M. Pelc, J. Marciak-Kozlowska and M. Kozłowski- Universal Relaxation Times for Electron and Nucleon Gases Excited by Ultra-Short Laser Pulses

8/3/2019 M. Pelc, J. Marciak-Kozlowska and M. Kozowski- Universal Relaxation Times for Electron and Nucleon Gases Excited by Ultra-Short Laser Pulses

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Lasers in Eng., Vol. 18, pp. 253262 2008 Old City Publishing, Inc.

Reprints available directly from the publisher Published by license under the OCP Science imprint,

Photocopying permitted by license only a member of the Old City Publishing Group

Universal Relaxation Times for Electronand Nucleon Gases Excited by

Ultra-Short Laser Pulses

M. Pelc1, J. Marciak-Kozlowska2 and M. Kozowski3,

1Institute of Physics, Maria Curie Sklodowska University, Lublin, Poland2 Institute of Electron Technology, Warsaw, Poland

3Institute of Experimental Physics, Warsaw University, Warsaw, Poland

In this paper we calculate the universal relaxation times for electron andnucleon fermionic gases. We argue that the universal relaxation time (i) is

equal (i)

=/mv(i)2 where v(i)

=a(i)c and a(1)

=0.15 for a nucleon

gas and a(2) = 1/137 for electron gas, c = light velocity. With the uni-versal relaxation time we derive the thermal Proca equation for fermionicgases.

Keywords: Universal relaxation time, thermal universal Proca equation.

INTRODUCTION

The differential equations of thermal energy transfer should be hyperbolic

so as to exclude action at a distance; yet the equations of irreversible

thermodynamics those of NavierStokes and Fourier are parabolic.

In our book [1] the new hyperbolic non the Fourier equation for heat

transport was derived and solved.

The excitation of matter on the quark nuclear and atomic level leads to

transfer of energy. The response of matter i.e. nucleus or an atom is governed

by the relaxation time.

In this paper, we develop the general universal definition of the relax-

ation time, which depends on coupling constants for electromagnetic or strong

interaction.

Corresponding author: E-mail: [email protected]

253


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254 M. Pelc et al.

It so happens that the general equation for the relaxation time can be

written as

i

=

mi (i c)2(1)

where mi is the heat carrier mass, i = (i = e, 1/137, i = N, m /mn) iscoupling constant for electromagnetic and strong interaction, c is the speed of

light in vacuum. As the c is the maximum speed all relaxation times fulfil the

inequality

> i (2)

Consequently i is the minimum universal relaxation time.

1 QUANTUM HEAT TRANSPORT EQUATION

Dynamical processes are commonly investigated using laser pump-probe

experiments with a pump pulse exciting the system of interest and a second

probe pulse tracking is temporal evolution. As the time resolution attainable in

such experiments depends on the temporal definition of the laser pulse, pulse

compression to the attosecond domain is a recent promising development.

After the standards of time and space were defined the laws of classicalphysics relating such parameters as distance, time, velocity, temperature are

assumed to be independent of the accuracy with which these parameters can

be measured. It should be noted that this assumption does not enter explicitly

into the formulation of classical physics. It implies that together with the

assumption of the existence of an object and really independently of any

measurements (in classical physics) it was tacitly assumed that there was a

possibility of an unlimited increase in the measurement accuracy. Bearing

in mind the atomicity of time i.e. considering the smallest time period, the

Planck time, the above statement is obviously not true. Attosecond laser pulsesare at the limit of laser time resolution.

Attosecond laser pulses belong to a new Nano World, where size becomes

comparable to atomic dimensions and transport phenomena follow differ-

ent laws from that in the macro world. This first stage of miniaturization,

from 103 m to 106 m is over and the new one, from 106 m to 109 m justbeginning. The Nano World is a quantum world with all the predicable and

non-predicable (yet) features.

In this paragraph, we develop and solve the quantum relativistic heat

transport equation for nanoscale transport phenomena where external forcesexist [2]. In paragraph 2, we develop the new hyperbolic heat transport equa-

tion which generalizes the Fourier heat transport equation for rapid thermal

processes. The hyperbolic heat transport equation (HHT) for the fermionic


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Electron and Nucleon Gases by Ultra-Short Laser Pulses 255

system has been written in the form (3):

1

(

1

3

2

F)

2T

t2+ 1

(

1

3

2

F)

T

t= 2T , (3)

where T denotes the temperature, the relaxation time for the thermal

disturbance of the fermionic system, and F is the Fermi velocity.

In what follows we present the derivation of the HHT, considering the

details of the two fermionic systems: electron gas in metals and the nucleon

gas [1].

In the electron gas in metals, the Fermi energy has the form

EeF = (3 )2 n2/32

2me, (4)

where n denotes the density and me electron mass. Considering that

n1/3 aB 2

me2, (5)

and aB = Bohr radius, one obtains

EeF n2/32

2me

2

ma2 2mec2, (6)

where c = the speed of light in a vacuum and = 1/137 is the fine-structureconstant for electromagnetic interaction. The Fermi momentum pF is

peF

aB mec, (7)

And the Fermi speed is F,

eF pF

me c. (8)

Equation (8) provides the theoretical background for the result presented in

paragraph 2.1. Considering Equation (8), the equation HHT can be written as

1

c2

2T

t2 +1

c2

T

t =2

3 2

T . (9)

As is seen from (9), the HHT equation is a relativistic equation, since it

takes into account the finite speed of light.


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256 M. Pelc et al.

For a nucleon gas, the Fermi energy equals

ENF =(9 )2/32

8mr20, (10)

where m denotes the nucleon mass and r0, which describes the range of strong

interaction, is given by

r0 = m c

, (11)

where m is the pion mass. Equation (11), enables one to obtain for the nucleon

Fermi energy

ENF

mm

2 mc2. (12)

In analogy to Equation (6), Equation (12) can be written as

ENF 2s mc2, (13)where s = mm = 0.15 is the fine-structure constant for strong interactions.Analogously, we obtain the nucleon Fermi momentum

peF

r0 s mc (14)and the nucleon Fermi velocity

NF pF

m s c, (15)

and HHT for nucleon gas can be written as

1

c2

2T

t2 +1

c2

T

t =2s

3 2T . (16)

In the following, the procedure for the discretiztion of temperature T (r , t ) ina hot fermion gas will be developed. First of all, we introduce the reduced

de Broglie wavelength

eB =

meeh

, eh =1

3c,

NB

=

m

N

h

, Nh

=

1

3s c,

(17)

and the mean free paths e and N

e = ehe, N = Nh N. (18)


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In view of formulas (17) and (18), we obtain the HHC for electron and nucleon

gases

eBeh

2T

t2 +eBe

T

t =

me 2

Te

, (19)

NB

Nh

2T

t2+

NB

NT

t=

m2TN. (20)

Equations (19) and (20) are the hyperbolic partial differential equations, which

are the master equations for heat propagation in Fermi electron and nucleon

gases. In the following, we will study the quantum limit of heat transport in

fermionic systems. We define the quantum heat transport limit as follows:

e = eB , N = NB . (21)

In which case, Equations (19) and (20) have the form

e 2Te

t2+ T

e

t=

me2Te, (22)

N 2TN

t2+ T

N

t=

m2TN, (23)

where

e = me(

eh)

2, N =

m(Nh )2

. (24)

Equations (22) and (23) define the master equation for quantum heat trans-

port (QHT). Knowing the relaxation times e and N, one can define the

pulsations eh and Nh

eh = (e)1, Nh = (N)1, (25)

or

eh =me(

eh)

2

, Nh =

m(Nh )2

,

i.e.,

eh = me(eh)2 = me23 c2,

Nh = m(Nh )2 =m2s

3c2.

(26)


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equation the Proca equation for thermal processes will be developed and

solved [2].

In paper [2] the relativistic hyperbolic transport equation was developed:

1

2

2T

t2+ m0

T

t= 2T . (32)

In Equation (32) is the velocity of heat waves, m0 is the mass of heat carrier

and is the Lorentz factor, = (1 2c2

)1/2. As was shown in paper [2] theheat energy (heaton temperature) Th can be defined as follows:

Th = m0 2. (33)

Considering that , the thermal wave velocity equals [2]

= c, (34)

where is the coupling constant for the interactions, which generates the

thermal wave ( = 1/137 and = 0.15 for electromagnetic and strongforces respectively), the heaton temperature is equal to

Th = m02

c2

1 2

. (35)

Based on Equation (35) one concludes that the heaton temperature is

a linear function of the mass m0 of the heat carrier. It is interesting to

observe that the proportionality of Th and the heat carrier mass m0 was

observed for the first time in ultra high energy heavy ion reactions measured

at CERN [3]. In paper [3] it was shown that the temperature of pions, kaons

and protons produced in Pb

+Pb, S

+S reactions are proportional to the

mass of particles. Recently, at Rutherford Appleton Laboratory (RAL), theVULCAN LASER was used to produce the elementary particles: electrons and

pions [4].

When an external force is present F(x, t) the forced damped heat transport

is obtained [2] (in the one dimensional case):

1

2

2T

t2+ m0

T

t+ 2V m0

2T

2T

x2= F(x,t). (36)

The hyperbolic relativistic quantum heat transport equation, (36), describesthe forced motion of heat carriers, which undergo scattering (

m0

Tt

term)

and are influenced by the potential term (2V mo

2T).


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260 M. Pelc et al.

Equation (36) is the Proca thermal equation and can be written as [2]:

2 + 2V m02

T + m0

T

t= F(x,t),

2 = 1

2

2

t2

2

x2.

(37)

We seek the solution of Equation (37) in the form

T ( x , t ) = et/2u(x, t), (38)

where i

=

m2is the relaxation time. After substituting Equation (38) in

Equation (37) we obtain a new equation

(2 + q)u(x, t) = et/2F(x, t) (39)

and

q = 2V m2

m

2

2, (40)

m = m0 . (41)In free space i.e. when F(x, t) 0 Equation (39) reduces to

(2 + q)u(x,t) = 0, (42)

which is essentially the free Proca type equation.

The Proca equation describes the interaction of laser pulses with the matter.

As was shown in book [1] the quantization of the temperature field leads to

heatons quanta of thermal energy with a mass mh = 2h [1], where is the relaxation time and h is the finite velocity for heat propagation. For

h , i.e. for c , m0 0, it can be concluded that in a non-relativistic approximation (c = infinite) the Proca equation is the diffusionequation for mass less photons and heatons.

3 SOLUTION OF THE PROCA THERMAL EQUATION

For the initial Cauchy condition:

u(x, 0) = f (x), ut(x, 0) = g(x) (43)


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M. Pelc, J. Marciak-Kozlowska and M. Kozłowski- Universal Relaxation Times for Electron and Nucleon Gases Excited by Ultra-Short Laser Pulses