Nobel Prize in Physics awarded for neutrino oscillations
October 6, 2015 4:49 AM   Subscribe

The 2015 Nobel Prize in Physics Takaaki Kajita of Japan and Arthur B. McDonald of Canada share the 2015 Nobel Prize in Physics for their work in neutrino oscillations ("metamorphosis in neutrinos" in the press release), in which neutrinos switch flavors as it propagates through space. The finding has a large impact on the standard model, as it requires neutrinos to have non-zero mass.
posted by oheso (22 comments total) 9 users marked this as a favorite
Last year my materials physics professor was talking about the development of the blue LED and how important it was exactly while the Nobel Prize announcement was going on. Sadly, I didn't have subatomic physics today, but I'm sure my professor will be pretty excited to talk about it on Friday.
posted by Karmeliet at 5:50 AM on October 6, 2015 [1 favorite]

Karmaliet, I remember when blue LEDs were an as-yet unreached goal. Enjoy your lecture.
posted by oheso at 5:53 AM on October 6, 2015 [2 favorites]

I was *wondering* when they'd get the prize. It almost always takes a few years.

But yeah, discovering that neutrinos in fact have mass and change flavors? Given how hard it is to detect the damn things? Nobel worthy and how.

As to Blue LEDS. I think the RSAS nailed it in the press release. "They succeeded where everyone else had failed. Akasaki worked together with Amano at the University of Nagoya, while Nakamura was employed at Nichia Chemicals, a small company in Tokushima. Their inventions were revolutionary. Incandescent light bulbs lit the 20th century; the 21st century will be lit by LED lamps."

The goal of the prize, as stated by Alfred Nobel, was to award work that provided the greatest benefit to mankind. In general nowadays in physics, this ends up being knowledge, like this work about neutrino oscillations.

Blue LEDs are one of the few recent cases where the benefit to mankind is in such a concrete form. But given the way LEDs are winning -- I'm slowly converting my incandescents and CF to LED, and I'm by far not the only one around here -- there was clearly a massive benefit to mankind with these devices.
posted by eriko at 6:45 AM on October 6, 2015 [2 favorites]

When I started fooling around with electronics, LEDs were not even in the conversation. Wiccuhpeedia sez that "until 1968, visible and infrared LEDs were extremely costly, on the order of $200 per unit, and so had little practical use."

That's when Monsanto began to mass-produce gallium arsenide phosphide (GaAsP) to produce red LEDs suitable for indicators and when Hewlett Packard started employing them in their products. By the mid-1970s, calculators with tiny red LED displays were cheap and ubiquitous. Amazing.

I've mentioned it here before, but when I last took physics, neutrinos were still said to have inertia but no mass (somehow), and were humorously described as "Nothing, spinning."

posted by Herodios at 6:57 AM on October 6, 2015 [4 favorites]

That is to say, angular momentum, not "inertia".
posted by Herodios at 7:16 AM on October 6, 2015

Wasn't the blue LED more of an invention rather than a scientific discovery? That neutrinos have mass proves that the Standard Model is incomplete and that's a pretty big damn deal even if it doesn't produce white light.
posted by three blind mice at 7:20 AM on October 6, 2015 which neutrinos switch flavors as it propagates through space.

They also display an ability to change grammatical number as they pass through a sentence.
posted by The Tensor at 7:46 AM on October 6, 2015 [17 favorites]

This poem is popular today...

Cosmic Gall
John Updike

Neutrinos they are very small.
They have no charge and have no mass1
And do not interact at all.
The earth is just a silly ball
To them, through which they simply pass,
Like dustmaids down a drafty hall
Or photons through a sheet of glass.
They snub the most exquisite gas,
Ignore the most substantial wall,
Cold-shoulder steel and sounding brass,
Insult the stallion in his stall,
And, scorning barriers of class,
Infiltrate you and me! Like tall
And painless guillotines, they fall
Down through our heads into the grass.
At night, they enter at Nepal
And pierce the lover and his lass
From underneath the bed – you call
It wonderful; I call it crass.

1Ok, very little mass.
posted by RedOrGreen at 8:20 AM on October 6, 2015 [4 favorites]

The earth is just a silly ball
To them, through which they simply pass . . .

Hmm. T'heck w' the blue LEDs*, we need to get cracking on that Oscillation Overthruster.

* Zo what. Beeg deel.
posted by Herodios at 8:27 AM on October 6, 2015 [2 favorites]

It's quantum.
posted by Zalzidrax at 8:27 AM on October 6, 2015

The 2002 Nobel Prize was also given in part for neutrino physics. Those scientists discovered the solar neutrino problem, which led directly to the work that won this year's Nobel.

(I'd link to something, but I'm on my phone and it's a pain in the ass)
posted by dirigibleman at 8:57 AM on October 6, 2015

The 2002 blurb from the APS.

Part of the 2002 award was also based on work in part at the Kamiokande (Japan) observatory, but SNO (in Canada) wasn't part of that prize.
posted by bonehead at 10:13 AM on October 6, 2015

I took a freshman seminar on neutrinos, with a professor who was working on MINOS, which generated a neutrino beam at Fermilab and fired it through the earth to the Soudan mine in northern Minnesota, with neutrino detectors on either end to detect the change in flavor. We even got to go down to Fermilab to take a tour and see the (smaller) near-end detector under construction, which was a bunch of steel plates with detectors sandwiched between them, because neutrinos.
posted by ckape at 11:49 AM on October 6, 2015 [1 favorite]

Just for fun, here's a picture of the Sun's core taken in neutrinos by the Super-Kamiokande detector.

Through the Earth's core.

With a 500-day exposure.

Using a 50,000-ton water pool 1 km underground.

Give the man his prize, Martha.

(OK, it was only through the Earth's core during the hours around midnight. But ain't no thang to dawg neutrino.)
posted by Devonian at 12:29 PM on October 6, 2015 [12 favorites]

There was some buzz on Twitter that this year's Physics Nobel would go for evidence for dark matter, to Vera Rubin and Kent Ford for their observational astronomy on rotating galaxies. (They rotate too fast to hold together with just the observed mass.) Some speculation that Rubin might get the Nobel alone, too. Her work is amazing, years of meticulous observation. She's 87 now, hope they don't wait much longer.
posted by Nelson at 1:14 PM on October 6, 2015

I've mentioned it here before, but when I last took physics, neutrinos were still said to have inertia but no mass (somehow), and were humorously described as "Nothing, spinning."

Herodios, since photons do have inertia but no mass, why are neutrinos being similar any more confusing?

And, having taken a lot of verbal punches in my day, I assure you that weightless objects can pack inertia.
posted by IAmBroom at 1:20 PM on October 6, 2015

If I recall correctly, the inertia of massless particles comes from the mass equivalent of their energy, courtesy of E = mc^2.
posted by oheso at 2:16 PM on October 6, 2015

Thanks, Tensor. Something I'd also noticed as soon as it was too late to do anything about it. Let's put our names in for next year's Nobel ..
posted by oheso at 2:17 PM on October 6, 2015

The estimated maximum rest mass of the neutrino (which the discovery of the oscillation confirms must not be zero) is just shockingly tiny, or maybe not so shocking since it's an order of magnitude smaller than the equivalent mass of the energy a single electron gains as it passes between the two electrodes of a single cell of the lithium battery of the laptop I'm typing this on, at less than 0.3 eV -- and it might be much lower than that:
The strongest upper limit on the masses of neutrinos comes from cosmology: the Big Bang model predicts that there is a fixed ratio between the number of neutrinos and the number of photons in the cosmic microwave background. If the total energy of all three types of neutrinos exceeded an average of 50 eV per neutrino, there would be so much mass in the universe that it would collapse.[39] This limit can be circumvented by assuming that the neutrino is unstable; however, there are limits within the Standard Model that make this difficult. A much more stringent constraint comes from a careful analysis of cosmological data, such as the cosmic microwave background radiation, galaxy surveys, and the Lyman-alpha forest. These indicate that the summed masses of the three neutrinos must be less than 0.3 eV.[40]
In other words, the rest mass energy of neutrinos is smaller than the energy associated with single instances of many chemical reactions, and that is so intriguing to me; if there are a lot of very low velocity neutrinos hanging around (and the big bang alone has apparently contributed like 100+/cc throughout the entire volume of space), I would really hope we might find some way they could interact in those chemical reactions.

And as enormous as the energies of some classes of supernovas are in terms of photons and the kinetic energies of less exotic massive particles, they are so much more energetic in neutrinos:
Colgate and White's theory of supernova neutrino production was confirmed in 1987, when neutrinos from supernova 1987A were detected.
The number of neutrinos counted was also consistent with a total neutrino energy of 2.2 x 1046 joules, which was estimated to be nearly all of the total energy of the supernova.[65] {later in the article described as 99% of the energy of the supernova}
And if neutrino detection could be sufficiently improved (it would take quite a few orders of magnitude, sadly, and the prospects are not at all good) and generation of neutrinos brought under very precise control, perhaps SETI could abandon the stale and so far unprofitable interrogation of the radio spectrum in favor of an attempt to find neutrino signals, because nothing short of black holes seem able to stop them:
In November 2012 American scientists used a particle accelerator to send a coherent neutrino message through 780 feet of rock. This marks the first use of neutrinos for communication, and future research may permit binary neutrino messages to be sent immense distances through even the densest materials, such as the Earth's core.[70]
posted by jamjam at 2:45 PM on October 6, 2015 [1 favorite]

photons have momentum p (what you're calling inertia) given by the dispersion relation
p = E/c,
where E is the energy and c is the speed of light. The dispersion relation for massive particles is
E^2 = (pc)^2+(mc^2)^2.

So for massless particles
E = pc or p = E/c.
For massive particles that aren't moving (p=0),
E = mc^2.
Now, there's additional information here, since for a moving massive particle with speed v,
E = gamma m c^2,
gamma = 1/sqrt(1-v^2/c^2). Expanding this relation for small velocities would give you the identification of kinetic energy as 1/2 m v^2, which is the relation you learn in classical mechanics.

Gamma goes towards infinity as v goes to c, which is why you sometimes hear of particles "gaining mass" as they go close to the speed of light. This is wrong. Mass is mass is mass, a particle always has the same rest mass. It gains kinetic energy as gamma increases, which one could identify as mass via
mass = gamma m,
but this is very misleading, and the source of a number of misapprehensions about special and general relativity. Best to keep mass as the property of the object (easiest to measure when it is at rest relative to you), and energy as the thing that increases as it goes faster.

Neutrinos, being nearly massless, are usually moving with enough kinetic energy that the dispersion relation is closer to p = E/c than that of a massive particle (take E^2 = (pc)^2 + (mc^2)^2 and set mc << p). We don't have effective ways to create or capture slow moving neutrinos, as they are generated by high energy processes and so typically have keV-scale energies. Since their masses are at most eV-scale (and likely much less), this means that slowest moving neutrinos around are moving with
gamma ~ keV/eV ~ 1000,
which translates to speeds of 0.9999995c.

Cosmic background neutrinos would be moving very slowly in intergalactic space, however as we live in a deep galactic potential well, they have sped up when they reach the Earth. So there's a wind of cosmic background neutrinos passing through us with speeds of approximately 250 km/s (~0.001c). We cannot yet see these neutrinos directly, and that's really too bad because we'll learn a great deal about an earlier epoch of the Universe than we can otherwise, but they are incredibly low energy and so incredibly weakly interacting. Experimentalists joke that "every particle experimentalist goes and tries to measure the cosmic neutrino background for a few years before giving up." It's an absolute lock on a Nobel for whomever does it.

As for the oscillation stuff, I'm writing something up elsewhere and will let you know when its done (assuming I get it done in a reasonable time).
posted by physicsmatt at 2:59 PM on October 6, 2015 [8 favorites]

I'm really pleased about this. In 1990 in undergrad physics I did a "Problem Solving in Physics" course (thanks Dr Swan, you were great!) where the idea was to do some back of the envelope type calculations to work something out using whatever methods you might think of. One of those problems was to calculate the neutrino flux you'd expect to get from the sun given what we knew about the nuclear fusion reactions. We were all quite stunned after an hour or two of playing with our pencils when the number we came of with was 1/3 short of the actual measured flux, and that much smarter people than us doing careful calculations still came up with about the same shortfall. It's kind of amazing when a major problem you've often wondered about for a couple of decades is solved.
posted by drnick at 7:37 PM on October 6, 2015

So what are neutrinos, and why is their oscillation so interesting and Nobel-worthy?

The neutrinos (often written as the Greek character nu: ν) are a fundamental particle. Like the electrons, quarks, muons, and taus, they are fermions, that is, spin-1/2 particles. This makes them matter particles, unlike the bosonic force carriers like the photon, W and Z bosons, the gluons, or the Higgs boson. Like the other matter particles, we can find that neutrinos come in three types that are almost identical (except, as it turns out, in mass). We call these types generations or flavors, and there is a natural way to pair them with the electron and its heavier generations (the muon and the tau), so we call the three flavors of neutrinos the electron-type, mu-type, and tau-type (νe, νμ, and ντ). Collectively, we refer to the electrons and its partners as the charged leptons and the neutrinos and the charged leptons are all called leptons.

But they are very unusual matter particles. As I will explain, they are so light that they tend to travel near lightspeed, and are so weakly interacting they cannot be bound into nuclei. So you and I aren't made of neutrinos, in the same way that we are made of quarks and electrons. Still, after their discovery, neutrinos as particles are part of the Standard Model of particle physics, the list of particles and their interactions that allow us to predict with unparalleled accuracy, the behavior of things in the Universe on the smallest scales.

The Standard Model is complete, in that it can be used to calculate everything we see in particle physics across the energies we can directly test. That is, if you give me the list of particles which we know to exist, and the list of forces that we know to exist, I can tell you how those particles will behave with very high accuracy (I'm skating over some very important issues to do with the strong nuclear force, but the general idea is here). The Standard Model doesn't tell you "whys": why this number of particles, why these forces. It tells you "hows": how electrons respond to photons, how gluons interact, and so on. You can think of the Standard Model as the rulebook for a sport: it doesn't explain why the playing field is set up the way it is, but it tells you how the players move within the confines of the game.

But even within this The Standard Model it can calculate everything with six major exceptions:

It can't explain what dark matter is.
It can't explain what dark energy is.
It can't explain how the Universe generated more matter than antimatter.
It can't explain how the "Charge-Parity" symmetry is respected by the strong nuclear force.
It can't explain how the Higgs boson mass is stabilized to the value we measure it to be.
It can't explain how neutrinos have mass.

Each of these holes then is of massive interest to physicists. As physicists, as scientists, we want the gaps. We want to find the parts of our theory where everything falls apart, because patching those gaps is where we discover new things. It's also where we gain fame and glory and Nobel Prizes, and clearly we are in it for the money.

The Nobel Prize awarded was thus for the discovery of one of these holes in our understanding of particle physics. Not the solution to the problem, but for the discovery of the problem itself. Kajita and McDonald, as leads of the Kamiokande and SnoLab respectively, discovered that neutrinos change between flavors, νe can become νμ or ντ, νμ can become νe or ντ, and so on. That is, neutrinos oscillate. As I will explain, this discovery is equivalent to the discovery that neutrinos are massive, and that cannot be explained by the Standard Model alone.

Thus, the discovery of neutrino oscillation is the discovery of new physics. We don't understand what that new physics entails yet, but that's where the fun is.

OK, so now let me talk about what neutrino oscillation is, and why that means that neutrinos have mass. Then I'll explain why that can't be explained in the Standard Model, given that every other fermion in the Standard Model has mass, and that's not a problem.

First, let me explain how the discovery was made, as that's what the Nobel prizes were awarded for. Neutrinos interact only through the weak nuclear force. Whereas a charged particle can interact (or scatter) off of photons, and strong nuclear interactions can scatter off of gluons, the weak nuclear force proceeds through the W and Z interactions. One of the key points is that the interactions preserve flavor: an interaction via the charged W± boson turns a neutrino of one flavor into a charged lepton of the same flavor (or vice versa), and a neutrino scattering off the neutral Z0 leaves with same flavor as it came in with.

As the name implies, weak forces are weak, so these scatterings are really rare. So if you want to look for neutrinos, you need to get a lot of target material, and be willing to sift through a lot of junk to get those rare events. Neutrinos were discovered in fact by Ray Davis and John Bahcall in the Homestake Mine using a tank of 100,000 gallons of cleaning fluid. Every week or so, 10 or so atoms of chlorine would absorbe a neutrino emitted by nuclear reactions in the Sun, and transmute into an atom of argon, which was collected and counted. So, very small rates. The process is: a W− is exchanged, simulataneously turning a νe into a e, and a neutron inside the chlorine atom into a proton, which turns chlorine (element 17) into argon (element 18). Due to the energy available from solar neutrinos, only the electron-type neutrino can participate. So Davis and Bahcall could only measure the number of νe passing through their experiment. But that seemed ok: the Sun should only emit νe, not νμ or ντ (as the Sun is made of matter which, like us, contains only electrons, not muons or taus).

The Homestake experiment, however, found that the flux of νe from the Sun was 1/3 the prediction from experiment. Eventually, this rate was confirmed, and Davis shared the 2002 Nobel Prize in Physics for this discovery. But this "Solar Neutrino Problem" remained. One possibility was that there were 1/3 the number of neutrinos emitted from the Sun as we thought, but that was difficult to explain, as that number of neutrinos was predicted by using the energy emitted by the Sun in visible light, and using the well-known physics of nuclear fusion.

As an aside, there was a bit of a fad among science-fiction authors at this time to use the Solar Neutrino Problem to postulate that the Sun was turning off: that nuclear fusion at the core (where the neutrinos come from) had massively decreased for Reasons, and eventually that shut-down would make its presence known at the surface (where the light is emitted) which would be Bad. The Arthur C. Clarke novel Songs of a Distant Earth is the best known one (to me) with this as a plot device.

The other alternative is that neutrinos have mass, and neutrinos emitted as electron-type turn into muon-type or tau-type by time they reach the Earth from their emission point in the Sun. Why would massive neutrinos do this?

The key point is that there are three types of neutrinos, and they are all identical up to this somewhat arbitrary label of flavor. Flavor matters for the weak interaction: as I said, weak interactions preserve flavor. However, there is no reason that the three particles we would identify as a neutrino with some mass would have to be identified with any one of the three flavors. That is, if you have a neutrino with a precise mass (what is called a mass eigenstate in the parlence of our times), it doesn't have to be 100% νe, or 100% νμ, or 100% ντ. It could be, but there's no reason it has to be.

In fact, this mixing of neutrino-types is what we find in Nature. In the Sun, neutrinos are emitted through weak interactions, so they are emitted as 100% electron-type νe. However, what we are calling an νe is really a combination of the three "mass eigenstate" neutrinos, ν1, ν2 and ν3, each with a slightly different mass. We can visualize this as a wave: the νe is build of waves of ν1, ν2 and ν3. Since these neutrinos have slightly different mass, each wave piece in the combination νe propagates at slightly different speeds (after all, having mass is just the statement that you aren't traveling at the speed of light). So, each part of the wave goes from peak, to trough, back to peak at a slightly different rate. So, by time the combination neutrino reaches the Earth, the neutrino that passes through your detector is not made up of the same combination of ν1, ν2, and ν3 that it started as, and that means that, at that particular moment when the neutrino could interact in your detector, it doesn't have to interact exactly as the νe it started life as: it's some combination of νe and νμ and ντ.

It turns out that the Sun is very far away compared to the distance over which νe oscillate into other types of neutrinos. So, on average, by time the Solar neutrinos reach us, they are on average an equal mix of each of the three flavors. So Davis and Bahcall measured 1/3 the rate they "should" have.

Similarly, neutrinos are produced in the atmosphere from the decay of charged muons (which are themselves produced by the collision of high energy cosmic rays and the atoms in our atmosphere). These neutrinos start life as νμ, but by time they reach our detectors, they've oscillated and contain other flavors. Here, the distances traveled can be close to the relevant oscillation lengths, so we can measure properties of the neutrino masses. Finally, we can produce beams of neutrinos (by producing muons and taus and waiting for them to decay in flight) and shoot them towards neutrino detectors and look at the oscillations.

It turns out the thing that matters in this oscillation is in fact the mass squared difference between the different mass states. So by measuring the oscillation of neutrinos into different flavors, you measure the quantity Δm2 between mass states, not the masses themselves. So while we know there are three neutrinos, we don't know if the lightest one is massive or not, just that two of the three neutrinos are massive.

Of course, to measure the oscillation, we need to be able to see the other flavors of neutrinos, not just electron-type. This is what this week's Nobel is for. The experiments led by Kajita and McDonald: Kamiokande and SNO, both are Cherenkov detectors. Cherenkov light is the eerie blue glow that gives nuclear reactors some of their creepy reputation. It's the equivalent of a sonic boom: light travels slower in air or water (or glass, or any transparent medium) than it does in a vacuum. If you take a charged particle and push it faster than the local speed of light in that medium, it builds up a wavefront that we see as light, just as a supersonic airplane builds up a wave that we hear as a sonic boom.

Neutrinos are neutral, so no Cherenkov light comes from them passing through a transparent medium. However, if they scatter off matter and emit a charged lepton, that charged particle can pick up so much energy that it moves faster than the local speed of light, and that light can be seen. So Kamiokande and SNO have giant tanks of water, and giant arrays of phototubes around the edge, looking inward for the light indicative of a neutrino interaction.

Kamiokande used regular water, and could see interactions of any flavor of neutrinos with electrons; however, the νμ and ντ could only interact via the Z, and the νe could interact via the Z and the W. This made the interaction of the electron-type much more likely than others. Given the rate of events, it was not possible for Kamiokande to see the small rate of νμ and ντ over the large νe.

However, SNO used heavy water as a target (borrowed from the Canadian strategic nuclear reserve, as the heavy water was just sitting there, and it might as well do something useful in the meantime). Heavy water has hydrogen atoms which are deuterium isotopes: containing both a proton and a neutron (rather than just a proton). When any type of neutrino comes through, it can dissociate the proton and neutron of deuterium, breaking the nucleus apart. This gives a direct measurement of the total number of νe+νμ+ντ, while measurements of the electron recoil gives the rate of νe. This measurement confirmed that the flux of neutrinos from the Sun was what we predicted, but that some of them had converted to νμ and ντ en route.

Finally, using the Super-Kamiokande upgrade of Kamiokande, we could confirm the direction of atmospheric neutrinos (coming from muon decays). These neutrinos are much higher energy than those coming from the Sun; high enough energy to produce muons in the detector when interacting via W's (Solar neutrinos just give pre-existing electrons a high energy kick). Muons give a different pattern of Cherenkov light than electrons, so the flavor can be measured in Super-K. The direction of the light also gives a clue of the incoming neutrino direction. Super-K sees neutrinos from all directions: the atmospheric neutrinos pass straight through rock, so it sees those produced from all over the planet. However, those coming up from underground are coming from the far side of the world, while those from above are coming from only a few tens or hundred kilometers above. The upgoing flux of neutrinos compared to downgoing indicated neutrino oscillation of muon-type neutrinos.
So, that's the experimental evidence: neutrinos oscillate, and that means at least two of the three neutrinos have mass. What's the big deal?

Well, the problem is that neutrinos can't have mass in the Standard Model. As I talked about previously, for fermions to have mass, the Standard Model needs to tie two different particles together: one that is "left-spinning" and one that is "right-spinning." For some bizarre reason, the Universe decided to treat the fermions differently depending on how they spin. Most importantly, the weak nuclear force only interacts with the left-spinners. The job of the Higgs boson is to equilize the different quantum numbers of the left- and right-spinners, allowing the fermions to have mass.

So the neutrinos we detect are the ones interacting with the weak nuclear force. That means they are all left-handed, by definition. The problem is that the Standard Model doesn't contain the equivalent right-handed neutrinos. It has right-handed charged leptons, and right-handed quarks, but not right-handed neutrinos. Such particles could exist, but since they don't interact via any of the forces we can directly probe (electromagnetism, strong nuclear, or weak nuclear), we can't see them.

So, we could imagine extending the Standard Model to include right-handed neutrinos, which would give us neutrino masses. Sometimes this extension I've seen written as the ν-Standard Model (the "nu"-Standard Model. Get it?). But it involves new particles.

The other issue is that neutrinos are neutral, unlike any other fermion we know about. This means that the neutrino could be its own antiparticle. Maybe the needed right-handed neutrinos are just the antiparticles of the left-handed ones, rather than adding new states. If this is true, that means that the Universe doesn't care to differentiate between neutrino and anti-neutrino. Since neutrinos can turn into charged leptons and neutrinos into charged anti-leptons, that means there's a way to violate the number of leptons in the Universe. So depending on how neutrinos gain mass, there could be important new interactions that are not present in the Standard Model. We don't know yet, but step 1 is always finding out that a problem exists. Then we worry about solving it.

(taken from here, in case you want the original with diagrams and pictures and such)
posted by physicsmatt at 11:01 AM on October 8, 2015 [6 favorites]

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