neutron resonance states in overdense crystals n.takibayev institute of experimental and theoretical...

Neutron Resonance States in Overdense Crystals

N.Takibayev

Institute of Experimental and Theoretical Physics, Kazakh National University, Almaty, Kazakhstan

[email protected]

New type resonances of light particle in systems of heavy centers have been obtained in the framework of model problems that admit exact solutions.See: Takibayev N.Zh. “Class of Model Problems in Three-Body Quantum Mechanics That Admit Exact Solutions”// Physics of Atomic Nuclei, V 71, No 3, p 460-468, 2008

The resonance energies and widths are shown depend on distances between heavy centers. Takibayev N.Zh., Exact Analytical Solutions in Three-Body Problems and Model of Neutrino Generator, EPJ Web of Conferences, V 3, 05028, p 1– 8, (2010); arXiv: 1002.2257v1 [nucl-th].

Seoul, 23-08-2011Takibayev N "Neutron

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In the case of neutron the new resonance states appear in subsystem of heavy fixed nuclei when these distances are many times more than the radius of nuclear forces and many times less than the atomic size.

Takibayev N.Zh., Neutron Resonances in Systems of Few Nuclei and Their Possible Role in Radiation of Overdense Stars, Few-Body Systems, Springer-Verlag, Springer ed., DOI: 10.1007/s00601-010-0207-4, 2011

Therefore, new neutron resonance states can take place in overdense crystals, for instance, inside of overdense star crusts or cores. Parameters of these resonances in two nuclei subsystems of Mn, Ti, Fe, Ni and some others and their radiation spectra have been calculated.

Problems whose solutions are known in an analytic form are of paramount importance since they permit performing a transparent investigation into phenomena.

Model problems can result in unusual and unexpected solutions. A generalization of such model solutions can provide an efficient tool for studying more complicated systems, this being of importance for a number of practical applications. Model problems are also of interest as tests of various numerical methods.Solutions to these problems may be of interest for a number of problems in atomic and molecular physics, as well as some problems in crystal physics, nuclear physics, and nuclear astrophysics.


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Class of Model Problems in Three-Body Quantum Mechanics That Admit Exact Solutions

Here we consider a very simple model of three particle system. This model is based on two simplifying approximations:

- two-particle interactions are taken in the form of separable potentials;

-in three-particle system Born-Oppenheimer approximation is used.

The first approximation gives the exact solution of the two- particle problem at once in an analytical form.

The second one leads to a simplification of the three-particle problem in the assumption that one particle is light, while the others are heavy.


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The first approximation assumes that two-body interactions have separable or “compound” form.

|| iiiit

where the enhancement factor η = η(E) depends on the initial energy only:

resi EE

Const

2/

2/

)(

1

22.

2

iEEE

tR

i

The amplitude of Breit-Wigner form corresponds to such definition

where 2/22. iEE Rres Index “2” marks parameters of two-body system


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The second (Born-Oppenheimer) approximation (the limit of ) results in simplification of three-body equations. And together two approximations leads to exact solutions of three-body problem.

0/ Mm

Then we introduce the Fourier transform of three-body equations (see in Physics of Atomic Nuclear, V 71, pp 460-468, 2008) and obtain:

)',()();()'();()',( 000 rrMpprJrrprJrrM kllklk

0

)()()exp(2);(

220

0 ipp

pppripdmprJ kl

lk

where

001 pp

- initial momentum of the light particle, and l,k = 2,3.


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The solutions of problem of light-particle scattering on two fixed centers can be represented in the analytical form. We can write out the two modes of solution:

where i,j = 2,3 – numbers of heavy centers, r, r’ - radius-vectors of the initial and final fixed scattering centers, respectively.

Elements of diagonal matrix are

ijijijii )r(J)r(J)r(B

)'rr()r(M)'rr()r(M)'r,r(M

where)(

)(

1),( 0 rJ

rBIprM ij

ii

ij

10 )(

)(

1),(

iii

ii

ii rBrBI

prM


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Now we consider the problem of neutron scattering on two fixed centers subsystem if the two-body scattering amplitudes have the Breit-Wigner resonant form:

2/

2/

)(

1

22.

2

iEEE

tR

i

The energy and width of resonance are determined with real and imaginary parts of resonance wave number:

mppE IRR 2/)( 22,

22,2, mpp IR /||2 2,2,2

Here, index 2 marks two-body parameters (below index 3 will mark three-body parameters).

The BW representation can be transformed to separable form if determine a form-factor as

)(2/)( 2 Epi


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Then we obtain )2/( 22,1 iEE Ri

So, three-body exact solutions can be obtained in case of Breit-Wigner pair t-matrices, too.

Let we have an isolated pair resonance acting in S-wave. Then we can write the elements of matrix J as

rp

riprJJJ

Rjiij

2,

02 )exp(

2)(

Zeros of D are determined by equation

0)r(J2

iEE)r(J2

iEE 22,R

22,R

And we mark values of resres rrpp ,0

33, iEE Rres

when 0),( 0 rpD

),Re( 03, ppR )det( ijBID


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There are two sets of three-body resonances:

rp

rpEE

R

RRR

2,

3,22,3,

)cos(

2

rp

rp

R

R

2,

3,23

)sin(1

rp

rpEE

R

RRR

2,

3,22,3,

)cos(

2

rp

rp

R

R

2,

3,23

)sin(1

Some of them are situated at energy scale above the energy of two-body resonance, and others - under this energy. Some of three-body resonances will have more narrowed width, others – more widened in comparison with the width of two-body resonance Note, there are no principal difficulties in including more complicated forms of two-body separable potentials and other partial components into the model.

Sapporo, 07-07-2011Takibayev N "Exact Analytic

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In common case of neutron-nucleus resonance scattering we have to take into account not only elastic scattering:

An,

*'AAn

where is exited nucleus of mass number bigger than . *'A

Then resonance part of two-body t-matrix can be represented in Breit-Wigner’s form, as:

2/

1)(

;;

0;

iRi

nini

n

Rinni iEEmp

EEt

2/

1)(

;;

0;

iRi

ini

n

Rini iEEmp

EEt

but inelastic channels too. First of all we take into account the reactions of gamma radiations via neutron resonance capture :

nAAn

A


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It is remarkable that our consideration of new three-body quantum states remains without modifications.

Really, the neutron capture by nucleus (as a two-body reaction) leads to disappearance of neutron as the main particle in three-body rescattering.

So, the process of three-body rescattering, which gives new quantum states, will stop after the act of neutron capture.

We guess that a probability of neutron appearance via inverse reaction will be very small:

*'AAn

AnA *'Therefore, the function ( - the enhancement factor of three-body amplitude) is the result of neutron elastic rescattering on subsystem of heavy centers.

D 1D


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However, three-body inelastic reactions will be connected with elastic one and all of them will have resonance behavior at the same resonance points depending on values of

It means that new type of gamma radiation will appear under the same conditions as three-body neutron resonances where function

...,, 21 resres rrr ...,, 210 resres ppp

0),( 0 rpD

,,0

;

1jijni

nnij M

mpT

We can write a coupled part of three-body inelastic amplitude in form

)',()(

1)',( rrJ

rBIrrM ij

ii

ij

where as above


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It is important to note that matrix is the effective potential created by the light particle (neutron) rescattering between two-heavy centers (ions in nodes of dense crystal).

)',( rrM ij

This effective potential with resonance behavior becomes very strong at resonance points and can be many times more than ordinary direct two-body interaction, for example, the Coulomb repulsive forces between ions.

We consider results of calculations for the neutron resonances inside of solid structure. Nucleus will be assumed fixed in the nodes of the crystal lattice. Our calculations are based on the well-known data of neutron-nucleus resonances. The lowest neutron-nucleus resonances, which are well isolated from each other, have been taken into account.


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The important quantity of the three-body problem is

because K is the universal factor in amplitudes

Moreover this factor determines the ratio of three-body and two-body amplitudes under the same conditions.

)()(

1),( 0 rJ

rBIprM ij

ii

ij

10 )(

)(

1),(

iii

ii

ii rBrBI

prM

DK

1


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The n+ Mn55 + Mn55 three-body system

The lowest resonances in two-body subsystems of n+Mn551)with J=3 ; l = 0 :-bound states with energy: Eb1= - 5.1 keV in S-wave,- the first resonance: ER = 1,098 keV; G = 18 eV ; Gn = 18 eV; - the second resonance : ER = 2,37 keV, G =Gn = 460 eV ;

2) with J=2 ; l = 0-bound states with energy: Eb2=- 2.4 keV in S-wave, - the first resonance: ER = 0,336 keV and Γ =(18,3+0.435 ) eV Γn = 18.3 eV ;


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Here the distance between ions of Mn55: d=r/2, r - in fm

Re (K) Im (K)

Three-body resonances in the case of J=3, l=0

k0 = kR*(1+y); the parameter y give a deviation y = k0/kR-1


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Here the distance between ions of Mn55: d=r/2, r - in fm

Re (K) Im (K)

Three-body resonances in the case of J=2, l=0

k0 = kR*(1+y); the parameter y give a deviation y = k0/kR-1


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The n+ Fe56 + Fe56 three-body system;

The lowest resonances in two-body subsystems n+Fe56:

1)bound states with energy: Eb= - 6.52 keV in S-wave,2)the first resonance : ER = 1,147 keV and G = 0.74 eV ; Gn = 0.056 eV ; 3) the second resonance: ER = 12,45 keV G = 0.0023 eV= Gn

Re (K) Im (K)


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The n+ Fe57 + Fe57 three-body system;

The lowest resonances in two-body subsystems n+Fe57:

1)bound states with energy: Eb= - 0.44 keV in S-wave, J=12)the first resonance : ER = 1,63 keV and G = 0.11 eV =Gn ; 3) the second resonance: ER = 3,95 keV G = 108 eV= Gn

Re (K) Im (K)


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The n+ Ni61 + Ni61 three-body system

The lowest resonances in two-body subsystems n+Ni61:

1)bound states with energy: Eb= - 0.0095 keV in S-wave, J=22)the first resonance : ER = 7,545 keV and G = 227,3 eV Gn = 225 eV ;

Re (K) Im (K)


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The three-body system: n+ Ni62 + Ni62

The lowest resonances in two-body subsystems n+Ni62:

- bound states with energy: Eb= - 0.077 keV in S-wave, J=1/2-the lowest resonance : ER = 4,54 keV ; G = 1880 eV =Gn

Re (K) Im (K)


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Re (K) Im (K)

The three-body system: n+ Ni62 + Ni62

A wider range of r : (10 – 500) fm.


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The resonant nature of the effective interaction may lead to the following interesting phenomena. If the distance between the nuclei in solid structure is close to the , the resonance will be evolving rapidly. The structure will collect plenty of structural resonance neutrons. Their multiple rescattering on nuclei of structure will be accompanied by the radiation capture of neutrons. There will be a resonant gamma radiation.

resres rd 2

resdd

,,; ~ jijninij MT

Effect resonance scattering of light particle on two heavy centers when distance between centers can be changed

d =2r

In the case of a single crystal under high pressure, the collective effects create a series of resonant states for neutrons in the crystal.


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As a result, the resonant structure will release energy. Released energy and radiation will spread inside the neutron star. But by raising the temperature the structure volume will increase. Therefore, the structure will leave the resonance property of energy generation.

resdd

Then the structure begins to cool down and returns to former state with the parameters close to the resonance.

resdd

The resonance cycle can be repeated over and over again. It will be a kind of "breathing" of star solid structures.

Acknowledgments. I thank Prof. Kiyoshi Kato and the group of “Nuclear Reaction Asia Database“ for useful discussions, the Hokkaido University for their hospitality and support, and Prof. M. W. Snow for his valuable advices.


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Thank you for your attention!

neutron resonance states in overdense crystals n.takibayev institute of experimental and theoretical...

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