- Email: [email protected]
The delocalization of the energy of the ionizing radiation in a molecular medium and its radiation-chemical features
Radial. Printed
Phys. Chem. Vol. 26, No. in the U.S.A.
1, pp. 53-56,
0146-5724/85 $3.00 + .OO Pergamon Press Ltd.
1985
THEc,DELOcALIZATION OF THE ENERGY OF THE aIONIZING RADIATION IN A MOLECULAR MEDIUM AND ITS RADIATION-CHEMICAL FEATURES I. G. KAPLAN and A.‘M. MITEREV L. Y. Karpov Physical-Chemical Institute, Moscow, USSR (Received
8 June 1984)
;Qbstr&t-The,problem of delocalization of the energy initially absorbed by the medium is considered. We present’s rigorous derivation of the formula for the delocalization due to the uncertainty principle for particles of arbitrary velocity. Using the value of the effective range of the electromagnetic field of a moving particle, we estimate the dimensions of the collective excitation area. We also discuss the possible radiationchemical consequences of the ‘initial delocalization of the energy absorbed by the medium.
1,. INTRODUCTION
in Ref. 7, since the subsequent localization is.of stochastic nature, the initial delocalization makes it WHEN PASSINGthrough a medium, a charged particle impossible to determine the point where the energy loses its energy through interaction with atoms and absorbtion took place more precisely than is almolecules. This results in the formation of disturblowed by the.uncertainty principle. ance areas along’the trajectory of the particle, conAs for the delocaliiation due to collective osciltaining excited molecules, ‘positive ions, and lations, the veryexistence of the latter has been the knocked out electrons and atoms. It is these dissubject of discussion for quite a long time (see Refs. turbance areas that ,make up the particle’s track. 7-9). Later on, it was established that, in the moBy ~simulating the track on a computer using the lecular medium, too, fast moving charged particles Monte-Carlo method, we can find the coordinates carrinduce states of a collective nature,(9-“) though of the space points where the energy dissipation the dimensions of the area of these collective extook place. If we are not interested in the averaged citations have not been determined. track (as was the case in Ref 1), but instead are In this study we consider the whole range of quesrecording successive instantaneous pictures of the tions connected with the delocalization of the iniparticle passing through the medium, each obtained tially absorbed energy. We present a rigorous dertrack will be a set of space points in which the ineivation of the formula for the delocalization due to scattering (excitation, lastic ionization) octhe uncertainty principle for particles of arbitrary curred.‘2”’ velocity. Using the value of the effective range of This approach is correct if, in every act of scatthe electromagnetic field of a moving particle, we tering, the energy transferred to the medium is estimate the dimensions of the collective excitation strictly localized. However, as far back as ‘1960, area. We also discuss the possible radiation-chemFanoc4’ had pointed out the two possible reasons ical consequences of the initial delocalization of the for delocalization of the energy absorbed by the meenergy absorbed by the medium. dium. The first one has to do with the quantummechanical nature of microparticles; the second one is connected with the possibility of excitation 2. DELOCALIZATION AND THE of the plasma-type collective oscillations.(5’ UNCERTAINTY PRINCIPLE The quantum-mechanical nature of microobjects manifests itself in the Heisenberg uncertainty prinConsider a particle with the energy E transferring ciple. It was common belief that the limitations im- part of its energy AE to the medium. At a moment posed by this principle are inessential. In Ref. 6, when the energy is transferred there is an uncerthis was substantiated by the fact that during the tainty AE in the energy of the particle and a corperiod of time -1O-‘6 s, the excitation is localized responding uncertainty Ax in its location. The reon one of the molecules. However, this substantilation between these two uncertainties has already ation cannot be considered correct. As was stressed been found for ultrarelativistic@’ and nonrelativis53
54
I. G. KAPLANand A. M. MITEREV
tic c7) particles. Here we will consider the general case. As in Refs. 6 and 7, we will start with the uncertainty principle formula, connecting the uncertainties in the coordinate and the momentum of microparticle "2) (1)
can see from this table, for relativistic electrons the delocalization due to ionization is no less than 13.1 nm, and if we are exciting the 7.5 eV electronic states, the delocalization will be twice as large. Thus, the area o f delocalization may contain up to several dozens of molecules o f medium size, and though later on the energy is localized on one of them, this localization is stochastic, and thus, the coordinates of the points of ionization or excitation cannot be determined more precisely than to within Ax. According to (4), the delocalization depends on the velocity of the particle, so for heavy particles the values of Ax presented in Table 1 correspond to greater energies, viz. E = (M/me)Ee. F o r example, a proton has the velocity v = c at E > 2 GeV.
Ax • APx -> h.
According to relativistic mechanics, for the energy of the particle, moving along the x-axis, and its momentum, we have the following relations cI3) (2)
E 2 = c 2(P2 + M~oc2)
(3)
p2 = Evx/C2.
F r o m (2) we have E A E = c2PxAPx or APx = (El c2px)AE. Substituting this into (1) and using (3), we get the following formula for the uncertainty in the coordinate o f the particle when it is transferring the energy AE. (4)
3. C O L L E C T I V E E X C I T A T I O N STATES IN A MOLECULAR MEDIUM The second reason for the delocalization of energy losses is the collective nature of excited states. This collectivity may exist even for excited states of a single molecule. The simplest example is the excitation of 7r-electron states, which are delocalized along the molecule. When a fast moving electron excites such a molecule, it transfers its energy to the whole ensemble of ~r-electrons. As a result, the energy absorption is delocalized along the molecule, and the latter can be a long one (e.g. a polymer molecule). Because of periodic symmetry, the electronic excitation states in a molecular crystal are also of collective nature. These are the well-known excition s t a t e s . °4'n5) Their energy is close to the energy of the discrete electronic states of an isolated molecule (4-8 eV), but the excitation envelops a large group of molecules, migrating up to 100 nm along the crystal. "5'16) In the same manner, because of the effective migration, the excitation of a fragment of a polymer chain spreads over the whole molecule. "7) Thus, the excitation of discrete electronic states (AEn < 11) in polymer molecules and molecular crystals is certainly delocalized. Collective excitation states exist not only in crystals, but also in liquids and polymer films, when the latter are irradiated by fast-moving charged parti-
A x >- hvxlAE.
This will also be the uncertainty in the location of the point where the energy was absorbed. Since at small velocities all relativistic relations are replaced by their nonrelativistic analogues, formula (4) holds for the whole range of velocities of the bombarding particle. The greater the transferred energy, the smaller the delocalization is. When we are knocking out I selectrons, the energy absorption is actually localized. Indeed, according to (4), even in the case of ultrarelativistic particles (v = c) when the energy transfer is 300 eV ( A E -> I(ls¢)), we have Ax = 0.66 nm, and for A E = 500 eV ( A E -> I (lso)) Ax = 0.4 rim. However, when the energy losses are small (and this is the case for most of the inelastic scattering processes) delocalization can be considerably large. In ionization processes most of the knocked out electrons have small energies: a most probable energy transfer is A E = 15 eV. The values of Ax for A E = 15 ev and the different energies of the bombarding electron are presented in Table 1. As one Table 1.
T H E DELOCALIZATION FOR ENERGY A E = 15 e V TRANSFERRED TO THE MEDIUM BY AN ELECTRON WITH THE ENERGY E e ,
Ee, eV
102
103
104
v, 10a cm/s
5.93
18.7
58.5
AX, nm
0.26
0.82
2.56
105 164 7.2
106 282 12.3
107 300 13.1
108 300 13.1
The delocalization of the energy of the ionizing radiation in a molecular medium cles. Such excitations are of the plasmon type, their energy, as a rule, being higher than the first ionization potential, i.e. somewhere about 15-25 eV.(9AoAs) The dielectric permitivity of a medium can be expressed through a set of eigen-frequencies of the medium W~, each o f which is characterized by its oscillator strength f~ and its damping constant Ti (~/i = I/T,')(19) (5)
¢(to) = 1 + ~f,~Op2(O)~ - to - i%~o)-1, i
where top is the plasma oscillation frequency depending only on the density of electrons (6)
top
=
(4"trNee2/Me) 1/2.
Since the energy of plasmons does not exceed 30 eV, the electrons of the inner shells do not participate in collective oscillations. The value of Ne in (6) is determined b y the density of the valent electrons. The plasma-type oscillations in a molecular medium represent a longitudinal polarized wave, causing oscillations o f the induced dipole moment of the molecule.(11) As in the case of the plasmons in metals, (2°) the frequency of these oscillations satisfies the condition (7)
where Ko('q) and K~('q) are the modified Bessel functions.(22) The electromagnetic field of a moving charged particles is isotropic at small velocities. F o r relativistic electrons we have a sharp anisotropy: the electric vector lies in a narrow angle near the plane perpendicular to the electron direction; (m the collective oscillations will have the same direction. Since the group velocity of collective oscillations is small, the period of time 10-16 s (the mean lifetime of a collective oscillation) is too short for energy to migrate outside the area o f the initial excitation. In order to estimate the maximum linear dimensions of this area, it is sufficient to find the ultimate value of p at which the polarization losses are close to zero. Since the functions K,(~q) and Kl('q) are noticeably large only in the region ~ -< (see Fig. 9.7 in Ref. 22), we can consider the area of collective oscillation to be limited by distances p from the axis of the track corresponding to ~ = ,ff
(10)
p
~ Bpe
= Tr'u/to r ~- q T h v / A E z .
The maximum value corresponds to v = c and is equal to B ~ x = ~rhclAEz. F o r water we have AEz = hto, = 21.4 eV, (9) which gives us B ~ X ( / / 2 0 ) = 29 nm.
~(to) = 0.
In the general case, it is a very hard to solve equation (7). F o r many molecules there exists a transition which has the highest f-value; let it be the first transition. If we confine ourselves to this single transition and assume that the corresponding damping constant is small, the resonance frequency satisfying equation (7) will be o0) (8)
55
tor :
( f l t o p 2 "{- 0.)2)1/2.
F o r molecular fluids flto 2 ~ to2 and the collective oscillations are not solely plasma-like, but are partially intra-molecular. In order to find the dimensions of the area where the plasma type collective oscillations can be excited, let us consider the dependence of the polarization losses o f a fast moving electron on the value of the impact parameter. If in formula (5) we keep only the term corresponding to the resonance frequency (8), the polarization losses Spo~for distances from p (the impact parameter) to oo will be given by the following formula ¢21) (9) Spol = (e2flto2/v2)'qgo(lq)
• Kl('q); "q = to,p/v,
4. C O N C L U S I O N Thus, at the initial moment (T -< 10- ~ s) the energy transferred by a fast moving charged particle is delocalized due to both the quantum-mechanical uncertainty principle and the cooperative effects. Because of the delocalization along the direction of motion due to the uncertainty principle, the coordinates of the points of ionization and excitation cannot be determined more precisely than to within Ax given by formula (4). F o r relativistic particles Ax is quite large and the corresponding area contains up to several dozens of molecules for example, for a 7.5 excited level we have Ax = 26 nm. This is especially important for biosystems which have a very unhomogeneous sensitivity to irradiation. The targets in radiobiology cannot be taken smaller than 10-25 nm (depending on the value of A E causing the bioeffect). As was shown in Sec. 3 the linear dimensions of the area of the plasmon-type collective oscillations may be as large as 30 nm. Since in the presence of collective peaks the oscillator strengths are concentrated near the maximum o f the peak, the outlet of plasmonic absorptions is considerably large. As it follows from the behaviour of the energy losses
56
I. G. KAPLANand A. M, MITEREV
function in water, t9) the existence of the plasmonlc absorption diminishes the share of the oscillator strefigths for transitions into discrete excited states in comparison with their distribution in the steam. Since the plasmon-type collective excitation decays m o s t l y by transferring its energy to one molecule, with the subsequent ionization, the energy transfer from discrete excited states to collective ones must lead to the increase of the share of the ionization in liquid systems in comparison with the gaseous ones. At the same time. as was stressed by Voltz, t23) the decay of collective excitations may be an additional source of formation of the superexcited states (SES), t24) see also Refs. 25 and 7. For SES there exists a competition between the predissociation and the autoionization. In order for the SES to be formed as a result of the decay of a collective excitation, it is necessary for the energy of the latter to be in the region of the SES formation for the considered molecules. In conclusion we will discuss the consequences of the delocalization of electronic excited states in long molecules and of the effective migration of the electronic excitation along the polymer chain, t~7) These processes lead to the rupture mainly of the weaker chemical bonds and also of those near the embedded atoms and the defects of the chain, since it is there that the excitation energy is localized. In such a polymer matrix there can be no track effects due to the energy transferred by the electrons of the medium, since the absorbed energy dissipates rapidly troughout the electron subsystem. The situation here is the same as in metals, where the electron gas scatters the absorbed energy throughout the volume of the metal, and no linear tracks are formed. REFERENCES
3
4. 5. 6. 7. 8. 9. 10. I 1.
12. 13 14. 15. 16. 17. 18. 19. 20. 21. 22.
23.
1. I. G. KAPLAN,A. M. MITEREV,and L. M. KHADZHI- 24. 25. BEKOVA, Khimiya Vysokikh Energiy 1977, 11,409. 2. H. G. PARETZKE,G. LEUTHOLD,G. BURGER,and W.
JACOBI, In Proc. 4th Symposium on Microdosimetry, Luxemburg, 1974, vol. 1, p. 123. V. V.'TREGUB,A. M. RAITSIMRING,and V. M. MORALOV. Preprint No. 7. 1980. and No. 8, 1981. Novosibirsk. Inst. Chem. Kinetics and Combustion, Acad. Sc. USSR. U. FANO,In Comparative Effects of Radiation (Edited by M. Burton et al.), p. 14. John Wiley, New York, 1960. U. FANO,Phys. Rev. 1960, 118, 451. A. MOZUMBERand J. L. MAGEE.Radiat. Phys. Chem. 1975.7, 83 I. G. KAPLAN.High Energy Chemistry 1983, 17, 159. I. G. KAPLANand V. E. SKURAT.Khimiya Vysokikh Energzy 1972.6, 252. C. J. KUTCHERand A. E. S. GREEN, Radiat. Res. 1976. 67, 408. M. W. WILLIAMS,R. N. HAMN, E. T. ARAKAWA,L. R. PAINTER. and R. D, BIRKHOFF. Radiat. Phys. Chem. 1975. 7, 95. H. RAETHER, Excitation of plasmons and interband transitions by electrons. In Springer Tracts in Modern Physics, vol. 88, p. 1~ Springer-Verlag, Berlin. 1980. L. D. LANDAUand E. M. LIFSHITS, Quantum Mechanics. Nanka, Moscow, 1974. L. D. LANDAUand E. M. LIFSHITS, Field Theory. Nauka, Moskow, 1967. A. S. DAVYDOV.The Theory of Solid-States. Nauka, Moscow, 1976. A. S. DAVYDOV,The Theory of Molecular Excitions. Plenum, New York, 1971. R. E. MERRIFIELD,J. chem. Phys. 1958, 28, 647. I. G. KAPLANand V. G. PLOTNIKOV.Khimiya Vysokikh Energiy 1967, 1, 507. N. SWANSONand C. J. POWELL,Phys. Rev. 1966, 145, 195. J. SLATER.Insulators, Semiconductors and Metals. McGraw-Hill, New York, 1967. D. PAINS, Elementary Excitations in Solids. Amsterdam, Ne w Y0rk, 1963. B. M. BOLOTOVSKY, Uspekhi Phys. Nauk 1957, 62, 201. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Edited by M. Abramowitz and I. A. Stegun). NBS, Washington, 1964. R, VOLTZ, in: Progress and Problems in Contemporary Radiation Chemistry, vol. 1, p. 139. Prague, 1971. R. L. PLATZMAN,Radiat. Res. 1962, 17, 419. V. I. MAKAROVand L. S. POLAK, Khimlya Vysokikh Energiy 1970, 4, 3.
Phys. Chem. Vol. 26, No. in the U.S.A.
1, pp. 53-56,
0146-5724/85 $3.00 + .OO Pergamon Press Ltd.
1985
THEc,DELOcALIZATION OF THE ENERGY OF THE aIONIZING RADIATION IN A MOLECULAR MEDIUM AND ITS RADIATION-CHEMICAL FEATURES I. G. KAPLAN and A.‘M. MITEREV L. Y. Karpov Physical-Chemical Institute, Moscow, USSR (Received
8 June 1984)
;Qbstr&t-The,problem of delocalization of the energy initially absorbed by the medium is considered. We present’s rigorous derivation of the formula for the delocalization due to the uncertainty principle for particles of arbitrary velocity. Using the value of the effective range of the electromagnetic field of a moving particle, we estimate the dimensions of the collective excitation area. We also discuss the possible radiationchemical consequences of the ‘initial delocalization of the energy absorbed by the medium.
1,. INTRODUCTION
in Ref. 7, since the subsequent localization is.of stochastic nature, the initial delocalization makes it WHEN PASSINGthrough a medium, a charged particle impossible to determine the point where the energy loses its energy through interaction with atoms and absorbtion took place more precisely than is almolecules. This results in the formation of disturblowed by the.uncertainty principle. ance areas along’the trajectory of the particle, conAs for the delocaliiation due to collective osciltaining excited molecules, ‘positive ions, and lations, the veryexistence of the latter has been the knocked out electrons and atoms. It is these dissubject of discussion for quite a long time (see Refs. turbance areas that ,make up the particle’s track. 7-9). Later on, it was established that, in the moBy ~simulating the track on a computer using the lecular medium, too, fast moving charged particles Monte-Carlo method, we can find the coordinates carrinduce states of a collective nature,(9-“) though of the space points where the energy dissipation the dimensions of the area of these collective extook place. If we are not interested in the averaged citations have not been determined. track (as was the case in Ref 1), but instead are In this study we consider the whole range of quesrecording successive instantaneous pictures of the tions connected with the delocalization of the iniparticle passing through the medium, each obtained tially absorbed energy. We present a rigorous dertrack will be a set of space points in which the ineivation of the formula for the delocalization due to scattering (excitation, lastic ionization) octhe uncertainty principle for particles of arbitrary curred.‘2”’ velocity. Using the value of the effective range of This approach is correct if, in every act of scatthe electromagnetic field of a moving particle, we tering, the energy transferred to the medium is estimate the dimensions of the collective excitation strictly localized. However, as far back as ‘1960, area. We also discuss the possible radiation-chemFanoc4’ had pointed out the two possible reasons ical consequences of the initial delocalization of the for delocalization of the energy absorbed by the meenergy absorbed by the medium. dium. The first one has to do with the quantummechanical nature of microparticles; the second one is connected with the possibility of excitation 2. DELOCALIZATION AND THE of the plasma-type collective oscillations.(5’ UNCERTAINTY PRINCIPLE The quantum-mechanical nature of microobjects manifests itself in the Heisenberg uncertainty prinConsider a particle with the energy E transferring ciple. It was common belief that the limitations im- part of its energy AE to the medium. At a moment posed by this principle are inessential. In Ref. 6, when the energy is transferred there is an uncerthis was substantiated by the fact that during the tainty AE in the energy of the particle and a corperiod of time -1O-‘6 s, the excitation is localized responding uncertainty Ax in its location. The reon one of the molecules. However, this substantilation between these two uncertainties has already ation cannot be considered correct. As was stressed been found for ultrarelativistic@’ and nonrelativis53
54
I. G. KAPLANand A. M. MITEREV
tic c7) particles. Here we will consider the general case. As in Refs. 6 and 7, we will start with the uncertainty principle formula, connecting the uncertainties in the coordinate and the momentum of microparticle "2) (1)
can see from this table, for relativistic electrons the delocalization due to ionization is no less than 13.1 nm, and if we are exciting the 7.5 eV electronic states, the delocalization will be twice as large. Thus, the area o f delocalization may contain up to several dozens of molecules o f medium size, and though later on the energy is localized on one of them, this localization is stochastic, and thus, the coordinates of the points of ionization or excitation cannot be determined more precisely than to within Ax. According to (4), the delocalization depends on the velocity of the particle, so for heavy particles the values of Ax presented in Table 1 correspond to greater energies, viz. E = (M/me)Ee. F o r example, a proton has the velocity v = c at E > 2 GeV.
Ax • APx -> h.
According to relativistic mechanics, for the energy of the particle, moving along the x-axis, and its momentum, we have the following relations cI3) (2)
E 2 = c 2(P2 + M~oc2)
(3)
p2 = Evx/C2.
F r o m (2) we have E A E = c2PxAPx or APx = (El c2px)AE. Substituting this into (1) and using (3), we get the following formula for the uncertainty in the coordinate o f the particle when it is transferring the energy AE. (4)
3. C O L L E C T I V E E X C I T A T I O N STATES IN A MOLECULAR MEDIUM The second reason for the delocalization of energy losses is the collective nature of excited states. This collectivity may exist even for excited states of a single molecule. The simplest example is the excitation of 7r-electron states, which are delocalized along the molecule. When a fast moving electron excites such a molecule, it transfers its energy to the whole ensemble of ~r-electrons. As a result, the energy absorption is delocalized along the molecule, and the latter can be a long one (e.g. a polymer molecule). Because of periodic symmetry, the electronic excitation states in a molecular crystal are also of collective nature. These are the well-known excition s t a t e s . °4'n5) Their energy is close to the energy of the discrete electronic states of an isolated molecule (4-8 eV), but the excitation envelops a large group of molecules, migrating up to 100 nm along the crystal. "5'16) In the same manner, because of the effective migration, the excitation of a fragment of a polymer chain spreads over the whole molecule. "7) Thus, the excitation of discrete electronic states (AEn < 11) in polymer molecules and molecular crystals is certainly delocalized. Collective excitation states exist not only in crystals, but also in liquids and polymer films, when the latter are irradiated by fast-moving charged parti-
A x >- hvxlAE.
This will also be the uncertainty in the location of the point where the energy was absorbed. Since at small velocities all relativistic relations are replaced by their nonrelativistic analogues, formula (4) holds for the whole range of velocities of the bombarding particle. The greater the transferred energy, the smaller the delocalization is. When we are knocking out I selectrons, the energy absorption is actually localized. Indeed, according to (4), even in the case of ultrarelativistic particles (v = c) when the energy transfer is 300 eV ( A E -> I(ls¢)), we have Ax = 0.66 nm, and for A E = 500 eV ( A E -> I (lso)) Ax = 0.4 rim. However, when the energy losses are small (and this is the case for most of the inelastic scattering processes) delocalization can be considerably large. In ionization processes most of the knocked out electrons have small energies: a most probable energy transfer is A E = 15 eV. The values of Ax for A E = 15 ev and the different energies of the bombarding electron are presented in Table 1. As one Table 1.
T H E DELOCALIZATION FOR ENERGY A E = 15 e V TRANSFERRED TO THE MEDIUM BY AN ELECTRON WITH THE ENERGY E e ,
Ee, eV
102
103
104
v, 10a cm/s
5.93
18.7
58.5
AX, nm
0.26
0.82
2.56
105 164 7.2
106 282 12.3
107 300 13.1
108 300 13.1
The delocalization of the energy of the ionizing radiation in a molecular medium cles. Such excitations are of the plasmon type, their energy, as a rule, being higher than the first ionization potential, i.e. somewhere about 15-25 eV.(9AoAs) The dielectric permitivity of a medium can be expressed through a set of eigen-frequencies of the medium W~, each o f which is characterized by its oscillator strength f~ and its damping constant Ti (~/i = I/T,')(19) (5)
¢(to) = 1 + ~f,~Op2(O)~ - to - i%~o)-1, i
where top is the plasma oscillation frequency depending only on the density of electrons (6)
top
=
(4"trNee2/Me) 1/2.
Since the energy of plasmons does not exceed 30 eV, the electrons of the inner shells do not participate in collective oscillations. The value of Ne in (6) is determined b y the density of the valent electrons. The plasma-type oscillations in a molecular medium represent a longitudinal polarized wave, causing oscillations o f the induced dipole moment of the molecule.(11) As in the case of the plasmons in metals, (2°) the frequency of these oscillations satisfies the condition (7)
where Ko('q) and K~('q) are the modified Bessel functions.(22) The electromagnetic field of a moving charged particles is isotropic at small velocities. F o r relativistic electrons we have a sharp anisotropy: the electric vector lies in a narrow angle near the plane perpendicular to the electron direction; (m the collective oscillations will have the same direction. Since the group velocity of collective oscillations is small, the period of time 10-16 s (the mean lifetime of a collective oscillation) is too short for energy to migrate outside the area o f the initial excitation. In order to estimate the maximum linear dimensions of this area, it is sufficient to find the ultimate value of p at which the polarization losses are close to zero. Since the functions K,(~q) and Kl('q) are noticeably large only in the region ~ -< (see Fig. 9.7 in Ref. 22), we can consider the area of collective oscillation to be limited by distances p from the axis of the track corresponding to ~ = ,ff
(10)
p
~ Bpe
= Tr'u/to r ~- q T h v / A E z .
The maximum value corresponds to v = c and is equal to B ~ x = ~rhclAEz. F o r water we have AEz = hto, = 21.4 eV, (9) which gives us B ~ X ( / / 2 0 ) = 29 nm.
~(to) = 0.
In the general case, it is a very hard to solve equation (7). F o r many molecules there exists a transition which has the highest f-value; let it be the first transition. If we confine ourselves to this single transition and assume that the corresponding damping constant is small, the resonance frequency satisfying equation (7) will be o0) (8)
55
tor :
( f l t o p 2 "{- 0.)2)1/2.
F o r molecular fluids flto 2 ~ to2 and the collective oscillations are not solely plasma-like, but are partially intra-molecular. In order to find the dimensions of the area where the plasma type collective oscillations can be excited, let us consider the dependence of the polarization losses o f a fast moving electron on the value of the impact parameter. If in formula (5) we keep only the term corresponding to the resonance frequency (8), the polarization losses Spo~for distances from p (the impact parameter) to oo will be given by the following formula ¢21) (9) Spol = (e2flto2/v2)'qgo(lq)
• Kl('q); "q = to,p/v,
4. C O N C L U S I O N Thus, at the initial moment (T -< 10- ~ s) the energy transferred by a fast moving charged particle is delocalized due to both the quantum-mechanical uncertainty principle and the cooperative effects. Because of the delocalization along the direction of motion due to the uncertainty principle, the coordinates of the points of ionization and excitation cannot be determined more precisely than to within Ax given by formula (4). F o r relativistic particles Ax is quite large and the corresponding area contains up to several dozens of molecules for example, for a 7.5 excited level we have Ax = 26 nm. This is especially important for biosystems which have a very unhomogeneous sensitivity to irradiation. The targets in radiobiology cannot be taken smaller than 10-25 nm (depending on the value of A E causing the bioeffect). As was shown in Sec. 3 the linear dimensions of the area of the plasmon-type collective oscillations may be as large as 30 nm. Since in the presence of collective peaks the oscillator strengths are concentrated near the maximum o f the peak, the outlet of plasmonic absorptions is considerably large. As it follows from the behaviour of the energy losses
56
I. G. KAPLANand A. M, MITEREV
function in water, t9) the existence of the plasmonlc absorption diminishes the share of the oscillator strefigths for transitions into discrete excited states in comparison with their distribution in the steam. Since the plasmon-type collective excitation decays m o s t l y by transferring its energy to one molecule, with the subsequent ionization, the energy transfer from discrete excited states to collective ones must lead to the increase of the share of the ionization in liquid systems in comparison with the gaseous ones. At the same time. as was stressed by Voltz, t23) the decay of collective excitations may be an additional source of formation of the superexcited states (SES), t24) see also Refs. 25 and 7. For SES there exists a competition between the predissociation and the autoionization. In order for the SES to be formed as a result of the decay of a collective excitation, it is necessary for the energy of the latter to be in the region of the SES formation for the considered molecules. In conclusion we will discuss the consequences of the delocalization of electronic excited states in long molecules and of the effective migration of the electronic excitation along the polymer chain, t~7) These processes lead to the rupture mainly of the weaker chemical bonds and also of those near the embedded atoms and the defects of the chain, since it is there that the excitation energy is localized. In such a polymer matrix there can be no track effects due to the energy transferred by the electrons of the medium, since the absorbed energy dissipates rapidly troughout the electron subsystem. The situation here is the same as in metals, where the electron gas scatters the absorbed energy throughout the volume of the metal, and no linear tracks are formed. REFERENCES
3
4. 5. 6. 7. 8. 9. 10. I 1.
12. 13 14. 15. 16. 17. 18. 19. 20. 21. 22.
23.
1. I. G. KAPLAN,A. M. MITEREV,and L. M. KHADZHI- 24. 25. BEKOVA, Khimiya Vysokikh Energiy 1977, 11,409. 2. H. G. PARETZKE,G. LEUTHOLD,G. BURGER,and W.
JACOBI, In Proc. 4th Symposium on Microdosimetry, Luxemburg, 1974, vol. 1, p. 123. V. V.'TREGUB,A. M. RAITSIMRING,and V. M. MORALOV. Preprint No. 7. 1980. and No. 8, 1981. Novosibirsk. Inst. Chem. Kinetics and Combustion, Acad. Sc. USSR. U. FANO,In Comparative Effects of Radiation (Edited by M. Burton et al.), p. 14. John Wiley, New York, 1960. U. FANO,Phys. Rev. 1960, 118, 451. A. MOZUMBERand J. L. MAGEE.Radiat. Phys. Chem. 1975.7, 83 I. G. KAPLAN.High Energy Chemistry 1983, 17, 159. I. G. KAPLANand V. E. SKURAT.Khimiya Vysokikh Energzy 1972.6, 252. C. J. KUTCHERand A. E. S. GREEN, Radiat. Res. 1976. 67, 408. M. W. WILLIAMS,R. N. HAMN, E. T. ARAKAWA,L. R. PAINTER. and R. D, BIRKHOFF. Radiat. Phys. Chem. 1975. 7, 95. H. RAETHER, Excitation of plasmons and interband transitions by electrons. In Springer Tracts in Modern Physics, vol. 88, p. 1~ Springer-Verlag, Berlin. 1980. L. D. LANDAUand E. M. LIFSHITS, Quantum Mechanics. Nanka, Moscow, 1974. L. D. LANDAUand E. M. LIFSHITS, Field Theory. Nauka, Moskow, 1967. A. S. DAVYDOV.The Theory of Solid-States. Nauka, Moscow, 1976. A. S. DAVYDOV,The Theory of Molecular Excitions. Plenum, New York, 1971. R. E. MERRIFIELD,J. chem. Phys. 1958, 28, 647. I. G. KAPLANand V. G. PLOTNIKOV.Khimiya Vysokikh Energiy 1967, 1, 507. N. SWANSONand C. J. POWELL,Phys. Rev. 1966, 145, 195. J. SLATER.Insulators, Semiconductors and Metals. McGraw-Hill, New York, 1967. D. PAINS, Elementary Excitations in Solids. Amsterdam, Ne w Y0rk, 1963. B. M. BOLOTOVSKY, Uspekhi Phys. Nauk 1957, 62, 201. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Edited by M. Abramowitz and I. A. Stegun). NBS, Washington, 1964. R, VOLTZ, in: Progress and Problems in Contemporary Radiation Chemistry, vol. 1, p. 139. Prague, 1971. R. L. PLATZMAN,Radiat. Res. 1962, 17, 419. V. I. MAKAROVand L. S. POLAK, Khimlya Vysokikh Energiy 1970, 4, 3.