# About Efimov States and Halo-Nuclei

Halo nuclei could be seen as special Efimov states, depending on the subtle definitions. (The last sentence in the second to last paragraph of this Wikipedia article.)

This does not seem trivial to me in the least. Can someone shed some light on this for me?

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Efimov states in general

Efimov states are special states of three praticle systems. Their existence is a purely quantum effect, because "size" (i.e. cross-section) of these states can be much greater than the range of underlying particle-particle interaction. (1,2)

For each system having an Efimov state there are in fact infinitely many of such states. Some properties of Efimov states are universal and do not depend on the nature of underlying particle-particle interaction: (1,2)

1) size of $n$-th Efimov state is $s_0 \approx 22.7$ times the size of $(n-1)$-th state

2) energy of $n$-th Efimov state is $s_0^2$ times the energy of $(n-1)$-th state

$s_0$ is the solution of the following transcedental equation (3):

$s_0 \cosh(\pi s_0/2) = \frac{8}{\sqrt{3}} \sinh(\pi s_0/6)$

There can be Efimov states, so bound three particle states, when there are no bound dimer states. This is often depicted with borromean rings, since if we take one of the praticles away, the resulting two particle state will be unbound.

Efimov states and halo nuclei

The original Efimov work treated three identical bosons. However, the particles need not be identical, they can have different mases. The Efimov theory can also be extended to describe fermions. (2)

Halo nuclei, that can be viewed as Efimov states, are the nuclei with two-neutron halos:

A two-neutron halo is exhibited by ${}^6He$, ${}^{11}Li$, ${}^{17}B$, ${}^{19}B$ and ${}^{22}C$. Two-neutron halo nuclei break into three fragments, never two, and are called Borromean because of this behavior (referring to a system of three interlocked rings in which breaking any ring frees both of the others). (Atomic nucleus-Wikipedia)

The two neutrons in the halo "do not stick together", so we indeed have a three-particle system. It is interesting, that

[lithium-11] has an exceptionally large cross-section of 3.16 fm, comparable to that of ${}^{208}Pb$. (4)

For an introduction to halo nuclei and Efimov states see the presentation Halo world: The story according to Faddeev,Efimov and Fano by Indranil Mazumdar: http://www.boseinst.ernet.in/capss/talks/2011/IndranilMazumder.ppt

I also advertise the freely available Physics.APS.org article on Efimov states: http://physics.aps.org/articles/v3/9

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this is excellent, thank you! – Dylan Sabulsky Nov 27 '12 at 15:12
The smallest positive root of the above transcendental equation is $s_0 \approx 1.0062378251027814891$, and then $\exp(\pi/s_0) \approx 22.69438259536669519$ (from Kraemer et al.) – Scott Centoni May 27 '14 at 23:20