Many physical phenomena that concern the research these days are basically complicated because of being multi-parametric. Thus, their study and understanding meets with big if not unsolved obstacles. Such complicated and multi-parametric is the plasmatic state as well, where the plasma and the physical quantities that appear along with it have chaotic behavior. Many of those physical quantities change exponentially and at most times they are stabilized by presenting wavy behavior. Mostly in the transitive state rather than the steady state, the exponentially changing quantities (Growth, Damping etc) depend on each other in most cases. Thus, it is difficult to distinguish the cause from the result. The present paper attempts to help this difficult study and understanding by proposing mathematical exponential models that could relate with the study and understanding of the plasmatic wavy instability behavior. Such instabilities are already detected, understood and presented in previous publications of our laboratory. In other words, our new contribution is the study of the already known plasmatic quantities by using mathematical models (modeling and simulation). These methods are both useful and applicable in the chaotic theory. In addition, our ambition is to also conduct a list of models useful for the study of chaotic problems, such as those that appear into the plasma, starting with this paper's examples.

Among the other subjects with great interest on the plasma research during the last two decades are the plasma's nonlinear dynamics, which cause the plasma chaotic state.1 Various methods were used to calm the plasma chaotic behaviour, such as the feed-back process,2 or more easily, the intervention on the signal's phase.3–5 

In addition, it is common experience that the plasma wave growth rate or damping has almost always a complicated form,6–8 as the involved physical quantities are multi-parametric and very hard to be considered as separate. Furthermore, in many instances these involved quantities influence one another through the feed-back process.1,9,10 Such plasma waves have been observed in the early 60's11–14 and their growth has been studied as well; at many times the absence of synchronization leads to plasma turbulences.15 As the time passed their wavy properties have been studied extensively and the plasma waves have been recognized and classified as electrostatic waves,16,17 drift waves,18,19 Alfven waves,20 short-wavelength electron plasma waves,21 long-wavelength waves,16–20,22 ion-sound waves,23 e.t.c. All the above mentioned cases have been researched and their results have been carried out by considering and finding an exponential change on time of the plasma quantities (plasma density, plasma potential, ions and electrons velocity e.t.c.). If the thoughts are extended in the other areas of Physics, then we can find many examples with exponential change on the time usually and, sometimes, on the space dimension as well. For the first case the extinguishing of the oscillations by considering a resistance proportional to the vibrated mass velocity,24,25 the charge and discharge law of the capacitor from a d.c. generator through a resistance and the establishment or interruption of the d.c. current on a wrapper (the known time-circuits), the radioactive conversions Law from Nuclear Physics26 are mentioned. Afterwards, for the second case, it is enough to mention the absorption law of the radiation from an absorbent material.

The need for an approaching solution of the differential equations for every problem, which describes the change on time or space, is obvious and the proposed mathematical models have the ambition to alleviate the problem. In the first approach, the solution of such kind of problems is limited to the exponential known forms- functions, where the equalization factor is considered as “constant”, and this convenient and easy acceptance results in direct deductions. However, the stability of this kind of “constants” must be put under scrutiny as some results of this acceptance may rise doubts. This is why the equalization factor must not be considered as a constant, but as lightly changeable in different ways.

In the present work three such examples have been given,27,28 although the purpose of our team is to shape a full list of the models, which may be useful and easy for the experimental and theoretical researchers. Thus, the completion of the model list is the immediate future work, since the experimental confirmation is the difficult part of the completed research; this difficulty is caused by the little amount of time for the growth establishment of the rising wave, whereas this amount of time is very large for the nuclear decays; for this reason the measurements must be taken very methodically.

These examples that are mentioned above were selected from the previous work on the plasma waves, which has been carried out at the our Plasma Laboratory29,31 and presented as the first involvement with the topic.

The paper is organized as following: A brief description of our experimental device, the plasma production and the wave appearance are given in Sec. II. In Sec. III the weakness of the simple radioactivity problem is given in detail as a known example. Afterwards, three characteristic models are studied in Sec. IV, whereas the discussions and conclusions are made in Sec. V. Finally, in the two Appendix sections more details of the mathematical elaboration are given.

A nearly 4 m long semi-Q machine was installed in the Plasma Physics Laboratory of the NCSR ‘Demokritos’ four decades ago and many studies on the rf produced plasma have been carried out.27–31 A steady steel cylindrical cavity of 6 cm internal diameter, with its’ length adaptable to any purpose, is used almost always, as it is preferable due to its’ cylindrical symmetry simplicity. The argon-plasma is usually produced due to the argon atoms inertia and its’ low penetration. A d.c. generator supplies the Q-machine with constant current into a wide value region and with high accuracy. So, the produced magnetic field along the cylindrical cavity axis has an inclination from the constant value smaller than 4% if the Q-machine electro-magnets are placed correctly.

A low power Magnetron generator operates at constant value of the signal frequency (2.45 GHz) and supplies the plasma production with the indispensable energy into a wide region of the external magnetic field values (Table I).

Table I.

The plasma parameters and plasma quantities ranging values.

ParametersMinimum valueMaximum value
Argon pressure p 0.001 Pa 0.1 Pa 
Argon number density, ng 2 × 1015m−3 2 × 1017m−3 
Magnetic field intensity, B 10 mT 200 mT 
Microwaves’ power, P 20 Watt 120 Watt 
Frequency of the rf power (standard value) 2.45 GHz   
Electron density, n0 2 × 1015m−3 4.6 × 1015m–3 
Electron temperature, Te 1.5 eV 10 eV 
Ion temperature, Ti 0.025 eV 0.048 eV 
Ionization rate 0.1% 90% 
Electron - neutral collision frequency, νe 1.2 × 107s–1 3 × 109s–1 
ParametersMinimum valueMaximum value
Argon pressure p 0.001 Pa 0.1 Pa 
Argon number density, ng 2 × 1015m−3 2 × 1017m−3 
Magnetic field intensity, B 10 mT 200 mT 
Microwaves’ power, P 20 Watt 120 Watt 
Frequency of the rf power (standard value) 2.45 GHz   
Electron density, n0 2 × 1015m−3 4.6 × 1015m–3 
Electron temperature, Te 1.5 eV 10 eV 
Ion temperature, Ti 0.025 eV 0.048 eV 
Ionization rate 0.1% 90% 
Electron - neutral collision frequency, νe 1.2 × 107s–1 3 × 109s–1 

Electrical probes, disk probes, double probes and probe arrays, which can be moved accordingly or not, provide the possibility of measuring the plasma quantities (plasma density, plasma temperature, plasma potential, plasma wave form, e.t.c.) in every point of the plasma column. Figure 1(a) presents a drawing of the Set-Up for better understanding and Fig. 1(b) shows a photograph of a similar experimental device.

FIG. 1.

The plasma cavity with probes is presented in (a), whereas a photo of the experimental device is shown in (b).

FIG. 1.

The plasma cavity with probes is presented in (a), whereas a photo of the experimental device is shown in (b).

Close modal

By using a combination of a rotary and a diffusion pump (Balzers type) connected with the cylindrical cavity, the argon pressure can be adjusted in order for the plasma to light within a wide region of values. In a previous publication,31 a complete study of the plasma external parameters, such as gas pressure, rf wave power and magnetic field intensity, has been given. In the present paper, the external parameters and the plasma quantities are summarized in Table I.

Among the other noteworthy findings of the thus produced plasma, are its’ stability, repetition, and the persistently rising low frequency electrostatic waves, many of which have become audible through the suitable conversion. The waves may have wave-vector component along the three axis originally, but, as the steady state is established, standing waves are formed at the radial and cylinder axis direction, and the waves propagate only azimouthally.

The study of these waves has been done theoretically27,28,30 by using the fluid mechanics equations and their dispersion relation, as well as the growth rate and damping, have been found. Thus, three types of dispersion relations and their growth rate are mentioned here; the first dispersion relation is the following,

\begin{equation*}\omega _l \cong l\Omega _i + \displaystyle\frac{l}{2}(\Omega _R - \Omega _D) - j\displaystyle\frac{{\nu _i }}{2} + j\left| s \right|C_s \sqrt {\displaystyle\frac{{U_R }}{{U_D }} - 1}\end{equation*}
ωllΩi+l2(ΩRΩD)jνi2+jsCsURUD1

with the growth rate,

\begin{equation}\omega _i = \left| s \right|C_s \sqrt {\displaystyle\frac{{U_R }}{{U_D }} - 1} - \displaystyle\frac{{\nu _i }}{2}\end{equation}
ωi=sCsURUD1νi2
(1)

where, Ωi, ΩR, ΩD are the angular velocities for ions due to d.c. potential gradient, the rf field and the plasma density gradient, respectively. In addition, there is |$C_s^2 \equiv \displaystyle\frac{{K_B T_e }}{{m_i }}$|Cs2KBTemi and |$s \equiv \displaystyle\frac{1}{{n_0 }}.\displaystyle\frac{{dn_0 }}{{d\rho }}$|s1n0.dn0dρ.

Afterwards, the second one is,

\begin{equation*}\omega \cong ku_e + j\nu _e \displaystyle\frac{{\omega _{pe}^2 }}{{\omega _{pi}^2 }}.\displaystyle\frac{{k^2 (u^2 - C_s^2) - \omega _{ci}^2 }}{{\omega _{ce}^2 }}\end{equation*}
ωkue+jνeωpe2ωpi2.k2(u2Cs2)ωci2ωce2

with the growth rate,

\begin{equation}\omega _i = \nu _e \displaystyle\frac{{\omega _{pe}^2 }}{{\omega _{pi}^2 }}.\displaystyle\frac{{k^2 (u^2 - C_s^2) - \omega _{ci}^2 }}{{\omega _{ce}^2 }}\end{equation}
ωi=νeωpe2ωpi2.k2(u2Cs2)ωci2ωce2
(2)

where, νe is the collision frequency between electrons and neutrals, |$\displaystyle\frac{{\omega _{pe}^2 }}{{\omega _{pi}^2 }} = \displaystyle\frac{{m_i }}{{m_e }}$|ωpe2ωpi2=mime the electron and ion plasma frequencies, and u = |uiue| the absolute drift velocity.

The third dispersion relation is,

\begin{equation*}\omega \cong k(u_i + C_s) - j\displaystyle\frac{{\nu _i }}{2} + j\displaystyle\frac{{m_e }}{{m_i }}\displaystyle\frac{{\nu _e }}{2}\displaystyle\frac{{u_e - (u_i + C_s)}}{{C_s }}\end{equation*}
ωk(ui+Cs)jνi2+jmemiνe2ue(ui+Cs)Cs

and the growth rate is expressed as,

\begin{equation}\omega _i = \displaystyle\frac{{m_e }}{{m_i }}\displaystyle\frac{{\nu _e }}{2}\displaystyle\frac{{u_e - (u_i + C_s)}}{{C_s }} - \displaystyle\frac{{\nu _i }}{2}\end{equation}
ωi=memiνe2ue(ui+Cs)Csνi2
(3)

The first kind of waves is caused by the radial rf-field gradient,27,29 since the second and third kind are identified as electron-neutral and ion-neutral collisional waves, respectively.28 

Figure 2 shows the wave-form and the frequency spectrum of two electrical plasma waves; each spectrum contains the fundamental frequency and its’ upper harmonics, in full accordance with the dispersion relations (1) and (2). Figures 2(a) and 2(b) is the wave-form and its’ spectrum for the wave caused by the rf-field radial gradient and Figs. 2(a) and 2(b) is the wave form and its’ spectrum for the collisional wave.

FIG. 2.

The wave forms are shown in (a) and (a), whereas the wave spectra are presented in (b) and (b) for rf-drift and collisional wave respectively.

FIG. 2.

The wave forms are shown in (a) and (a), whereas the wave spectra are presented in (b) and (b) for rf-drift and collisional wave respectively.

Close modal

Although many phenomena appear on the plasma waves, most of which have been presented in the previous publications, in the present paper only the influence of the gas pressure on the wavy frequency and amplitude is mentioned; this is considered to be enough for the first fitting between an experimental given fact and a suitable model.

The indispensable measurements were taken using an electrical probe placed in the middle of the cylinder radius and the argon was lit in the following values of the external plasma parameters; magnetic field intensity B = 72 mT and microwave power P = 45 Watts. The examined wave is the collisional one, which is described by the dispersion relation (2), and its’ frequency and amplitude was taken from the spectrum on every pressure value. So, Table II is completed and the graphic is presented in Fig. 3.

Table II.

The wave Frequency and Amplitude with Pressure values B = 72 mT, P = 45 Watts.

Gas Pressure (Pa)Wave Frequency (kHz)Wave Amplitude (Arbitrary Units)
0.001 122 2.9 
0.01 102 2.6 
0.02 85 2.3 
0.03 77 2.0 
0.04 65 1.8 
0.05 58 1.6 
0.06 50 1.4 
0.07 46 1.2 
0.08 46 1.2 
0.09 46 1.2 
0.1 46 1.1 
Gas Pressure (Pa)Wave Frequency (kHz)Wave Amplitude (Arbitrary Units)
0.001 122 2.9 
0.01 102 2.6 
0.02 85 2.3 
0.03 77 2.0 
0.04 65 1.8 
0.05 58 1.6 
0.06 50 1.4 
0.07 46 1.2 
0.08 46 1.2 
0.09 46 1.2 
0.1 46 1.1 
FIG. 3.

The wave frequency, and the wave amplitude by the gas pressure increase, are presented in (a) and (b) curves, respectively.

FIG. 3.

The wave frequency, and the wave amplitude by the gas pressure increase, are presented in (a) and (b) curves, respectively.

Close modal

Many examples have been taken from other areas of Physics and not only to state the models for the exponentially changed quantities, which is the topic of the present study; the known Radioactive Conversion (Change) Law is taken from Nuclear Physics and the mortality problem is a clearly statistical subject.

An easily perceptible example is the solution of the radioactivity law problem. Although the solution of this problem is known since the early university lessons, let us repeat its’ solution here, for two basic reasons: i) to give the physical interpretation of every mathematical hypothesis or elaboration, and ii) to study the terms of this simple problem, such as the conversion rate, sub-duplication time, semi-life time e.t.c.

The problem situation:

At the time t = 0, the unbroken radioactive nucleuses are N0. How many unbroken nucleuses N will still exist after the passing of the time t?

Starting by the given fact that in the moment of the time t the remaining unbroken nucleuses are N, an infinitesimal increase of the time by dt is considered. A consequence of this is the breaking off dN from the unbroken nucleuses (the infinitesimal increase of the time causes infinitesimal decrease of the unbroken nucleuses).

The next step is the seeking of the dependence of the dN change of the unbroken nucleuses on the other physical quantities. (the whole physical interest of the issue is focused on this point of the solution proceedings). These influences are the following: i) the dN change is proportional to the time increase dt (this is caused by the infinitesimal quantities), ii) the dN change is proportional to the available quantity of the unbroken nucleuses N in that moment t. The change dN is proportional to the product of these two factors consequently and in accordance with the following relation,

\begin{equation}dN \propto N.dt\end{equation}
dNN.dt
(4)

If it is considered that there are no other changeable physical quantities that influence the dN, an analogy constant λ (for the quantities’ units equalization) must be introduced to the above relation (4). So, the following differential equation is resulted, which fits the problem,

\begin{equation}dN = - \lambda.N.dt\end{equation}
dN=λ.N.dt
(5)

The constant λ, is named “breaking off constant”, depends on the breaking nuclear material, and its’ unit is the |$\sec ^{ - 1}$|sec1. To sign (-) is simply put due to the decrease of the remained unbroken nucleuses.

Although the differential equation (5) is solved very easily, at the end of the paper Appendix  A gives more details; its’ solution is the known relation,

\begin{equation}N = N_0.e^{ - \lambda.t}\end{equation}
N=N0.eλ.t
(6)

1. Sub-duplication time

As sub-duplication time is defined the time t = t1/2 at which the remaining unbroken nucleuses are half the original ones, |$N = \displaystyle\frac{{N_0 }}{2}$|N=N02. With the replacement of the pair of the values (⁠|$t_{1/2},\displaystyle\frac{{N_0 }}{2})$|t1/2,N02) on the Eq. (6) it is found that,

\begin{equation*}\displaystyle\frac{{N_0 }}{2} = N_0.e^{ - \lambda.t_{1/2} } \quad \text{or} \quad 2 = e^{\lambda.t_{1/2} }\end{equation*}
N02=N0.eλ.t1/2or2=eλ.t1/2

and finally,

\begin{equation}t_{1/2} = \displaystyle\frac{{\ln 2}}{\lambda }\end{equation}
t1/2=ln2λ
(7)

In the same way the time of the sub-quadruplication t1/4, for which the remaining unbroken nucleuses are |$N = \displaystyle\frac{{N_0 }}{4}$|N=N04, can be found. With the same mathematical thoughts, the following is resulted,

\begin{equation}t_{1/4} = \displaystyle\frac{{\ln 4}}{\lambda } = \displaystyle\frac{{2\ln 2}}{\lambda } = 2.t_{1/2}\end{equation}
t1/4=ln4λ=2ln2λ=2.t1/2
(8)

For the sub-eight time t1/8 it is found that,

\begin{equation}t_{1/8} = \displaystyle\frac{{\ln 8}}{\lambda } = \displaystyle\frac{{3\ln 2}}{\lambda } = 3.t_{1/2}\end{equation}
t1/8=ln8λ=3ln2λ=3.t1/2
(9)

Thinking that going from |$\displaystyle\frac{{N_0 }}{4}$|N04 unbroken nucleuses to |$\displaystyle\frac{{N_0 }}{8}$|N08 is actually a sub-duplication, it is valid that,

\begin{equation}t_{1/8} - t_{1/4} = 3.t_{1/2} - 2.t_{1/2} = t_{1/2}\end{equation}
t1/8t1/4=3.t1/22.t1/2=t1/2
(10)

2. Broken nucleuses

The broken nucleuses N′ are: N′ = N0N = N0N0.e−λ.t = N0.(1 − e−λt) or

\begin{equation}N' = N_0.(1 - e^{ - \lambda.t})\end{equation}
N=N0.(1eλ.t)
(11)

The drawing of the relations N = N(t), (Eq. (6)) and N′ = N′(t) (Eq. (11)) is presented in Fig. 4.

FIG. 4.

The N = N(t) and N′ = N′(t) drawing is presented.

FIG. 4.

The N = N(t) and N′ = N′(t) drawing is presented.

Close modal

3. Conversion rate

The quotient |$\displaystyle\frac{{dN}}{{dt}}$|dNdt is defined as conversion rate. Consequently, the derivative of the relation (6) gives the conversion rate as following,

\begin{eqnarray}&& \displaystyle\frac{{dN}}{{dt}} = N_0.( - \lambda).e^{ - \lambda.t} = - \lambda.N \nonumber \\&& \text{or}\quad \displaystyle\frac{{dN}}{{dt}} = - \lambda.N\end{eqnarray}
dNdt=N0.(λ).eλ.t=λ.NordNdt=λ.N
(12)

In Fig. 5 the conversion rate versus the time is presented graphically.

FIG. 5.

The conversion rate |$\displaystyle\frac{{dN}}{{dt}}$|dNdt versus the time t is shown.

FIG. 5.

The conversion rate |$\displaystyle\frac{{dN}}{{dt}}$|dNdt versus the time t is shown.

Close modal
  1. The sub-duplication time remains constant regardless of the quantity of the unbroken radioactive nucleuses.

  2. In accordance with the radioactivity law (relation 6), when t = ∞, the remaining unbroken nucleuses are nullified.

  3. The drawings of the remaining nucleuses N = N0.e−λ.t and the already broken ones N′ = N0(1 − e−λ.t) are symmetrical to the straight line |$\psi = \displaystyle\frac{{N_0 }}{2}$|ψ=N02 (Fig. 4).

In most cases the factor λ is not constant but changeable by the time (quantities changeable by the time), sometimes in a small rate and other times in a big one. Let us consider the radioactivity conversion again: two disputes of the results found from the previous solution can be placed here: i) the stability of the sub-duplication time t1/2, regardless of the available number of the unbroken nucleuses N, and ii) the total breaking off of all the available nucleuses.

The physical perception obtained from the observation of related physical phenomena expects the sub-duplication time to decrease as the available unbroken nucleuses diminish, while the conversion proceedings have to stop leaving a small quantity of unbroken nucleuses.

Nuclear breaking off with decreased factor λ

Let us now consider that the factor λ is not constant, but it has the following influence from the time,

\begin{equation}\lambda = \lambda _0 - \mu t\end{equation}
λ=λ0μt
(13)

where μ is a constant measured in |$\sec ^{ - 2}$|sec2.

Repeating the formulation of the previous problem, where λ is considered as a constant, and, if at the moment t the remaining unbroken nucleuses are N, then, within the infinitesimal time dt, the change of the unbroken nucleuses dN is given from the following relation,

\begin{eqnarray}dN = - \lambda.N.dt \quad \text{or} \quad dN = - (\lambda _0 - \mu t).N.dt \quad \text{or} \quad \displaystyle\frac{{dN}}{N} = - (\lambda _0 - \mu t).dt\end{eqnarray}
dN=λ.N.dtordN=(λ0μt).N.dtordNN=(λ0μt).dt
(14)

The integration of the relation (14) gives the influence of time for the unbroken nucleuses evolution,

\begin{equation}N = N_0.e^{ - \lambda _0 t + \frac{\mu }{2}.t^2 }\end{equation}
N=N0.eλ0t+μ2.t2
(15)

1. The law's (15) study

A. Sub-duplication time

By putting t = t1/2 when |$N = \displaystyle\frac{{N_0 }}{2}$|N=N02, the equation |$\mu.t_{1/2}^2 - 2\lambda _0.t_{1/2} + 2\ln 2 = 0$|μ.t1/222λ0.t1/2+2ln2=0 is obtained and its’ solution gives the semi-life time,

\begin{equation}t_{1/2} = \displaystyle\frac{{\lambda _0 - \sqrt {\lambda _0^2 - 2\mu \ln 2} }}{\mu }\end{equation}
t1/2=λ0λ022μln2μ
(16)

If it is put that t = t1/4 when |$N = \displaystyle\frac{{N_0 }}{4}$|N=N04, in the same way as above the sub-quadruplication time is obtained,

\begin{equation}t_{1/4} = \displaystyle\frac{{\lambda _0 - \sqrt {\lambda _0^2 - 4\mu \ln 2} }}{\mu }\end{equation}
t1/4=λ0λ024μln2μ
(17)

From the last two relations (16) and (17) and by using the mathematical inducement method, it is easily proved that,

\begin{equation*}t_{1/4} > 2.t_{1/2}\end{equation*}
t1/4>2.t1/2
B. Broken nucleuses

The broken nucleuses N′ are calculated from the difference N′ = N0N or

\begin{equation}N' = N_0 (1 - e^{ - \lambda _0.t + \frac{\mu }{2}.t^2 })\end{equation}
N=N0(1eλ0.t+μ2.t2)
(18)

The drawing of the relations N = N(t) (Eq. (15)) and the N′ = N′(t) (Eq. (18)) is presented in Fig. 6.

FIG. 6.

The N = N(t) (relation 15) and N′ = N′(t) (relation 18) drawings are presented.

FIG. 6.

The N = N(t) (relation 15) and N′ = N′(t) (relation 18) drawings are presented.

Close modal
C. Conversion rate

The conversion rate |${\textstyle{{dN} \over {dt}}}$|dNdt is defined from the derivative of the relation (15). This derivative of the time is,

\begin{equation}\displaystyle\frac{{dN}}{{dt}} = N_0 ( - \lambda _0 + \mu.t).e^{ - \lambda _0.t + \frac{\mu }{2}.t^2 } \quad \text{or} \quad \displaystyle\frac{{dN}}{{dt}} = - (\lambda _0 - \mu.t).N\end{equation}
dNdt=N0(λ0+μ.t).eλ0.t+μ2.t2ordNdt=(λ0μ.t).N
(19)
D. The relation (15) study

The derivative of the relation (15) gives the conversion rate, which is,

\begin{equation*}\displaystyle\frac{{dN}}{{dt}} = N_0 ( - \lambda _0 + \mu.t).e^{ - \lambda _0.t + \frac{\mu }{2}.t^2 }\end{equation*}
dNdt=N0(λ0+μ.t).eλ0.t+μ2.t2

If it is put that |$\displaystyle\frac{{dN}}{{dt}} = 0$|dNdt=0, then t = λ0/μ, which is the duration time of the phenomenon, the relation (15) has an extremity value as well. The kind of the extremity value is found from the relation |$\left( {\displaystyle\frac{{d^2 N}}{{dt^2 }}} \right)_{t = {\lambda _0 / \mu }}$|d2Ndt2t=λ0/μ, and its’ value from the relation N0/μ).

For the second derivative it is concluded that,

\begin{eqnarray}&& \displaystyle\frac{{d^2 N}}{{dt^2 }} = N_0 \mu.e^{\frac{\mu }{2}t^2 - \lambda _0 t} + N_0 (\mu.t - \lambda _0)(\mu.t - \lambda _0).e^{\frac{\mu }{2}t^2 - \lambda _0 t} \,\text{or} \nonumber \\[6pt]&& \displaystyle\frac{{d^2 N}}{{dt^2 }} = N_0 \left[ {\mu + (\mu t - \lambda _0)^2 } \right].e^{\frac{\mu }{2}t^2 - \lambda _0 t}\end{eqnarray}
d2Ndt2=N0μ.eμ2t2λ0t+N0(μ.tλ0)(μ.tλ0).eμ2t2λ0tord2Ndt2=N0μ+(μtλ0)2.eμ2t2λ0t
(20)

By setting t = λ0/μ the relation (20) gives,

\begin{eqnarray}&& \displaystyle\frac{{d^2 N}}{{dt^2 }}(t = {\lambda _0 / \mu }) = N_0 \left[ {\mu + (\lambda _0 - \lambda _0)^2 } \right].e^{\frac{\mu }{2}.\frac{{\lambda _0^2 }}{{\mu ^2 }} - \lambda _0.\frac{{\lambda _0 }}{\mu }} \ \quad {\rm and,}\,{\rm finally,}\nonumber \\[6pt]&& \displaystyle\frac{{d^2 N}}{{dt^2 }}(t = {\lambda _0 / \mu }) = N_0.\mu..e^{ - \frac{{\lambda _0^2 }}{{2\mu }}} > 0\end{eqnarray}
d2Ndt2(t=λ0/μ)=N0μ+(λ0λ0)2.eμ2.λ02μ2λ0.λ0μ and , finally ,d2Ndt2(t=λ0/μ)=N0.μ..eλ022μ>0
(21)

It is resulted from the relation (21) that the remaining unbroken nucleusesN have a minimum value, which is,

\begin{equation}N(t = {\lambda _0 / \mu } ) = N_0.e^{ - \frac{{\lambda _0^2 }}{{2\mu }}}\end{equation}
N(t=λ0/μ)=N0.eλ022μ
(22)

In Fig. 7 the change by the time of the factor λ(t), the unbroken nucleusesN(t) and the conversion rate |$\displaystyle\frac{{dN}}{{dt}}$|dNdt is presented.

FIG. 7.

The factor λ(t), the unbroken nucleuses N(t) and the conversion rate |$\displaystyle\frac{{dN}}{{dt}}$|dNdt versus the time t is shown.

FIG. 7.

The factor λ(t), the unbroken nucleuses N(t) and the conversion rate |$\displaystyle\frac{{dN}}{{dt}}$|dNdt versus the time t is shown.

Close modal

2. Comments:

By considering the conversion factor λ not constant but changeable by the time, the following advantages arise from the solution of the problem:

  • The sub-duplication time t1/2 does not remain constant, but it increases as the unbroken nucleuses diminish.

  • The initially available nucleuses N0 are not broken in total, but there is a remaining quantity |$N_0.e^{ - \lambda _0^2 / 2\mu }$|N0.eλ02/2μ.

  • The solution of the problem and its’ results are general and include the results of the solution with λ = constan t, if it is set on the solution, where μ = 0.

  • The suggested change of the factor λ is linear, which results to the solution being relatively simple, although slightly more complicated from what it is considered to be λ = constan t.

  • In the problem the change factor μ appears, which is experimentally determinable.

The problem situation-solution

Now, let us consider that the constant λ is influenced by the remaining unbroken nucleuses N (and consequently, indirectly from the time t), in accordance with the relation,

\begin{equation}\lambda = \lambda _0 + \mu N\end{equation}
λ=λ0+μN
(23)

Then the differential equation is written as following:

\begin{equation*}dN = - (\lambda _0 + \mu N).N.dt \quad \text{or} \quad \displaystyle\frac{{dN}}{{(\lambda _0 + \mu N).N}} = - dt\end{equation*}
dN=(λ0+μN).N.dtordN(λ0+μN).N=dt

Integrating the last one, it is obtained that,

\begin{equation}\int {} \displaystyle\frac{{dN}}{{(\lambda _0 + \mu N).N}} = - \int {} dt + C\end{equation}
dN(λ0+μN).N=dt+C
(24)

The above relation (24) has the solution:

\begin{equation}N = \displaystyle\frac{{N_0.\Psi }}{{\Psi + \mu (1 - e^{ - \lambda _0 t})}}.e^{ - \lambda _0.t}\end{equation}
N=N0.ΨΨ+μ(1eλ0t).eλ0.t
(25)

where is,

\begin{equation*}\Psi = {\lambda _0 /{{\rm N}_0 }}\end{equation*}
Ψ=λ0/N0

1. The law's (25) study

A. Sub-duplication time

By setting into the (25)t = t1/2 when |$N = \displaystyle\frac{{N_0 }}{2}$|N=N02, the next equation is obtained,

\begin{equation*}2 = \displaystyle\frac{{\Psi + \mu (1 - e^{ - \lambda _0.t_{1/2} })}}{\Psi }.e^{\lambda _0.t_{1/2} }\end{equation*}
2=Ψ+μ(1eλ0.t1/2)Ψ.eλ0.t1/2

the solution of which gives the sub-duplication time,

\begin{equation}t_{1/2} = \displaystyle\frac{1}{{\lambda _0 }}\ln \displaystyle\frac{{2\lambda _0 + \mu N_0 }}{{\lambda _0 + \mu N_0 }}\end{equation}
t1/2=1λ0ln2λ0+μN0λ0+μN0
(26)

If it is set that t = t1/4 when |$N = \displaystyle\frac{{N_0 }}{4}$|N=N04, in the same way as above the following result is obtained again

\begin{equation}t_{1/4} = \displaystyle\frac{1}{{\lambda _0 }}\ln \displaystyle\frac{{4\lambda _0 + \mu N_0 }}{{\lambda _0 + \mu N_0 }}\end{equation}
t1/4=1λ0ln4λ0+μN0λ0+μN0
(27)

From the last two relations (26) and (27) and by using the mathematical inducement method it is easily proved that,

\begin{equation*}t_{1/4} > 2.t_{1/2}\end{equation*}
t1/4>2.t1/2
B. Broken nucleuses

The broken nucleuses N′ are found from the difference N′ = N0N or

\begin{equation}N^{\prime} = N_0 \left(1 - \displaystyle\frac{\Psi }{{\Psi + \mu (1 - e^{ - \lambda _0 t})}}e^{ - \lambda _0.t}\right)\end{equation}
N=N01ΨΨ+μ(1eλ0t)eλ0.t
(28)
C. Conversion rate

The conversion rate |${\textstyle{{dN} \over {dt}}}$|dNdt is calculated from the derivative of the relation (25). This derivative on the time is,

\begin{equation}\displaystyle\frac{{dN}}{{dt}} = - \lambda _0^2.{{{\Psi + \mu } \over {\left[ {\Psi - \mu (1 - e^{ - \lambda _0 t})} \right]^2 }}}.e^{ - \lambda _0.t}\end{equation}
dNdt=λ02.Ψ+μΨμ(1eλ0t)2.eλ0.t
(29)
D. The study of the relation (25)

The derivation on time of the relation (25) is the relation (29), which is not zero at any moment except the point t = ∞. The N(t) does not have extreme values consequently.

2. Comments:

By considering the conversion factor λ not constant but changeable by the time, the following advantages arise from the solution of the problem:

  • The sub-duplication time t1/2 does not remain constant, but it increases as the unbroken nucleuses diminish.

  • The initially available nucleuses N0 are broken in total here.

  • The solution of the problem and its’ results are general and include the results of the solution with λ = constan t, if it is set on the solution, where μ = 0.

  • The suggested change of the factor λ is not linear, which results to the solution being relatively more difficult from the previous example.

  • In the problem the change factor μ appears, which is experimentally determinable.

Another example of the exponentially changed quantities is the radiation absorption law, which is a phenomenon that occurs in the space; it is known that the radiation intensity decreases by the increase of the absorbent diaphanous material thickness; this is a good representative problem of an exponentially changed quantity (the radiation intensity) by the space dimension (here is the absorbent material depth).

The known exponential relation by its’ simple form is,

\begin{equation}I = I_0.e^{ - \mu.x}\end{equation}
I=I0.eμ.x
(30)

where I0 is the intensity of the incident radiation, I the intensity at the depth xand μ is the absorption factor considered constant.

The mathematical proof of the relation (30) is based on the same presuppositions of the relation (6) proof and the same question rises again: is the factor μ really constant or is it changeable by the absorbent material thickness?

Changeable factor

As the factor μ expresses the radiation absorption rate |${\raise0.7ex\hbox{{dI}} \!\mathord{\left/ {\vphantom {{dI} {dx}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{{dx}}}$|dI/dx, per intensity unit, it must increase by the x since the radiation intensity decreases.

Therefore, the following mathematical form for the factor μ has been proposed,

\begin{equation}\mu = \mu _0 + \nu x\end{equation}
μ=μ0+νx
(31)

where ν is constant and measured in m−2.

By using the same presupposition of the previous examples, the next differential equation is valid,

\begin{equation}dI = - \mu.I.dx\end{equation}
dI=μ.I.dx
(32)

A combination of the (31) and (32) leads to the differential equation,

\begin{equation}dI = - (\mu _0 + \nu.x).I.dx\end{equation}
dI=(μ0+ν.x).I.dx
(33)

Equation (33) is solved by using the separate variables’ method and following simple steps similar to the A Case of Sec. III and Appendix  B; then the next solution is obtained,

\begin{equation}I = I_0.e^{ - \mu _0 x - \frac{\nu }{2}.x^2 }\end{equation}
I=I0.eμ0xν2.x2
(34)

1. The relation (34) study

A. Sub-duplication thickness

By putting on the Eq. (34)I = I0/2 and x = x1/2, the following result for the sub-duplication thickness is obtained,

\begin{equation}x_{1/2} = \displaystyle\frac{{ - \mu _0 + \sqrt {\mu _0^2 + 2\ln 2\nu } }}{\nu }\end{equation}
x1/2=μ0+μ02+2ln2νν
(35)

If it is put that x = x1/4 when |$I = \displaystyle\frac{{I_0 }}{4}$|I=I04, in the same way as above the sub-quadruplication thickness is obtained,

\begin{equation}x_{1/4} = \displaystyle\frac{{ - \mu _0 + \sqrt {\mu _0^2 + 4\nu \ln 2} }}{\nu }\end{equation}
x1/4=μ0+μ02+4νln2ν
(36)

From the last two relations (35) and (36) and by using the mathematical inducement method, it is easily proved that,

\begin{equation*}x_{1/4} < 2.x_{1/2}\end{equation*}
x1/4<2.x1/2
B. Absorbent radiation

The absorbent radiation I′ is calculated from the difference I′ = I0I or

\begin{equation}I' = I_0 (1 - e^{ - \mu _0.x - \frac{\nu }{2}.x^2 })\end{equation}
I=I0(1eμ0.xν2.x2)
(37)

The drawing of the relations I = I(x) (Eq. (34)) and I′ = I′(x) (Eq. (37)) is presented in Fig. 8. It is evident that the sub-duplication thickness, when μ is changeable, is perceptibly smaller than the sub-duplication thickness, when μ is constant.

FIG. 8.

The I = I(x), (relation 34) and I′ = I′(x), (relation 37) drawings are presented.

FIG. 8.

The I = I(x), (relation 34) and I′ = I′(x), (relation 37) drawings are presented.

Close modal
C. Absorption rate

The absorption rate |${\textstyle{{dI} \over {dx}}}$|dIdx is defined from the derivative of the relation (34). This derivative on the depth x is,

\begin{equation}\displaystyle\frac{{dI}}{{dx}} = I_0 ( - \mu _0 - \nu.x).e^{ - \mu _0.x - \frac{\nu }{2}.x^2 } \quad \text{or} \quad \displaystyle\frac{{dI}}{{dx}} = - (\mu _0 + \nu.x).I\end{equation}
dIdx=I0(μ0ν.x).eμ0.xν2.x2ordIdx=(μ0+ν.x).I
(38)
D. The study of the relation (34)

If the Eqs. (30) and (34) are presented graphically, in the same drawing together as in Fig. 9, many ideas for profound study and conclusions can be carried out.

FIG. 9.

The I = I(x), from Eqs. (30) and (34) drawings are presented.

FIG. 9.

The I = I(x), from Eqs. (30) and (34) drawings are presented.

Close modal

The difference of the above two Eqs. (30) and (34) is written,

\begin{equation}I' = \Delta I = I_0.e^{ - \mu _0.x}.(1 - e^{ - \frac{\nu }{2}.x^2 })\end{equation}
I=ΔI=I0.eμ0.x.(1eν2.x2)
(39)

Because the radiation intensity difference I′ is nullified at two points, for x = 0 and x = ∞, there is a maximum value between 0 and ∞, as the two functions (30) and (34) are monotonous within the interval (0, ∞).

So, the problem is presented; which should be the depth of the absorbent material so that the difference in the absorption by the model of the Eq. (31) becomes more evident?

The answer will be given from the nihilism of the function I′ = f(x), which is the first derivative of the Eq. (34).

It is valid that,

\begin{equation}\displaystyle\frac{{dI'}}{{dx}} = I_0.e^{ - \mu _0 x}.\left[ { - \mu _0 + (\mu _0 + \nu x).e^{ - \frac{\nu }{2}.x^2 } } \right]\end{equation}
dIdx=I0.eμ0x.μ0+(μ0+νx).eν2.x2
(40)

and the condition |$\displaystyle\frac{{dI^\prime }}{{dx}} = 0$|dIdx=0 leads to |$(\mu _0 + \nu x).e^{ - \frac{\nu }{2}.x^2 } = \mu _0$|(μ0+νx).eν2.x2=μ0, or better,

\begin{equation}(\mu _0 + \nu.x) = \mu _0.e^{\frac{\nu }{2}.x^2 }\end{equation}
(μ0+ν.x)=μ0.eν2.x2
(41)

The last Eq. (41), may be solved graphically, but for now there are not available experimental values for μ0 and ν, and the graphical solution will be given in our future work.

2. Comments

By considering the sub-duplication thickness factor μ not constant but changeable by the thickness, the following advantages arise from the solution of the problem:

  • The sub-duplication thickness x1/2, does not remain constant, but it decreases as the radiation diminish.

  • The initially available radiation intensity I0 is absorbent in total.

  • The solution of the problem and its’ results are general and include the results of the solution with μ = constan t, if it is set on the solution, where ν = 0.

  • The suggested change of the factor μ is linear, which results to the solution being relatively simple, although slightly more complicated from what it is considered to be μ = constan t.

  • In the problem the change factor ν appears, which is experimentally determinable.

In Sec. II, Fig. 3 represents the plasma wave frequency and wave amplitude decrease by the gas pressure increase; with a first look these two changes have exponential form, since the scrutiny leads to two significant observations; firstly, the required change of the pressure amount for the sub-duplication is not constant, but it increases along with the pressure increase; secondly, the wave frequency and amplitude are not nullified, but remain a sufficient quantity until the plasma is put out. The above results mean that the “extinguishing factor” λ is not constant, but changeable in some way. The curves of Fig. 3 are similar enough to those of Figs. 6 and 7(b).

If the thoughts are limited on the plasma wave rising only, the following syllogism may be useful for the understanding of the growth rate or damping role.

It is known that the plasma wave appearance becomes evident by the plasma potential fluctuation, in accordance with the relation,

\begin{equation}V = V_0.e^{ - j(\omega t - {\mathop{k}\limits^{\rightharpoonup}}.{\mathop{r}\limits^{\rightharpoonup}} )}\end{equation}
V=V0.ej(ωtk.r)
(42)

When the circular frequencyωincludes an imaginary part ωi, the above wavy expression becomes,

\begin{equation}V = V_0.e^{ - j(\omega + \omega _i j)t}.e^{ + j{\mathord{\buildrel{\rightharpoonup}\over k}}.\vec r} \quad \text{or} \quad V = V_0.e^{\omega _i t}.e^{ - j(\omega t - \vec k.\vec r)}\end{equation}
V=V0.ej(ω+ωij)t.e+jk.rorV=V0.eωit.ej(ωtk.r)
(43)

where ω is now the real part ωr of the circular frequency.

It is evident from the relation (43), that the wave amplitude is described by the factor |$V = V_0.e^{\omega _i t}$|V=V0.eωit, and is an exponentially changed quantity by the time; furthermore, the sign of the ωi, defines the wave rising (growth), the wave extinguishing (damping), or the wave stability. The wave growth rate occurs by positive ωi, the wave damping by negative value, whereas, the ωi nihilism gives the wave stability. The imaginary part of the wave circular frequency has been actually calculated via the steady state hydrodynamic equations.27,28 On the other hand, the wave stability on the steady state demands ωi to be near zero, whence it must be considered that the extinguishing factors press for ωi → 0.

When a steady plasma wave appears, the final amplitude becomes constant by starting from one initial value V, and completed at V0, at the time t0 (transit).

There is always the transition time t0, which is the required time for the wave establishment. If the ωi remains variable but is positive within this time, the inequality |$e^{\varpi _i t_0 } > 1$|eϖit0>1 is valid and the wave amplitude increases at the value |$V_0.e^{\varpi _i t_0 }$|V0.eϖit0. By thinking, for example, that the ωi is varied as ωi = ωi0 − μt, the t0 is defined by the form t0 = ωi0/μ.

Figure 10 represents the wave amplitude establishment for different values of the ωi

FIG. 10.

The wave amplitude V0(t) versus the time t is presented for different values of the ωi.

FIG. 10.

The wave amplitude V0(t) versus the time t is presented for different values of the ωi.

Close modal

Although the mechanism of the wave rising is very complicated and in most cases impossible to understand, the difficulty is treated partially by following the thoughts below.

Every wave existence is caused by two antagonism factors. The first one is the cause for which the wave rises and is expressed by the growth rate. In the low frequency waves, for example, the drift waves are caused in different gradients of the plasma quantities (plasma density, plasma temperature, d.c. potential e.t.c.). The second antagonism factor involves the wave damping and expresses the different “resistances”, which may interfere with the wave transmission, as the collisions between the plasma particles (collision frequency).

The above mentioned two factors appear together into the imaginary part ωiof the wave frequency ω in the previous three examples. In the Equations (1) and (3) it is expressed with a sum,

\begin{equation}\omega _i = \left| s \right|C_s \sqrt {\displaystyle\frac{{U_R }}{{U_D }} - 1} - \displaystyle\frac{{\nu _i }}{2}\end{equation}
ωi=sCsURUD1νi2
(44)
\begin{equation}\omega _i = \displaystyle\frac{{m_e }}{{m_i }}\displaystyle\frac{{\nu _e }}{2}\displaystyle\frac{{u_e - (u_i + C_s)}}{{C_s }} - \displaystyle\frac{{\nu _i }}{2}\end{equation}
ωi=memiνe2ue(ui+Cs)Csνi2
(45)

since in the relation (2) it is formed as a product.

\begin{equation}\omega _i = \nu _e \displaystyle\frac{{\omega _{pe}^2 }}{{\omega _{pi}^2 }}.\displaystyle\frac{{k^2 (u^2 - C_s^2) - \omega _{ci}^2 }}{{\omega _{ce}^2 }}\end{equation}
ωi=νeωpe2ωpi2.k2(u2Cs2)ωci2ωce2
(46)

The balance of the two factors secures the wave stability and the inclination from the equilibrium gives the growth or the damping, respectively.

The problem rises as the calculated imaginary part of the wave frequency ωi is not constant but changeable on the time, at least during the wave establishment or extinguishing. The mutual-dependence of the plasma quantities, which are involved in the ωi, is impossible to find and express in detail, so their modeling becomes necessary.

In the present work such a modeling is set out with the ambition to be completed in the immediate future in a full list of models applicable on any actual experimental data. This approaching fitting between the model and the experimental data must be confirmed by using delay-time methods, as the wave establishment time is in most cases very limited.

With the examples, which are included in the paper and have been taken from the other areas of Physics (Nuclear Physics), the results are much more satisfactory and acceptable than those believed until now.

In the end the conclusion is that, although the experimental confirmation of the present study's usefulness is feeble now, the effort for the models’ development must continue and a list of those models must be composed. This means that the ‘Demokritos’ team has to do theoretical future work on the same topic and experimental confirmation of the mathematic models.

In any case, the experimental measurements are very difficult to be carried out; firstly, because of the very little time required for the establishment of the steady state of the plasma waves, and, secondly, due to the great amount of time required for a perceptible physical nuclear decay.

The authors wish to thank Dr. A.J.Anastassiades for his valuable help in theoretical, as well as, experimental subjects. They are also grateful to Dr. Y. Bassiakos, Dr. E. Filippaki and other members of the Plasma Laboratory of NCSR “Demokritos” for their assistance in various ways. In addition, the present work is dedicated to the memory of Professor Padma Kant Shukla, who suddenly passed away during the last year. We have known him since the early years through his publications on the modern Plasma Physics, whereas we have had a very good cooperation during the last decade.

Solution of the differential Equation (5)

The Equation (5) is the simplest form of a differential equation with two changeable quantities (N, t), which can be divided into its’ two parts. So, the following is resulted,

\begin{equation}\displaystyle\frac{{dN}}{N} = - \lambda.dt\end{equation}
dNN=λ.dt
(A1)

The relation (A1) is integrated by parts in two ways: i) by defined integrals, if the changeable quantities’ limits are known, or ii) by indefinite integrals, adding the integration constant C. If the second method is prefered, the following is resulted,

\begin{eqnarray}& \displaystyle \int \displaystyle\frac{{dN}}{N} = - \lambda.\int {dt} + C & \nonumber\\& \text{or}\qquad \ln N = - \lambda.t + C &\end{eqnarray}
dNN=λ.dt+CorlnN=λ.t+C
(A2)

For the finding of the integration constant C, one pair of values of the changeable quantities N and t is enough to be known. One known pair of values in this problem is the original conditions, where, for t = 0, it is N = N0. The replacement of the quantities t and N on the Equation (A2) with the above known values, gives the value of the constant as,

\begin{equation}C = \ln N_0\end{equation}
C=lnN0
(A3)

By the substitution on the relation (A2), the following relation is resulted,

\begin{eqnarray}&& \ln N = - \lambda.t + \ln N_0 \quad \text{or} \nonumber\\&& \ln \displaystyle\frac{N}{{N_0 }} = - \lambda.t\end{eqnarray}
lnN=λ.t+lnN0orlnNN0=λ.t
(A4)

And, finally, the known law of the radioactivity is obtained,

\begin{equation}N = N_0.e^{ - \lambda.t}\end{equation}
N=N0.eλ.t
(A5)

Solution of the differential Euation (24).

By dividing the integral function of the first part of the (24) into smaller additives, two factors α and β are seeked for the following equality to be valid,

\begin{equation}\displaystyle\frac{1}{{(\lambda _0 + \mu.N).N}} = \displaystyle\frac{\alpha }{N} + \displaystyle\frac{\beta }{{\lambda _0 + \mu.N}}\end{equation}
1(λ0+μ.N).N=αN+βλ0+μ.N
(B1)

Finally, the two factors have the values, α = 1/λ0 and β = −μ/λ0, and the last relation is written,

\begin{equation}\displaystyle\frac{1}{{(\lambda _0 + \mu.N).N}} = \displaystyle\frac{1}{{\lambda _0 N}} - \displaystyle\frac{\mu }{{\lambda _0 (\lambda _0 + \mu.N)}}\end{equation}
1(λ0+μ.N).N=1λ0Nμλ0(λ0+μ.N)
(B2)

With the substitution of the relation (B2) into the (B1) one, it is obtained that,

\begin{eqnarray}&& \int {\displaystyle\frac{{dN}}{{(\lambda _0 + \mu.N).N}}} = \displaystyle\frac{1}{{\lambda _0 }}.\int {\displaystyle\frac{{dN}}{N}} - \displaystyle\frac{\mu }{{\lambda _0 }}.\int {\displaystyle\frac{{dN}}{{\lambda _0 + \mu.N}}} = - \int {dt} + C \nonumber\\&& \text{or}\qquad \ln N - \ln (\lambda _0 + \mu.N) = - \lambda _0 t + C'\end{eqnarray}
dN(λ0+μ.N).N=1λ0.dNNμλ0.dNλ0+μ.N=dt+CorlnNln(λ0+μ.N)=λ0t+C
(B3)

The initial condition (t = 0, N = N0) determines the integration constant C, which takes the value, C′ = ln N0 − ln (λ0 + μ.N0)

With substitution into the relation (B3) and by using suitable mathematical elaboration the following is obtained,

\begin{equation}N = \displaystyle\frac{{N_0.\Psi }}{{\Psi + \mu (1 - e^{ - \lambda _0 t})}}.e^{ - \lambda _0.t}\end{equation}
N=N0.ΨΨ+μ(1eλ0t).eλ0.t
(B4)

where is,

\begin{equation*}\Psi = {\lambda _0 /{{\rm N}_0 }}\end{equation*}
Ψ=λ0/N0
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