Investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects
The new algorithm for the determinations of the characteristics functions of the energy distribution, decay probability, decay intensities and life times of the excited levels is proposed. Quantum-mechanical motivation of the needing of the time characteristics revising for the nuclear-chronometers...
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| Date: | 2004 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2004
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| Cite this: | Investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects / N.L. Doroshko, M.E. Dolinska // Вопросы атомной науки и техники. — 2004. — № 5. — С. 89-92. — Бібліогр.: 4 назв. — англ. |
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| author | Doroshko, N.L. Dolinska, M.E. |
| author_facet | Doroshko, N.L. Dolinska, M.E. |
| citation_txt | Investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects / N.L. Doroshko, M.E. Dolinska // Вопросы атомной науки и техники. — 2004. — № 5. — С. 89-92. — Бібліогр.: 4 назв. — англ. |
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| description | The new algorithm for the determinations of the characteristics functions of the energy distribution, decay probability, decay intensities and life times of the excited levels is proposed. Quantum-mechanical motivation of the needing of the time characteristics revising for the nuclear-chronometers it is given. Calculations for concrete decay events of the nuclei ²³⁸U, ²³2Th, ²³⁵U at room and stars temperatures, with and without account ofthe Doppler effect were conducted.
Запропоновано новий алгоритм для визначення характеристичних функцій розподілу по енергії, ймовірностей розпаду, інтенсивності розпаду і часу життя збуджених станів рівнів. Зроблено квантовомеханічне обгрунтування необхідності перегляду часових характеристик ядер-хронометрів. Проведені розрахунки для конкретных випадків розпаду ядер ²³⁸U, ²³2Th, ²³⁵U при кімнатних і зоряних температурах з урахуванням ефекту Доплера та за його відсутності
Предложен новый алгоритм для определения характеристических функций распределения по энергии, вероятности распада, интенсивности распада и времени жизни возбужденных состояний уровней. Дано квантовомеханическое обоснование необходимости пересмотра временных характеристик ядер-хронометров. Проведены расчеты для конкретных случаев распада ядер ²³⁸U, ²³2Th, ²³⁵U при комнатных и звездных температурах с учетом эффекта Доплера и при его отсутствии.
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INVESTIGATION OF NUCLEAR CHRONOMETER
TIME DECAY CHARACTERISTICS FOR REVISION
OF AGE OF ASTROPHYSICS OBJECTS
N.L. Doroshko, M.E. Dolinska
Institute for Nuclear Research, Kiev, Ukraine
e-mail: kinr.kiev.ua
The new algorithm for the determinations of the characteristics functions of the energy distribution, decay probabil-
ity, decay intensities and life times of the excited levels is proposed. Quantum-mechanical motivation of the needing of
the time characteristics revising for the nuclear-chronometers it is given. Calculations for concrete decay events of the
nuclei 238U, 232Th, 235U at room and stars temperatures, with and without account of the Doppler effect were conducted.
PACS: 25.40. Lw, 25.40 Ny, 28.41.Kw, 89.60 Ec
1.INTRODUCTION
Since the statistic low of the radioactive decay it is con-
sidered absolutely exact, work of the "nuclear clock",
which are used for measurement gap time in the geologies,
archeologies, astrophysicist, is founded on its principle.
Main principle of the technology nuclear chronome-
ters consists in correlations measurement of the mother
and daughter nuclear in the greater volume of the mat-
ter. Such principle is founded on tacit suggestion that
velocity of the radioactive decay is constant and does
not depend on physical and chemical states of the ambi-
ence, in which are found radioactive nuclei. However it
was currently realized, that thereof rules in some cases
there are exceptions.
So, change of the chemical state of the decay atom
and the thermodynamic influences bring to the observ-
able change of the electron seizure velocity and internal
conversion. And in unusual states of the strong ionizing,
for instance, in depths of the stars, more strong effects
of the velocities increase can exist even for α- and β-
decay. This is necessary to take into account in radioac-
tive chronology.
Besides, up to recently time, in all known method
nucleus chronometry took into consideration only life
time of the main levels of the decay nuclear. But dura-
tion of the nucleus radiation processes seizes not only
main, but also many excited states of the synthesized
nuclear, which were formed. Account these factor can
greatly change the final results estimation for all time
interval, characterizing evolution decay chain, and up-
dated factors of "nuclear clock" can correspond to great-
ly smaller values of the real processes duration for the
nuclear-chronometers decay. Signifies that the "age" of
object in which occurs decay should be smaller [1].
2. DECAY EVOLUTIONS
Purpose given work was a development of the algo-
rithm for the account of the decay of the necessary
amount excited states of the radioactive nuclei and their
heat motion.
For description of the decay evolution and determi-
nations its time characteristics (probability, intensities,
life time) in these work quantum-mechanical approach,
founded on Krylov-Fock theorem [2], generalized for
the mixed states (when decay process of the ensemble
of the particles simultaneously goes with its formation
by the nucleus syntheses or decay of the previous state)
[3] is used. Such approach allows taking into account
mentioned above factors.
For simplification of the analysis limit ideal event of
the long life α-active nuclear, which in determined ini-
tial time moment (t = 0) portioned in two states (main
and first excited).
Then, in accordance with Krylov-Fock theorem de-
cay functions L(t) and L0(t), characterizing decay of cer-
tain initial (first excited) and following (main) states ac-
cordingly, are presented in the manner of
L t p t p( ) ( ) ( )=
2 2
0 , (1)
L t p t p0 0
2
0
2
0( ) ( ) ( )= , (2)
were
p t G i t d0 0
2
0
( ) ( ) exp( )= −
∞
∫ ε ε ε , (3)
p t G i t d( ) ( ) exp( )= −
∞
∫ ε ε ε2
0
- (4)
characteristics functions of the energy distribution ener-
gy at the main and first states. Marking width of the
main and first excited levels of the α-decay by Гα
0 and Г
α
1 accordingly, and width of the γ decay by Гγ, the full
width of the main and the first excited state are present-
ed by
Г0 = Гα
0; Г1 = Гα
1 + Гγ .
Then characteristics functions for the main and first ex-
cited states will be determined by appropriation energy
spectrum of a distributions
[ ] ×Γ+−=
− 12
1
2
1
2
0 4)()( εεε constG
[ ] 12
0
2
0 4)( −
Γ+−× εε , (5)
[ ] 12
1
2
1
2 4)()(
−
Γ+−×= εεε constG . (6)
Here ε 0 , ε 1 и ε - energy of the main and first excit-
ed states and actual system energy, which consists of the
internal motion energy of the mother nuclear and kinetic
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2004, № 5.
Series: Nuclear Physics Investigations (44), p. 89-92. 89
energy of the heat motion. Probability of the decay will
be defined by
)(1)( tLtW nn −= , (7)
and velocities of the decay with provision of the each
channel contribution -
dttdWt nn
n
in )()( ΓΓ=ρ ,
were n = 0,1. (8)
Intensity of the decay it is possible to define as
∫ ′−′=
t
ttttdtI
0
10 )()()( ρρ (9)
For determination of the radioactive nuclear life time
and period of its half-life Т1/2, it is possible to use
known formulas
∫∫= dttIdtttI )()(τ , Т1/2 = τln2 . (10)
So, offered approach allows taking into account de-
cay not only main, but also previous state of the nuclear.
Such decay can occur on two channels. As can be seen
from energy distribution (5) and (6), decay of the first
exited state has purely exponential nature. Decay of the
main state already has not an exponential nature since it
is assigned by multiplying of the two exponents. This
allows expecting that account of the decay each previ-
ous exited level in decay chains will contribute certain
additional distortion in the exponential nature of result-
ing energy distribution. So, final estimations of the time
intervals can be changed.
3. DESCRIPTION OF THE DATA
First this approach (for the approbation reason) was
used for theoretical description of the γ-quantum time
distribution, which is resonance scattering by nuclei 57Fe
at Messbauer experiments [4].
Since according to scheme of such experiment [4],
the decay process of the scattering with excitation for
nuclear 57Fe goes simultaneously with their resonance
excitement by the γ-quantum, which flies from source,
for description of the decay evolutions we use general-
ized Krylov-Fock theorem (1) – (4), where
[ ] 122
0
2
0 4)()(
−
Γ+−= εεε constG
is spectrum of the excitement energy distribution at the
nuclear of a source, and
[ ]
[ ] 122
1222
4)(
4)()(
−
−
Γ+−∆−×
×Γ+−=
γ
γ
εεε
εεε constG
is spectrum of the excitement energy distribution at the
57Fe nuclear of the scattering matter; the energy shifting
∆ε is adjusted by the velocities driftage device of the
source; Γ = / τ .
Calculating )(tp and )(0 tp on formulas (3) and (4)
by means of theorems about deduction, we shall get in
accordance with (9)
{ }
[ ]ϕϕεε
ττ
sin)/sin()/(
/()/exp()(
−+∆∆Γ+
+−=
ht
ttconsttI
{ },cos)/cos( ϕϕε ++∆− t (9а)
where
)2/( εϕ ∆Γ−= arctg .
Fig. 1 express results of the calculations by the for-
mula (9а), with ∆ε= 3Г (curves 1), which are normal-
ized on upper points of the experimental data in com-
parison with results of the theoretical calculation [4]
(curves 2,3), which are executed within the framework
of classical theory of the electromagnetic radiation in-
teraction with a matter. As it is seen, results of our cal-
culation well agree with experiment that is indicative
about the correctness using of such approach for deter-
mination of the time decay characteristics.
Fig. 1. Time distribution of the γ-quantum, resonance
scattered by 57Fe nuclear (∆ε = 3Г)
4. DOPPLER EXPANSION
As is well known, Messbauer effect occurs without
Doppler expansion of the lines, but under usual decay of
the excited nuclear levels within matter with a very high
temperature (for instance, in depths of the stars),
Doppler effect can play observable role. So, for more
strict description of the decay evolutions it is necessary
to take into account the Maxwell function of the distri-
bution on energy
ε
εε
π
εεε γ d
DD
df r )
)(
exp(1),( 2
2−
−−= ,
where ε ε µγr c= 2 22 - kinetic energy of the return of
the nuclear after the releases or the absorptions of the γ-
quantum, ε γ - γ-quantum energy; kTD rε2= is the
Doppler width for the resonance release or absorption.
Then real functions of the g-quantum energy distribu-
tion with provision of the Doppler expansion we shall
present in the manner of
( ) εεεεεε γγ ddfGG ndopn ∫ ∫
+ ∞ ∞
=
0 0
22 ),()()(
where ε ε ε εγ = − − +0 r D , n = 0,1. As a result for
G0(ε) and G1(ε) shall get the following expressions:
90
−
+−+−
= ∫
−−ε εε
εεεε
ε
π
ε
0
2
0
2
0
2
1
2
1
)(
2
0 )))(()((
)(
22
aa
de
D
constG
rb
dop
−
−
+
++
− ∫
−ε ε π
εε
ε
0 01
2
1
2
0
2
0
22
1
2
2
0
22
1
222
)(2))(( a
e
a
e
aaaa
de ababb
,
G
const
D
e d
adop
b r
1
2
2 2
2
1
2
( )
( )
ε
π
ε
ε
ε ε εε
=
′
′ +
−
− ′ + −
− ∞
∫ e d
a
e
a
b b a− ′ ′
′ +
+
∫
2 2 2 2
2 2
0 2
εε ε
ε
.
The offered method possible to generalize on event greater amount excited states. For this matter function Gn(e)
shall present as [ ]G constn i i
i
N
( ) ( )ε ε ε
2 2 2
1
1
4= − +
=
−
∏ Γ , where [ ]i n∈ 1, , i is considered level, and similar image
we shall average on energy. As a result of multiple transformations we shall get following expression:
.
)(2)(
)(2)())((
)(
0 1
,0
22
0
22
0 0 1
,0
22
0
22
0
22
)(
2
2222
222222
∫ ∑
∏∏
∫ ∫ ∑
∏∏∏
=
≠
==
−
=
≠
==
−
=
−−
−
+
+
×
×
−
+
+
−
+−
=
ε ε
ε ε εεε
π
ε
ε
π
ε
ε
εε
ε
π
ε
N
i
N
ij
i
jii
ab
N
i
i
b
N
i
N
ij
i
jii
ab
N
i
i
b
N
i
ii
b
dopn
aaa
e
a
de
aaa
e
a
de
a
de
D
constG
i
ir
Here Г1
2 / 4 = a2, 1 / D2 = b2, Г0
2 / 4 = a0
2 ,
′ = −ε ε ε 1 - changes at calculation integral; a0, a1
and a are special points ( ε 0 0= ia ; ε 1 1= ia ; ε i iia0 = )
.
As it is seen, the generalized Krylov-Fock theorem
allows to take into account not only Doppler effect, but
also necessities amount of the excited states of the ra-
dioactive nuclear, appearing in nuclear-syntheses pro-
cess, that, certainly, must positively influence upon ac-
curacy of the estimation of the intensities and decay ve-
locities of the radioactive nuclear-chronometers, and,
signifies, upon estimation of the different objects age by
the methods of nuclear chronometry.
5. RESULTS OF THE CALCULATIONS
To realize, what influence renders Doppler effect on
the main features of decay, they were organized corre-
sponding calculations under room and under stars tem-
peratures for the radioactive nuclear 238U, 235U and 232Th,
which are broadly used in large-scale nucleus chronom-
etry for the dating of the astrophysical objects age. For
simplification of the calculation was considered event of
the consequent decay only from the first excited and the
main states. Calculations were conducted by numerical
methods and have shown that in the case of, when
Doppler effect was not taken into account, the graphics
of the intensities of the decay nuclei have maxima in
pointes, corresponding to the table values of the given
level life-time. This is indicative of that the offered
method is correct. As example on Fig. 2 is brought
graphic of the decay intensities dependencies from the
time for the first excited level of the 238U nuclear disre-
garding of the Doppler effect (τeksp = 2,93⋅10−10с, τteor =
2,92⋅10−10s).
I ( t ),
R e l a t.
u n i t s
t,c 0 0.1 0.3 0.5 0.7 0.9 1.1 1.3
0
0.5
0.1
Fig. 2. Decay intensities vs time for the excited nuclear
238U disregarding Doppler effect
Calculations conducted with provision of the
Doppler effect have demonstrated that under room tem-
perature it practically does not render influences on the
decay velocity and life time of the excited level (τeksp ≈ τ
teor). But under stars temperature (Т = 3⋅108), as was ex-
pected, probability and intensity of decay noticeably in-
creased and, accordingly, decreased the life time. As ex-
ample on Fig. 3 is brought the graphic of the decay in-
tensities dependencies from the time for the first excited
level for the 238U nuclear with provision of the Doppler
effect under stars temperature. In this case, τteor = 4,6⋅10-
11 s, when turning to the stars temperatures, the life time
of the first excited level of the 238U decreased approxi-
mately in 6,4 times.
91
I ( t ) D,
R e l a t.
u n i t s
0 0.5 1.0 1.5 2 2.5 3.0 3.5
0
3
6
t⋅ 10-10c
Fig. 3. Decay intensities vs time for the excited nuclear
238U with accounted of the Doppler effect
The similar calculations of the life time for the first
excited level 235U and 232Th, called on for with provision
of the Doppler effect, have demonstrated its reduction
under stars temperature in compare with experimental
data in 5,7 times for 235U and in 7,2 times for 232Th. Fol-
lows to expect that account of the greater number of the
excited states will give else greater reduction of the life
time.
Given work is a first stage of the complex quantum
theoretical research of the decay time features. Analyti-
cal expressions are received in this work for the time
characteristics of decay for the long life α-active nucle-
ar, which in determined initial time moment portioned
in two states (main and first excited) with provision of
the Doppler effect. Generalization of the getting expres-
sions is made on event greater amount of the excited
states. Calculations called on for concrete nuclear under
room and stars temperatures, have demonstrated signifi-
cant increase of the decay velocities under stars temper-
atures and possibility of the observable speedup of de-
cay because of presence of the excited states.
Designed method allows at decay processes to take
into account Doppler effect and necessities amount of
the excited states. Such method conducts corresponding
calculations for the different nuclear and for the differ-
ent temperature, that permits its practical application in
nucleus chronometry, for instance for more exact deter-
mination of the astrophysical objects age and decay time
of the nuclear waste.
6. CONCLUSIONS
Modern methods of the nucleus chronometry found-
ed on classical belief about constancy of the decay ve-
locities and not taking into account influences upon de-
cay time features of the different factors (in particular,
the heat motion of the nuclear and decay of the previous
states), give only possible upper estimation limites of
the objects age. So such estimations must be revised
within the framework of more general quantum theory.
REFERENCES
1. V.S. Ol’khovsky, I. Grantsev. Is it Essential for
Nuclear Chronometry of Astrophysi cal Processes to
Consider the Decay of the Excited Radioactive
Nuclei? Scientific Papers of the Institute for Nuclear
Research, Kiev, 1999, p. 53 56.
2. Н.С. Крылов, В.А. Фок. On two main interpreta-
tions of the unserttainte relation for energy and time
// Soviet Journal of Experimental and Theor. Phys.
1947, v. 17, p. 93-107.
3. V.S. Ol’khovsky. Effects of compaund-nuclears and
direct prosesses in the averaged crosscection and du-
ration s of nuclear reactions // Izv. AN SSSR. Ser.
Fiz. 1985, v. 49, №5, p. 993-1002.
4. P. Thieberger, J.A Moragues, A.W. Sunyar Time de-
pendence of Mossbauer scattered radiation // Phys.
Rev. 1968, v. 171, №2, p. 425-435.
ИССЛЕДОВАНИЕ ВРЕМЕННЫХ ХАРАКТЕРИСТИК РАСПАДА ЯДЕР-ХРОНОМЕТРОВ
С ЦЕЛЬЮ УТОЧНЕНИЯ ВОЗРАСТА АСТРОФИЗИЧЕСКИХ ОБЪЕКТОВ
Н.Л. Дорошко, М.Э. Долинская
Предложен новый алгоритм для определения характеристических функций распределения по энергии,
вероятности распада, интенсивности распада и времени жизни возбужденных состояний уровней. Дано
квантовомеханическое обоснование необходимости пересмотра временных характеристик ядер-хрономет-
ров. Проведены расчеты для конкретных случаев распада ядер 238U, 232Th, 235U при комнатных и звездных
температурах с учетом эффекта Доплера и при его отсутствии.
ДОСЛІДЖЕННЯ ЧАСОВИХ ХАРАКТЕРИСТИК РОЗПАДУ ЯДЕР-ХРОНОМЕТРІВ
З МЕТОЮ УТОЧНЕННЯ ВІКУ АСТРОФІЗИЧНИХ ОБ'ЄКТІВ
Н.Л. Дорошко, М.Е. Долинська
Запропоновано новий алгоритм для визначення характеристичних функцій розподілу по енергії,
ймовірностей розпаду, інтенсивності розпаду і часу життя збуджених станів рівнів. Зроблено
квантовомеханічне обгрунтування необхідності перегляду часових характеристик ядер-хронометрів.
Проведені розрахунки для конкретных випадків розпаду ядер 238U, 232Th, 235U при кімнатних і зоряних
температурах з урахуванням
ефекту Доплера та за його відсутності.
92
|
| id | nasplib_isofts_kiev_ua-123456789-80554 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T13:20:56Z |
| publishDate | 2004 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Doroshko, N.L. Dolinska, M.E. 2015-04-18T20:26:01Z 2015-04-18T20:26:01Z 2004 Investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects / N.L. Doroshko, M.E. Dolinska // Вопросы атомной науки и техники. — 2004. — № 5. — С. 89-92. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 25.40. Lw, 25.40 Ny, 28.41.Kw, 89.60 Ec https://nasplib.isofts.kiev.ua/handle/123456789/80554 The new algorithm for the determinations of the characteristics functions of the energy distribution, decay probability, decay intensities and life times of the excited levels is proposed. Quantum-mechanical motivation of the needing of the time characteristics revising for the nuclear-chronometers it is given. Calculations for concrete decay events of the nuclei ²³⁸U, ²³2Th, ²³⁵U at room and stars temperatures, with and without account ofthe Doppler effect were conducted. Запропоновано новий алгоритм для визначення характеристичних функцій розподілу по енергії, ймовірностей розпаду, інтенсивності розпаду і часу життя збуджених станів рівнів. Зроблено квантовомеханічне обгрунтування необхідності перегляду часових характеристик ядер-хронометрів. Проведені розрахунки для конкретных випадків розпаду ядер ²³⁸U, ²³2Th, ²³⁵U при кімнатних і зоряних температурах з урахуванням ефекту Доплера та за його відсутності Предложен новый алгоритм для определения характеристических функций распределения по энергии, вероятности распада, интенсивности распада и времени жизни возбужденных состояний уровней. Дано квантовомеханическое обоснование необходимости пересмотра временных характеристик ядер-хронометров. Проведены расчеты для конкретных случаев распада ядер ²³⁸U, ²³2Th, ²³⁵U при комнатных и звездных температурах с учетом эффекта Доплера и при его отсутствии. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Применение ядерных методов Investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects Дослідження часових характеристик розпаду ядер-хронометрів з метою уточнення віку астрофізичних об'єктів Исследование временных характеристик распада ядер-хронометров с целью уточнения возраста астрофизических объектов Article published earlier |
| spellingShingle | Investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects Doroshko, N.L. Dolinska, M.E. Применение ядерных методов |
| title | Investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects |
| title_alt | Дослідження часових характеристик розпаду ядер-хронометрів з метою уточнення віку астрофізичних об'єктів Исследование временных характеристик распада ядер-хронометров с целью уточнения возраста астрофизических объектов |
| title_full | Investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects |
| title_fullStr | Investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects |
| title_full_unstemmed | Investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects |
| title_short | Investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects |
| title_sort | investigation of nuclear chronometer time decay characteristics for revision of age of astrophysics objects |
| topic | Применение ядерных методов |
| topic_facet | Применение ядерных методов |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80554 |
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