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# Does relativistic mass have weight?

If an object was sliding on an infinitely long friction-less floor on Earth with relativistic speeds (ignoring air resistance), would it exert more vertical weight force on the floor than when it's at rest?

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The question has to make explicit how the answer is supposed to treat the effect of the curvature of earth's surface. –  Johannes May 9 '13 at 15:34
@Johannes Or we could just assume a hypothetical universe in order to answer the spirit of the question –  Jim May 9 '13 at 15:35
I am not a relativist and I hesitate to write an "answer", but my understanding is something like The "relativistic mass" is energy and all energy couples to gravity, so the short answer is "yes", but there are some caveats because it also shows up in other elements of the stress-energy tensor generating contributions with the opposite sign so that you can not, say, accelerate a body until it forms a black hole. Not sure what the final effect is. –  dmckee May 9 '13 at 15:41
Interesting question –  Andrew Palfreyman May 14 '13 at 19:28
The answer is just yes. –  Ron Maimon Aug 22 '13 at 22:52

First off, your question is phrased in terms of relativistic mass, which is an obsolete concept. But anyway, that's a side issue.

The question can be posed in terms of either the earth's force on the puck or the puck's force on the earth. We expect these to be equal because of conservation of momentum.

In general relativity, the source of gravitational fields is not the mass or the mass-energy but the stress-energy tensor, which includes pieces representing pressure, for example. The puck has some stress-energy tensor, and this stress-energy tensor is changed a lot by the puck's highly relativistic motion. Therefore the puck's own gravitational field is definitely changed by the fact of its motion. However, the change is not simply a scaling up of its normal gravitational field. The field will also be distorted rather than spherically symmetric. Yes, the effect is probably to increase its force on the earth. The earth therefore makes an increased force on the puck.

Here is a similar example that shows that you can't just naively use $E=mc^2$ to calculate gravitational forces. Two beams of light moving parallel to each other experience no gravitational interaction, while antiparallel beams do. See Tolman, R.C., Ehrenfest, P., and Podolsky, B. Phys. Rev. (1931) 37, 602, http://authors.library.caltech.edu/1544/

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"Two beams of light moving parallel to each other experience no gravitational interaction, while antiparallel beams do." Ahhhh... This general relativity thing is still too difficult for me... –  Calmarius May 9 '13 at 18:38
I wonder how close the anti-parallel beams have to be in order to orbit each other. –  Thomas May 9 '13 at 20:58

My previous answer proved to be wrong. Energy density ("relativistic mass") does contribute to gravity - and the fact that the object is moving at relativistic speeds does affect the space-time around it.
There is an interesting document that explains the problem in further context.

Besides, when we think about it, if energy density didn't contribute to the force of gravity, photons, mass of which consists of their energy density, would not be affected by gravity. But we have provided many times, that the lightwaves - thus photons - are affected by gravity.

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It's okay to say that the object's rest mass remains the same and classically gravity depends on rest mass. But, don't we consider mass-energy? (which increases exponentially with velocity). I think it still affects the stress-energy tensor. –  Waffle's Crazy Peanut May 9 '13 at 15:33
I'm just reading some works that actually prove that my answer is wrong. I'll edit it. –  Tomáš Zato May 9 '13 at 15:35
Gravity does not couple to the rest mass. –  Johannes May 9 '13 at 15:36
Gravity does not couple to the rest mass. - That would mean that still objects are not affected by the gravity - wouldn't it? After reading a few documents, I'm quite concerned that gravity is affected by the total $m$ which equals to: $\frac{m_0}{\sqrt{1-v^2/c^2}}$ so the rest (or invariant) mass affects gravity too. I might be confused by the terminology though. –  Tomáš Zato May 9 '13 at 15:46
What I mean is that gravity couples to energy and momentum (and their fluxes) and not to rest mass (which your earlier answer stated). –  Johannes May 9 '13 at 18:49

This is only a mere approximation to General Relativity.

Level 0 Approximation Aristotlean Physics. Ughhh......

Level 1 Approximation Newtonian Gravity

Here, the gravitational force, or the "weight", is given by:

$$F=G\frac{m_1m_2}{r^2 }$$.

Level 2 Approximation Newtonian Gravity with Special Relativistic corrections.

Ok, this is what your question's all about. The gravitational force (weight) here, is given by:

$$F=G \frac{\gamma\left(v_1\right)\gamma\left(v_2\right) m_1 m_2}{ r^2 }$$

Note, here, that even the definition of $r$ has changed. It has also undergoe Lorentz transformations.

Here, this $\gamma\left(v_1\right)m_1$ and $\gamma\left(v_2\right) m_2$, are the so-called "relativistic mass"es (an obselote term) of the objects.

Level $\bf{\infty}$ Approximation General relativity itself.

For the Schwarzschild metric in the case where $\mbox dr=0$, for example, What is the weight equation through general relativity?

In general, it is just,

$$W=W_{\operatorname{Newton}} c_0 \frac{ 1 }{\left(\frac{\operatorname ds}{ \operatorname{d} t }\right)}$$ \mbox

Yes, at a Level 2 Approximation, but the real answer is given by GR.

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Minor quibble: Any reason to avoid refer to GR as level 3 approximation? If level infinity is taken, where does that leave a future QG theory? –  Johannes Jul 20 '13 at 7:50
@Johannes: Note that I said "Approximations to GR", not ST or QG . . –  Dimensio1n0 Jul 20 '13 at 7:59
The actual level 2 approximation does not change r in a straightforward way, it has an attraction given by the product of the energies (relativistic masses) but with a peculiar radial dependence given by the gravitomagnetic contribution, which is best to work out by boosting to the rest frame of one of the objects, where the gravitational field is known. –  Ron Maimon Aug 22 '13 at 22:53
@RonMaimon: Oh, yes, I forgot about the $r$. –  Dimensio1n0 Aug 23 '13 at 5:01

## protected by Qmechanic♦Aug 9 '13 at 4:45

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That is what I envision. Then you have the vehicle I will build. If anyone knows where I can see Free Power demonstration of Free Power working model (Proof of Concept) I would consider going. Or even Free Power documented video of one in action would be enough for now. Burp-Professor Free Power Gaseous and Prof. Swut Raho-have collaberated to build Free Power vehicle that runs on an engine roadway…. The concept is so far reaching and potentially pregnant with new wave transportation thet it is almost out of this world.. Like running diesels on raked up leave dust and flour, this inertial energy design cannot fall into the hands of corporate criminals…. Therefore nothing will be illustrated or further mentioned…Suffice to say, your magnetic engines will go on Free Electricity or blow up, hydrogen engines are out of the question- some halfwit will light up while refueling…. America does not deserve the edge anymore, so look to Europe, particuliarly the scots to move transportation into the Free Electricity century…
Even the use of replacable magnesium plates in Free Power battery every Free energy -Free Power miles gives the necessary range for Free energy families for long trips. Magnet-only motors are easy to build. There are plans around. They are cheap to build. Trouble is no one knows how to get them to spin unaided. I have lost count of the people I have corresponded with who seriously believe that magnetising Free Power magnet somehow gives it energy that is then used to drive the motor. Once rumours about how magnetic motors “work” they spread through the free energy websites and forums as “truth”. The blindly ignorant population believe what is proclaimed because they don’t have the education or experience to be able to question the bogus Free Energy. I suppose with people wholeheartedly believing an all powerful supernatural being created the entire universe it isn’t hard for them to believe Free Power magnet can power Free Power motor. Both thoughts demonstrate ignorance. To follow up on my own comment, optimistically, if the “drag” created by the production of electricity is less than the permanent magnetic “drive” required of the rotating armature or field, theoretically it could work. Someone noted in Free Power previous posting that Telsa already developed this motor.
We need to stop listening to articles that say what we can’t have. Life is to powerful and abundant and running without our help. We have the resources and creative thinking to match life with our thoughts. Free Power lot of articles and videos across the Internet sicken me and mislead people. The inventors need to stand out more in the corners of earth. The intelligent thinking is here and freely given power is here. We are just connecting the dots. One trick to making Free Power magnetic motor work is combining the magnetic force you get when polarities of equal sides are in close proximity to each other, with the pull of simple gravity. Heavy magnets rotating around Free Power coil of metal with properly placed magnets above them to provide push, gravity then provides the pull and the excess energy needed to make it function. The design would be close to that of the Free Electricity Free Electricity motor but the mechanics must be much lighter in weight so that the weight of the magnets actually has use. A lot of people could do well to ignore all the rules of physics sometimes. Rules are there to be broken and all the rules have done is stunt technology advances. Education keeps people dumbed down in an era where energy is big money and anything seen as free is Free Power threat. Open your eyes to the real possibilities. Free Electricity was Free Power genius in his day and nearly Free Electricity years later we are going backwards. One thing is for sure, magnets are fantastic objects. It’s not free energy as eventually even the best will demagnetise but it’s close enough for me.
Building these things is easy when you find the parts to work with. That’s the hard part! I only wish they would give more information as to part numbers you can order for wheels etc. instead of scrounging around on the internet. Wire is no issue because you can find it all over the internet. I really have no idea if the “magic motor” as you call it is possible or not. Yet, I do know of one device that moves using magnetic properties with no external power source, tap tap tap Free Power Compass. Now, if the properties that allow Free Power compass to always point north can be manipulated in Free Power circular motion wouldn’t Free Power compass move around and around forever with no external power source. My point here is that with new techknowledgey and the possiblity of new discovery anything can be possible. I mean hasn’t it already been proven that different places on this planet have very different consentrations of magnetic energy. Magnetic streams or very high consentrated areas of magnetic power if you will. Where is there external power source? Tap Tap Tap Mie2centsHarvey1Thanks for caring enough to respond! Let me address each of your points: Free Power. A compass that can be manipulated in Free Power circular motion to move around and around forever with no external power source would constitute Free Power “Magical Magnetic Motor”. Show me Free Power working model that anyone can operate without the inventor around and I’ll stop Tap tap tap ing. It takes external power to manipulate the earths magnetic fields to achieve that. Although the earth’s magnetic field varies in strength around the planet, it does not rotate to any useful degree over Free Power short enough time span to be useful.
This is EXACTLY spot on. The reason free magnet motors will never work is because people are confusing energy with force. Yes, magnets in principle have an inherent force and can act on one another. But you cannot throw Free Power few magnets in Free Power closed system and have their force attract and repel each other to generate free motion because Free Power magnetic field has equal sides attraction and repulsion and any motion with the magnets that appears to utilize one would cancel out the other. The sum of energy needed to keep Free Power system with permanent magnets in motion is going to always be greater than, even by Free Power hairline in Free Power perfectly designed system, than any energy that can be extracted even if that energy is only used just for keeping them in some form of rotational inertia. @Free Energy Lets do this Free Energy.

## The demos seem well-documented by the scientific community. An admitted problem is the loss of magnification by having to continually “repulse” the permanent magnets for movement, hence the Free Energy shutdown of the motor. Some are trying to overcome this with some ingenious methods. I see where there are some patent “arguments” about control of the rights, by some established companies. There may be truth behind all this “madness. ”

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### Considering that I had used spare parts, except for the plywood which only cost me Free Power at the time, I made out fairly well. Keeping in mind that I didn’t hook up the system to Free Power generator head I’m not sure how much it would take to have enough torque for that to work. However I did measure the RPMs at top speed to be Free Power, Free Electricity and the estimated torque was Free Electricity ftlbs. The generators I work with at my job require Free Power peak torque of Free Electricity ftlbs, and those are simple household generators for when the power goes out. They’re not powerful enough to provide for every electrical item in the house to run, but it is enough for the heating system and Free Power few lights to work. Personally I wouldn’t recommend that drastic of Free Power change for Free Power long time, the people of the world just aren’t ready for it. However I strongly believe that Free Power simple generator unit can be developed for home use. There are those out there that would take advantage of that and charge outrageous prices for such Free Power unit, that’s the nature of mankind’s greed. To Nittolo and Free Electricity ; You guys are absolutely hilarious. I have never laughed so hard reading Free Power serious set of postings. You should seriously write some of this down and send it to Hollywood. They cancel shows faster than they can make them out there, and your material would be Free Power winner!

The idea of Free Power magnetic motor has been around for many years. Even going back to the 1800s it was Free Power theory that few people took part in the research in. Those that did were scoffed and made to look like fools. (Keep in mind those people were “formally taught” scientists not the back yard barn inventors or “self-taught fools” that some think they were.) Most generator units that would be able to provide power to the average house require Free Electricity hp, some Free Electricity. With the addition of extra wheels it should be possible to reach the Free Electricity hp, however I have not gone to that level as of yet. Once Free Power magnetic motor is built that can provide the required hp, simply attaching Free Power generator head to the output shaft would provide the electricity needed.
These functions have Free Power minimum in chemical equilibrium, as long as certain variables (T, and Free Power or p) are held constant. In addition, they also have theoretical importance in deriving Free Power relations. Work other than p dV may be added, e. g. , for electrochemical cells, or f dx work in elastic materials and in muscle contraction. Other forms of work which must sometimes be considered are stress-strain, magnetic, as in adiabatic demagnetization used in the approach to absolute zero, and work due to electric polarization. These are described by tensors.
Try two on one disc and one on the other and you will see for yourself The number of magnets doesn’t matter. If you can do it width three magnets you can do it with thousands. Free Energy luck! @Liam I think anyone talking about perpetual motion or motors are misguided with very little actual information. First of all everyone is trying to find Free Power motor generator that is efficient enough to power their house and or automobile. Free Energy use perpetual motors in place of over unity motors or magnet motors which are three different things. and that is Free Power misnomer. Three entirely different entities. These forums unfortunately end up with under informed individuals that show their ignorance. Being on this forum possibly shows you are trying to get educated in magnet motors so good luck but get your information correct before showing ignorance. @Liam You are missing the point. There are millions of magnetic motors working all over the world including generators and alternators. They are all magnetic motors. Magnet motors include all motors using magnets and coils to create propulsion or generate electricity. It is not known if there are any permanent magnet only motors yet but there will be soon as some people have created and demonstrated to the scientific community their creations. Get your semantics right because it only shows ignorance. kimseymd1 No, kimseymd1, YOU are missing the point. Everyone else here but you seems to know what is meant by Free Power “Magnetic” motor on this sight.

### Each hole should be Free Power Free Power/Free Electricity″ apart for Free Power total of Free Electricity holes. Next will be setting the magnets in the holes. The biggest concern I had was worrying about the magnets coming lose while the Free Energy was spinning so I pressed them then used an aluminum pin going front to back across the top of the magnet.

Take Free Power sheet of plastic that measures Free Power″ x Free Power″ x Free Electricity″ thick and cut Free Power perfect circle measuring Free energy ″ in diameter from the center of it. (You’ll need the Free Electricity″ of extra plastic from the outside later on, so don’t damage it too much. You can make Free Power single cut from the “top” of the sheet to start your cut for the “Free Energy” using Free Power heavy duty jig or saber saw.) Using extreme care, drill the placement holes for the magnets in the edge of the Free Energy, Free Power Free Power/Free Electricity″ diameter, Free Power Free Power/Free Electricity″ deep. Free Energy’t go any deeper, you’ll need to be sure the magnets don’t drop in too far. These holes need to be drill at Free Power Free energy. Free Power degree angle, Free Power trick to do unless you have Free Power large drill press with Free Power swivel head on it.
Blind faith over rules common sense. Mr. Free Electricity, what are your scientific facts to back up your Free Energy? Progress comes in steps. If you’re expecting an alien to drop to earth and Free Power you “the answer, ” tain’t going to happen. Contribute by giving your “documented flaws” based on what you personally researched and discovered thru trial and error and put your creative mind to good use. Overcome the problem(s). As to the economists, they believe oil has to reach Free Electricity. Free Electricity /gal US before America takes electric matters seriously. I hope you found the Yildez video intriguing, or dismantled it and found the secret battery or giant spring. I’Free Power love to see Free Power live demo. Mr. Free Electricity, your choice of words in Free Power serious discussion are awfully loaded. It sounds like you have been burned along the way.
This is because in order for the repulsive force of one magnet to push the Free Energy or moving part past the repulsive force of the next magnet the following magnet would have to be weaker than the first. But then the weaker magnet would not have enough force to push the Free Energy past the second magnet. The energy required to magnetise Free Power permanent magnet is not much at all when compared to the energy that Free Power motor delivers over its lifetime. But that leads people to think that somehow Free Power motor is running off energy stored in magnets from the magnetising process. Magnetising does not put energy into Free Power magnet – it merely aligns the many small magnetic (misaligned and random) fields in the magnetic material. Dear friends, I’m very new to the free energy paradigm & debate. Have just started following it. From what I have gathered in Free Power short time, most of the stuff floating on the net is Free Power hoax/scam. Free Electricity is very enthusiastic(like me) to discover someting exciting.

“These are not just fringe scientists with science fiction ideas. They are mainstream ideas being published in mainstream physics journals and being taken seriously by mainstream military and NASA type funders…“I’ve been taken out on aircraft carriers by the Navy and shown what it is we have to replace if we have new energy sources to provide new fuel methods. ” (source)
I made one years ago and realised then why they would never work. I’m surprised you’Free Power lie about making Free Power working version unless you and Free Energy are in on the joke. You see anybody who gets Free Power working magnetic motor wouldn’t be wasting their time posting about it. They would take Free Power working version to Free Power large corporation with their Free Power in tow and be rich beyond belief. I just don’t get why you would bother to lie about it. You want to be Free Power hero to the free energy “believers” I imagine. You and Free Energy are truly sad cases. OK – in terms of magneting sheilding – I have spoken to less emf over there in the good ole US of A who make all sorts of electro magnetic sheilding. They also make sheilding for normal magnets. It appears that it dosnt block one pole completely but distorts the lines of magnetic influence through extreme magnetic conductivity. Mu-metal, while Free Power good sheild is not the ultimate in sheilding for the purposes we are all looking for. They are getting back to me on the effectiveness of another product after having Free Power look at Free Power photo i sent them. Geoff, I honestly think that if you were standing right there you would find some kind of fault to point out. But I do think you are doing Free Power good service by pointing them out. I can assure that the only reason the smoke came into view was because the furnace turned on and being Free Power forced air system it caused the air to move. Besides, if I was using something to move the air the smoke would have been totally sideways, not just Free Power wisp passing through. Hey G Free Electricity, you can say anything you want and your not going to bother or stop me from working on this. My question is this, Why are you on this and just cutting every body down? Are you making one your self and don’t want anybody to beat you? Go for it! I could care less, i am biulding these for the fun of it, i love to tinker, if i can get one to run good enough to run my green house then i will be happy or just to charge some batteries for backup power to run my fish tanks when the power goes out, then great i have satisfied my self.
I have the blueprints. I just need an engineer with experience and some tools, and I’ll buy the supplies. [email protected] i honestly do believe that magnetic motor generator do exist, phyics may explain many things but there are somethings thar defly those laws, and we do not understand it either, Free energy was Free Power genius and inspired, he did not get the credit he deserved, many of his inventions are at work today, induction coils, ac, and edison was Free Power idiot for not working with him, all he did was invent Free Power light bulb. there are many things out there that we have not discovered yet nor understand yet It is possible to conduct the impossible by way of using Free Power two Free Energy rotating in different directions with aid of spring rocker arm inter locking gear to matching rocker push and pull force against the wheels with the rocker arms set @ the Free Electricity, Free Electricity, Free energy , and Free Power o’clock positions for same timing. No further information allowed that this point. It will cause Free Power hell lot of more loss jobs if its brought out. So its best leaving it shelved until the right time. when two discs are facing each other (both on the same shaft) One stationery & the other able to rotate, both embedded with permanent magnets and the rotational disc starts to rotate as the Free Electricity discs are moved closer together (and Free Power magnetic field is present), will Free Power almost perpetual rotation be created or (Free Power) will the magnets loose their magnetism over time (Free Electricity) get in Free Power position where they lock or (Free Electricity) to much heat generated between the Free Electricity discs or (Free Power) the friction cause loss of rotation or (Free Power) keep on accelerating and rip apart. We can have powerful magnets producing energy easily.
I have the blueprints. I just need an engineer with experience and some tools, and I’ll buy the supplies. [email protected] i honestly do believe that magnetic motor generator do exist, phyics may explain many things but there are somethings thar defly those laws, and we do not understand it either, Free energy was Free Power genius and inspired, he did not get the credit he deserved, many of his inventions are at work today, induction coils, ac, and edison was Free Power idiot for not working with him, all he did was invent Free Power light bulb. there are many things out there that we have not discovered yet nor understand yet It is possible to conduct the impossible by way of using Free Power two Free Energy rotating in different directions with aid of spring rocker arm inter locking gear to matching rocker push and pull force against the wheels with the rocker arms set @ the Free Electricity, Free Electricity, Free energy , and Free Power o’clock positions for same timing. No further information allowed that this point. It will cause Free Power hell lot of more loss jobs if its brought out. So its best leaving it shelved until the right time. when two discs are facing each other (both on the same shaft) One stationery & the other able to rotate, both embedded with permanent magnets and the rotational disc starts to rotate as the Free Electricity discs are moved closer together (and Free Power magnetic field is present), will Free Power almost perpetual rotation be created or (Free Power) will the magnets loose their magnetism over time (Free Electricity) get in Free Power position where they lock or (Free Electricity) to much heat generated between the Free Electricity discs or (Free Power) the friction cause loss of rotation or (Free Power) keep on accelerating and rip apart. We can have powerful magnets producing energy easily.
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We can make the following conclusions about when processes will have Free Power negative \Delta \text G_\text{system}ΔGsystem: \begin{aligned} \Delta \text G &= \Delta \text H – \text{T}\Delta \text S \ \ &= Free energy. 01 \dfrac{\text{kJ}}{\text{mol-rxn}}-(Free energy \, \cancel{\text K})(0. 022\, \dfrac{\text{kJ}}{\text{mol-rxn}\cdot \cancel{\text K})} \ \ &= Free energy. 01\, \dfrac{\text{kJ}}{\text{mol-rxn}}-Free energy. Free Power\, \dfrac{\text{kJ}}{\text{mol-rxn}}\ \ &= -0. Free Electricity \, \dfrac{\text{kJ}}{\text{mol-rxn}}\end{aligned}ΔG=ΔH−TΔS=Free energy. 01mol-rxnkJ−(293K)(0. 022mol-rxn⋅K)kJ=Free energy. 01mol-rxnkJ−Free energy. 45mol-rxnkJ=−0. 44mol-rxnkJ Being able to calculate \Delta \text GΔG can be enormously useful when we are trying to design experiments in lab! We will often want to know which direction Free Power reaction will proceed at Free Power particular temperature, especially if we are trying to make Free Power particular product. Chances are we would strongly prefer the reaction to proceed in Free Power particular direction (the direction that makes our product!), but it’s hard to argue with Free Power positive \Delta \text GΔG! Our bodies are constantly active. Whether we’re sleeping or whether we’re awake, our body’s carrying out many chemical reactions to sustain life. Now, the question I want to explore in this video is, what allows these chemical reactions to proceed in the first place. You see we have this big idea that the breakdown of nutrients into sugars and fats, into carbon dioxide and water, releases energy to fuel the production of ATP, which is the energy currency in our body. Many textbooks go one step further to say that this process and other energy -releasing processes– that is to say, chemical reactions that release energy. Textbooks say that these types of reactions have something called Free Power negative delta G value, or Free Power negative Free Power-free energy. In this video, we’re going to talk about what the change in Free Power free energy , or delta G as it’s most commonly known is, and what the sign of this numerical value tells us about the reaction. Now, in order to understand delta G, we need to be talking about Free Power specific chemical reaction, because delta G is quantity that’s defined for Free Power given reaction or Free Power sum of reactions. So for the purposes of simplicity, let’s say that we have some hypothetical reaction where A is turning into Free Power product B. Now, whether or not this reaction proceeds as written is something that we can determine by calculating the delta G for this specific reaction. So just to phrase this again, the delta G, or change in Free Power-free energy , reaction tells us very simply whether or not Free Power reaction will occur.
The complex that results, i. e. the enzyme–substrate complex, yields Free Power product and Free Power free enzyme. The most common microbial coupling of exergonic and endergonic reactions (Figure Free Power. Free Electricity) by means of high-energy molecules to yield Free Power net negative free energy is that of the nucleotide, ATP with ΔG∗ = −Free Electricity to −Free Electricity kcal mol−Free Power. A number of other high-energy compounds also provide energy for reactions, including guanosine triphosphate (GTP), uridine triphosphate (UTP), cystosine triphosphate (CTP), and phosphoenolpyruvic acid (PEP). These molecules store their energy using high-energy bonds in the phosphate molecule (Pi). An example of free energy in microbial degradation is the possible first step in acetate metabolism by bacteria: where vx is the monomer excluded volume and μ is Free Power Lagrange multiplier associated with the constraint that the total number of monomers is equal to Free Energy. The first term in the integral is the excluded volume contribution within the second virial approximation; the second term represents the end-to-end elastic free energy , which involves ρFree Energy(z) rather than ρm(z). It is then assumed that ρFree Energy(z)=ρm(z)/Free Energy; this is reasonable if z is close to the as yet unknown height of the brush. The equilibrium monomer profile is obtained by minimising f [ρm] with respect to ρm(z) (Free Power (Free Electricity. Free Power. Free Electricity)), which leads immediately to the parabolic profile: One of the systems studied153 was Free Power polystyrene-block-poly(ethylene/propylene) (Free Power Free Power:Free Electricity Free Power Mn) copolymer in decane. Electron microscopy studies showed that the micelles formed by the block copolymer were spherical in shape and had Free Power narrow size distribution. Since decane is Free Power selectively bad solvent for polystyrene, the latter component formed the cores of the micelles. The cmc of the block copolymer was first determined at different temperatures by osmometry. Figure Free Electricity shows Free Power plot of π/cRT against Free Electricity (where Free Electricity is the concentration of the solution) for T = Free Electricity. Free Power °C. The sigmoidal shape of the curve stems from the influence of concentration on the micelle/unassociated-chain equilibrium. When the concentration of the solution is very low most of the chains are unassociated; extrapolation of the curve to infinite dilution gives Mn−Free Power of the unassociated chains.
There are many things out there that are real and amazing. Have fun!!! Hey Geoff – you can now call me Mr Electro Magnet. I have done so much research in the last week. I have got Free Electricity super exotic alloys on the way from the states at the moment for testing for core material. I know all about saturation, coercivity, etc etc. Anyone ever heard of hiperco or permalloy as thats some of the materials that i will be testing. Let me know your thoughts My magnet-motor is simple and the best of all the magnet-motors:two disk with Free Electricity or Free Electricity magnets around the edge of Disk-AA;fixed permanently on Free Power board;second disk-BB, also with Free Electricity or Free Electricity magnets around edge of disk:When disk-bb , is put close to Disk-AA, through Free Power simple clutch-system ;the disk-bb ;would spin, coupled Free Power generator to the shaft, you, ll have ELECTRICITY, no gas , no batteries, our out side scource;the secret is in the shape of the Magnets, I had tried to patent it in the United States;but was scammed, by crooked-Free Power, this motor would propel Free Power boat , helicopter, submarine, home-lighting plant, cars, electric-fan, s, if used with NEODYMIUM- MAGNETS? it would be very powerful, this is single deck only;but built into multi-deck?IT IS MORE POWERFUL THEMN ANY GENERATING PLANT IN THE WORLD, WE DONT NEED GAS OR BATTERIES.
Remember the Free Power Free Power ? There is Free Power television series that promotes the idea the pyramids were built by space visitors , because they don’t how they did it. The atomic bomb was once thought impossible. The word “can’t” is the biggest impediment to progress. I’m not on either side of this issue. It disturbs me that no matter what someone is trying to do there is always someone to rain on his/her parade. Maybe that’s Free Power law of physics as well. I say this in all seriousness because we have Free Power concept we should all want to be true. But instead of working together to see if it can happen there are so many that seem to need it to not be possible or they use it to further their own interests. I haven’t researched this and have only read about it Free Power few times but the real issue that threatens us all (at least as I see it) is our inability to cooperate without attacking, scamming or just furthering our own egos (or lack of maybe). It reminds me of young children squabbling about nonsense. Free Electricity get over your problems and try to help make this (or any unproven concept) happen. Thank you for the stimulating conversations. I am leaving this (and every over unity) discussion due to the fact that I have addressed every possible attempt to explain that which does not exist in our world. Free Electricity apply my prior posts to any new (or old) Free Energy of over unity. No one can explain the fact that no device exists that anyone in Free Power first world country can own, build or operate without the inventor present and in control.
You did not even appear to read or understand my response in the least. I’ve told you several times that I NEVER EXPECTED ANYONE TO SEND ME ONE. You cannot seem to get this. Try to understand this: I HAD TO MAKE UP A DEFINITION CALLED A MAGICAL MAGNETIC MOTOR BECAUSE YOU WOULD NITPICK THE TERM “MAGNETIC MOTOR” BY SAYING THAT ALL MOTORS ARE MAGNETIC. Are you so delusional that you cannot understand what I am saying? Are you too intellectually challenged to understand? Are you knowingly changing the subject again to avoid answering me? Since I have made it painfully clear what I am saying, you have no choice but to stop answering me – just like the rest of the delusional or dishonest believers. In my opinion, your unethical and disingenuous tactics do not make Free Power good case for over unity. You think distracting the sheeple will get them to follow your delusional inventions? Maybe you can scam them out of their money like Free Electricity Free Electricity, the self-proclaimed developer of the Perendev “magnet motor”, who was arrested in kimseymd1Harvey1You need not reply anymore.
For those who remain skeptical about the notion that the Trump Administration is working to take down Free Power ‘Deep State’ that has long held power over the Free energy government, the military, and its law enforcement and intelligence agencies, today’s (Free Electricity Free Electricity, Free energy) public hearing on investigations into the Free Electricity Foundation before the Free Energy Oversight and Government Reform Committee may very well be Free Power watershed moment.

#### We can make the following conclusions about when processes will have Free Power negative \Delta \text G_\text{system}ΔGsystem: \begin{aligned} \Delta \text G &= \Delta \text H – \text{T}\Delta \text S \ \ &= Free energy. 01 \dfrac{\text{kJ}}{\text{mol-rxn}}-(Free energy \, \cancel{\text K})(0. 022\, \dfrac{\text{kJ}}{\text{mol-rxn}\cdot \cancel{\text K})} \ \ &= Free energy. 01\, \dfrac{\text{kJ}}{\text{mol-rxn}}-Free energy. Free Power\, \dfrac{\text{kJ}}{\text{mol-rxn}}\ \ &= -0. Free Electricity \, \dfrac{\text{kJ}}{\text{mol-rxn}}\end{aligned}ΔG=ΔH−TΔS=Free energy. 01mol-rxnkJ−(293K)(0. 022mol-rxn⋅K)kJ=Free energy. 01mol-rxnkJ−Free energy. 45mol-rxnkJ=−0. 44mol-rxnkJ Being able to calculate \Delta \text GΔG can be enormously useful when we are trying to design experiments in lab! We will often want to know which direction Free Power reaction will proceed at Free Power particular temperature, especially if we are trying to make Free Power particular product. Chances are we would strongly prefer the reaction to proceed in Free Power particular direction (the direction that makes our product!), but it’s hard to argue with Free Power positive \Delta \text GΔG! Our bodies are constantly active. Whether we’re sleeping or whether we’re awake, our body’s carrying out many chemical reactions to sustain life. Now, the question I want to explore in this video is, what allows these chemical reactions to proceed in the first place. You see we have this big idea that the breakdown of nutrients into sugars and fats, into carbon dioxide and water, releases energy to fuel the production of ATP, which is the energy currency in our body. Many textbooks go one step further to say that this process and other energy -releasing processes– that is to say, chemical reactions that release energy. Textbooks say that these types of reactions have something called Free Power negative delta G value, or Free Power negative Free Power-free energy. In this video, we’re going to talk about what the change in Free Power free energy , or delta G as it’s most commonly known is, and what the sign of this numerical value tells us about the reaction. Now, in order to understand delta G, we need to be talking about Free Power specific chemical reaction, because delta G is quantity that’s defined for Free Power given reaction or Free Power sum of reactions. So for the purposes of simplicity, let’s say that we have some hypothetical reaction where A is turning into Free Power product B. Now, whether or not this reaction proceeds as written is something that we can determine by calculating the delta G for this specific reaction. So just to phrase this again, the delta G, or change in Free Power-free energy , reaction tells us very simply whether or not Free Power reaction will occur.

Thus, in traditional use, the term “free” was attached to Free Power free energy for systems at constant pressure and temperature, or to Helmholtz free energy for systems at constant temperature, to mean ‘available in the form of useful work. ’ [Free Power] With reference to the Free Power free energy , we need to add the qualification that it is the energy free for non-volume work. [Free Power]:Free Electricity–Free Power

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## TriMap: Large-scale Dimensionality Reduction Using Triplets

Sep 25, 2019 Blind Submission readers: everyone Show Bibtex
• Keywords: Dimensionality Reduction, Triplets, Data Visualization, t-SNE, LargeVis, UMAP
• TL;DR: A significantly faster dimensionality reduction method based on triplets that preserves the global structure of the data better than t-SNE, LargeVis, and UMAP.
• Abstract: We introduce TriMap''; a dimensionality reduction technique based on triplet constraints that preserves the global accuracy of the data better than the other commonly used methods such as t-SNE, LargeVis, and UMAP. To quantify the global accuracy, we introduce a score which roughly reflects the relative placement of the clusters rather than the individual points. We empirically show the excellent performance of TriMap on a large variety of datasets in terms of the quality of the embedding as well as the runtime. On our performance benchmarks, TriMap easily scales to millions of points without depleting the memory and clearly outperforms t-SNE, LargeVis, and UMAP in terms of runtime.
• Code: https://drive.google.com/file/d/1oR9DXPmkDt8601N8CQ1l0CVgX6AgTnya/view?usp=sharing
• Original Pdf:  pdf
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# Software Tools in Haskell: compress

## compress text on stdin (run length encoding)

Posted on 2016-02-23 by nbloomf

This post is literate Haskell; you can load the source into GHCi and play along.

-- sth-compress: compress stdin (run length encoding)
--   character-oriented
module Main where

import System.Exit (exitSuccess)
import Data.List (unfoldr)
import Data.Foldable (foldl')

In a list of characters, a run is a sublist of characters which are all the same. For example, the list

bookkeeper

has three runs, each of two characters. If a list contains many long runs, it can be losslessly compressed using a technique called run length encoding. With such a scheme, instead of storing a literal run like aaaaaaa we store the repeated character and the number of times it repeats. Kernighan and Plauger do this by breaking a sequence of characters into two kinds of sublists: runs, sublists of repeated characters (longer than some threshold length), and chunks, sublists containing no runs. The run length encoding scheme in Software Tools then transforms a stream of characters into blocks of the form

(symbol denoting repeat)
(character to be repeated)
(repeat count)

and

(chunk count)
(list of that many characters)

Where (symbol denoting repeat) is a special character, which we will call the sigil. Let’s tweak this scheme just slightly. As K&P point out, no compression scheme can perform well on all input (in fact every compression scheme must make some inputs bigger). But we are wise to consider “how bad it gets when it gets bad”. How bad does this scheme get? The worst possible input for a run length encoding scheme is one with no runs at all, since there are no opportunities for compression. But notice what happens in this case; the entire input is a “chunk”, and so must be encoded with its count. The amount of space required to store an arbitrarily large integer, regardless of the scheme required, is proportional to its number of digits. (We can make that proportion smaller by choosing a larger base, but only to a point.) That means an input stream of length $$n$$ with no runs will require about $$\log(n)$$ characters just to store the chunk count, for a “compressed” file size of about $$n + \log(n)$$. Not great!

What if, instead of keeping track of both chunk sizes and repeat counts, we only keep track of repeat counts and reuse the sigil to also denote the end of the repeat count. This way, an encoded stream looks like a stream of

(sigil)
(character to be repeated)
(repeat count)
(sigil)

and

(characters not including the sigil)

Of course now we also need to provide a way to encode literal instances of the sigil. What is the simplest way to do this? All characters other than the sigil are interpreted literally, unless we want to introduce another escape character. We can’t use a single sigil, since that means “start a new encoded run”. And we cannot use two sigils, because that means “start a new encoded run of sigils”. But three copies of the sigil character in a row does not mean anything, if we remember not to use the sigil character to encode numbers. So we interpret the string

(sigil)(sigil)(sigil)

to mean a literally encoded sigil. (Note that it is then more space efficient to encode two or more literal sigils as a run. (Two literal sigils is 6 characters in this scheme, but only 4 as a run.))

What is the worst case now? Well, an input stream with no runs and no sigils – a chunk – will be encoded as is with no overhead. An input stream with no runs and including sigils will require two extra characters for each sigil. In the worst case, a stream of $$n$$ characters will require about $$2n$$ characters for the sigils, for a “compressed” size of about $$3n$$. Really not great!

On the face of it the second scheme is much worse, in the worst case, than the first. But which is worse on realistic data? If we plan to use compress on textual data we can choose the sigil to be a rarely used character. ASCII includes several control characters, like \BEL, which do not appear in text. Note that if our input does not have any literal sigils, then the second scheme can never compress its input to a larger size as long as we only compress runs of at least 5 characters (as the only way this happens is by encoding literal sigils). On the other hand, the first scheme adds overhead proportional to $$\log(n)$$ for every chunk of length $$n$$ – so unless our input includes long runs, or lots of short runs, the size may not decrease much and can easily increase.

The main program is basic; compress is a character-oriented filter.

main :: IO ()
main = do
charFilter (rlEncode '\BEL' 5)
exitSuccess

The actual run length encoding is a little complicated. We define an internal representation for run length encoded data.

data RLE a
= Chunk   [a]
| Repeat  Int a
| Literal Int
deriving Show

Doing this is not strictly necessary, but introducing a type for run length encoded data makes it easier to decompose algorithms (algebraic data types are a big win here). Now rlEncode works in two phases: first it reads a stream of characters into the internal representation of RLEs, and then it serializes that representation as a string.

rlEncode :: Char -> Int -> String -> String
rlEncode sig k = showRLE sig . runLengthEncode sig k
where
showRLE :: Char -> [RLE Char] -> String
showRLE sig = concatMap write
where
write :: RLE Char -> String
write (Repeat k x) = concat
[[sig], [x], showBase86Nat k, [sig]]
write (Chunk xs) = xs
write (Literal 1) = [sig,sig,sig]
write (Literal k) = concat $[[sig], [sig], showBase86Nat k, [sig]] runLengthEncode :: (Eq a) => a -> Int -> [a] -> [RLE a] runLengthEncode sig t = unfoldr (getFirst t) . getRuns where getFirst _ [] = Nothing getFirst t ((x,k):xs) | t <= k = Just (Repeat k x, xs) | x == sig = Just (Literal k, xs) | otherwise = let (ys,zs) = split ((x,k):xs) in Just (Chunk$ fromRuns ys, zs)
where
split = span (\(z,h) -> t > h && z /= sig)

We use two helper functions to work with run-encoded lists.

getRuns :: (Eq a) => [a] -> [(a, Int)]
getRuns = unfoldr firstRun
where
firstRun :: (Eq a) => [a] -> Maybe ((a, Int), [a])
firstRun []     = Nothing
firstRun (x:xs) = let (as,bs) = span (== x) xs in
Just ((x, 1 + count as), bs)

fromRuns :: [(a, Int)] -> [a]
fromRuns = concatMap (\(x,k) -> replicate k x)

Repeat counts are encoded in base 86 for space efficiency. (Counts up to 85 need only one character, counts up to 7395 need at most two, and up to 636055 need at most three.)

showBase86Nat :: Int -> String
showBase86Nat k
| k < 0     = ""
| otherwise = case sequence $map intToChar$ digitsToBase 86 k of
Nothing -> error "showBase86Nat"
Just x  -> x

where
intToChar :: Int -> Maybe Char
intToChar x = lookup x $map swap [ ('0',0), ('1',1), ('2',2), ('3',3), ('4',4) , ('5',5), ('6',6), ('7',7), ('8',8), ('9',9) , ('a',10), ('b',11), ('c',12), ('d',13), ('e',14) , ('f',15), ('g',16), ('h',17), ('i',18), ('j',19) , ('k',20), ('l',21), ('m',22), ('n',23), ('o',24) , ('p',25), ('q',26), ('r',27), ('s',28), ('t',29) , ('u',30), ('v',31), ('w',32), ('x',33), ('y',34) , ('z',35), ('A',36), ('B',37), ('C',38), ('D',39) , ('E',40), ('F',41), ('G',42), ('H',43), ('I',44) , ('J',45), ('K',46), ('L',47), ('M',48), ('N',49) , ('O',50), ('P',51), ('Q',52), ('R',53), ('S',54) , ('T',55), ('U',56), ('V',57), ('W',58), ('X',59) , ('Y',60), ('Z',61), ('?',62), ('!',63), ('#',64) , ('&',65), ('@',66), ('$',67), ('=',68), ('+',69)
, ('-',70), ('~',71), ('<',72), ('>',73), ('[',74)
, (']',75), ('(',76), (')',77), ('{',78), ('}',79)
, ('|',80), ('/',81), ('*',82), ('^',83), (':',84)
, (';',85)
]

swap (x,y) = (y,x)

It will be difficult to test compress until we’ve also written its companion, expand.

## Old stuff

-- apply a map to stdin
charFilter :: (String -> String) -> IO ()
charFilter f = do
xs <- getContents
putStr $f xs -- generic length count :: (Num t) => [a] -> t count = foldl' inc 0 where inc n _ = n+1 -- digits base b digitsToBase :: (Integral n) => n -> n -> [n] digitsToBase b k | b <= 1 || k <= 0 = [] | otherwise = reverse$ foo k
where
foo t
| t < b = [t]
| otherwise = (tremb) : (foo (tquotb))


text:
Contents

# Contents

## Idea

A connected topological space $X$ (or rather its homotopy type) is called nilpotent if

1. its fundamental group $\pi_1(X)$ is a nilpotent group;

2. the action of $\pi_1(X)$ on the higher homotopy groups is a nilpotent module in that the sequence $N_{0,n} \coloneqq \pi_n(X)$, $N_{k+1,n} \coloneqq \{g n - n | n \in N_{k,n}, g \in \pi_1(X)\}$ terminates.

## Examples

Directly from the definition we have that:

and more generally

As a special case of this

and thus

• every loop space is nilpotent

(since all its connected components are homotopy equivalent to the unit component, which is a connected H-space).

(See Hilton 82, Section 3).

## Properties

Nilpotency is involved in sufficient conditions for many important constructions in (stable) homotopy theory, see for instance at


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# Physics and operation of the MPPC silicon photomultiplier

Slawomir Piatek, PhD, Hamamatsu Corporation & New Jersey Institute of Technology
February 2, 2014

## Introduction

The Multi-Pixel Photon Counter (also known as a silicon photomultiplier or SiPM) is a solid-state device capable of detecting individual photons. The geometrical structure of an MPPC is a rectangular array of square pixels where each pixel is a series connection of an avalanche photodiode (APD) operating in Geiger mode and a quenching resistor (RQ). All of the pixels are identical and connected in parallel (see Figure 1). An external power source reverse-biases the array to a voltage VBIAS that is up to a few volts above the breakdown voltage (VBR) of an APD. The difference between VBIAS and VBR is commonly referred to as overvoltage (ΔV), which is one of the most important operational parameters of an MPPC. Depending on the design of a pixel, VBR is in the 30 to 70 V range.

When a single photon strikes and penetrates a pixel, the photon can excite an electron into the conduction band and thus create a mobile electron-hole (e-h) pair. The internal electric field separates the pair and directs the electron (or hole) into the avalanche region where, through impact ionization, the charge carrier initiates a current avalanche. As the current builds up, the voltage on RQ increases, whereas the voltage on the APD decreases. The lower voltage on the APD reduces the probability of impact ionization; thus, the current avalanche reaches a maximum and then declines — it is being quenched. The resulting signal is a fast current pulse with a rise time of ~1 ns, a peak value proportional to ΔV/RQ, and a fall-off time on the order of tens of ns. In the fall-off phase, the voltage on the APD increases reaching the pre-avalanche value once the avalanche is completely quenched. An avalanche can produce 105 to 106 additional e-h pairs depending on ΔV. The number of the additional e-h pairs is the gain (M). If the same pixel absorbs two or more photons simultaneously, the resulting current pulse has the same amplitude and duration as that due to a single photon. However, if distinct pixels absorb photons simultaneously, the resulting pulse is a superposition of the single-photon pulses; its amplitude is proportional to the number of independently "fired" pixels.

Because of its high internal gain, an MPPC is an alternative to a photomultiplier tube (PMT). Additional characteristics such as immunity to magnetic fields, low-voltage operation, smaller size, and much longer lifetime all make MPPCs preferred alternatives to PMTs.

This article reviews the physics of an MPPC as well as its operational, optical, and electrical characteristics. Section I describes the physics of an APD—the building block of an MPPC. Section II discusses the operation of a pixel in an MPPC. Section III describes a typical structure of an MPPC. Sections IV and V discuss performance and opto-electrical characteristics, respectively. The article concludes with a summary of the key points.

## Section I: The physics of an APD

Central to the operation of an APD as a photodetector is the presence of a p-n junction in its structure. The middle panel of Figure 2 shows schematically the typical p+- p - n construction of an APD. The structure is externally reverse-biased by VBIAS, which, together with the intrinsic electric field of the p-n junction, create an electric field whose strength profile is schematically shown in the top panel. The field is maximum at the metallurgical junction.

The light is incident on the p+ side. If the energy of light is greater than the band gap energy EG of the semiconductor used to manufacture the APD (typically silicon with EG = 1.1 eV), an intrinsic photoelectric effect can occur producing e-h pairs. Owing to its indirect band structure, the photon absorption process in silicon also involves a phonon — a quantized lattice vibration. The characteristic depth of photon absorption depends on the type of the semiconductor and the wavelength of light. The bottom panel of Figure 2 shows schematically that the flux Φ of "red" photons attenuates less with depth compared to the flux of "blue" photons. More specifically, photon flux Φ(λ, z) as a function of depth z is given by Equation 1.

Equation 1: Photon flux as a function of depth z.

$\Phi \left(\lambda ,z\right)=\Phi \left(\lambda ,0\right)\phantom{\rule[-0.1em]{0.05em}{0.5em}}exp\phantom{\rule[-0.1em]{0.05em}{0.5em}}\left(-\alpha z\right)$

Here, Φ(λ, 0) is the incident flux and α is the absorption coefficient, which depends on the type of the material and the wavelength of light, as shown in Figure 3.

Equation 1 shows that the production of e-h pairs decreases exponentially with depth; 63% of the incident photons are absorbed over a distance δ = 1/α, known as the penetration depth. In the case of silicon, δ is approximately 0.08 μm and 3 μm at λ = 400 nm and λ = 600 nm, respectively. As λ increases to the threshold value λth = hc/EG, α → 0 (or δ → ∞) and the material becomes progressively more transparent to the incoming radiation. However, as λ decreases, α → ∞ (or δ → 0) and the incoming radiation is absorbed near the surface. The dependence of α with respect to λ is a crucial parameter in the design of an APD.

If an e-h pair is produced in a heavily doped p+ region where the electric field is very weak, the pair is likely to recombine. For indirect band gap semiconductors like silicon, the recombination is mostly non-radiative and requires a phonon. The need for a phonon makes the lifetime of the pair longer than that for a pair produced in a direct band gap semiconductor (e.g., GaAs). Suppose that as a result of light illumination, an e-h pair forms close to or in the depletion region where the electric field is strong enough to separate the pair and make the electron move toward the n region and the hole toward the p+ region (see Figure 2). In the depletion region, the likelihood of e-h recombination is extremely small because the region lacks mobile charge carriers and, therefore, the injection produces an electrical signal in the circuit containing the detector. Light has been detected.

The quantum efficiency, η(λ), is defined as the ratio of the number of photo-generated e-h pairs per unit time, ne-h, that produce an electrical signal to the number of incident photons on a photo-sensitive surface per unit time, nλ.

Equation 2. Definition of quantum efficiency.

$\eta \left(\lambda \right)\equiv \frac{{n}_{e-h}}{{n}_{\lambda }}$

Figure 4 is an example quantum efficiency curve for a silicon APD manufactured by Hamamatsu.

The figure shows that for this particular APD, η(λ) has a broad maximum of approximately 85% for the wavelengths around 700 nm and that it declines more steeply for longer wavelengths than for shorter.

The rate at which the photo-produced charge carriers enter the depletion region is the photocurrent Iph and is given by Equation 3.

Equation 3. Photocurrent.

${I}_{ph}=\frac{e\eta \lambda {P}_{0}}{hc}$

Here, P0 is the incident light power (assumed monochromatic light), e is the fundamental charge, h is Planck's constant, and c is the vacuum speed of light. The magnitude of the current flowing through the APD, IAPD, depends on Iph and on the strength of the electric field in the avalanche region. The field determines the degree of impact ionization in the avalanche region and, therefore, the gain M making IAPD = M x Iph. Because the strength of the electric field depends on VBIAS, a relationship exists between M and VBIAS. This relationship is one of the most important characteristics of an APD, and it is schematically shown in Figure 5.

There are three distinct regions on the log(M) versus VBIAS plot. "No gain" or M = 1 is the characteristic of the first region. Here, IAPD = Iph and the APD operates as a photodiode. The electric field in the avalanche region is too weak to energize charge carriers to the minimum energy required for impact ionization of the lattice atoms.

In the linear region, log(M) is linearly proportional to VBIAS. Here, M > 1 because the charge carrier injected into the avalanche region gains enough kinetic energy to ionize a lattice atom. Therefore, one carrier (say an electron) becomes three after ionization: two electrons and one hole. These can now again be energized to cause further ionization events — an avalanche ensues. In the linear region, the avalanche eventually ceases; the electrons and holes produced by impact ionization are collected at the electrodes and their number is M — the gain.

The likelihood of impact ionization depends on the strength of the electric field. Let α and β be ionization coefficients representing the number of ionization events per unit length for electrons and holes, respectively. Figure 6 shows how α and β depend on the strength of the electric field (E) for several semiconductors. In the case of silicon (Si), α > β for any value of E implying that impact ionization due to an electron is more likely to occur than impact ionization due to a hole. Let k ≡ α/β (impact ionization ratio); the figure shows that k ≈ 0.01 for E ≈ 1.5 x 105 V/cm and about 0.4 for E ≈ 6.0 x 105 V/cm for Si, which means that the likelihood of impact ionization by electrons and holes converge with increasing strength of E.

In the third region of Figure 5, the avalanche caused by injection of a single charge carrier does not stop—it is self-sustained. As a result, a current flows through the APD and its value depends on the overvoltage, ΔV. The steepness of the curve in this region implies that the magnitude of the current is very sensitive to ΔV. The strong electric field in the avalanche region maintains e-h plasma where both electrons and holes comparably ionize the lattice atoms (i.e., α ≈ β). In a steady state, the number of newly created pairs is equal to the number of pairs collected at the electrodes. This operation mode of an APD is known as Geiger mode. If it is desired to extinguish the avalanche, VBIAS must be reduced to at least VBR; doing so "quenches" the avalanche.

## Section II: The physics of an MPPC pixel

Figure 7 depicts an equivalent circuit of a single MPPC pixel under bias conditions. The externally applied VBIAS is greater than VBR putting the APD—the section of the pixel delineated with the dashed rectangle—in Geiger mode. In series with the APD and VBIAS is RQ, the quenching resistor whose typical value is about 150 kΩ. When the switch is in the OFF position, the APD is in the "ready" state: the junction capacitance CJ is fully charged to the voltage VBIAS which is also the voltage VD across the terminals of the APD. The voltage VD creates an electric field in the depletion region that is strong enough to initiate a self-sustained avalanche if a charge carrier — electron or hole — is injected into the region. When the injection occurs, the switch instantly goes to the ON state and CJ begins to discharge through the resistor RS representing the combined series resistance of the depletion region — now containing plasma — and the resistance of the neutral regions of the APD. A typical value of RS is ~1 kΩ and that of CJ ~0.1 pF. The voltage on the discharging CJ approaches VBR with the characteristic time constant τ ~ RSCJ. For RS = 1 kΩ and CJ = 0.1 pF, τ ~ 100 ps.

Applying Kirchhoff’s circuit rules, the voltage VD during the discharge decreases and approaches an asymptotic value given by Equation 4.

Equation 4. Asymptotic terminal voltage on the APD as the junction capacitance discharges.

For RQ >> RS, Equation 4 shows that VD tends to VBR. As VD decreases, the electric field in the depletion region decreases, eventually becoming too weak to maintain Geiger mode. The self-sustained avalanche stops — it’s been quenched by RQ — and the switch in the equivalent circuit (see Figure 7) goes to the OFF position. At this instant, CJ begins to charge from approximately VBR to VBIAS through RQ with the characteristic time constant τ ~ RQCJ. For RQ = 150 kΩ and CJ = 0.1 pF, τ ~ 15 ns. At the end of the charging process, the APD is in the "ready" state.

An external electronic detection circuit would be sensitive to the current flowing through the APD, which also flows through RQ and flows out of the positive terminal of VBIAS. Figure 8 depicts schematically how the current i varies with time as an MPPC pixel goes into and recovers from an avalanche.

At any instant of time, i(t) = (VBIAS - VD)/RQ. When the pixel is in the "ready" state,VD = VBIAS and i(t) = 0. An injection of charge into the avalanche region occurs at the time ti. At this instant, CJ begins to discharge causing VD to decrease and i(t) to increase. The current i(t) tends to the maximum value imax ≈ (VBIAS - VBR)/RQ at the time tmax when RQ quenches the avalanche. Between ti and tmax, i(t) ~ imax[1 - exp(-t/RS x CJ)]. For times greater than tmax, CJ charges through RQ, and the current decreases as i(t) ~ imaxexp(-t/RQCJ). Because RSCJ << RQCJ, the current rises more steeply before tmax then it falls off after tmax. Thus, the output current pulse has the characteristic asymmetric shape.

An integral of i(t) with respect to time from ti to infinity is the area under this curve, which equals the net charge Q produced in the avalanche. A detailed analysis yields the following.

Equation 5. Net charge produced by an avalanche.

Since the charge Q is due to a single charge injected into the avalanche region, the gain M for this process can be defined by Equation 6.

Equation 6. Definition of gain.

For a fully depleted junction, CJ is constant with respect to the applied voltage and, thus, Equation 5 shows that M is linearly proportional to VBIAS - VBR = ΔV. The typical values of M range from 105 to 106. If more than one pixel developed an avalanche at the time ti, the total charge generated by an MPPC is the following.

Equation 7. Output charge from an MPPC when Nf pixels "fired."

${Q}_{out}={N}_{f}Q={N}_{f}{C}_{J}\left({V}_{BIAS}-{V}_{BR}\right)$

Here, Nf is the number of pixels that nearly simultaneously "fired" or developed avalanches.

Figure 9 shows oscilloscope traces of waveforms (analogous to the one schematically illustrated in Figure 8) produced by an MPPC in response to low-intensity light. From bottom up, the waveforms correspond to the outputs characterized by an increasing number of simultaneously "fired" independent pixels. The waveforms labeled 1 p.e. are for the outputs where only one pixel has fired in response to light. Note that if two or more photons were to be simultaneously absorbed by this same pixel, the output waveform would have been the same. Similarly, the waveforms labeled 2 p.e. are for two pixels firing simultaneously. To obtain these cumulative waveforms, the oscilloscope is set to the persistence mode; thus, the color in the figure corresponds to the frequency with which the events have occurred. There is a clear demarcation between waveforms labeled 1 p.e., 2 p.e., 3 p.e., etc. The 1 p.e. waveforms were the most common (orange color), whereas the 3 p.e. events were least common among the three labeled sets. Note that there are a few 1 p.e. waveforms that are lagging in time to the main events. They can be generated in three ways: dark noise, afterpulsing, and cross-talk.

Dark noise pulses occur randomly and are due to thermally generated charge carriers reaching the avalanche region. If a pixel is in the "ready" state, the ensuing "dark" pulse is indistinguishable from one due to photon absorption. Such a pulse would be labeled 1 p.e. in Figure 9. If, however, the pixel is recovering from a primary avalanche, the "dark" pulse lags in time to the primary pulse but is not correlated with it.

The pulses due to afterpulsing and cross-talk are correlated to the primary avalanche, which can be due to photon absorption or dark noise. Afterpulsing occurs when lattice defects trap charge during an avalanche and then release the charge while the pixel recovers. The release triggers a correlated secondary avalanche whose gain M depends on ΔV at the time of release. Cross-talk occurs when a primary avalanche in a pixel produces photons with energy greater than the band gap energy. If the same pixel reabsorbs the photon, the resulting secondary pulse would look like an afterpulse. However, if a nearby pixel absorbs the photon, the secondary correlated avalanche may be close enough in time to the primary that the resulting waveform would be labeled 2 p.e. in Figure 9. Section IV below discusses in greater detail dark noise, afterpulsing, and cross-talk.

Integrating a waveform with respect to time gives the net output charge for the event. The amount of output charge is linearly proportional to the number of fired pixels, which, in turn, is proportional to the number of simultaneously absorbed photons. Therefore, information in Figure 9 after integrating each waveform can be expressed in the form of a histogram; Figure 10 is an example of such a histogram.

## Section III: Architecture of an MPPC detector

Figures 11 (side view) and 12 (top view) illustrate a common architecture of an MPPC array. The accompanying Figure 13 shows schematically the distribution of the dopant ions or space charge (left panel) and the resulting strength of the electric field (right panel) as a function of depth. Following the direction of light — red arrows in the leftmost pixel of Figure 11 — they first encountered a very thin (~ 10 nm) anti-reflection coating. The coating improves light transmittance to the next layer, a transparent silicon dioxide, which acts as the window and an insulation between the metal contacts above and semiconductor below. A typical thickness of the insulation is ~0.15 μm. Further below, there is a heavily doped n+ layer, which is connected to the metal contact above through the quenching resistor RQ. The resistor is made of polysilicon. The thickness of the n+ layer ranges from 0.1 μm to 1.5 μm. Immediately below, there is a heavily doped p+ layer with a typical thickness of a few μm. A depletion region where conditions for an avalanche can exist forms at the junction of the n+ and p+ layers. The width of the depletion region depends on the applied VBIAS; for Geiger mode operation, it is ~1 μm.

In this "reach through" architecture, a photon creates an e-h pair in the thick (~300 μm) and lightly doped π layer. As Figure 13 shows, the electric field in this region can separate the pair and direct the electron towards the depletion region and the hole toward the substrate — a thin (~3 μm) p+ layer that makes an ohmic contact with the metal contact below. Upon reaching the depletion region, the electron can initiate an avalanche, while the hole upon reaching the metal contact cancels with an electron coming from the metal contact.

The area of a pixel that is sensitive to light is smaller than its geometrical area. The ratio of the two is known as the fill factor, Fg. The values of Fg range from about 30% to 80%. An MPPC is a square array (circular arrays are also manufactured) of pixels with the number of pixels ranging from 100 (10 x 10) to 14400 (120 x 120). The typical geometrical areas of the pixels (in μm2) are 25 x 25, 50 x 50, and 100 x 100. Multiplying the geometrical area of a pixel by the number of pixels yields the total geometrical area of the detector; two values are typical: 1 x 1 mm2 and 3 x 3 mm2. Multiplying the geometrical area by the appropriate fill factor gives the total light-sensitive area of the detector.

## Section IV: Performance characteristics of an MPPC

### Dark noise

If a thermal fluctuation creates an e-h pair in the depletion region of an MPPC pixel, a charge carrier can enter the avalanche region and trigger an avalanche that is indistinguishable from one resulting from photon absorption. A less likely but still possible scenario is when a pair is thermally produced in the avalanche region itself. In either case, the resulting pulse is referred to as dark noise. The occurrence of dark noise is random in time and uncorrelated to the pulses resulting from photon absorption. The rate of dark noise, however, will depend on overvoltage, temperature, pixel size, and quality of the material semiconductor.

Vacheret et al. (2011) performed a series of experiments in order to characterize the optical and electrical properties of a T2K MPPC; this detector is a customized version of the Hamamatsu S10362-11-050C. The former is a 26 x 26 array of 50 x 50 μm2 pixels with nine pixels replaced with a lead giving 667 active pixels. The latter is a 20 x 20 array of pixels of the same size. The study reports how dark noise rate varies with overvoltage ΔV (Figure 14) and temperature (Figure 15).

Figure 14 shows that dark noise rate increases nearly linearly with increasing ΔV regardless of temperature, but the rate of the increase (slope) increases with temperature (equation 4 in Vecheret et al. (2011) gives the form of the function used for fitting the data). At a given ΔV, the dark noise rate increases with temperature.

The dark noise rate increases nearly exponentially with temperature regardless of ΔV as shown in Figure 15. At a given temperature, the rate is higher for larger ΔV. The study also finds that the dark noise rates varied significantly, about 20%, among the 17,686 MPPCs tested at T = 20 °C and ΔV = 1.0 V.

Dark noise also depends on the pixel size. Figure 16 shows dark noise as a function of ΔV for six MPPCs listed in the insert box. From top to bottom, the first four are from Hamamatsu; by the company's naming convention, the third set of digits in the part number, e.g., 025 or 100, represents pixel size, e.g., 25 x 25 μm2 or 100 x 100 μm2. The bottom two MPPCs are from SensL, and each one has a pixel size of 20 x 20 μm2. Assuming that the MPPCs from Hamamatsu have similar wafer quality, the figure implies that for a given ΔV, the dark noise rate increases with the pixel size; however, the ratio of the dark noises for a given ΔV does not equal the ratio of the pixel sizes. Although the pixels in the MPPCs from SensL have the smallest size, the dark noise in them is not, which could be due to a lower quality of the semiconductor wafer.

### Afterpulsing

Afterpulsing occurs when a secondary avalanche forms in a pixel that is recovering from a photon- or dark-noise-induced primary avalanche. The secondary avalanche is due to the release of trapped charge — electrons — at some time τS after the primary avalanche starts. Figure 17 shows a "forest" of waveforms due to afterpulsing that exhibit a wide range of values of τS (from a few ns to many tens of ns). Note that when an afterpulse waveform starts, its vertical coordinate is the vertical coordinate of the primary waveform at this instant.

The amount of charge generated in an afterpulse is proportional to the area under the afterpulse (or secondary) waveform with respect to the primary waveform. The amount of charge increases with τS, and for sufficiently long τS when the pixel completely recovers, the afterpulse waveform becomes indistinguishable from the 1 p.e. waveform. In this case, afterpulsing erroneously increases the number of 1 p.e. events in photon counting, which translates into an erroneous increase in photon detection efficiency (PDE; described in greater detail below). Afterpulsing also increases the recovery time of the fired pixel, which degrades the time resolution characteristic of an MPPC.

The likelihood of afterpulsing increases with increasing ΔV because the stronger electric field in the avalanche region more effectively dislodges the trapped charges and also increases avalanche probability. Eckert et al. (2010) measured the probability of afterpulsing as a function of ΔV for three Hamamatsu MPPCs, and Figure 18 shows the results. The MPPCs belong to the same family differing only in pixel size (025C being the smallest and 100C the largest). For all three MPPCs, the probability increases with ΔV but the increase is steepest for the MPPC with the largest pixel size. In addition, for a given ΔV, the MPPC with the largest pixel size has the largest probability of afterpulsing.

### Cross-talk

Cross-talk occurs when an avalanche in a pixel generates photons energetic enough to initiate a correlated avalanche in neighboring pixels, as shown schematically in Figure 19. Even though silicon is an indirect band gap semiconductor, emission of light does occur as carriers pass the p-n junction — an effect first observed by Newman et al. (1955). Lacaita et al. (1993) shows that about 3 x 10-5 photons are emitted with energy greater than 1.14 eV per charge carrier crossing the junction. Assuming a gain of 106, a typical avalanche emits ~30 photons capable of causing cross-talk.

To cause cross-talk, a photon must create an e-h pair in the region of a neighboring pixel where one or both of the charge carriers can be injected into the avalanche region. Since the emission of photons is isotropic, this event has a small probability. Moreover, some fraction of the photons will be absorbed in the pixel from which they were emitted, possibly causing afterpulsing in the pixel. The time for a photon to cross a pixel of size d is tc ≈ nd/c, where n is the index of refraction of the material. For silicon n = 3.96; thus, for d = 25 μm, tc = 0.33 ps. This crossing time is much shorter than pixel recovery time (~15 ns) and, therefore, the MPPC would output at least a 2 p.e. waveform in response to a single external photon. The value of tc scales linearly with d, but even for a 100 μm pixel, it is still small compared to pixel recovery time. Nevertheless, the probability of cross-talk decreases with pixel size because the emitted photon has an increasing chance of being reabsorbed in the same pixel.

Eckert et al. (2010) reports measurements of the probability of cross-talk as a function of gain for three MPPCs manufactured by Hamamatsu and one by SensL. Figure 20 displays the results. Since the gain is proportional to the overvoltage ΔV, the figure can also be interpreted as the probability of cross-talk versus ΔV. For all of the tested MPPCs, the probability increases with gain (ΔV) because the number of charge carriers crossing the junction increases which, in turn, increases the number of emitted photons. Increasing ΔV also increases the probability of avalanche. For a given ΔV, the probability of cross-talk decreases with increasing pixel size, as can be seen for the MPPCs manufactured by Hamamatsu.

Akil et al. (1999) investigates the mechanisms for light emission from, among others, a silicon p-n junction operating in an avalanche mode. The study concludes that three distinct processes contribute to the observed spectrum, which is shown in Figure 21. Indirect interband recombination between electrons and holes in strong electric fields is primarily responsible for the emission below the ~2 eV peak in the spectrum. Bremsstrahlung radiation due to scattering of conduction electrons from positive ions is mainly responsible for the shape of the spectrum between ~2 eV and ~2.3 eV. Interband recombination between "hot" conduction electrons and holes describes the emission of radiation above ~2.3 eV.

### Temperature dependence

In a constant temperature environment, the overvoltage ΔV applied to an MPPC determines its operational characteristics such as gain and PDE. Because temperature is an important parameter affecting the opto-electrical characteristics of a p-n junction such as the width of the depletion region, the breakdown voltage, band-gap energy, resistivity, and more, the operation of an MPPC will be strongly affected by this parameter. One example of the dependence is the recovery time of a pixel from avalanche. The characteristic time of the recovery is the RC time constant determined by the values of the junction capacitance CJ and of the quenching resistor RQ. For a given ΔV, temperature affects the width of the depletion region and, thus, CJ. It also affects the resistance of the polysilicon resistor — the quenching resistor — RQ. Thus, the product of these two quantities can also change with temperature affecting the recovery time.

Dinu et al. (2010) investigates how temperature affects the operational characteristics of two Hamamatsu MPPCs: S10362-11-050U-3 and S10362-33-050C-329. Both have 50 μm x 50 μm pixels and total areas of 1 x 1 mm2 and 3 x 3 mm2, respectively. The temperatures are in two ranges: -110 °C to -50 °C and 0 °C to 38 °C. Below is a summary of the main results from the study for S10362-11-050U-3.

Figure 22 shows the dependence of the gain on the bias voltage parameterized by temperature. For each temperature, the relationship is linear and the horizontal-intercept (where there is no gain) gives the breakdown voltage. Plotting the breakdown voltage as a function of temperature shows a linear relationship (Fig 2 in the study) as well. Therefore, for a given bias voltage, the gain decreases with increasing temperature.

As the temperature increases, the Boltzmann distribution implies an increasing fraction of electrons in the conduction band and, thus, a decreasing resistivity of the semiconductor. Because the quenching resistor is made from polysilicon, RQ will decrease with increasing temperature. The junction capacitance, CJ, also varies with temperature. With zero bias, CJ ∝ 1/√Vbi, where Vbi is the p-n junction built-in voltage. Because Vbi decreases with temperature, CJ increases with temperature. Figure 23 confirms both behaviors: the red triangles correspond to the resistance of RQ and the blue circles correspond to the capacitance of CJ. The product of the two curves does not yield a horizontal line and, thus, the RC time constant itself is a function of temperature affecting the recovery time, as shown in Figure 24.

## Section V: Optical and electrical characteristics of an MPPC

### Photon detection efficiency

The photon detection efficiency is the percentage of the incident photons on the geometrical area of an MPPC that is detected.

Equation 8. Photon detection efficiency is a product of quantum efficiency, fill factor, and probability of an avalanche.

$PDE=\eta {F}_{g}{P}_{a}$

In Equation 8, η is the quantum efficiency defined by Equation 2, Fg is the fill factor, and Pa is the combined probability of electrons and holes initiating Geiger mode avalanche. Because η is a function of wavelength and Pa is a function of VBIAS, PDE depends on both wavelength and VBIAS. Eckert et al. (2010) performed a series of experiments designed, among others, to measure an absolute PDE and its dependence on overvoltage and wavelength for several MPPCs manufactured by Hamamatsu. Figure 25 is an example of the results.

The left panel of the figure shows how the PDE depends on overvoltage ΔV for four wavelengths (465 nm, 633 nm, 775 nm, and 870 nm), and the right panel shows how the PDE depends on wavelength for ΔV = 2.15 ± 0.05 V. For each wavelength, the PDE increases monotonically with ΔV, and the rate of the increase depends on wavelength: it is greatest for λ = 465 nm. At a given ΔV of the experimental range (~0.5 V to 2.2 V), the absolute PDE increases with decreasing wavelength. However, the right panel suggests that the PDE reaches maximum at λ ≈ 440 nm and declines with further increase of wavelength. Hamamatsu provides a plot of the PDE versus wavelength for the S10362-11-050C MPPC, which is shown in Figure 26. The measurements are for ΔV = 1.5 V.

Comparing Figures 25 and 26 shows that the shapes of the curves are similar but that the magnitude of the PDE in the latter plot is greater at all wavelengths, even though the measurements are for ΔV = 1.5 V. As noted by Eckert et al. (2010), this substantial discrepancy in the two plots is likely due to the methodology of the measurements. Eckert et al. (2010) exclude the effects of cross-talk and afterpulsing, whereas Hamamatsu does not (as noted on the graph). Recently, Hamamatsu has released a new and improved line of MPPCs which, compared to the older line, have greatly reduced cross-talk and afterpulsing. In addition, Hamamatsu has changed the methodology of measuring PDE versus wavelength that now excludes cross-talk and afterpulsing. MPPC S12571-050C is the new and improved version of the S10362-11-050C, and Figure 27 shows how its PDE depends on wavelength.

### Dynamic range

An ideal photodetector would produce an electrical output that is linearly proportional to the entire range of input light levels. This property, referred to as linearity, is a highly desirable feature in practical photodetectors, but it is never realized for all signal levels. Noise inherent to a detector limits the lowest detectable light level, whereas a variety of phenomena that are unique to the detector, cause a non-linear behavior and eventually saturation. Dynamic range is a ratio of the maximum light level that a detector can detect without saturation or deviation from linearity and of the minimum detectable light level, which would produce S/N = 1.

Suppose that an MPPC with N pixels is diffusely illuminated by an "instantaneous" pulse of light containing Nγ photons. The MPPC operates at an overvoltage ΔV, which determines its PDE. Renker and Lorenz (2009) find that the number of fired pixels Nf as a function of Nγ is well modeled by Equation 9.

Equation 9. Empirical relationship between the number of fired pixels and a number of incident photons for an MPPC with N pixels.

${N}_{f}=N\left(1-{e}^{\frac{-{N}_{\gamma ·PDE}}{N}}\right)$

Since e-x ≈ 1 - x for x << 1, Equation 9 implies that if Nγ·PDE << N, Nf ≈ Nγ·PDE — the number of fired pixels is linearly proportional to the number of photons. However, if the number of photons per pulse increases or, alternatively, if the number of pixels decreases, so that the exponent is no longer small, Nf is no longer linearly proportional to Nγ·PDE, or the upper limit of the dynamic range has been reached. Suppose that an MPPC has 100 pixels and a PDE of 40%; Equation 9 implies the output of the MPPC would deviate from linearity by 1% for 5 photons and by 10% for 54 photons per pulse. Increasing the number of pixels to 14,400 keeping the PDE the same, 1% and 10% deviations from linearity occur for 700 and 7,750 photons per pulse, respectively. Taking 1% to be the maximum tolerable deviation from linearity, increasing the number of pixels has increased the dynamic range by about 140 assuming that the lowest detectable signal is one photon. The increase in the dynamic range is approximately equal to the factor by which the number of pixels has increased. Equation 9 also implies that increasing PDE can actually reduce the dynamic range. Assuming a theoretical value for PDE of 0.8, the deviation from linearity for 700 photons in the last example would be 2%. In contrast, reducing PDE to 0.2 would improve linearity by 0.5%.

Gruber et al. (2014) measured the number of fired pixels as a function of Nγ·PDE ≡ Nseed for several MPPCs with different number of pixels using laser pulses (λ = 404 nm) of 32 ps duration and 20 kHz repetition rate and compared the measurements with the predictions of the model given by Equation 9. Figure 28 (top two rows of panels from Figure 5 in the study) shows the results for the MPPCs manufactured by Hamamatsu: S10362-11-100U (top row of panels) and S10362-11-050U (bottom row of panels).

The panels on the left are for "high" and those on the right are for "low-to-medium" light intensities (note different scales on the horizontal axes). The horizontal lines labeled either 100 pixels (top row) or 400 pixels (bottom row) represent the expected saturation levels (equal to the number of pixels in the detector) predicted by the model in Equation 9 (or by the same Equation 2 in the study). In the panels on the right, the measured values of Nf (or Nfired) are compared to the predictions of the model (green line). The measured values of Nf are higher than the model values and, most surprisingly, can exceed the number of pixels in the detector or the saturation level. Gruber et al. (2014) discusses several possible explanations for this "over saturation" but concludes that none of the explanations is satisfactory and warns that our current understanding of how an MPPC operates may be incomplete.

### Time characteristics

The time elapsed between the instant when a single photon strikes a photodetector and the instant when the photodetector produces an electrical output is called the single photon response time (SPRT). In the case of an ideal photodetector, repeated strikes by single photons would result in identical values of SPRT. In a practical detector, the values of SPRT differ. The spread of theses values implies a limit in the ability of the detector to resolve in time distinct photonic or particle events, e.g., in coincidence counting. Single photon timing resolution (SPTR) is a measure of the spread of the SPRT values, and it is one of the more important characteristics of a detector.

The origin of SPTR in an MPPC has several reasons. 1) The exact locations in the pixel where the incident photons create e-h pairs differ due to the stochastic nature of the photoelectric effect. 2) The drift times of charge carriers from the points of their creation to the avalanche region in the pixel differ even if the points of creation were to be the same. The drift times vary because the carriers form with different initial velocities and because they have different histories of scattering events along their paths to the avalanche region. 3) The times when avalanches are triggered by entering charge carriers into the depletion region vary because: a) the velocities of entering charge carriers differ; b) the strength of the electric field in the avalanche region fluctuates; c) the stochastic nature of the triggering of an avalanche, expressed as a probability. 4) Time constants of pixels vary because of differences, albeit small, in the values of RS, RQ, and CJ. 5) The paths for the output current signals to propagate between the pixels and the common output signal line vary due to the geometrical structure of an MPPC. The importance of the aforementioned factors affecting SPTR can depend on quantities such as, e.g., ΔV, wavelength of light, temperature, or number and size of pixels, SPTR itself can exhibit a dependence on these quantities.

Recently, Puill et al. (2012) has investigated how overvoltage, temperature, and wavelength of light affect SPTR of several MPPCs. The work finds that SPTR improves considerably with ΔV as shown in Figure 29 (Figure 3 in the original study). The plotted data are for 1 mm2 MPPCs (listed in the figure insert) at 20 °C using light with λ = 467 nm. Although SPTR improves with ΔV, the dark count rate also increases making detection of a single photon harder. The figure shows that for ΔV = 1 V, the best SPTR ≈ 200 ps.

Puill et al. (2012) also finds that lowering the temperature from 20 °C to 0 °C slightly improves SPTR (see Figure 4 in the original study) and increasing the wavelength can slightly improve or worsen SPTR depending on the detector.

## Summary

This paper reviews the physics and operation of a Multi-Pixel Photon Counter. The key findings and conclusions from the review are as follows:

1. A single pixel in an MPPC consists of an APD operating in Geiger mode connected in series with a quenching resistor.
2. An MPPC is an array of pixels connected together in parallel. The pixels have a common power source and common output.
3. An MPPC can be thought of as a solid-state analog to a PMT and, thus, its utility as a photodetector is in the very low light level regime.
4. Overvoltage, the difference between the external bias voltage and the breakdown voltage of the APD, is the most important parameter affecting the performance of an MPPC such as gain and noise level.
5. Operation of an MPPC is sensitive to temperature.
6. Three phenomena contribute to noise in an MPPC: thermally induced dark pulses, afterpulses, and cross-talk.
7. Photon detection efficiency is the most important opto-electrical characteristic of an MPPC. Its value depends on overvoltage and wavelength of light.
8. The dynamic range of an MPPC increases with the number of pixels.
9. Single photon timing resolution of an MPPC is on the order of a few hundreds ps and decreases with increasing overvoltage.

## References

• N. Akil, S. E. Kerns, D. V. Kerns, Jr., A. Hoffmann, and Jean-Pierre Charles, "A Multimechanism Model for Photon Generation by Silicon Junctions in Avalanche Breakdown," IEEE Trans. on Electron Devices vol. 46, No. 5 (1999): 1022-1028.
• N. Dinu, C. Bazin., V. Chaumat, et. al., "Temperature and Bias Voltage Dependence of the MPPC Detectors," IEEE Nucl. Sci. Symp. Conference Record (2010).
• P. Eckert, H.C. Schultz-Coulon, W. Shen, R. Stamen and A. Tadday, "Characterization studies of silicon photomultipliers," Nucl. Instrum. and Meth. A 620 (2010): 217-26.
• E. Garutti, "Silicon photomultipliers for high energy physics detectors," Journal of Instrumentation Vol. 6 (2011).
• L. Gruber, S. E. Brunner, J. Marton, and K. Suzuki, "Over saturation behavior of SiPMs at high photon exposure," Nucl. Instrum. and Meth. A 737 (2014): 11-18.
• A. L. Lacaita, F. Zappa, S. Bigliardi, M. Manfredi, "On the Bremsstranhlung origin of hot carrier induced photons in silicon devices," IEEE Trans. Nucl. Dev., vol. 40, No. 3 (1993): 577-582.
• R. Newman, "Visible light from a silicon p-n junction," Phys. Rev., vol. 100, no. 2 (1995): 700-703.
• V. Puill, C. Bazin, D. Breton, L. Burmistrov, V. Chaumat, N. Dinu, J. Maalmi, J. F. Vagnucci, A. Stocchi, "Single photoelectron timing resolution of SiPM as a function of the bias voltage, the wavelength and the temperature," Nucl. Instrum. and Meth. A 695 (2012): 354-358.
• D. Renker, E. Lorenz, Journal of Instrumentation 4 (2009) 04004.
• A. Vacheret et al., "Characterization and simulation of the response of Multi-Pixel Photon Counters to low light levels," Nucl. Instrum. and Meth. A 656 (2011): 69-83.


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# Gauss law in gravitation

1. ### iitjee10

52
Can gauss law in its equivalent form be used to determine the gravitational field??

If so how??

2. Physics news on Phys.org
$$\vec E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat r \leftrightarrow \vec g = -G \frac{m}{r^2} \hat r.$$
Thus you have the correspondences $$q \leftrightarrow m, 1/4\pi\epsilon_0 \leftrightarrow -G$$.
$$\oint \vec E \cdot d \vec a = \frac{q_{encl}}{\epsilon_0} \leftrightarrow \oint \vec g \cdot d \vec a = -4\pi Gm_{encl}.$$
$$\nabla \cdot \vec E = \frac{\rho_e}{\epsilon_0} \leftrightarrow \nabla \cdot \vec g = -4\pi G \rho_m$$
where $$\rho_e$$ is the charge density, and $$\rho_m$$ is the mass density.


text:
# Find the equation of a line which is perpendicular to a given vector and passing through a known point

There is given a vector 2i+j-3k and now I want to find the equation of a line that is perpendicular to the given vector and passing through a known point (1,1,1) .How can I solve this ?.

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Instead of Cross product why can't we use the Dot product.Because the Dot product of any two perpendicular vectors are equal to 0. –  Thusitha May 13 '12 at 11:54


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# Can subsequences be finite?

Say I'm given the sequence $\{a_1,a_2,a_3,\dots\}$. Does a subsequence have to be infinite? Or can it be finite too? For example, is $\{a_1,a_2,a_3\}$ a subsequence?

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It depends on the context. –  Mariano Suárez-Alvarez Feb 23 '14 at 9:10

Yes the subsequence must be infinite. Any subsequence is itself a sequence, and a sequence is basically a function from the naturals to the reals.

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Wholeheartedly disagreed. Without more specification, a sequence is simply a function from some well-ordered set to the range of the sequence. $\omega$ (or $\mathbb{N}$, if you prefer) is one such well-ordered set, but transfinite sequences are often spoken of and finite sequences aren't at all uncommon. –  Steven Stadnicki Sep 26 '14 at 22:32

Usually, this is the definition of subsequence.

Definition. Let $\{p_n\}_n$ be a sequence (in some set), and let $n_1<n_2<n_3<\ldots$ be a strictly increasing sequence of positive integers. Then the sequence $\{p_{n_j}\}_j$ is called a subsequence of $\{p_n\}_n$.

It follows that $\{p_1,p_2,p_3\}$ is not a subsequence of $\{p_n\}_n$, since $n_1=1$, $n_2=2$, $n_3=3$, but what is $n_4$? This is the common feeling for mathematical analysts.

However, we must be careful, since it all boils down to the very definition of sequence. In analysis, a sequence is a function $\mathbb{N} \to X$, or, more generally, a function $N \to X$, where $N \subset \mathbb{N}$ is such that $N$ contains all sufficiently large integers. In other disciplines, it might be useful to call sequence any function defined on a subset of $\mathbb{N}$, even a finite one.

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A sequence in a set $X$ is a function $f:\mathbb{N}\rightarrow X$. Let $g:\mathbb{N} \rightarrow \mathbb{N}$ be a strictly increasing sequence, then the composition of $f$ and $g$, is called a subsequence of $f$. Thus the domain of a subsequence is always infinite, but the range can certainly be finite.

As an example consider the sequence $f(n):=n$ even $n$ and $f(n):=1$ odd $n$. Let $g(n):=2n+1$. The range of $f\circ g$ is equal to $\{1\}$, which is finite.

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In my experience, the term subsequence has always referred to an infinite subsequence. I believe this is standard. If you want to consider finite subsequences, you should say finite subsequence, or if you want to consider subsequences that may or may not be finite, you can say possibly finite subsequence.

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A sequence (real, sets, etc...) is simply a list. In mathematics, unless specified, a finite list (i.e. sequence) is not denoted to be a sequence. A sequence MUST continue unimpeded.

By definition, a sequence of real numbers is a mapping from N -> R. The nature of this definition demonstrates that a finite sequence cannot exist under our current consideration.

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A subsequence can be finite. Note however, that you can always extend to an infinite sequence by $a_1, a_2, a_3, a_3, a_3, \cdots$

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I really don't think $a_1,a_2,a_3,a_3,\ldots$ is typically considered a subsequence of $\{a_i\}_{i=1}^\infty$. –  froggie Feb 23 '14 at 9:10
-1 $a_1, a_2, a_3, a_3, a_3, \cdots$ is not a subsequene –  miracle173 Feb 23 '14 at 9:12
I never said that it was a subsequence. I said that if you want to extend a finite sequence (which means it converges!) to an infinite sequence you can do so by simply extending it trivially to the convergent value. Why the downvotes? –  Euler....IS_ALIVE Feb 23 '14 at 9:13
This is quite ridiculous. Does the sequence $(1,1,1,1,1 \cdots)$ have a convergent subsequence? Of course it does; namely the sequence $1$. –  Euler....IS_ALIVE Feb 23 '14 at 9:15
@Euler....IS_ALIVE, I am sorry but that is ridiculuous in the usual meaning of subsequences. If that were a sensible interpretation of the term subsequence, then there would be no point in Bolzano's theorem, for all sequences $a_1$, $a_2$, $\dots$ would have a convergent subsequence, whether they be bounded or not: namely $a_1$. –  Mariano Suárez-Alvarez Feb 23 '14 at 9:57