Mathematical modeling of radiolysis process of water under the impact of low-energy electrons
The radiolysis process of water (liquid phase) under the impact of low - energy electrons (E = 1, 2.5, 5, 10 keV ) was mathematically modeled using Monte-Carlo, single collision and pacing methods on the base of Mathcad program. The radiation-chemical yields of the physical (single ionized molecular...
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irk-123456789-1114732017-01-11T03:02:43Z Mathematical modeling of radiolysis process of water under the impact of low-energy electrons Jafarov, Y.D. Ядерно-физические методы и обработка данных The radiolysis process of water (liquid phase) under the impact of low - energy electrons (E = 1, 2.5, 5, 10 keV ) was mathematically modeled using Monte-Carlo, single collision and pacing methods on the base of Mathcad program. The radiation-chemical yields of the physical (single ionized molecular orbitals - H₂O⁺j (1a₁, 2a₁, 1b₁, 3a₁, 1b₁), e⁻sub electron - lost its energy up to a primary electron - excited energy and electron-excited states: H₂O* (A¹B₁, B¹A₁, Rydberg state, diffusion band, dissociative excitation and plasmon-H₂O**) and physicochemical (OH, e⁻aq, H, H₃O+, H₂, H₂O₂, HO₂, O₂, OH⁻, O⁻₂, HO⁻₂ ) phase products of the non-elastic collision of electrons and water molecules were determined. Проведено математичне моделювання радіолізу води (фаза рідини) під дією електронів з малою енергією (E = 1, 2.5, 5, 10 кеВ) за допомогою програми Mathcad з використанням Монте-Карло, однократних зіткнень та шагового методів. На основі цієї моделі обраховані радіаційно-хімічні виходи слідуючих продуктів непружних зіткнень електронів з молекулами води: фізичні (одноіонізовані молекулярні орбіталі - H₂O⁺j (1a₁, 2a₁, 1b₁, 3a₁, 1b₁) електрони з енергією, пониженою до рівня з першим електронно-збудженим станом - e⁻sub, і електронно-збуджені стани - H₂O* (A¹B₁, B¹A₁, рідберговскі стани, полоса дифузії, дисоціативне збудження і плазмон - H₂O**) і фізико-хімічна стадія (OH, e⁻aq, H, H₃O+, H₂, H₂O₂, HO₂, O₂, OH⁻, O⁻₂, HO⁻₂). Проведено математическое моделирование радиолиза воды (жидкая фаза) под действием электронов с малой энергией (E=1, 2.5, 5, 10 кэВ) при помощи программы Mathcad с использованием Монте-Карло, однократных столкновений и шагового методов. На основе этой модели вычислены радиационно-химические выходы следующих продуктов неупругих столкновений электронов с молекулами воды: физические (одноионизированные молекулярные орбитали -H₂O⁺j (1a₁, 2a₁, 1b₁, 3a₁, 1b₁), электроны с энергией, снизившейся до уровня с первым электронно-возбужденным состоянием - e-sub, и электронно-возбужденные состояния - H₂O*(A¹B₁, B¹A₁, ридберговские состояния, полоса диффузии, диссоциативное возбуждение и плазмон - H₂O**) и физико-химическая стадия (OH, e⁻aq, H, H₃O+, H₂, H₂O₂, HO₂, O₂, OH⁻, O⁻₂, HO⁻₂). 2011 Article Mathematical modeling of radiolysis process of water under the impact of low-energy electrons / Y.D. Jafarov // Вопросы атомной науки и техники. — 2011. — № 5. — С. 42-47 — Бібліогр.: 44 назв. — англ. 1562-6016 PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk http://dspace.nbuv.gov.ua/handle/123456789/111473 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерно-физические методы и обработка данных Ядерно-физические методы и обработка данных |
| spellingShingle |
Ядерно-физические методы и обработка данных Ядерно-физические методы и обработка данных Jafarov, Y.D. Mathematical modeling of radiolysis process of water under the impact of low-energy electrons Вопросы атомной науки и техники |
| description |
The radiolysis process of water (liquid phase) under the impact of low - energy electrons (E = 1, 2.5, 5, 10 keV ) was mathematically modeled using Monte-Carlo, single collision and pacing methods on the base of Mathcad program. The radiation-chemical yields of the physical (single ionized molecular orbitals - H₂O⁺j (1a₁, 2a₁, 1b₁, 3a₁, 1b₁), e⁻sub electron - lost its energy up to a primary electron - excited energy and electron-excited states: H₂O* (A¹B₁, B¹A₁, Rydberg state, diffusion band, dissociative excitation and plasmon-H₂O**) and physicochemical (OH, e⁻aq, H, H₃O+, H₂, H₂O₂, HO₂, O₂, OH⁻, O⁻₂, HO⁻₂ ) phase products of the non-elastic collision of electrons and water molecules were determined. |
| format |
Article |
| author |
Jafarov, Y.D. |
| author_facet |
Jafarov, Y.D. |
| author_sort |
Jafarov, Y.D. |
| title |
Mathematical modeling of radiolysis process of water under the impact of low-energy electrons |
| title_short |
Mathematical modeling of radiolysis process of water under the impact of low-energy electrons |
| title_full |
Mathematical modeling of radiolysis process of water under the impact of low-energy electrons |
| title_fullStr |
Mathematical modeling of radiolysis process of water under the impact of low-energy electrons |
| title_full_unstemmed |
Mathematical modeling of radiolysis process of water under the impact of low-energy electrons |
| title_sort |
mathematical modeling of radiolysis process of water under the impact of low-energy electrons |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2011 |
| topic_facet |
Ядерно-физические методы и обработка данных |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/111473 |
| citation_txt |
Mathematical modeling of radiolysis process of water under the impact of low-energy electrons / Y.D. Jafarov // Вопросы атомной науки и техники. — 2011. — № 5. — С. 42-47 — Бібліогр.: 44 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT jafarovyd mathematicalmodelingofradiolysisprocessofwaterundertheimpactoflowenergyelectrons |
| first_indexed |
2025-07-08T02:13:00Z |
| last_indexed |
2025-07-08T02:13:00Z |
| _version_ |
1837043057178968064 |
| fulltext |
MATHEMATICAL MODELING OF RADIOLYSIS PROCESS
OF WATER UNDER THE IMPACT OF LOW-ENERGY
ELECTRONS
Y.D. Jafarov∗
Institute of Radiation Problems, AZ1143, Baku, F.Aghayev 9
(Received April 29, 2011)
The radiolysis process of water (liquid phase) under the impact of low - energy electrons (E = 1, 2.5, 5, 10 keV )
was mathematically modeled using Monte-Carlo, single collision and pacing methods on the base of Mathcad pro-
gram. The radiation-chemical yields of the physical (single ionized molecular orbitals- H2O
+
j (1a1, 2a1, 1b2, 3a1, 1b1),
e−sub electron - lost its energy up to a primary electron - excited energy and electron-excited states: H2O
∗
(A1B1, B1A1, Rydberg state, diffusion band, dissociative excitation and plasmon-H2O
∗∗) and physicochemical
(OH, e−aq, H, H3O
+, H2, H2O2, O2, OH−, O−2 , HO−2 ) phase products of the non-elastic collision of electrons and
water molecules were determined.
PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk
1. INTRODUCTION
Low-(0.05...10 keV ), moderate-(10...100 keV ) and
high-energy (100...5000 keV ) electrons gradually lose
their kinetic energies while passing through water
in elastic and non-elastic collisions with water mole-
cules. According to the mechanism of radiation en-
ergy loss at the physical phase of the non-elastic
collision process (< 10−15 sec.) intermediate parti-
cles such as H2O
+
j -direct single ionization of sev-
eral molecular orbitals (MO), e−sub electron-lost its
energy up to a primary electron-excited energy and
electron-excited states: intermediate particles such
as A1B1, B1A1 Rydberg state (Ry), diffusion band
(ab), dissociative excitation (de) and plasmon (ce-
collective excitation) are generating. As these parti-
cles have a strong effect on the physical, chemical, bi-
ological processes progressing in water and water so-
lutions, it’s possible to predict events which are likely
to happen in the future by studying their roles in dif-
ferent fields of science (atomic, nuclear and plasma
physics, astrophysics, modelling of atmospheric phe-
nomena, radiochemistry, radiobiology etc.). There-
fore, to theoretically and experimentally study the
generation and consumption of these products is one
of the main problems.
In the world literature the reliable values of the
effective cross-section of a water molecule ionization
under an electron impact (e−, 2e−) was determined
by authors [1-6] using different experimental meth-
ods and by authors [7-16] using different theoretical
approaches.
There’s no reference regarding water molecule ex-
citation under an electron impact yet. Only au-
thors [17-19] determined water electron-excitation
states using a photoabsorption spectroscopy method.
At the present time authors [20-24] have theoreti-
cally calculated the effective cross-section of water
electron-excitation states using different polyempiri-
cal methods on the base of the experimental results.
The total effective cross-section of electron-water
molecule scattering was identified by authors [25-30].
While comparing the results of the experimental and
theoretical calculations it becomes evident that at the
values of energy more than 30 eV they coincide but
at lower values some deviations are observed.
The radiation-chemical yields of the products gen-
erated at both phases (physical and physicochemical)
of the water radiolysis process were theoretically cal-
culated by authors [31-41] according to different ap-
proaches. There are some conformities with some er-
rors between the theoretical and experimental results
of different authors and our results.
In the given work the radiolysis process of wa-
ter (liquid phase) under the impact of low-energy
(T = 1.0; 2.5; 5.0; 10 keV ) electrons were mathemat-
ically modeled. The radiation - chemical yields of
the primary products which were likely to generate
in the physical (< 10−15 sec.) and physicochemical
(10−15...10−12 sec.) phases of the process were calcu-
lated. The more improved formula of Mott equation
[7] was used to determine the effective cross - section
of ionization process of molecular orbitals (MO), and
the equations proposed by different authors were used
to determine the effective cross - sections of electron
- excited states of different types [13] in the model.
Calculation was made on the base of Mathcad pro-
∗Corresponding author E-mail address: azirp@rambler.ru
42 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2011, N5.
Series: Nuclear Physics Investigations (56), p.42-47.
gram using Monte-Carlo, single collision and pacing
methods.
2. THEORETICAL METHODS AND
OBTAINED RESULTS
The energy balance between low-energy electrons and
water molecules during a non-elastic collision can be
simply expressed by T = Es+4E, here T and Es are
accordingly kinetic energies of an electron before and
after collision, 4E is the energy transferred to water
molecule by an electron during collision. This energy
is used for an electron-excitation of water molecule
(1) and direct single ionization (2) of molecular or-
bitals (MO) (1a1, 2a1, 1b2, 3a1, 1b1).
e−0 + H2O →
→
{
H2O
∗(H2O
∗∗) + e−s , ∆E = E0n,
H2O
+
j + e−s + e−e , ∆E = W + Bj .
(1)
(2)
Here e−0 , and e−e are accordingly incident, scattered
and ejected electrons, H2O
+, H2O
∗, H2O
∗∗ are cor-
respondingly ionization, excitation and extreme ex-
citation (plasmon) states. The transferred energy
∆E during these processes equals to an excitation
energy of a molecule Eon in (1) case and the sum
of the kinetic energy of an ejected electron W from
j-molecular orbital and bound energy Bj of the corre-
spondent orbital in (2) case. Kinetic energies of scat-
tered and ejected electrons may accordingly change
from Es = 0 up to Emax
s = T −Bj or from W = 0 up
to Wmax = (T − Bj)/2. A more improved formula
[3] of Mott equation was used to calculate the effec-
tive cross - section of the single ionization of water
molecule’s j −MO:
dσj(W,T )
dW
=
S
Bj(t + u + 1)
{
1
(w + 1)2
+
1
(w − 1)2
1
(w + 1)(t + w)
+
4u
3
[
1
(w + 1)3
+
1
(t− w)3
]}
, (3)
here, t = T
Bj
, w = W
Bj
, u = Uj
Bj
, S = 4πa2
0Nj
(
R
Bj
)2
,
Uj and Nj are the kinetic energy and number of an
electron correspondent to j − MO, a0-Bohr radius,
R = 13.61 eV is Rydberg energy. The values of Bj ,
Nj and Uj according to different molecular orbitals
were given in Table 1. If we integrate the (3) ex-
pression for all possible values of ejected electrons’
energies, i.e. from W = 0 up to Wmax, we can get
the expression describing the dependence of ioniza-
tion effective cross - section on the kinetic energy of
a primary (Fig.1) electron during non-inelastic colli-
sison corresponding to j −MO:
σj (T ) =
∫ W max
0
dσj(W,T )
dW
dW . (4)
Fig.1. Dependence of the effective cross - section of
a direct single ionization corresponding to j − MO
(σ1 − 1a1, σ2 − 2a1, σ3 − 1b2, σ4 − 3a1, σ5 − 1b1)
of water molecules under the impact of low - energy
electrons
In order to get the total effective cross - section of
ionization (Fig.1) we should to sum the (4) expres-
sion according to the number of molecular orbitals
(MO) [11]:
σion (T ) =
NMO∑
j=1
σj (T ) . (5)
Here NMO = 5 is the number of molecular orbitals
(MO) of a water molecule. During non-elastic col-
lision between electrons and water molecules the ef-
fective cross-section of electron transformation from
a ground state (0) to (n) excited state was calculated
on the base of [13] expression:
σ0n (T ) =
4πa2
0R
T + B + E0n
[
a ln
(
T
R
)
+ b + c
R
T
]
.
(6)
Here a, b and c are constants corresponding to 0 → n
junction and E0n [13] is an excitation energy corre-
spondent to that junction. Constants for each junc-
tion were chosen according to the values of radiation-
chemical yield obtained from the experiments. The
values of the constants and E0n used in calculations
were given in Table 2. In Fig.2 the dependence of the
effective cross-sections of electron-excitation states
(σ1 − A1B1, σ1 − B1A1, σ3-Rydberg state (Ry), σ4-
diffusion band (db), σ5-dissociative excitation (de)
and σ6-plasmon (ce)) in water under the impact of
low-energy electrons on their energies were described.
In order to obtain the totally effective cross-
section of electron excitation (Fig.1) (6) expression
should be summed according to the number of ex-
cited states:
σexc (T ) =
∑
n
σ0n (T ) . (7)
Primary electrons or δ-electrons of a new generation
formed by them lose part of their energy during each
43
non-elastic collision and this process continues till
the next non-elastic collision of an electron energy.
Fig.2. The dependence of the effective cross- sec-
tions of electron-excitation states (σ1 − A1B1,σ1 −
B1A1,σ3-Rydberg state (Ry), σ4-diffusion band
(db), σ5-dissociative excitation (de) and σ6-plasmon
(ce)) in water under the impact of low-energy
electrons on their energies
The amount of the moderate energy ∆E(T ) lost
during each non-elastic collision of the electron with
T energy with a water molecule was taken as an
equivalent to the following expression [38, 39, 41]:
∆E(T ) =
∑
n
P0n(T )En +
∑
j
Pj(T )εj(T ) . (8)
Here P0n(T ) = σ0n(T )
σtot(T ) is the probability of trans-
formation into n-excited state, Pj(T ) = σj(T )
σtot(T )
is the probability of the event occurrence
corresponding to the ionization of j − MO,
σtot(T ) =
∑
n σ0n(T ) +
∑
j σj(T ) - a total effec-
tive cross-section (Fig.3), εj(T ) =
∫ T
Bj
ε
dσj(T )
dε dε
- a moderate energy calculated for form-
ing one electron-positive ion pair in j −
MO by the electron with T energy.
Fig.3. The dependence of electron-excitation
(σexc (T ) =
∑
n σ0n (T )), MO ionization (σion (T ) =
=
∑
j σj (T )) states in water under the impact of
low-energy electrons and the full effective cross-
section σtot(T ) = σexc(T ) + σion(T ) on their
energies
Table 1. The radiation-chemical yield of a direct single ionization
corresponding to molecular orbitals (MO) of water molecule (1a1, 2a1,
1b2, 3a1, 1b1) under the impact of low-energy electrons
Molecular
orbits
N Bj , eV Uj , eV T, keV
1 2.5 5 10
1a1 2 539.7 793.4 1.685 1.687 1.689 1.691
2a1 2 36.88 70.71 0.951 0.950 0.954 0.957
1b2 2 19.83 48.36 0.697 0.701 0.704 0.704
3a1 2 15.57 59.52 0.259 0.260 0.263 0.265
1b1 2 12.61 61.91 0.0003 0.0005 0.0007 0.0009
Table 2. The radiation-chemical yields of electron-excited states (A1B1, B1A1, Ry-
dberg states (Ry), diffusion band (db), dissosiative excitation (de) and plasmon (ce)
states) formed in water under the impact of low-energy electrons
Electron-
excited state
Excitation
energy, eV
Constants T, keV
a b c 1 2.5 5 10
A1B1 8.4 0.7532 0.4751 -0.0675 0.870 0.877 0.872 0.875
B1A1 10.1 0.3900 0.1500 0.0015 0.412 0.407 0.413 0.409
Ry 12.26 0.0465 0.0282 -0.010 0.052 0.053 0.057 0.055
db 12.93 0.2380 0.0010 0.0265 0.267 0.269 0.264 0.260
de 14.1 0.2473 -0.010 0.0150 0.218 0.212 0.208 0.214
ce 21.4 1.2951 0.0120 -0.7532 1.089 1.086 1.087 1.094
44
On the base of the model calculations were made
for the single ionization of five molecular orbitals
(MO) and six electron-excited states of water mole-
cules. The calculated values of the single ionization
correspondent to 1a1, 2a1, 1b2, 3a1, 1b1 molecular
orbitals (MO) were given in Table 1, and in Table 2
the calculated values of the radiation-chemical yields
of A1B1, B1A1, Rydberg state (Ry), diffusion band
(db), dissosiative excitation (de) and plasmon (ce)
electron-excited states were given.
The products of the physical phase gradually (in
the course of 10−15...10−12 sec.) lose their energies at
the next physicochemical phase: at the result of e−sub-
electrons’ elastic collision and dipole relaxation and
transform into thermal electrons and consequently
solvates (e−sub → e−aq ), H2O
+ ions transform into
H3O
+ ion and OH radical at the result of an ion-
molecular reaction, and electron-excited molecules
generate their next products due to relaxation, auto-
ionization and dissociation. The next products [40]
which were likely to be generated by the products of
the physical phase in the course of ∼ 10−12 sec. were
given in Table 3.
Table 3. Percentage of the products which are likely to be generated
by the products of the physical phase
Products of the G Generation Percentage,
physical phase 100 eV channels %
e−sub 3.592 e−aq 100
H2O
+ 3.592 H3O
+ + OH 100
A1B1 0.870
{
H2O
H2 + OH
{
25
75
B1A1 0.412
{
H2O
H2 + H2O2
{
45
55
Ry
db
de
0.487
H2O
H + OH
H3O
+ + OH + e−aq
23
20
57
ce 1.089
{
H3O
+ + OH
H+ + OH
{
92.2
7.8
In the calculation carried out by us the radiation-
chemical yields of the products generated in the
physical and physicochemical phases of the radiolysis
process progressing in the course of ∼ 10−12 sec. were
determined and the obtained results (Table 4) were
given in comparison with the theoretical [31, 42] and
experimental ones [43,44] obtained by the authors.
Table 4. Radiation-chemical yields of the products generated at the physic-
ochemical phase (10−15...10−12 sec.) of the radiolysis process progressing in
water under the impact of low-energy electrons
Primary J.E.Turner I.Q.Kaplan Experimental Our
products [42] [31] results [43,44] results
OH 8.4 5.85 5.9 5.624
e−aq 6.3 5.16 4.7 4.874
H 2.1 0.61 0.7 0.75
H3O
+ 6.3 5.16 4.8 4.874
H2 0.3 0.39 0.45 0.227
H2O2 0.3 0.39 - 0.227
HO2 - - - -
O2 - - - -
OH− - - - -
O−2 - - - -
HO−
2 - - - -
45
The results obtained for the radiation-chemical
yields of the active intermediate products generated
in the physical and physicochemical phases of the ra-
diolysis process progressing in water, calculated on
the base of our model from the theoretical calcula-
tions and experiments conducted by different authors
according to different approaches conform with some
errors.
3. CONCLUSIONS
The processes progressing in water and water solu-
tions under the impact of ionizing radiation (electron
and ?-quantum) can be calculated on the base of this
model. The model can be used to determine the
processes progressing in aerosols of the atmosphere
under the effect of space rays, changes in water and
water solutions used in atomic and nuclear power-
engineering as an energy-carrier or for different pur-
poses, the nano-, micro and total absorption dose
during radiating ancological patients by electrons and
?-quantum in the same and different directions as well
as for the minimum selection of by-effects.
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МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ПРОЦЕССА РАДИОЛИЗА ВОДЫ ПОД
ДЕЙСТВИЕМ НИЗКОЭНЕРГЕТИЧЕСКИХ ЭЛЕКТРОНОВ
Я.Д. Джафаров
Проведено математическое моделирование радиолиза воды (жидкая фаза) под действием электронов
с малой энергией (E = 1, 2.5, 5, 10 кэВ) при помощи программы Mathcad с использованием Монте-
Карло, однократных столкновений и шагового методов. На основе этой модели вычислены радиационно-
химические выходы следующих продуктов неупругих столкновений электронов с молекулами воды:
физические (одноионизированные молекулярные орбитали - H2Oj (1a1, 2a1, 1b2, 3a1, 1b1) электроны с
энергией, снизившейся до уровня с первым электронно-возбужденным состоянием - e−sub, и электронно-
возбужденные состояния - H2O
∗ (A1B1, B1A1, ридберговские состояния, полоса диффузии, диссоциа-
тивное возбуждение и плазмон - H2O
∗∗) и физико-химическая стадия ( OH, eaq, H, H3O
+, H2, H2O2,
HO2, O2, OH−, O−2 , HO−
2 ).
МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ПРОЦЕСУ РАДIОЛIЗУ ВОДИ ПIД ДIЄЮ
НИЗЬКОЕНЕРГЕТИЧНИХ ЕЛЕКТРОНIВ
Я.Д. Джафаров
Проведено математичне моделювання радiолiзу води (фаза рiдини) пiд дiєю електронiв з малою енер-
гiєю (E = 1, 2.5, 5, 10 кеВ) за допомогою програми Mathcad з використанням Монте-Карло, однократ-
них зiткнень та шагового методiв. На основi цiєї моделi обрахованi радiацiйно-хiмiчнi виходи слiдуючих
продуктiв непружних зiткнень електронiв з молекулами води: фiзичнi (одноiонiзованi молекулярнi ор-
бiталi - H2Oj (1a1, 2a1, 1b2, 3a1, 1b1) електрони з енергiєю, пониженою до рiвня з першим електронно-
збудженим станом - e−sub, i електронно-збудженi стани - H2O
∗ (A1B1, B1A1, рiдберговскi стани, полоса
дифузiї, дисоцiативне збудження i плазмон - H2O
∗∗) i фiзико-хiмiчна стадiя ( OH, eaq, H, H3O
+, H2,
H2O2, HO2, O2, OH−, O−
2 , HO−
2 ).
47
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