Bremsstrahlung in alpha-decay: angular analysis of spectra
A quantum mechanical method of calculation of bremsstrahlung spectra in alpha-decay of heavy nuclei with taking into account an angle between directions of the alpha-particle motion and the photon emission is presented. Dependence between the bremsstrahlung spectrum and the angle is obtained in a...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-805142015-04-19T03:02:22Z Bremsstrahlung in alpha-decay: angular analysis of spectra Maydanyuk, S.P. Belchikov, S.V. Ядерная физика и элементарные частицы A quantum mechanical method of calculation of bremsstrahlung spectra in alpha-decay of heavy nuclei with taking into account an angle between directions of the alpha-particle motion and the photon emission is presented. Dependence between the bremsstrahlung spectrum and the angle is obtained in a simple analytical form. The method can be used for a comparative analysis of experimental data, obtained at different angles. Представлено квантово-механічний метод розрахунку спектрів гальмівного випромінювання при альфа-розпаді важких ядер, що враховує величину кута між напрямками поширення альфа-частинки і випромінювання фотона. Залежність спектра від величини кута отримано у явному аналітичному виді. Метод може бути використаний для порівняльного аналізу експериментальних даних, отриманих для різних величин кута. Представлен квантово-механический метод расчета спектров тормозного излучения при альфа-распаде тяжелых ядер, учитывающий величину угла между направлениями распространения альфа-частицы и излучения фотона. Зависимость спектра от величины угла получена в явном аналитическом виде. Метод может быть использован для сравнительного анализа экспериментальных данных, полученных для разных величин угла. 2004 Article Bremsstrahlung in alpha-decay: angular analysis of spectra / S.P. Maydanyuk, S.V. Belchikov // Вопросы атомной науки и техники. — 2004. — № 5. — С. 19-21. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 03.40.Kf, 03.65.Xp, 12.20.-m, 23.60.+e, 24.10.-i, 41.60.-m http://dspace.nbuv.gov.ua/handle/123456789/80514 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы Maydanyuk, S.P. Belchikov, S.V. Bremsstrahlung in alpha-decay: angular analysis of spectra Вопросы атомной науки и техники |
| description |
A quantum mechanical method of calculation of bremsstrahlung spectra in alpha-decay of heavy nuclei with
taking into account an angle between directions of the alpha-particle motion and the photon emission is presented.
Dependence between the bremsstrahlung spectrum and the angle is obtained in a simple analytical form. The
method can be used for a comparative analysis of experimental data, obtained at different angles. |
| format |
Article |
| author |
Maydanyuk, S.P. Belchikov, S.V. |
| author_facet |
Maydanyuk, S.P. Belchikov, S.V. |
| author_sort |
Maydanyuk, S.P. |
| title |
Bremsstrahlung in alpha-decay: angular analysis of spectra |
| title_short |
Bremsstrahlung in alpha-decay: angular analysis of spectra |
| title_full |
Bremsstrahlung in alpha-decay: angular analysis of spectra |
| title_fullStr |
Bremsstrahlung in alpha-decay: angular analysis of spectra |
| title_full_unstemmed |
Bremsstrahlung in alpha-decay: angular analysis of spectra |
| title_sort |
bremsstrahlung in alpha-decay: angular analysis of spectra |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2004 |
| topic_facet |
Ядерная физика и элементарные частицы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80514 |
| citation_txt |
Bremsstrahlung in alpha-decay: angular analysis of spectra / S.P. Maydanyuk, S.V. Belchikov // Вопросы атомной науки и техники. — 2004. — № 5. — С. 19-21. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT maydanyuksp bremsstrahlunginalphadecayangularanalysisofspectra AT belchikovsv bremsstrahlunginalphadecayangularanalysisofspectra |
| first_indexed |
2025-07-06T04:31:33Z |
| last_indexed |
2025-07-06T04:31:33Z |
| _version_ |
1836870578989957120 |
| fulltext |
BREMSSTRAHLUNG IN ALPHA-DECAY: ANGULAR ANALYSIS OF SPECTRA
S.P. Maydanyuk, S.V. Belchikov
Institute for Nuclear Research, National Academy of Sciences of Ukraine, Kiev, Ukraine
e-mails: maidan@kinr.kiev.ua, sbelchik@kinr.kiev.ua
A quantum mechanical method of calculation of bremsstrahlung spectra in alpha-decay of heavy nuclei with
taking into account an angle between directions of the alpha-particle motion and the photon emission is presented.
Dependence between the bremsstrahlung spectrum and the angle is obtained in a simple analytical form. The
method can be used for a comparative analysis of experimental data, obtained at different angles.
PACS: 03.40.Kf, 03.65.Xp, 12.20.-m, 23.60.+e, 24.10.-i, 41.60.-m
1. INTRODUCTION
Research of bremsstrahlung in α-decay of heavy
nuclei has caused an increased interest last years. We
note main purposes for study of this phenomenon:
•study of properties of the α-decay dynamics on the
basis of analysis of the experimental spectra of the
bremsstrahlung; development of a non-stationary
model of detailed description of this process
(including a space barrier region), tested on the basis
of the experimental data;
•investigation of the subbarrier bremsstrahlung in
the alpha-decay, i.e. the photons emission during
tunneling of the alpha-particle through decay barrier;
•construction of a method of tunneling time
determination of the alpha-particle through the
decay barrier (which approximately equals to
nuclear times values 10-20…10-24 sec) on the basis of
analysis of the experimental spectra of the
bremsstrahlung.
For successful realization of all these researches the
theoretical model is needed, which allows describing the
alpha decay with bremsstrahlung and calculating their
main characteristics, and which is tested by
experimental data.
The experiments were fulfilled with such nuclei:
210Po, 214Po, 226Ra and 244Cm. Note, that in the behavior
of the experimental spectra for the nucleus 210Po,
obtained by Russian-Italian group [1] and Japanese
group [2] independently, there is a qualitative
difference, which had caused discussions in some
papers. These experiments were fulfilled for the
different values of the angle between the directions of
the alpha-particle motion and the photon emission
(which equal to 90° and 25°) and, perhaps, by this fact
one can explain the difference between their spectra.
But we note that more accurate analysis can be fulfilled
on the basis of the unified model which allows
calculating the bremsstrahlung spectra with taking into
account the different values of such angle, and this
analysis had not done else.
Constructed theoretical models are differed also in
their description of the bremsstrahlung spectra. Here,
the instant accelerated model [3], developed on the basis
of classical electrodynamics and used such
characteristics as a velocity of the alpha-particle during
its leaving outside from the barrier region, gives enough
good description of the experimental bremsstrahlung
spectra. But one can consider these characteristics as
additional parameters, which introduction allows
moving the calculated bremsstrahlung spectrum curve
near the experimental data. Quantum mechanical
models, proposed both by T. Papenbrock and
G.F. Bertsch [4], and by E.V. Tkalya [5], do not use
such additional parameters and their descriptions of
experimental spectra are less satisfactory (from our
point of view).
However, the models, constructed on the basis of
quantum electrodynamics and without semi-classical
approach, are the most effective in study of the alpha-
decay dynamics in the space barrier region and also for
detailed study of the subbarrier bremsstrahlung effect in
the alpha-decay. And the model, in which the method of
calculation of the bremsstrahlung spectra takes into
account the angle value, will be useful for analysis of
the experimental spectra, obtained for different angles
(at present, the proposed models [2-5] are isotropic).
2. SHORT REVIEW OF QUANTUM
MECHANICAL MODELS
In paper [6] we proposed the multipolar quantum
mechanical model of the alpha-decay with the
bremsstrahlung, which allows calculating the
bremsstrahlung spectra in dependence on the angle
value between directions of the alpha-particle motion
and the photons emission. In accordance with this
model, the probability of the spontaneous photon
emission in the alpha-decay is (ωfi = Ei –Ef):
2
fi4
iff
22
eff )k,k(p
m)2(
keZ
d
dW
π
ω
=
Ω ν
,
,mE21k f,if,i
=
(1)
where p(ki,kf) has the following form:
∑ ∫ ∫
=α
+ ∞
−α ψ
∂
∂ψΩ=
2,1 0
i
i*
f
2*)(
fi ).(e)(rddre)k,k(p r
r
r rk
(2)
Here Zeff is the effective charge, m is reduced mass of
the composite system (the alpha-particle and the
daughter nucleus), Ei and Ef are the total energy of the
system in initial i-state (i. e. the state of the system
before the photon emission) and final f-state (i.e. the
state of the system after the photon emission), ki and kf
are the wave vector of the system in the initial i- and
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2004, № 5.
Series: Nuclear Physics Investigations (44), p. 19-21. 19
mailto:maidan@kinr.kiev.ua
final f-states, )(r
iψ and )(r
fψ are the wave function
of the system in the initial i- and final f-states, e(α) is the
unit polarization vector of the emitted photon, k is the
wave vector of the photon, |k|kw f,i
== . Vector )(αe
is perpendicular to k in Coulomb calibration. We use the
unit system when 1= and c = 1.
Main difference between the methods, in which the
calculation of the bremsstrahlung spectrum is based on
quantum electrodynamics, consists in different
approaches for calculation of the value p(ki,kf) (on our
view, the best review of such methods is present in [5]).
Here, we use the multipolar expansion of the vector
potential of electromagnetic field of the daughter
nucleus (see [7], p.57, 51, 49). Our result is:
( )
)1l(JI
1l2
l)1l(JI
1l2
1l)k,k(p
),l(JI)k,k(p
,)]k,k(ip)k,k(p[)i(1l2
2)k,k(p
32fi
1fi
1
fifi
l
fi
+
+
+−
+
+−=
=
−−+×
π=
∑
=
ll
lE
l
lM
l
lElM
(3)
where
∫
+ ∞
=
0
*
,
2 )(
)(
)()( drkrj
dr
rd
rrnJ n
i
f
ψ
ψ ll
(4)
Ω=
Ω=
Ω=
∫∑
∑ ∫
∑ ∫
+
−=
−=
−
−=
dnnYI
dnnYI
dnnYI
i
r
f
rLM
i
r
f
rLM
i
r
f
rLM
)()()(
,)()()(
,)()()(
**
1
1
2
3
1
1
**2
2
1
1
**
1
ν
µ
µ
µ
νµ
µ
νµ
µ
µ
µ
μ1,ll01,0
μ1,ll01,0
μl,l01,0
TTn
TTn
TTn
(5)
and jn(kr) is the spherical Bessel function of order n (n is
a natural number), )( f
rLMY n are the normalized
spherical functions, )(nm
μ,lT are the vector spherical
harmonics (see [7], p. 45).
Calculations of the bremsstrahlung spectrum in the
alpha-decay of the nucleus 210Po on the basis of such a
model give enough good description of the experimental
data [1], obtained by Russian-Italian group for the angle
90°. Evaluation of the spectrum at angle 25° on the basis
of this models gives a monotonous behavior (without
appearance of the “hole” like experimental data [2]).
However, there are some problems in the described
model, concerned with the restriction of calculations
accuracy of wave functions in the asymptotic area in the
states before and after the photon emission. As the
subintegral expression in the integral (4) is the oscillated
and slowly damped function, that calculations
convergence of the spectra is limited. As a result, the
numerical calculation of the spectra becomes difficult
sufficiently and in this sense the model [6] is not
suitable enough. The convergence of the spectra
calculation by the models [4, 5] is higher and these
models are not exposed to necessity of calculations of
the wave functions in the asymptotic area with high
precision. But, probably, they give more approximated
calculation of the bremsstrahlung spectra (see [6]).
The second important point in the analysis of
quantum mechanical models is their possibility to
calculate the bremsstrahlung spectra in dependence on
the value of the angle between directions of the alpha-
particle motion and the photons emission, and, as result,
a possibility to fulfill a comparative analysis of the
experimental data for the nucleus 210Po [1, 2], obtained
for the angles 90° and 25°. Here, the model [6] allows
doing this at the first time. In this model the dependence
of the bremsstrahlung spectra on the angle value between
directions of the alpha-particle motion and the photon
emission is in the angular integrals (5). However, such
angular dependence is not suitable for the speed angular
qualitative analysis of experimental spectra.
Further, we present a new alternative approach for
the calculation of the bremsstrahlung spectra, where
such angular dependence is shown more obviously and
more simple.
3. SIMPLIFIED APPROACH FOR THE
ANGULAR CALCULATIONS OF THE
SPECTRA
Let's consider k
and r . Vector k
is the photon
impulse and points to a direction of the photon motion.
Vector r is the radius-vector, which points to a space
position of the alpha-particle relatively to the center of
mass of the daughter nucleus. We suppose, that as the
mass of the daughter nucleus is sufficiently more then
the alpha-particle mass, then the direction of the radius-
vector of the alpha-particle position coincides with the
direction of alpha-particle velocity. Then the angle
between the vectors k
and r is the angle between the
directions of the alpha-particle motion and the photon
emission. One can write:
)cosexp()exp( βikri −=− rk
(6)
where ||k k
= , ||r r= , β is the angle between the
direction of the alpha-particle motion kk
and the
direction of the propagation of the emitted photon rr .
Find the expression for the bremsstrahlung
spectrum. Let's write polarization vectors αe in terms
of circular polarization vectors ξ with opposite
directions of rotation (see [7], p. 42):
)(
2
1),(
2
1 21
1
21
1 ieeiee +−=−= +− ξξ
(7)
We obtain:
),()(
),(
*
1,1
2
0
*
r
r
r kr
i
i
f
fi
e
rddrhkkp
ϕϕ
ξ
µ
µµ
∂
∂×
×Ω=
−
+−=
+ ∞
∑ ∫∫
(8)
where
.2/)i1(h,2/)i1(h 11 −−=−=− (9)
We use the following property (see [7], p. 44-46; [6]):
20
,3/1)0,1,1|110()0,1,1|110(
,)(Y)0|110()(
),(
dr
)r(d)(
1
1
i
r,1
i
r0,01
i
r0,01
i
i
=−=−
ξµ µ−=
ψ−=ψ
∂
∂
∑
−=µ
µµ− nnT
nTr
r
(10)
where )0|110( µµ− are the Clebsch-Gordan
coefficients. Taking into account Eq. (10), (6) and the
orthogonal property of the polarization vectors 1−ξ and
1ξ , we find:
.e)(Y)(Yd
r
)r(
)r(rdrh3/1)k,k(p
cosikri
r,1
f
r
*
'm'l
1,1
i*
f
2
0
fi
β−
µ−
+−=µ
+ ∞
µ
∫
∑ ∫
Ω×
∂
ϕ∂
ϕ−=
nn
(11)
Further, we suppose that the process of the photon
creation does not change the direction of the alpha-
particle motion, i.e.:
f
r
i
r nn = (12)
Taking into account this approximation, and also the
orthogonal property of the functions )( rlmY n , we obtain
the following expression for p(ki,kf):
,
)0,(
),1,(
3
1),(
*
1,1 0
cos2
dr
r
mlr
mlr
erhkkp
i
f
ikr
fi
∂
==∂
−==×
×−= ∑ ∫
+−=
+ ∞
−
ϕ
µϕ
µ
β
µ
(13)
Note, that we have the quantum numbers l=1, m=µ for
final state and l=m=0 for initial state. Such deter-
mination follows from the supposition (12). From here
we can also conclude, that in multipolar approach of
quantum mechanical calculation of the bremsstrahlung
spectra the E1 multipole gives the most important
contribution into the total spectrum (such idea of
estimation of multipole contributions into a total
bremsstrahlung spectrum is obtained at the first time).
In such a form of p(ki,kf) the dependence of the
bremsstrahlung spectrum on the angle is more obvious.
In new approach one can calculate the bremsstrahlung
spectrum on the basis of (1), where one can use Eq. (13)
for p(ki,kf) value. Note that the model [6] allows
calculating the bremsstrahlung spectra only with taking
into consideration of the selected electrical and
magnetic multiples, whereas the method, proposed in
this paper, gives the calculation of the bremsstrahlung
spectrum as a whole.
4. CONCLUSIONS
We propose new approach for calculation of the
bremsstrahlung spectra in alpha-decay, where the angle
between the directions of the alpha-particle motion and
the photon emission is taken into account, and the
dependence of the bremsstrahlung spectra on the values
of such angle is more obvious and simple then in model
[6]. This method can be useful for a comparative analysis
of the experimental data [1,2], obtained for different
angles.
ACKNOWLEDGMENTS
The authors are grateful to Prof. V.S. Olkhovsky for
setting of the problem of the bremsstrahlung in the
alpha-decay and for productive discussions concerning
of study of this phenomena and calculations of
tunneling times.
REFERENCES
1. N.V. Eremin, G. Fazio, G. Giardina. Comment on
“Bremsstrahlung in α-decay of 210Po: do α-particles
emit photons in tunneling?” // Physical Review Letters.
2000, v. 85, №14, p. 3061-3062.
2. J. Kasagi, H. Yamazaki, N. Kasajima, T. Ohtsuki,
H. Yuki. Bremsstrahlung in α-decay of 210Po: do α-
particles emit photons in tunneling? // Physical Review
Letters. 1997, v. 79, №3, p. 371-374.
3. A. D'Arrigo, N.V. Eremin, G. Fazio, G. Giardina
M.G. Glotova, T.V. Klochko, M. Sacchi, A. Taccone.
Investigation of bremsstrahlung emission in α-decay of
heavy nuclei // Phys. Lett. B. 1994, v. 332, p. 25-30.
4. T. Papenbrock, G.F. Bertsch. Bremsstrahlung in
α-decay // Phys. Rev. Let. 1998, v. 80, p. 4141-4144.
5. E.V. Tkalya. Bremsstrahlung in α-decay and
“interference of space regions” // Physical Review C.
1999, v. 60, №5, p. 0446XX (1-4).
6. S.P. Maydanyuk, V.S. Olkhovsky. Does sub-
barrier bremsstrahlung in α-decay of 210Po exist? //
Progr. Theor. Phys. 2003, v. 109, №2, p. 203-211.
7. I.A. Aizenberg, V. Grainer. Nuclear Theory.
Vol. 2. M: “Atomizdat”, 1973, p. 347 (in Russian).
ТОРМОЗНОЕ ИЗЛУЧЕНИЕ ПРИ АЛЬФА-РАСПАДЕ: УГЛОВОЙ АНАЛИЗ СПЕКТРОВ
С. П. Майданюк, С. В. Бельчиков
Представлен квантово-механический метод расчета спектров тормозного излучения при альфа-распаде
тяжелых ядер, учитывающий величину угла между направлениями распространения альфа-частицы и
излучения фотона. Зависимость спектра от величины угла получена в явном аналитическом виде. Метод
может быть использован для сравнительного анализа экспериментальных данных, полученных для разных
величин угла.
ГАЛЬМІВНЕ ВИПРОМІНЮВАННЯ ПРИ АЛЬФА-РОЗПАДІ: КУТОВИЙ АНАЛІЗ СПЕКТРІВ
21
С. П. Майданюк, С. В. Бельчиков
Представлено квантово-механічний метод розрахунку спектрів гальмівного випромінювання при альфа-
розпаді важких ядер, що враховує величину кута між напрямками поширення альфа-частинки і
випромінювання фотона. Залежність спектра від величини кута отримано у явному аналітичному виді.
Метод може бути використаний для порівняльного аналізу експериментальних даних, отриманих для різних
величин кута.
22
PACS: 03.40.Kf, 03.65.Xp, 12.20.-m, 23.60.+e, 24.10.-i, 41.60.-m
ACKNOWLEDGMENTS
REFERENCES
С. П. Майданюк, С. В. Бельчиков
С. П. Майданюк, С. В. Бельчиков
|