In velocity map imaging (photo-dissociation and photo-emission), the ejected particles form a newton sphere. I didn't really get the concept why it is called a "newton sphere" and also why at the poles the distribution is $\cos^2(\theta)$?

# [Physics] Why it is called a Newton Sphere? (Velocity map imaging)

atomic-physicsmolecules

#### Related Solutions

I've never heard of this before and it seems rather weird, but I'll have a guess that seems like the only reasonable thing that could be going on. The kinetic energy of one particle definitely does not depend (fundamentally) on what another particle is doing. It only indirectly depends on the other particles through interactions (i.e. potentials or gauge fields). The author must have in mind some sort of Born-Oppenheimer-like approximation where some subset of interactions are taken care of or fixed before hand, leaving correlations between the remaining degrees of freedom. So here is my guess about what is happening.

If you have two equal mass particles with momenta $\vec{p}_1,\vec{p}_2$ then the total energy is

$$ H = \frac{\vec{p}_1^2}{2m} + \frac{\vec{p}_2^2}{2m} + V(\vec{r}_1,\vec{r}_2).$$

Now introduce the total $\vec{P}=\vec{p}_1+\vec{p}_2$ and relative momentum $\vec{p}=(\vec{p}_1-\vec{p}_2)/2$ then the Hamiltonian becomes

$$ H = \frac{\vec{P}^2}{2M} + \frac{\vec{p}^2}{2\mu} + V(\vec{r}_1,\vec{r}_2),$$

where $M=2m$ is the total mass and $\mu=m/2$ the reduced mass. Now if you neglect the relative momentum because the two particles are rigidly bound and move together always, then you get

$$ H \approx \frac{\vec{P}^2}{2M} + V(\vec{r}_1,\vec{r}_2). $$

Finally substituting back in you get a cross term $\propto \vec{p}_1\cdot\vec{p}_2$ of the kind mentioned in the article, and the mass $M$ indeed refers to the mass of a conglomerate of several particles. You can also do this if the potential is such that you can separate the centre of mass and relative motion problems and solve the relative motion problem on its own. And this idea should straightforwardly extend to the $N$ particle case. I leave that to you.

There is a problem answering your question - the neutrino's mass, which is not known. This means that what kinetic energy it has is difficult to say. For a long time neutrinos were thought to be massless

Kinetic energy is defined as energy beyond rest mass due to motion, so using :

$$E^2 = p^2c^2 + m^2c^4$$

and

$$E_{kinetic} = E - mc^2$$

You see that whether a particle has mass and what that rest mass is changes the kinetic energy you can attribute to it for given momentum and energy values.

The problem is that as we do not know that mass (or masses as there could be multiple "flavors" of each neutrino with different masses for each flavor), we cannot effectively break the energy down into kinetic and rest mass without picking a theory as a basis, something your book won't want to get into and some of which would not have been known when it was printed (1999 ?).

So when textbooks talk about neutrinos in Beta decay they are limited by not being able to make definitive statements. They can stil talk about momentum and energy conservation as that is not affected, but kinetic energy is a problem.

## Best Answer

I think what has happened here is a bit of terminology drift.

One of the many topics that Newton explored was how to understand rotational motion by examining the behaviors of a pair of masses bound together by a coupling force (a string). A key point of his analysis was that if the string is under tension, it means that the two masses must be rotating around a common center of mass (located somewhere along the string). His analysis amounts to an intriguing early exploration of the

nonrelativity of rotation.Now think about all of that in terms of velocity map imaging, where the ideas is to use lasers to disassociate two ions whose masses are roughly comparable (versus emitting a super-light electron), e.g. methyl chloride. You again have two masses, which are the two component ions. They are again bound together, this time by a combination of ionic and covalent bonding. And finally, the two ions can rotate around each other in various well-defined (quantized of course) vibration modes.

The incoming photon essentially snips that bond between the ions, allowing all sorts of interesting data to be collected as the two masses sail off in different directions.

So, I'm pretty sure that Newton got invoked into the terminology of velocity map imaging because he was the first one to do an in-depth classical analysis of the behaviors of two masses behaving like bound ions rotating around a common center of mass.

So where is the terminology drift? Well, Newton didn't actually call the two objects at the ends of the binding string

masses, he called themspheres. So, his analysis is informally referred to as "Newton's spheres" -- note the plural! -- and from that, my suspicion is that the singular version of the phrase, "Newton's sphere," came to be interpreted as the sphere ofproductsproduced by clipping the string between two "Newton's spheres" (the bound ions).Such odd little drifts are actually fairly common in science and technology. One of my favorites was in early personal computers, where people would refer to the central semiconductor memory of a small computer as its "core" (as in "central" or "fundamental") memory. But the term "core" actually originated in a completely different early storage technology where a "core" was a quite literal little disk of magnetic material with a hole in the middle.