# Even neutrinos want some love sometimes

**What about that “almost” you mentioned earlier? What happens when neutrinos do interact with matter?**

Earlier I mentioned that almost all neutrinos that course through your body don’t interact with you in any way – they just plow right on through, passing between the gaps inside and between your atoms. That they hardly ever bump into anything is a testament to how much empty space a typical atom has. Very crudely speaking, we feel solid and substantial to ourselves because our electrons repel each other – they get nowhere near each other to touch or collide, preferring to keep a large distance from other like charges. Since neutrinos don’t have an electrical charge, they can get right up close to an electron (or any other particle) without feeling that strong repulsive force. And so a neutrino is free to pass through the 99.9999…% of volume of an atom that is empty space. Almost all of the time, it makes it all the way through the atom (and through many other atoms) without hitting one of those rare spots already occupied by a subatomic particle.

As with our earlier lessons, below I give you my full derivation — if you really don’t care, skip the “show your work” section and jump straight down to the “bottom line” below.

*Show your work*

Let’s try to calculate the likelihood of a single neutrino interacting with a human body. Let’s say it’s your human body. To estimate this, we’ll be using the typical math you use in collisions – collision cross sections, mean free paths, and transmissions through material.

First off, what is the collision cross section of a typical solar neutrino with the atoms in our bodies? We’ll neglect the chance of a neutrino to hit an electron, focusing only on protons and neutrons (as they’re much larger and more likely to be hit). The cross section of a neutrino depends on its energy, and we know the typical energies of solar neutrinos because we can measure them. The famous graph below shows the energy spectra of solar neutrinos, and shows that by far most of them come off the sun with energies of about 0.3 – 0.4 MeV. The cross-section of a neutrino depends on the energy, with cross-sections getting larger as energies get larger. But a famous calculation of the neutrino cross section at 2 MeV energy gave a value of 6 x 10^-44 cm^2. I can’t manage to find a value for energies at 0.3 – 0.4 MeV, but this is pretty close to 2 MeV for my estimation purposes. (An order of magnitude counts as “close” in this analysis, considering that neutrinos can come in energies well above 1 GeV.) Let’s go with the rough estimate that the solar neutrinos we see most of all have a cross section of about 10^-44 cm^2.

So, given the cross-section, we can calculate the mean free path of a neutrino with this cross section passing through our bodies. (Remember what “mean free path” is? It’s the average distance a particle will plow through another material before hitting anything.) From collision theory, the mean free path is approximately 1 / (collision cross section * density of things it can hit). Since we’re considering neutrinos hitting the protons and neutrons in our bodies, we’ve got to figure out the number of protons & neutrons per unit volume in our bodies. We’re all remarkably close to water in density – water is about 1000 kg per m^2, while the average human is about 1060 kg / m^2. So we can pretty safely assume we’re just water for this analysis. Given that water has a molecular weight of 18.015 grams per mole, it’s easy to calculate that the number of protons & neutrons per unit volume in our bodies is 3.55 x 10^28 per cubic meter. Put these together, and we get the mean free path of solar neutrinos through human bodies:

Mean free path = 1 / (10^-48 m^2 * 3.55 particles per m^{3})

= 2.818 x 10^19 meters

= almost **3000 light years!**

Well sheee-it, a neutrino will typically make it through 3000 light years of water (or human tissue) before it manages to hit something!

So given the size of our bodies, what’s the likelihood of a neutrino colliding with one of our protons or neutrons? We again can use collision theory for this. The transmission is defined to be the fraction of particles that make it through a block of material of a given length without a collision. And given the mean free path, calculating the transmission is easy – it’s just exp ( – length of the block of material / mean free path). So, we can just use (1 – transmission) as the likelihood of a single neutrino colliding with something in our bodies before it can make it through. Let’s assume a body length of 0.5 meters (about 20 inches), and do the calculation:

Likelihood of collision = 1 – transmission through water

= 1 – exp ( – body length / mean free path in water)

= 1 – exp ( – 0.5 m / 2.818 x 10^{19} m )

After a quick trip over to wolfram alpha to calculate this (as it defeats the machine precision limit on my computer), we get

Likelihood of collision = 1.77 x 10 ^{-20}

Finally, we multiply this likelihood by the total number of neutrinos that pass through our bodies in a typical lifetime (2.87 x 10^23), we get:

**Average # of collisions with our body per lifetime = 5092**

So, over the course of our lifetimes, about 5,000 neutrinos will wind up smacking into one of our atoms in our bodies. That’s about once a week!

*The Bottom Line*

So there is a staggering amount of neutrinos passing through your body every second, and each one has a staggeringly small chance of actually colliding with your body. The staggeringly large and the staggeringly small cancel each other out, so to speak, resulting in a surprisingly non-staggering number (in either direction) of neutrinos that will collide with you in your lifetime – about one a week or so.

I should say I’m not the first to do this kind of calculation. John Bahcall, who was probably the father of solar neutrino theory, also estimated this decades ago, and came up with a slightly lower number – that over a typical lifetime, we could expect to interact with about one neutrino. (I’m giving myself credit here, that being off from a Nobel Prize winner’s estimate by three orders of magnitude counts as “close”.)

More recently, over at the very cool Science in Real Life blog, *spatialrift47* made an estimate from a completely different direction, by starting with the observed number of collisions seen in current neutrino detectors. Turns out that modern neutrino detectors use giant tanks of water, and as noted earlier our bodies are pretty much water, to a first approximation. So, as he (she?) notes, you can envision emptying out the giant tank of water and packing it full of human flesh, and it ought to pick up just about the same number of neutrino-collision events per year. Given how many neutrinos the detector actually picks up per year, and accounting for the volume of water, he/she calculates that we ought to see just about 1 neutrino collision per lifetime, on average, matching Bahcall’s earlier estimate.

And over at Physics professor Calvin Johnson’s webpage, he points out that “one can estimate that out of the population of the Earth, one would expect one to five thousand people to have had a neutrino “event” in their body from the 1987 supernova, and perhaps one or two to have had an event— a blue flash—in their eye!”

Recently we heard some buzz in the news about neutrinos potentially going faster than the speed of light. What was that all about?

*Next Up: Do neutrinos really go faster than light? >> *