- Email: [email protected]
Energy loss rate and thermalization of subionizing positrons and electrons
Nuclear Instruments and Methods in Physics Research B 221 (2004) 235–238 www.elsevier.com/locate/nimb
Energy loss rate and thermalization of subionizing positrons and electrons Sergey V. Stepanov *, Vsevolod M. Byakov Institute of Theoretical and Experimental Physics, Bolshaya Cheremushkinskaya 25, Moscow 117218, Russia
Abstract Energy loss rate of low-energy eþ and e (with the energy below the ionization and electronic excitation thresholds) is calculated. Energy absorption in the medium is treated in terms of the complex frequency-dependent dielectric permittivity. Present consideration is based on the classical electrodynamics, but analytically goes further than previous ones because of utilizing of a cute mathematical procedure, which holds for particles moving in a diffusive way. It is just the case of thermalization of low-energy eþ and e . Account for the gaussian form factor (spatial charge distribution) of the moving particle, leads to additional simplifications. Numerical estimations are made for light and heavy water. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Slowing down; Energy loss; Thermalization; Dielectric friction; Vibrational excitations; Stopping power
The energy of the subionizing charged particles (eþ , e ) lies below the first ionization potential and electronic excitation threshold of molecules. So during thermalization in condensed medium, energized particles lose their energy by exciting molecular vibrations and (in polar media) due to reorientation of dipole moments. The problem of calculation of the dielectric energy loss rate, W_ , became classical since the well-known papers of Fr€ ohlich and Platzman [1] and Zwanzig [2]. Being exactly solvable, this problem has important tutorial aspect. Numerical estimations of W_ have certain practical interest in view of calculation of thermalization time and displacement, for example, of the positron, intratrack or Auger electrons
*
Corresponding author. Fax: +7-95-123-7124. E-mail address: [email protected] (S.V. Stepanov).
[3, Chapter 5]. Present calculation analytically goes further than the previous one [4,5], because of utilizing of a very cute mathematical procedure, which holds for the particles moving in a diffusive way (just the case of subionizing eþ and e ). With a help of the expression for the electromagnetic flow S ¼ 4pc ½EH (the Pointing vector), which holds in dispersive media [6], and using the Maxwell equations, one may write down the energy conservation law for the particle moving in a dispersive medium 1 oD oB E div S ¼ þH qðrÞvE: ð1Þ 4p ot ot Here Eðr; xðtÞÞ and Dðr xðtÞÞ are the electric field and the displacement at point r, produced by the particle, placed at xðtÞ. qðrÞ describes the charge distribution of the particle and v is its velocity. The magnetic term is very small when the velocity v is
0168-583X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.03.061
236
S.V. Stepanov, V.M. Byakov / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 235–238
much less than that of light and will be further omitted. We also assume that the motion of the light particle does not perturb the structure of the medium and therefore no hydrodynamical friction arises. Integration of div S over the whole space gives zero, because variations of the energy of the field, the medium and the kinetic energy of the particle compensate each other (the particle and the medium compose an isolated system). Integral from the last term in Eq. (1) gives variation of the kinetic R energy of the particle W_ ¼ d3 r qðrÞvEðr; xðtÞÞ. Therefore, as a result of the energy conservation, we obtain Z 1 o d3 r Eðr; xðtÞÞ Dðr xðtÞÞ: ð2Þ W_ ¼ 4p ot Polarization Pðr; tÞ of the dispersive polar medium cannot adjust rapidly enough to the instantaneous position of the moving particle. It results in an electrical drag force on the particle. Such a retarded response of Pðr; tÞ may be described in terms of the memory (aftereffect) function cðtÞ: Z 1 1 Pðr; tÞ ¼ dt1 cðt1 ÞDðr xðt t1 ÞÞ; ð3Þ 4p 0 where e is the electric charge of the moving particle. Eqs. (2) and (3) were used by Zwanzig [2] for calculation of the drag force (straightforwardly in time representation). Preceding equations assume that Dðr; tÞ ¼ Eðr; tÞ þ 4pPðr; tÞ and in the Fourier transform Dðr; xÞ ¼ Eðr; xÞ þ 4pPðr; xÞ eðxÞ Eðr; xÞ. Taking Fourier transform of Eq. (3), we can relate dielectric permittivity and response function cðtÞ: Z 1 1 ; cðtÞeixt dt ¼ 1 eðxÞ 0 Z 1 2 Im eðxÞ cr ðtÞ ¼ dx sinðxtÞ : ð4Þ p 0 jeðxÞj2 The response function cðtÞ consists of two parts: cðtÞ ¼ cs dðtÞ þ cr ðtÞ. It is intuitively clear (details see in [4]) that the singular term gives no contribution to the retardation. Therefore, further we will keep only the regular part, cr ðtÞ, of the mem-
ory function. After simple transformations W_ may be expressed in the following form [4] 1: Z 1 ow ðDðtÞÞ dt _ ; W ¼ cr ðtÞ DD ot 4p 0 Z where wDD ðDÞ ¼ d3 r DðrÞDðr DÞ: ð5Þ Let us note, the changing of wDD versus time is due to variation of DðtÞ, which denotes the position of the particle only. This time dependence can be extracted from wDD and seperated in a d-function multiplier: Z t 0 0 wDD ðDðtÞÞ ¼ wDD ðDÞ d DðtÞ vðt Þ dt : ð6Þ 0
In the case of diffusion motion we will assume that jvðt0 Þj is constant, while the direction of v randomly changes during each collision. So we may simulate fvðt0 Þg by a normal (gaussian) dcorrelated random process: hvl ðtÞvm ð0ÞifvðtÞg ¼ D dlm dðtÞ, where D ¼ vltr =6 is the energy dependent diffusion coefficient of the particle and ltr is its transport path. The average over fvðt0 Þg is calculated as follows [4]: Z Z t 3 0 0 h i ¼ d D d D vðt Þ dt ¼
Z
0
fvðt0 Þg
d3 D Pdiff ðD; tÞ :
ð7Þ
Here Pdiff ðD; tÞ is the Green function of the diffusion equation oto Pdiff ¼ Dr2 Pdiff . Taking the average of W_ over a diffusion type of motion leads to the following relationship: hW_ i ¼
Z 0
owdiff DD ðtÞ ¼ ot
1
cr ðtÞ Z
owdiff DD ðtÞ dt ; ot 4p
o Pdiff ðD; tÞ wDD ðDÞ d3 D: ot
ð8Þ
1 To obtain Eq. (5) we have to rearrange integration over r and differentiation over time. In our case it is valid, because the integration range over r (whole space) does not depend on the current position of the particle. It was not the case in [5], where W_ was calculated only outside a sphere surrounding moving particle. Correct expressions were obtained by Stepanov [4].
S.V. Stepanov, V.M. Byakov / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 235–238
Now let us calculate this quantity. The first step is the replacement of oto Pdiff ðD; tÞ by Dr2 Pdiff ðD; tÞ and subsequent integrations by parts: Z ðr2 Pdiff ðD; tÞÞ wDD ðDÞ d3 D Z ¼ Pdiff ðD; tÞ ðr2 wDD ðDÞÞ d3 D: ð9Þ Thus we arrive to Z Z o diff 3 w ðtÞ ¼ D d DPdiff ðD; tÞ d3 rDðrÞr2 Dðr DÞ: ot DD ð10Þ Here we may consider that the Laplace operator, r2 , acts on r instead of coordinate D, namely 2 r2 Dðr DÞ ¼ oro 2 Dðr DÞ. a At the second step we shall write down field– field scalar product as Db ðrÞDb ðr DÞ and express electric displacement through the electrostatic potential: Db ðrÞ ¼ orob /ðrÞ. Then we obtain Z o2 d3 r Db ðrÞ 2 Db ðr DÞ ora Z o/ðrÞ o2 o/ðr DÞ ¼ d3 r orb ora2 orb Z 2 o/ðrÞ o o /ðr DÞ ¼ d3 r : orb orb ora2 The third step consists in transposition of the derivative orob from the last term to the previous one (again by means of the integration by parts) and 2 ¼ making use of the Poisson’s equation o or/ðrÞ 2 a 4pqðrÞ. Continuing the preceding relationship, we have Z o2 /ðrÞ o2 /ðr DÞ d3 r ora2 orb2 Z ¼ 16p2 d3 r qðrÞqðr DÞ )
16p2 e2 expðD2 =2a2 Þ ð2pÞ3=2 a3
:
Using this distribution function, it is easy to integrate over D in Eq. (10): o diff 16p2 e2 D wDD ðtÞ ¼ : 3=2 ot ½pð2a2 þ 4DtÞ
ð11Þ
Finally, substituting this relationship into Eq. (8) we obtain Z 1 00 Z 8e2 D e ðxÞ dx 1 sinðxtÞ dt hW_ i ¼ 2 3=2 3=2 jeðxÞj ð2pÞ ða2 þ 2DtÞ 0 0 ! r ffiffiffiffiffiffiffiffi pffiffiffiffi Z 23=2 e2 1 xe00 ðxÞ a2 x ¼ pffiffiffiffi g dx: 2 pD p D 0 jeðxÞj Here gðzÞ ¼
R1 0
ð12Þ cost ffiffiffiffi ð2 þ 4:142z þ 3:492z2 þ pdt pffiffiffiffiffiffiffiffiffiffiffi tþpz2 =2 2p
3 1
6:67z Þ is the frequently used approximation of the Fresnel integrals [7] and e00 ðxÞ is the imaginary part of the dielectric permittivity. So far we have considered the stationary motion of the particle. However its excess kinetic energy is finite, and permanently decreases. Obviously the particle cannot produce excitation with the energy higher then W . It implies that the infinite upper limit in the integral, Eq. (12), has to be replaced by a finite frequency, which satisfies the condition x ¼ h=W . To our knowledge Eq. (12) is new and represents strict analytical result for the energy loss rate of the charged particle with the gaussian form-factor and moving in a diffusion way. It is very convenient for numerical calculations, when the energy loss function, e00 =jej2 , is experimentally known (Fig. 1). Some uncertainty of this approach consists in the choose of a and the transport length ltr , entering the diffusion coefficient. The simplest and, therefore, widespread approximation is a ltr RWS (where RWS is the Wigner–Seitz radius of the molecule; 4pR3WS =3 equals to the average volume per molecule of the medium). However, even Fr€ ohlich and Platzman [1] noted that at low energies more adequate choice should be a; ltr maxðk; RWS Þ, qffiffiffiffiffiffiffi 2
Thus we succeeded to express time derivative from the field–field correlator wdiff DD ðtÞ to the density– density correlator. At the end of the last relationship we presented the result of the integration for 2 =a2 Þ gaussian charge distribution qðrÞ ¼ e expðr . p3=2 a3
237
h where k ¼ 2ph=p ¼ 2p 2mW is the de Broglie wave length of the particle. Fig. 2 demonstrates calculation of the energy loss rate hW_ i in these two cases for light and heavy water at room temperature. The last choice of a and ltr leads to more
238
S.V. Stepanov, V.M. Byakov / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 235–238
0.1
intramolecular stretch
0.2
intramolecular bend
- Im ε -1(ω)
0.3
room temperature
librations, rotations
dipole reorientation
0.4
D2O
H2O
Acknowledgements We thank Prof. F.S. Dzheparov for useful suggestions. Acknowledgment is made to the Russian Foundation of Basic Research (Grants 01-03-32786 and 01-03-32407).
D2O H2O
0 0
3000 2000 1000 wave number, cm-1
4000
Fig. 1. Imaginary part of the inverse dielectric permittivity 00 Im 1e ¼ jeje 2 in H2 O ( , } and solid line) and D2 O (triangles and
dashed line) [8–11]. For electro-magnetic radiation with higher wave numbers (when hx is more then 0.5 eV and less than 7.4 eV) liquid water is transparent and ImeðxÞ ¼ 0 [12]. Conversion hc of the units: 2p ð1 cm1 Þ ¼ h ð1:8837 1011 rad=sÞ ¼ 1:2398 104 eV.
10
8
6
e-, e+ λ = R WS
13 < - dW/dt > , 10 eV/s
µ
4
2
0 0.01
0.1
1
10
100
realistic values of thermalization displacement and time. Anyway independent estimation of a and ltr for slowing down eþ and e is an important task in view of reliable calculation of the energy loss rate.
1000
W , eV Fig. 2. Energy dependence of hdW =dti of subionizing electrons and muons versus their kinetic energy W in case of diffusion-like motion (solid lines – slowing down in light water, dashed ones – in D2 O). Upper lines for e and eþ correspond to the ansatz a ¼ RWS , the lower lines a ¼ maxðk; RWS Þ. For all reasonable energies of muons their wave length is less then RWS .
References [1] H. Fr€ ohlich, R. Platzman, Energy loss of moving electrons to dipolar relaxation, Phys. Rev. 92 (1953) 1152. [2] R. Zwanzig, Dielectric friction of moving atom, J. Chem. Phys. 38 (7) (1963) 1603. [3] S.V. Stepanov, V.M. Byakov, in: Y.C. Jean, P.E. Mallone, D.M. Schrader (Eds.), Principles and Applications of Positron and Positronium Chemistry, World Scientific Publications, Singapore, 2003, Chapter 5. [4] S.V. Stepanov, Energy losses of subexcitation charged particles in polar media, Radiat. Phys. Chem. 46 (1995) 29. [5] M. Tachiya, H. Sano, Energy loss of electron in random motion, J. Chem. Phys. 67 (11) (1978) 5111. [6] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, New York, 1960. [7] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, NBS, Applied Mathematics Series-55, 1964, Chapter 7. [8] B.A. Mikhailov, V.M. Zolotarev, in: M.F. Vuks, A.I. Sidirova (Eds.), Structure and Role of Water in Living Organizm, Leningrad State Univ, 1963, p. 43. [9] M.N. Afsar, J.B. Hasted, J. Opt. Soc. Am. 67 (1977) 902. [10] M.R. Querry, I.L. Tyier, J. Chem. Phys. 72 (1980) 2495. [11] Y. Marechal, J. Phys. II Fr. 3 (1993) 557. [12] J.M. Heller Jr., R.N. Hamm, M.W. Williams, L.R. Painter, Collective oscillation in liquid water, J. Chem. Phys. 60 (1974) 3483.
Energy loss rate and thermalization of subionizing positrons and electrons Sergey V. Stepanov *, Vsevolod M. Byakov Institute of Theoretical and Experimental Physics, Bolshaya Cheremushkinskaya 25, Moscow 117218, Russia
Abstract Energy loss rate of low-energy eþ and e (with the energy below the ionization and electronic excitation thresholds) is calculated. Energy absorption in the medium is treated in terms of the complex frequency-dependent dielectric permittivity. Present consideration is based on the classical electrodynamics, but analytically goes further than previous ones because of utilizing of a cute mathematical procedure, which holds for particles moving in a diffusive way. It is just the case of thermalization of low-energy eþ and e . Account for the gaussian form factor (spatial charge distribution) of the moving particle, leads to additional simplifications. Numerical estimations are made for light and heavy water. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Slowing down; Energy loss; Thermalization; Dielectric friction; Vibrational excitations; Stopping power
The energy of the subionizing charged particles (eþ , e ) lies below the first ionization potential and electronic excitation threshold of molecules. So during thermalization in condensed medium, energized particles lose their energy by exciting molecular vibrations and (in polar media) due to reorientation of dipole moments. The problem of calculation of the dielectric energy loss rate, W_ , became classical since the well-known papers of Fr€ ohlich and Platzman [1] and Zwanzig [2]. Being exactly solvable, this problem has important tutorial aspect. Numerical estimations of W_ have certain practical interest in view of calculation of thermalization time and displacement, for example, of the positron, intratrack or Auger electrons
*
Corresponding author. Fax: +7-95-123-7124. E-mail address: [email protected] (S.V. Stepanov).
[3, Chapter 5]. Present calculation analytically goes further than the previous one [4,5], because of utilizing of a very cute mathematical procedure, which holds for the particles moving in a diffusive way (just the case of subionizing eþ and e ). With a help of the expression for the electromagnetic flow S ¼ 4pc ½EH (the Pointing vector), which holds in dispersive media [6], and using the Maxwell equations, one may write down the energy conservation law for the particle moving in a dispersive medium 1 oD oB E div S ¼ þH qðrÞvE: ð1Þ 4p ot ot Here Eðr; xðtÞÞ and Dðr xðtÞÞ are the electric field and the displacement at point r, produced by the particle, placed at xðtÞ. qðrÞ describes the charge distribution of the particle and v is its velocity. The magnetic term is very small when the velocity v is
0168-583X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.03.061
236
S.V. Stepanov, V.M. Byakov / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 235–238
much less than that of light and will be further omitted. We also assume that the motion of the light particle does not perturb the structure of the medium and therefore no hydrodynamical friction arises. Integration of div S over the whole space gives zero, because variations of the energy of the field, the medium and the kinetic energy of the particle compensate each other (the particle and the medium compose an isolated system). Integral from the last term in Eq. (1) gives variation of the kinetic R energy of the particle W_ ¼ d3 r qðrÞvEðr; xðtÞÞ. Therefore, as a result of the energy conservation, we obtain Z 1 o d3 r Eðr; xðtÞÞ Dðr xðtÞÞ: ð2Þ W_ ¼ 4p ot Polarization Pðr; tÞ of the dispersive polar medium cannot adjust rapidly enough to the instantaneous position of the moving particle. It results in an electrical drag force on the particle. Such a retarded response of Pðr; tÞ may be described in terms of the memory (aftereffect) function cðtÞ: Z 1 1 Pðr; tÞ ¼ dt1 cðt1 ÞDðr xðt t1 ÞÞ; ð3Þ 4p 0 where e is the electric charge of the moving particle. Eqs. (2) and (3) were used by Zwanzig [2] for calculation of the drag force (straightforwardly in time representation). Preceding equations assume that Dðr; tÞ ¼ Eðr; tÞ þ 4pPðr; tÞ and in the Fourier transform Dðr; xÞ ¼ Eðr; xÞ þ 4pPðr; xÞ eðxÞ Eðr; xÞ. Taking Fourier transform of Eq. (3), we can relate dielectric permittivity and response function cðtÞ: Z 1 1 ; cðtÞeixt dt ¼ 1 eðxÞ 0 Z 1 2 Im eðxÞ cr ðtÞ ¼ dx sinðxtÞ : ð4Þ p 0 jeðxÞj2 The response function cðtÞ consists of two parts: cðtÞ ¼ cs dðtÞ þ cr ðtÞ. It is intuitively clear (details see in [4]) that the singular term gives no contribution to the retardation. Therefore, further we will keep only the regular part, cr ðtÞ, of the mem-
ory function. After simple transformations W_ may be expressed in the following form [4] 1: Z 1 ow ðDðtÞÞ dt _ ; W ¼ cr ðtÞ DD ot 4p 0 Z where wDD ðDÞ ¼ d3 r DðrÞDðr DÞ: ð5Þ Let us note, the changing of wDD versus time is due to variation of DðtÞ, which denotes the position of the particle only. This time dependence can be extracted from wDD and seperated in a d-function multiplier: Z t 0 0 wDD ðDðtÞÞ ¼ wDD ðDÞ d DðtÞ vðt Þ dt : ð6Þ 0
In the case of diffusion motion we will assume that jvðt0 Þj is constant, while the direction of v randomly changes during each collision. So we may simulate fvðt0 Þg by a normal (gaussian) dcorrelated random process: hvl ðtÞvm ð0ÞifvðtÞg ¼ D dlm dðtÞ, where D ¼ vltr =6 is the energy dependent diffusion coefficient of the particle and ltr is its transport path. The average over fvðt0 Þg is calculated as follows [4]: Z Z t 3 0 0 h i ¼ d D d D vðt Þ dt ¼
Z
0
fvðt0 Þg
d3 D Pdiff ðD; tÞ :
ð7Þ
Here Pdiff ðD; tÞ is the Green function of the diffusion equation oto Pdiff ¼ Dr2 Pdiff . Taking the average of W_ over a diffusion type of motion leads to the following relationship: hW_ i ¼
Z 0
owdiff DD ðtÞ ¼ ot
1
cr ðtÞ Z
owdiff DD ðtÞ dt ; ot 4p
o Pdiff ðD; tÞ wDD ðDÞ d3 D: ot
ð8Þ
1 To obtain Eq. (5) we have to rearrange integration over r and differentiation over time. In our case it is valid, because the integration range over r (whole space) does not depend on the current position of the particle. It was not the case in [5], where W_ was calculated only outside a sphere surrounding moving particle. Correct expressions were obtained by Stepanov [4].
S.V. Stepanov, V.M. Byakov / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 235–238
Now let us calculate this quantity. The first step is the replacement of oto Pdiff ðD; tÞ by Dr2 Pdiff ðD; tÞ and subsequent integrations by parts: Z ðr2 Pdiff ðD; tÞÞ wDD ðDÞ d3 D Z ¼ Pdiff ðD; tÞ ðr2 wDD ðDÞÞ d3 D: ð9Þ Thus we arrive to Z Z o diff 3 w ðtÞ ¼ D d DPdiff ðD; tÞ d3 rDðrÞr2 Dðr DÞ: ot DD ð10Þ Here we may consider that the Laplace operator, r2 , acts on r instead of coordinate D, namely 2 r2 Dðr DÞ ¼ oro 2 Dðr DÞ. a At the second step we shall write down field– field scalar product as Db ðrÞDb ðr DÞ and express electric displacement through the electrostatic potential: Db ðrÞ ¼ orob /ðrÞ. Then we obtain Z o2 d3 r Db ðrÞ 2 Db ðr DÞ ora Z o/ðrÞ o2 o/ðr DÞ ¼ d3 r orb ora2 orb Z 2 o/ðrÞ o o /ðr DÞ ¼ d3 r : orb orb ora2 The third step consists in transposition of the derivative orob from the last term to the previous one (again by means of the integration by parts) and 2 ¼ making use of the Poisson’s equation o or/ðrÞ 2 a 4pqðrÞ. Continuing the preceding relationship, we have Z o2 /ðrÞ o2 /ðr DÞ d3 r ora2 orb2 Z ¼ 16p2 d3 r qðrÞqðr DÞ )
16p2 e2 expðD2 =2a2 Þ ð2pÞ3=2 a3
:
Using this distribution function, it is easy to integrate over D in Eq. (10): o diff 16p2 e2 D wDD ðtÞ ¼ : 3=2 ot ½pð2a2 þ 4DtÞ
ð11Þ
Finally, substituting this relationship into Eq. (8) we obtain Z 1 00 Z 8e2 D e ðxÞ dx 1 sinðxtÞ dt hW_ i ¼ 2 3=2 3=2 jeðxÞj ð2pÞ ða2 þ 2DtÞ 0 0 ! r ffiffiffiffiffiffiffiffi pffiffiffiffi Z 23=2 e2 1 xe00 ðxÞ a2 x ¼ pffiffiffiffi g dx: 2 pD p D 0 jeðxÞj Here gðzÞ ¼
R1 0
ð12Þ cost ffiffiffiffi ð2 þ 4:142z þ 3:492z2 þ pdt pffiffiffiffiffiffiffiffiffiffiffi tþpz2 =2 2p
3 1
6:67z Þ is the frequently used approximation of the Fresnel integrals [7] and e00 ðxÞ is the imaginary part of the dielectric permittivity. So far we have considered the stationary motion of the particle. However its excess kinetic energy is finite, and permanently decreases. Obviously the particle cannot produce excitation with the energy higher then W . It implies that the infinite upper limit in the integral, Eq. (12), has to be replaced by a finite frequency, which satisfies the condition x ¼ h=W . To our knowledge Eq. (12) is new and represents strict analytical result for the energy loss rate of the charged particle with the gaussian form-factor and moving in a diffusion way. It is very convenient for numerical calculations, when the energy loss function, e00 =jej2 , is experimentally known (Fig. 1). Some uncertainty of this approach consists in the choose of a and the transport length ltr , entering the diffusion coefficient. The simplest and, therefore, widespread approximation is a ltr RWS (where RWS is the Wigner–Seitz radius of the molecule; 4pR3WS =3 equals to the average volume per molecule of the medium). However, even Fr€ ohlich and Platzman [1] noted that at low energies more adequate choice should be a; ltr maxðk; RWS Þ, qffiffiffiffiffiffiffi 2
Thus we succeeded to express time derivative from the field–field correlator wdiff DD ðtÞ to the density– density correlator. At the end of the last relationship we presented the result of the integration for 2 =a2 Þ gaussian charge distribution qðrÞ ¼ e expðr . p3=2 a3
237
h where k ¼ 2ph=p ¼ 2p 2mW is the de Broglie wave length of the particle. Fig. 2 demonstrates calculation of the energy loss rate hW_ i in these two cases for light and heavy water at room temperature. The last choice of a and ltr leads to more
238
S.V. Stepanov, V.M. Byakov / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 235–238
0.1
intramolecular stretch
0.2
intramolecular bend
- Im ε -1(ω)
0.3
room temperature
librations, rotations
dipole reorientation
0.4
D2O
H2O
Acknowledgements We thank Prof. F.S. Dzheparov for useful suggestions. Acknowledgment is made to the Russian Foundation of Basic Research (Grants 01-03-32786 and 01-03-32407).
D2O H2O
0 0
3000 2000 1000 wave number, cm-1
4000
Fig. 1. Imaginary part of the inverse dielectric permittivity 00 Im 1e ¼ jeje 2 in H2 O ( , } and solid line) and D2 O (triangles and
dashed line) [8–11]. For electro-magnetic radiation with higher wave numbers (when hx is more then 0.5 eV and less than 7.4 eV) liquid water is transparent and ImeðxÞ ¼ 0 [12]. Conversion hc of the units: 2p ð1 cm1 Þ ¼ h ð1:8837 1011 rad=sÞ ¼ 1:2398 104 eV.
10
8
6
e-, e+ λ = R WS
13 < - dW/dt > , 10 eV/s
µ
4
2
0 0.01
0.1
1
10
100
realistic values of thermalization displacement and time. Anyway independent estimation of a and ltr for slowing down eþ and e is an important task in view of reliable calculation of the energy loss rate.
1000
W , eV Fig. 2. Energy dependence of hdW =dti of subionizing electrons and muons versus their kinetic energy W in case of diffusion-like motion (solid lines – slowing down in light water, dashed ones – in D2 O). Upper lines for e and eþ correspond to the ansatz a ¼ RWS , the lower lines a ¼ maxðk; RWS Þ. For all reasonable energies of muons their wave length is less then RWS .
References [1] H. Fr€ ohlich, R. Platzman, Energy loss of moving electrons to dipolar relaxation, Phys. Rev. 92 (1953) 1152. [2] R. Zwanzig, Dielectric friction of moving atom, J. Chem. Phys. 38 (7) (1963) 1603. [3] S.V. Stepanov, V.M. Byakov, in: Y.C. Jean, P.E. Mallone, D.M. Schrader (Eds.), Principles and Applications of Positron and Positronium Chemistry, World Scientific Publications, Singapore, 2003, Chapter 5. [4] S.V. Stepanov, Energy losses of subexcitation charged particles in polar media, Radiat. Phys. Chem. 46 (1995) 29. [5] M. Tachiya, H. Sano, Energy loss of electron in random motion, J. Chem. Phys. 67 (11) (1978) 5111. [6] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, New York, 1960. [7] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, NBS, Applied Mathematics Series-55, 1964, Chapter 7. [8] B.A. Mikhailov, V.M. Zolotarev, in: M.F. Vuks, A.I. Sidirova (Eds.), Structure and Role of Water in Living Organizm, Leningrad State Univ, 1963, p. 43. [9] M.N. Afsar, J.B. Hasted, J. Opt. Soc. Am. 67 (1977) 902. [10] M.R. Querry, I.L. Tyier, J. Chem. Phys. 72 (1980) 2495. [11] Y. Marechal, J. Phys. II Fr. 3 (1993) 557. [12] J.M. Heller Jr., R.N. Hamm, M.W. Williams, L.R. Painter, Collective oscillation in liquid water, J. Chem. Phys. 60 (1974) 3483.