ion moving independently in a background potential describing the distributed charge of the other ions, occur at high temperatures but not for a non-deforming crystal.)  Molecular dynamics (MD) simulations can find the COM frequency under finite temperature conditions, but this approach is computationally intensive for a large crystal. To understand the effects of anharmonicity with less computational overhead, we introduce the approximation of a non-deforming crystal in an anharmonic trap.  This approach is computationally simple enough to be useful in designing traps for large Coulomb crystal experiments.

A single ion in a zero temperature crystal is held in place by the cancellation of two forces: that due to the trap potential $F_{trap}$ and that due to the Coulomb repulsion of all other ions in the trap $F_{ions}$.  In the approximation that the crystal does not deform during oscillation, $F_{ions}$ remains constant while $F_{ions}+F_{trap}$ is, to leading order, a restoring force resulting in simple harmonic motion.

$F_{trap}$ can be calculated from $\phi_{trap}$, evaluated on a grid $(x_i,y_j,z_k)$ where $x_i=ih, y_j=jh$, and $z_k=kh$; $i$, $j$, $k$ are integers; $h$=50$\mu$m is the step size used in our calculation.  To illustrate our approach, we discuss calculation of the restoring force for small displacements in the $\hat{z}$ direction, at fixed $x_i$ and $y_j$.  By geometrical symmetry $\phi_{trap}$ is an even function of $z$, so the numerically calculated values can be fitted to a polynomial series of even terms:
\begin{equation}
\phi_{trap} (z) \Big |_{x_i, y_j}=C_0+C_2z^2+...+C_{10}z^{10}
\end{equation}
The axial force on an ion in a potential of this form is
\begin{equation}\label{eq-ftrap}
F_{trap}(z)=-q( 2C_2z + ... + 10C_{10}z^9).
\end{equation}
We now consider an ion whose equilibrium location in a crystal is $z=z_k$.  An expansion of $F_{trap}$ about $z_k$ gives
\begin{equation}\label{eq-fexp}
F_{trap}(z)=F_{trap}(z_k)+(z-z_k)F'_{trap}(z_k)+...
\end{equation}
Since $z=z_k$ is an equilibrium position for this ion, the Coulomb force due to all other ions $F_{ions}$ must exactly cancel the first term of this expansion.  In the approximation of a non-deforming crystal, $F_{ions}$ is a constant, and the total force on the ion is
\begin{align}\label{eq-ftot}
F_{trap}+F_{ions}&=-q\left( 2C_2 + ... + 90C_{10}z_k^8 \right)\left( z-z_k \right)
\\ &\equiv -k_z\Big |_{x_i, y_j, z_k} (z-z_k),
\end{align}
where $k_z|_{x_i, y_j, z_k}$ is a local spring constant describing the strength of the axial restoring force on an ion about its equilibrium location $(x_i, y_j, z_k)$.  As seen from Eqs.~\eqref{eq-ftrap} and \eqref{eq-ftot}, when $z_k\not=0$, anharmonic terms of the trap potential contribute to the spring constant $k_{z_k}$.

The restoring force on a crystal is the sum of the restoring forces on the individual ions.  For a non-deforming crystal, the result is oscillatory motion at a frequency set by the averaged local spring constants, with the average taken over the ion locations.  In our calculation we average instead over the grid locations, for points interior to the crystal volume. In this way, we can estimate the axial COM frequency for a crystal, given $\phi_{trap}$ and a specified crystal geometry.  We compute the COM frequencies for radial motion by the same method, again using a fitting polynomial of the same order.

Fig.~\ref{fig-wz} shows the change in axial COM frequency in each trap design as crystal volume grows.  Crystal extent is determined by evaluation of $\phi_{trap}$ on a Cartesian grid with spacing $h$=50$\mu$m, with all points below a fixed cutoff included, and ion number is then determined by the ion density \cite{dubin99},
\begin{equation}\label{eq-ion_density}
\rho = \frac{m\epsilon_0}{q^2}(\omega_z^2+2\omega_r^2),
\end{equation}
where $\omega_z$ and $\omega_r = \omega_x = \omega_y$ are the single particle secular frequencies.  We take this density as a constant; the anharmonicity-induced position dependence is negligible.  Implications of anharmonicity-induced frequency shifts for nonlinear resonance heating are discussed in Sec.\,\ref{sec-discussion}.

\begin{figure}
\includegraphics[width=85mm]{zlsf_figure_6-eps-converted-to.pdf}
\caption{\label{fig-wz} (Color online.) COM axial secular frequency as a function of ion number.  Curves for Designs $A$ and $B$ are overlapping.  Crystals are grown until their axial extent reaches ${\pm z_0/2}$; for equal length, design $D$ grows an appreciably smaller crystal.  Ripples result from finite evaluation grid size.  (See Sec.~\ref{sec-discussion} for discussion of both ion number in $D$ as well as finite grid effects.)}
\end{figure}

Radial COM shifts are also computed, with fittings made to radial rather than axial traces of $\phi_{trap}$.  Over the volume occupied by the largest crystals in Fig.~\ref{fig-wz}, no difference among the designs can be determined at our level of computational accuracy.  By varying all calculation parameters, we bound radial COM shifts at no more than tenths of a percent.

\section{Discussion}
\label{sec-discussion}

No difference in micromotion heating rate is expected among the different designs, since total micromotion amplitudes are comparable.  Indeed, in all cases axial micromotion amplitudes are much smaller than total amplitudes, indicating that rf-fringing should not limit crystal growth for these designs.  The actual limit on crystal size due to micromotion heating is difficult to estimate, but it can in principle be found from molecular dynamics simulations.

Axial micromotion, quantified by RMS averaging over a volume bounding any realistic crystal, is at least than an order of magnitude smaller  in designs $B$ and $D$ (in which the rf electrodes extend to $z > z_0$) than in $A$ and $C$ (in which the rf electrodes end at $z = z_0$).  Since axial micromotion results from fringing of the rf fields, these differences are expected, given the different rf boundary conditions at $z=\pm z_0$.  Given a conservative interpretation of the observed convergence of this average, $B$ and $D$ may be said to have comparable axial micromotion, while the situation in $A$ is much worse and in $C$ worse still.

The size of COM frequency shifts during crystal growth, found here to differ among the considered designs, is an indicator for the effect of nonlinear resonance heating on trap operation.  As the experimenter grows a crystal, adding ions continuously from an ion source, small COM frequency shifts (say a few percent) are not expected to present significant challenge.  The COM frequencies are unlikely to shift onto a nonlinear resonance during the intermediate stages of crystal growth, and the experimenter can load the largest crystal possible before checking if different voltages increase the maximum crystal size.  For larger size-dependent shifts, trapping voltages could, in principle, be dynamically tuned to avoid resonances as the crystal grows, but dynamic tuning would become increasingly challenging for larger shifts.  Thus the designs exhibiting large COM frequency shifts are expected to be more problematic for growing large crystals.

As an example of these concerns, we note that for the frequencies considered in this study, excitation of the nonlinear resonance $|n_x|+|n_y|=7$, $n_z=3$ is expected when the axial COM frequency increases by 5-10\%, a shift which occurs in design $D$ only for a crystal containing tens of thousands more ions than in other designs.  (Resonances of this order and higher have been observed experimentally in a linear trap \cite{drakoudis06}.)  The expected ability of $D$ to load more ions than the other designs before exciting this resonance is a direct result of the smaller axial COM shifts in $D$.  In general, since the rotated endcap design $D$ has the smallest axial COM shifts it is expected to be the least susceptible to nonlinear resonance heating.

\begin{figure}
\includegraphics[width=85mm]{aspect_ratio_vs_crystal_size_5-eps-converted-to.pdf}
\caption{{\label{fig-aspect_ratio}}(Color online.)  Aspect ratio as a function of crystal size, with aspect ratio estimated by comparison of the heights of the radial and axial potentials along $\hat{x}$ and $\hat{z}$ respectively.  $A$ is overlapped by $B$ and not visible.}
\end{figure}
The plots in Fig.~\ref{fig-wz} terminate at crystals of equal axial length.   For a perfectly harmonic trapping potential, the aspect ratio of a crystal will be independent of size and equal to the ratio of the secular frequencies, which in these traps were tuned to be equal in the harmonic trapping region.  Accordingly, eventual differences in aspect ratio are due to different degrees of anharmonicity sampled by the trapped ions.  In Fig.~\ref{fig-aspect_ratio}, we plot the expected change in aspect ratio during crystal growth in each design, calculated by a simple comparison of the heights of the radial and axial potentials along $\hat{x}$ and $\hat{z}$ respectively.  The tendency of design $D$ to maintain larger aspect ratio than other designs explains the ion numbers seen in Fig.~\ref{fig-wz}; at equal length, the crystals in designs $A$, $B$ and $C$ are radially wider than $D$ and so contain more ions.  Intuitively, the tendency of $D$ toward radially narrower crystals helps explain the smaller axial COM shifts in $D$, since the regions nearer the $\hat{z}$ axis are expected to be more harmonic than those radially further away.

The ripples in Fig.~\ref{fig-wz} caused by the finite size of the evaluation grid do not interfere with the conclusion that the COM frequency shift in $D$ is smaller than in the other designs, as the trend is clear at much sparser grids, where the ripples are even more pronounced.  The Laplace solutions are found to converge sufficiently to determine the COM shifts to within a few percent, and all other parameters--the associated Legendre and polynomial series fitting orders and the evaluation grid density--are adjusted until the resulting convergence in COM frequency shifts is not limiting.

\section{Conclusion}
\label{sec-conclusion}

We have compared the suitability of four trap designs for experiments on large Coulomb crystals.  The comparison is based on the desire to minimize micromotion heating and anharmonicity-induced nonlinear resonance heating.

We have compared total micromotion as a function of position within the trap volume. Micromotion heating, related to the total micromotion amplitude, is expected to be comparable in all designs.

We quantify axial micromotion by RMS averaging over a large trapping volume, finding that the resulting limit on spectroscopic accuracy in $B$ and $D$ to be much better than in either $A$ or $C$, with $C$ expected to be least suitable of all for precision spectroscopy on large crystals.

In order to compare the susceptibility of the different trap designs to nonlinear resonance heating, we have used a non-deforming crystal approximation to map the anharmonicity-induced axial and radial COM frequency shifts as a function of ion number.  Shifts of COM frequencies, as well as other normal mode frequencies for which COM shifts set the scale, are of concern since a growing crystal in an anharmonic trap will eventually cross a nonlinear heating resonance, leading to instantaneously elevated temperatures and potentially halting further crystal growth.  We find that a novel rotated endcap trap has the smallest axial COM shifts and observe that the relative stability of aspect ratio with increasing crystal size in this trap makes the result intuitive.  We suggest that traps aiming to grow large Coulomb crystals ought to minimize axial COM shifts, and we provide a method for quantifying these shifts.

The analysis presented here is simple to implement and should be predictive of trap behavior, so it is especially useful for taking a conservative approach to designing a trap for experiments on crystals of a desired size.  However, our analysis is not capable of predicting precisely when crystal growth will be limited by micromotion or by a given nonlinear heating resonance being crossed during crystal growth. To make detailed predictions of these limitations on ultimate crystal size and temperature, MD simulations could be used.  It would also be quite interesting to experimentally investigate the upper limits on crystal size set by anharmonicity-induced nonlinear heating resonances.

\begin{acknowledgments}
The authors gratefully thank Caroline Champenois, Eric Hudson, David Kielpinski, Joan Marler, Steven Schowalter, and Stephan Schiller for sharing their expertise in illuminating discussions.  This work is sponsored by NSF Grant No. PHY-0847748 and by NSF IGERT Grant No. 0801685.
\end{acknowledgments}

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