SELF-CONSISTENT DESCRIPTIONSELF CONSISTENT DESCRIPTION OF ALPHA DECAY PROCESS AND ATTENDANT ELECTRON SHELLATTENDANT ELECTRON SHELL

EVOLUTIONYu.M. Tchuvil’sky

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University Moscow RussiaMoscow State University, Moscow, Russia

In co-authorship withp

S.Yu. Igashov

V( )

ALPHA DECAY OF A NUCLEUSV(r) ( / )S Pα α αω πΓ = h

( )2 2exp ( )

extr

nuclP V r Q drαμ⎡ ⎤

= − −⎢ ⎥⎢ ⎥⎣ ⎦

∫h

nuclQ

r( )

intr⎢ ⎥⎣ ⎦∫h

Experiments fix the energy of α decay of anExperiments fix the energy of α-decay of anatom. The potential of the electron shelldecreases this energy compare to initial one.

The energy loss ΔE is about 40 keV for actinides. Due to that Pα changes by few tens per cent.

decreases this energy compare to initial one.

α g y p1. V. A. Erma, Phys. Rev. 103, 1784 (1952).An example of the formula:

7/5 2/564.3 80 ( )E Z Z keVΔ = −

SETTING UP OF THE PROBLEM

To consider the alpha-decay of the atom and therespective bare nucleus and to calculate the relationsbetween the characteristics of these processes, namelythe difference of the energy yields

E Q QΔ = −bn atE Q QΔ = −and the ratio of the alpha decay widths or moreexpressive value

/ ( )/( )at nucl at nuclδΓ Γ= Γ −Γ Γ +Γp

It is also interesting for the proton and the cluster decayIt is also interesting for the proton and the cluster decay.A supporting result2 V L Mikheev et al PEPAN Lett 5 623 (2008)is the difference between the widths of 212Po decay inPb and Ni matrixes:

2. V. L. Mikheev et al, PEPAN Lett. 5, 623 (2008).

Pb and Ni matrixes:

1/2 1/2 1/2( ) ( ) ( 0.66 0.25) ; 0.3T Pb T Ni ns T sμ− = − ± =

THE PRESENT STATE OF THE ART

Th di d bl i f i i A b fThe discussed problem is of certain interest. A number ofpapers are published in the last decade:3 W Rubinson and M L Perlman Phys Lett B 40 3523. W. Rubinson and M. L. Perlman, Phys. Lett. B 40, 352(1972).4. N. T. Zinner, Nucl. Phys. A 781, 81 (2007)., y , ( )5. Z. Patyk, H. Geissel, Y. A. Litvinov, A. Musumarra, and C.Nociforo, Phys. Rev. C 78, 054317 (2008).6 S Y I h Y M T h il’ k Ph At N l 76 14526. S.Yu. Igashov, Yu.M. Tchuvil’sky, Phys. At. Nucl. 76, 1452,(2013).7 F F Karpeshin Phys Rev C 87 054319 (2013)7. F. F. Karpeshin, Phys. Rev. C 87, 054319 (2013).8. A.Ya. Dzyublik, Phys. Rev. C 90, 054619 (2014).9. S.Yu. Igashov, Yu.M. Tchuvil’sky, Proceedings of IASEN-g y g2013. World Scientific .Singapore (2013) p. 191.All these papers contain some elements of accuratef li b t f th t ll Sformalism but no one of them aggregates all. Somesignificant details are skipped in all these papers.

STANDARD APPROACH. IMPACT OF THE ADDITIONAL POTENTIALADDITIONAL POTENTIAL

In the WKB approximation:

( ) ( )2 2/ exp ( ) ( ) ( )

ext extr r

at bn bn at e bn bnP P V r E V r dr V r E dr⎡ ⎤

= − − + − −⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫h

μ ( ) ( )int intr r⎢ ⎥⎣ ⎦∫ ∫h

here./ / 1at bnP PδΓ Γ= −

QUESTIONS CALL FOR ANALYSIS

1. Is WKB approximation correct?2. Is the reflection effect significant in the classicallyll d ?allowed area?

3. Does the form of the strong potential play a role?4 Is the Thomas-Fermi model satisfactory?4. Is the Thomas-Fermi model satisfactory?5. Can one use non-relativistic approach?6. Does finite-range distribution of nuclear charge play ag g p yrole?7. Is the process diabatic or adiabatic one in relation toth ti f l t ?the motion of electrons?8. Does the rearrangement of the electron shell play animportant role?important role?9. What is the probability of the electron to be knockedout?10. Is the alpha particle bare or dressed?11. Is the effect measurable? How to detect the effect?

THE PROCESS PATTERN

Th l b d l i b d d hThe strongly bound electrons remain bound under thechange of the nuclear charge and the weakly boundelectrons are too slow to leave the shell before the α-electrons are too slow to leave the shell before the α-particle. Indeed,the equivalent electron energy

( )e ev v E≈α

q gy

is equal

The probability ofa hard α-electroncollision with theenergy transferE 500 V tE~500 eV turnsout to be ~10-1.

There is a partial confirmation of that by theexperimental data – precisely measured energies of theemitted particles in the α decay of 226Ra (theemitted particles in the α-decay of 226Ra (theexperimental uncertainty of the energy yield is about250 eV) is steady but not scattered. High speed of the250 eV) is steady but not scattered. High speed of thealpha makes it possible to consider the interaction ofthe alpha with two-folded negative ion of the daughternucleus.

For the proton decay the probability of the electronFor the proton decay the probability of the electronknock-out or pick-up is much smaller.

So the charge of the residual system (daughter nucleusand electrons) turns out to be –2 (or –1 for the protondecay).

QUANTUM-MECHANICAL APPROACH

A pure quantum-mechanical statement of the problemof the heavy-charged-particle (proton, alpha) decay ofan atom requires the following asymptotics:

( ) ( ) ( ) 22~ , , , /(2 )r l l resr G kr iF kr where Z Z c Eχ η η η α μ+ =% %% % %( ) ( )

Where Zres is the charge of residual system (daughternucleus and residual electrons). For the bare nucleus ηi l d h h f h d h l Zis related to the charge of the daughter nucleus Z1.Consequently starting from this asymptotic formbackward to the short distances (deep sub-barrier areabackward to the short distances (deep sub-barrier area,the strong forces are disappeared, the electron chargeinside is negligible) one can obtain:

but not

g g )( ) ( ) ( )~ , ,r l lr AG kr BF krχ η η+

but not ( ) ( ) ( )~ , ,r l lr G kr iF krχ η η+

At the same time at the short distances :

( ) ( ) ( ) ( )~ , , .,r l l lr AG kr BF kr AG krηχ η η+ ≅

because

( ) ( ) 2 2, / , ( ) , ;

extr

l l extr

F kr G kr P V Ed as r rμ

η η ρ ρ≅ = − <<∫h

17(int) 1/2and ~10 for 1 .

r

rP t ms− =h

So the mathematical problem is to determine the multiplier A.

To do that two pairs of independent solutions of aHamiltonian in two regions of the variation of r are usedHamiltonian in two regions of the variation of r are used.They are:

( ) ( ), ; ,l lG kr F krη η −% %% %( ) ( ), ; ,l lG kr F krη η

at the short distances and ( ) ( );G kr F krη η( ) ( ), ; ,l lG kr F krη η −

in the outer asymptotic region. Because of that thel ti f th ti ithi th i t di t i t l fsolution of the equation within the intermediate interval of r

is equivalent to a linear transformation of the coefficientsdetermining the weights of related functions in one anddetermining the weights of related functions in one andanother regions. This transformation is described by theunimodular matrix ||M|| of the rank 2. Thus it is convenient|| ||to solve equation

( ) ( )H r E rχ χ=( ) ( ).r rH r E rχ χ=

where2 2dh ( ) ( ) ( )2 2

. .2 .2 Coul shell c f

d V rH V r V rdrμ

= − + + +h

ith h t di t b dwith short-distance boundary

( ) ( ), .reg lr F krχ η=( ) ( )g

Its asymptotics at infinity takes the form:

( ) ( ) ( )~ , , .reg l lr G kr F krχ α η β η+% %% %

This solution is equivalent to the transformation:This solution is equivalent to the transformation:

10|| || ; /M k k

αβ

−⎛ ⎞ ⎛ ⎞=Δ Δ=⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠%|| || ;

1 β⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

i. e.

/ /β12 22/ ; / .M Mα β= Δ = Δ

The inverse transformations

1 21 11 0|| || || ||

0A A

M and MB Bi

− −⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠1 20 B Bi⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

allow one to obtain the multiplier A and the final result:

2 21 2/ ; / ; / .A A Aα β α β=− Δ = Δ = + Δ

2 2 2 2 2/ ( )/ ( )k kAΓ Γ Δβ β%

2 2 2 2 2/ ( )/ ( ).bn at Akk

Γ Γ = = + Δ = +α β α β%

In the precise quantum-mechanical approachp q pp

/ 0.99812423.bn atΓ Γ =

In the WKB approach

/ 0 99812444Γ Γ/ 0.99812444.bn atΓ Γ =

The accurate quantum-mechanical calculationsqperformed here demonstrate that in spite of greatdifference of the boundary conditions that is nottaken into account by WKB approach thisapproximations turns out to be perfect.

DYNAMICS OF THE ALPHA PARTICLE PROPOGATION AND ENERGY FUNCTIONALAND ENERGY FUNCTIONAL

The models using for calculations of the energy functionalThe models using for calculations of the energy functionalVshell (r) in different papers vary in:1. Methods of calculation of the electron wave functions(Thomas-Fermi, Hartree-Fock, Hartree-Fock-Dirac).2. Scenarios of the process (diabatic – wave function ofthe electron shell is fixed and thus remains the same asthe electron shell is fixed and thus remains the same asthat of the parent atom, adiabatic – it is considered to bethe same as that of daughter one, two outer electrons turnst e sa e as t at o daug te o e, t o oute e ect o s tu sout to be forgotten.Papers [7,9] demonstrate that the results obtained by useof Hartree-Fock-Dirac model differ essentially from theones obtained using Thomas-Fermi or Hartree-Fock ones.So the former approach is only validSo the former approach is only valid.

In reality the motion of the alpha particle is adiabaic asrelated to fast electrons (ve > vα) and diabatic as related

h S h f h fto others. So the rearrangement of the former onesshould be taken into account.

Strongly bound (and consequently fast) electrons makea dominating impact on the outgoing alpha-particleg p g g p pwave. The influence of two outer electrons is negligible.That is why adiabatic model looks better than diabatic.

Nevertheless, accurately speaking, a hybrid modeldescribing fast and slow electrons in different ways isdescribing fast and slow electrons in different ways ispreferable.

But there is a problem of adiabatic approach. Itwas indicated in [5] that the energy difference

( ) ( 2) (2) (0)eE B Z B Z B VΔΔ = − − − −exists, where B(Z) is the binding energy of arespective atomic shell and Ve(0) is the potential ofealpha – electron shell interaction in the initial state.The origin is the rearrangement of the electron shellf th th t t th h ll f d ht Itof the mother atom to the shell of daughter one. It

must be taken into account both for the calculation ofΔE and as a consequence indirectly Γ The authorsΔE and, as a consequence, indirectly, Γ. The authorsproposed to add this energy to the alpha-particleenergy.

Unfortunately in reality the switching of the regime isnot instantaneous the electron shell evolves duringnot instantaneous – the electron shell evolves duringthe propagation of the alpha particle through it.

FORMALISM Th H ilt i f th t h th fThe Hamiltonian of the system has the form:

,( 2) ,( 2) , ,( ) .Z e e Z e e eH T V r T V V V− −= + + + + +α α α α

The interaction between the alpha and the electron(point-like volume with the charge 1) is expressed as(point like volume with the charge 1) is expressed as

( ) ( ) ( )2 2 *(1) (1) 0 0 1 21

1 42 2 .2 1

LL

e e Lm LmLr

V r e e Y YLα απ∞

<+

=− =− ⋅ ⋅ Ω Ω∑∑( ) ( ) ( ), (1) , (1) 0 0 1 21, (1) 2 1e e Lm LmLe L m Lr L rα α

α+

=− >+∑∑

In the monopole approximation

( ) (1)2 22 21 1

1 1 er r⎛ ⎞− <⎛ ⎞ ⎜ ⎟⎜ ⎟Δ

α( ) ( )2 2(1)(1) 0 0

(1) (1)(1)

, 2 21 1max ,

0ee

e ee

r rrr r r

r rV r e e ⎜ ⎟− =⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠Δ =−

⎟−

⎜ >⎝ ⎠

α

αα

α

Because of that theelectrons which arelocated outside thesphere rα remain to bein their initial states.The wave functions ofthese electrons andthese electrons andtheir desity turn out tobe unchanged. Theg

wave functions of the internal electrons are brought tothe orbitals of the daughter atom. So the density is

k i h ftaken in the form( )

( )final r r r

r⎛ ⎞< ⎟⎜ ⎟⎜= ⎟⎜

αρρ

2∑

( )( )initial

rr r r

= ⎟⎜ ⎟⎜ ⎟⎜ >⎝ ⎠α

ρρ

2

( )( )

( ) ( )ii

r rρ = φ∑

The electron wave functions are calculated by useof Hartree-Fock-Dirac approach The Hamiltonian

1 1ˆ ˆZ Z

DH H +∑ ∑

of Hartree Fock Dirac approach. The Hamiltonianis written in the form:

2ˆ ( ) ( 1) ( )D βh

1,

2ii i j ij

H Hr= ≠

= +∑ ∑where

2( ) ( 1) ( ).DenH c p mc V rα β= + − +h

Its matrix element ˆ ˆ| | | |

a b a b

D Da b ra H b a H bμ μ κ κμ μ δ δ=

wherewhereˆ ( )

ˆen

D

dV r cdr r

H

κ⎛ ⎞⎛ ⎞+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟=⎜ ⎟

h

1⎛ ⎞

2ˆ ( ) 2r

en

Hdc V r mcdr r

κ⎜ ⎟⎜ ⎟⎛ ⎞− + −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠h

12

Jκ ⎛ ⎞= ± +⎜ ⎟⎝ ⎠

It is convenient to present the wave function of an

( ) ( ),mn r⎛ ⎞Ω θ ϕ

⎜ ⎟k kG

orbital by the form of bi-spinor:

( ) ( )( ) ( )

,

,

,,

r

r

r

nn Jm m

ni r −

ϕ⎜ ⎟φ =⎜ ⎟Ω θ ϕ⎝ ⎠

k kk

k k

G

Fwhere

,1/ 2( , ) ( , ) ( ).m Jlm m lm mC Y

σ σ

μκΩ θ ϕ = θ ϕ χ σ∑∑ ,

m mσ σ

σ

∑∑

The expression of the electron density reads

( ) ( )( )2 22 1 .i i ir J= + +∑ρ G F( )i

The total nucleus-electron interaction and theinteraction of the alpha and the shell as a whole takeinteraction of the alpha and the shell as a whole takethe following forms:

( ) ( )final initial⎛ ⎞

f l⎛ ⎞

( ) ( ) ( ) ( )2 3 3,( 2) 02 ;

final initial

e Zr r r r

r rV r Z e d r d r

r rα α

αρ ρ

−< >

⎛ ⎞⎜ ⎟= − +⎜ ⎟⎝ ⎠∫ ∫

( ) ( ) ( )2 3 3, 0

12 .final

initiale

r r r r

rV r e d r r d r

r rα α

α αα

ρρ

< >

⎛ ⎞⎜ ⎟= +⎜ ⎟⎝ ⎠∫ ∫

⎝ ⎠

The formula of the kinetic energy of the electrons reads:

( ) 2* * 2( ) 2 1 2 ,r

i ie i i i i i e i

d dT r J dr c P Q c Q P mc Qd d

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎟⎜ ⎟ ⎟⎜ ⎜ ⎟= + ⋅ − + + ⋅ + −⎟ ⎟⎜ ⎜ ⎜ ⎟⎟ ⎟⎜ ⎟ ⎟⎜ ⎜ ⎟⎜∑ ∫α

αk k

( )0

e i i i i i e ii dr r dr r⎟ ⎟⎜ ⎟ ⎟⎜ ⎜ ⎟⎜ ⎝ ⎠ ⎝ ⎠⎝ ⎠∑ ∫α

where

, .i ii i

P Qr r

= =G F

Finally the alpha-particle motion in the electron shell isdescribed by the following two-body Hamiltonian:

,( 2) ,( ) ( ) ( ),Z eH T V r V r W rα α α α α α−= + + +

described by the following two body Hamiltonian:

,( 2) ,( ) ( ) ( ) ( ).e e Z e eW r T r V r V rα α α α−= + +where

S th ti f thi ti h t i d b th

(0) (0)Q Q E E V W= + Δ Δ = +

So the properties of this motion are characterized by theenergy

⎡ ⎤

,, (0) (0)at nucl eQ Q E E V Wα= + Δ Δ = +

where ΔE is the energy transfer; and the penetrability1

int1

,( 2) ,2 2

( ) ( ) ( )

/

extrat

Z e atr

V r V r W r Q dr

P P−

⎡ ⎤⎢ ⎥+ + − +⎢ ⎥⎢ ⎥

∫ α α α α αμh

2

,( 2)

/ exp .2 2

( )ext

at bn rbn

Z bn

P P

V r Q dr−

= − ⎢ ⎥⎢ ⎥

−⎢ ⎥⎢ ⎥⎣ ⎦

∫ α αμh

int 2r⎢ ⎥⎣ ⎦h

QUALITATIVE PROPERTIES OF THE EFFECT

While the interaction potential of the alpha and theelectron shell increases monotonically thef i l d d d h id h

, ( )eV rα α

functional drops down and thus provides thegrows of the penetrability.

( )W rαΔ

The energy transfer form the shell to the alpha-particle is relatively small, ΔΔE =713 eV butp y ,observable because for 226Ra decay the energyyield is measured with the higher accuracy4870 63 0 25 k V4870.63±0.25 keV

QUANTITATIVE RESULTS. PHYSICAL EFFECTSOF THE ELECTRON SHELLTHE ELECTRON SHELL

“Initial” modelInitial model

1. The continuous evolution of the electron shell of theatom and permanent energy between it and the alphaparticle as this particle propagates through the shell istaken into accounttaken into account.

2. Hartree-Fock-Dirac formalism considering the atomica t ee oc ac o a s co s de g t e ato cnucleus as a finite-range object is explored to describethis evolving system.

3. The alpha particle is considered to be bare at themoment of fly out form the electron shellmoment of fly out form the electron shell.

4 Th bi di i f t d d t l t i th4. The binding energies of two redundant electrons in thedaughter atom are believed to be close to zero. At the sametime these electrons are slow and therefore the alphatime these electrons are slow and therefore the alphaparticle leaves the interaction area earlier than them.

5. The reduced mass difference peculiar to the decay of anatom and the respective bare nucleus is neglected.

6. Twofold character of the regime of alpha-particlepenetration through the electron shell which is adiabaticpenetration through the electron shell which is adiabaticrelative to the motion of strongly bound electrons anddiabatic relative to the motion of loosely bound ones is notytaken into account and purely adiabatic scenario isassumed in the approximation.

The values of Eat (MeV), ΔE (keV) and δΓ/Γ(10-3 )

In the case that the nuclei are considered as a point-likepobjects the values ΔE and δΓ/Γ turn out to be equal to20.854 keV and 2.789·10-3 for 144Nd and 38.874 keV

3and 2.627·10-3 for 226Ra decay respectively.

In the case that non relativistic Hartree Fock model isIn the case that non-relativistic Hartree-Fock model isexplored, the values ΔE and δΓ/Γ are equal to 19.553keV and 1.608·10-3 for 144Nd alpha decay and to 33.874p ykeV and 0.662·10-3 for 226Ra respectively. Thus non-relativistic approach is in fact inadequate.

BASIC RESULTSThe values of E (MeV) ΔE (keV) and δΓ/Γ(10-3 )The values of Eat (MeV), ΔE (keV) and δΓ/Γ(10 3 )and mean excitation energy E* (eV)

⎡ ⎤1

int1

,( 2) ,2 2

( ) ( ) ( )

/

extrat

Z e atr

V r V r W r Q dr

P P−

⎡ ⎤⎢ ⎥− + + − +⎢ ⎥⎢ ⎥

∫ α α α α αμh

2

,( 2)

/ exp .2 2

( )ext

at bn rbn

Z bn

P P

V r Q dr−

= ⎢ ⎥⎢ ⎥

−⎢ ⎥⎢ ⎥⎣ ⎦

∫ α αμh

int 2r⎢ ⎥⎣ ⎦h

The electron shell make the reduced mass larger forThe electron shell make the reduced mass larger forthe “atomic” case so this effect make the decayslower.slower.

The process is not purely adiabatic – the velocity ofthe alpha particle is higher than the velocity of theouter electrons. So the energy yield calculated in theadiabatic approach is in fact the upper limitadiabatic approach is in fact the upper limit.

Because of the conservative behavior of the slowelectrons the final state of the daughter atom is not aneigenstate of the atomic Hamiltonian so this atomejects two or more electrons and turns out to beexcited after that. Therefore one can expect theemission of soft X rays and/or ultraviolet radiationemission of soft X-rays and/or ultraviolet radiation.

CONCLUSIONS.THE QUESTIONS AND THE ANSWERSTHE QUESTIONS AND THE ANSWERS

1. Is WKB approximation correct? Yes.

2. Is the reflection effect significant in the classicallyallowed area? The effect is negligibleallowed area? The effect is negligible.

3. Does the form of the strong potential play a role? No,3. Does the form of the strong potential play a role? No,the result is not affected by a reasonable variation of theinternal turning point.

4. Is the Thomas-Fermi model satisfactory? Directcalculations demonstrate that the answer is nocalculations demonstrate that the answer is no.

5. Can one use non-relativistic approach? No.5. Can one use non relativistic approach? No.

6. Does finite-range distribution of nuclear chargeg gplay a role? Yes.

7. Is the process diabatic or adiabatic one in relationto the motion of electrons? The truth is somewherein between but the accurate adiabatic approachin between but the accurate adiabatic approachprovide a good description of the quantitativecharacteristics of the decay.y

8. Does the rearrangement of the electron shell playi l ? Y h l f han important role? Yes, the role of the rearrangement

and the energy transfer accompanying it in theaggregate electron shell impact on alpha-decayaggregate electron shell impact on alpha-decaywidths is decisive.

9. What is the probability of the electron to be knockedout? Calculations demonstrate that the probability is

h llrather small.

10 Is the alpha particle bare or dressed? The10. Is the alpha particle bare or dressed? Theprobability to pick-up an electron by alpha particle issmall. In addition a neutral He atom can not be detected.

11. Is the effect measurable? How to detect the effect?P b blProbably yes.

There is an example in which proton and alpha are incompetition In that case the branching ratio but not thecompetition. In that case the branching ratio but not thehalf-life can be the object of measurements. Thisexample is 160Re isotope.example is Re isotope.

Decay Qat(MeV) Percentage ΔQ(keV)160Re→p 1.290 91 15.19160Re→α 6 699 9 30 38

A th i t th b hi ti f th

160Re→α 6.699 9 30.38

Another way is to measure the branching ratio of thealpha decay fine structure.

0 Is not the subject matter of the discussed0. Is not the subject matter of the discussedinvestigations of solely academic interest?

Basic prospects of the developed approach may befound beyond the examples of the alpha decay ofi i l b t ditiions in laboratory conditions.

The large scale effects of such a type may appear inThe large scale effects of such a type may appear inhigh electron density mediums: stars, etc.

Besides that the accumulated experience may be ofadvantage for the description of other nuclearprocesses in ol ing the electron shellprocesses involving the electron shell.

THANK YOU FOR ATTENTION!