# Plasmon Oscillations and de Broglie’s Matter Waves Instabilities

M. Akbari-Moghanjoughi  Faculty of Sciences, Department of Physics, Azarbaijan
Shahid Madani University, 51745-406 Tabriz, Iran

###### Abstract

In this research we study the effect of matter wave instability on propagation
of electron beam with arbitrary degree of degeneracy. Particular class of
solutions of the Schrödinger-Poisson system is used to model the electron-beam
propagation. It is shown that the electron-beam propagating at a constant
average speed is described by a coupled driven pseudoforce system solution of
which leads to plasmon excitations with dual wave-particle character. The
fundamental quantum mechanical de Broglie relation is found to be due to the
resonant interaction of particle-like plasmon excitation branch with the
electron beam drift. We further obtain a generalized double lengthscale de
Broglie wave-particle relation through which various beam-plasmon instability
is studied in this model. The effects of charge screening and periodic lattice
structure on the matter wave interaction and instabilities are discussed.
Current research may further illuminate the origin of matter wave in quantum
mechanics and may lead to clear understanding of novel wave-particle
interactions.

###### pacs:

52.30.-q,71.10.Ca, 05.30.-d

##  I Introduction

Plasma oscillations play fundamental role in many physical phenomena
associated with collective interaction of charged species. Due to complex
nature of electromagnetic interactions between variety of charges in
multispecies plasmas a wide spectrum of interesting linear and nonlinear
phenomena such as different instabilities chen , wave-particle interaction
krall , solitons, double layer, shock waves drazin , etc. can exist in these
environments. Plasma theories such a kinetic and hydrodynamic models have been
developed over the past decade in order to study various interesting
collective effects stenf1 ; stenf2 ; stenf3 ; stenf4 . In plasmas the electron
fluid is almost the main ingredient of dielectric response to electromagnetic
waves and dynamic structure factor ichimaru1 ; ichimaru2 ; ichimaru3 ; sturm
leading to many interesting phenomena such as Thomson, Compton jung1 ; jung2 ;
jung3 , stimulated Raman and Brillouin son ; roz  scattering which are used as
efficient plasma diagnostic tools. Electrons are also responsible for other
important physical properties of plasmas such as charge shielding zhao ; mold0
; hong ; ydj1 ; ydj2 ; yoon ; elak , ionic bound states yu ; sahoo ; kar ;
paul1 ; paul2 ; paul3 ; paul4 ; paul5 ; soy1 ; soy2 , electrical and energy
transport phenomena, heat capacity akbheat  and many others. Dielectric
response of electron gas in solid state plasmas plays the prominent role in
optical properties of metals kit ; ash , semiconductors hu1 ; seeg ,
nanometallic structures, low dimensional system and liquid crystals lucio ;
mark ; haug ; gardner ; man1 ; maier ; yofee . However, in order to understand
many of the physical properties of plasmas one has to have detailed
information on collective aspects of electron species in these environments.
In hydrodynamic plasma theories the electron dynamic is simply coupled to
electromagnetic fields through Maxwell equations. Therefore, hydrodynamic and
magnetohydrodynamic theories constitute a cost effective and analytic means to
investigate a wide range of plasma phenomena in a straightforward manner haas1
. These theories, if properly formulated, are also believed to be able to
capture many collective aspect such as collisionless or collisional damping
previously known to be purely kinetic effects mannew .

Due to evergrowing necessity of developments in miniaturized low-dimensional,
semiconductor and nanostructured devices for futuristic electronic purposes, a
renewed effort has been devoted to discover physical properties of dense
quantum electron gas over the past few years. Although the theories of
collective quantum electron gas dielectric response has been fully developed
more than half a century ago madelung ; fermi ; hoyle ; chandra ; bohm ; bohm1
; bohm2 ; pines ; levine ; klimontovich ; lind ; stern ; fetter ; mahan , many
interesting properties of quantum electron fluid is yet to be discovered. Many
renewed and enhanced quantum kinetic and quantum hydrodynamic theories have
emerged over the recent years. Application of these theories predicts a very
insightful future for the science of quantum plasmas and have helped to
discover a broad range of new interesting collective phenomena of quantum
electron gas unknown up until the recent decade se ; sten ; ses ; brod1 ;
mark1 ; man3 ; mold2 ; sm ; hurst ; fhaas ; scripta ; kim . Application of
Schrödinger-Poisson system manfredi  and coupled pseudoforce method has shown
a dualtone wave- and particle-like dynamics of the quantum electron gas in a
unique picture akbquant ; akbdirac ; akbgap ; akbtrans ; akbint ; akbfano ;
akbdual . We strongly believe that this dual-lengthscale theory of quantum
plasmas provides a better picture for wave-particle phenomena in collective
environments such as plasmas and fluids. The quantum fluid theory may also
provide appropriate answer to some of the most fundamental quantum mechanical
questions regarding the matter waves and mysterious duality nature of
particles zm  and nonlocality effect bohmian . It has been recently shown that
the Bohmian mechanics and the quantum potential appearing in the hydrodynamic
formulations is more fundamental than the single-particle Schrödinger equation
atiq . It has been therefore concluded that the Bohm’s quantum potential makes
a basis for the Schrödinger equation instead of being a consequence of this
equation. Therefore, the proof of de Broglie’s hypothetical matter wave
relation given in current research is original in the sense that it does not
rely on the fundamental equation of the quantum mechanics.

In current research starting from quantum hydrodynamic theory of an electron
gas a one dimensional model of pseudoforce for arbitrary degenerate electron
beam is developed and the de Broglie’s hypothetical matter wave equation is
mathematically described as a pseudoresonance effect between the particle-like
branch of the plasmon excitation in the beam and the electron drift. A
generalized dual wavenumber de Broglie equation is obtained which reduces to
the original equation the classical dilute electron beam limit. Some
interesting spatial matter wave instability in the beam propagation is
introduced for the first time and discussed in terms of the average beam speed
and electron screening effect.

##  II Mathematical Model

Dynamic properties of a free electron gas can be studied through the
hydrodynamic model to a great extent. Quantum hydrodynamic model may be used
to study the dynamic properties of electron gas with arbitrary degree of
degeneracy. A closed set of conventional quantum hydrodynamic equations for
isothermal electron gas read

$\displaystyle\frac{{\partial n}}{{\partial t}}+\frac{{\partial nv}}{{\partial
x}}=0,$ (1a) $\displaystyle\frac{{\partial v}}{{\partial t}}+v\frac{{\partial
v}}{{\partial x}}=\frac{e}{m}\frac{{\partial\phi}}{{\partial
x}}-\frac{1}{m}\frac{{\partial\mu}}{{\partial
x}}+{\rm{}}\frac{{{\hbar^{2}}}}{{2{m^{2}}}}\frac{\partial}{{\partial
x}}\left({\frac{1}{{\sqrt{n}}}\frac{{{\partial^{2}}\sqrt{n}}}{{\partial{x^{2}}}}}\right),$
(1b) $\displaystyle\frac{{{\partial^{2}}\phi}}{{\partial{x^{2}}}}=4\pi
e\left({n-{n_{0}}}\right),$ (1c)

where the dependent variables $n$, $v$, $\mu$ and $\phi$ refer to the number
density, average speed, chemical potential and electrostatic potential,
respectively. The last term in the momentum equation is due to the Bohm
potential which is the origin of nonlocal effects in de Broglie-Bohm pilot
wave theory bohmian . The parametric equation of state (EoS) for isothermal
electron gas with arbitrary degree of nonrelativistic degeneracy is given in
terms of the Fermi integrals

$\displaystyle
n(\eta,T)=\frac{{{2^{7/2}}\pi{m^{3/2}}}}{{{h^{3}}}}{F_{1/2}}(\eta)=-\frac{{{2^{5/2}}{{(\pi
m{k_{B}}T)}^{3/2}}}}{{{h^{3}}}}{\rm{L}}{{\rm{i}}_{3/2}}[-\exp(\eta)],$ (2a)
$\displaystyle
P(\eta,T)=\frac{{{2^{9/2}}\pi{m^{3/2}}}}{{3{h^{3}}}}{F_{3/2}}(\eta)=-\frac{{{2^{5/2}}{{(\pi
m{k_{B}}T)}^{3/2}}({k_{B}}T)}}{{{h^{3}}}}{\rm{L}}{{\rm{i}}_{5/2}}[-\exp(\eta)],$
(2b)

where $\eta=\beta\mu$ with $\beta=1/k_{B}T$ and $F_{k}$ is the Fermi integral
of order $k$

${F_{k}}(\eta)=\int_{0}^{\infty}{\frac{{{x^{k}}}}{{\exp(x-\eta)+1}}}dx.$ (3)

The function ${\rm{Li}}_{k}$ is called the polylog function defined as

${F_{k}}(\eta)=-\Gamma(k+1){\rm{L}}{{\rm{i}}_{k+1}}[-\exp(\eta)],$ (4)

in which $\Gamma$ is the gamma function. Note that the isothermal EoS is not
the only choice and for fast processes the adiabatic EoS elak  may be
employed. However, for illustrative simplified purpose we have chosen to use
the isothermal electron EoS in current model. It is well known that the
quantum hydrodynamic model (1) can be easily transformed manfredi  to the
following Schrödinger-Poisson system using the Madelung transformations
madelung

$\displaystyle i\hbar\frac{{\partial\cal N}}{{\partial
t}}=-\frac{{{\hbar^{2}}}}{{2m}}\frac{{\partial^{2}{{\cal
N}}}}{{\partial{x^{2}}}}-e\phi{\cal N}+V_{0}{\cal N}+\mu(n,T){\cal N},$ (5a)
$\displaystyle\frac{{\partial^{2}{\phi}}}{{\partial{x^{2}}}}=4\pi e(|{\cal
N}|^{2}-n_{0}),$ (5b)

where ${\cal N}=\sqrt{n(x,t)}\exp[iS(x,t)/\hbar]$ is the state function
characterizing spatiotemporal evolution of the electron gas with ${\cal
N}{\cal N^{*}}=n(x,t)$ being the number density and $v(x,t)=(1/m)\partial
S(x,t)/\partial x$ the electron fluid speed. Also, $V_{0}$ represents a
constant ambient potential. The coupled system (5) can be used to study the
propagation of an electron beam of arbitrary degeneracy with arbitrary degree
of degeneracy in an equilibrium temperature. Let us consider a beam of
electron moving with a constant average speed, $v$ through an electron gas
with the chemical potential $\mu$. The value of chemical potential varies
depending the degeneracy degree of electron fluid. In the classical limit it
can have the negative values up to zero electronvolts for dilute electron gas
and positive few electronvolts for typical metals. Therefore, Schrödinger-
Poisson system (5) can model a wide range of phenomenon for electron gas of
arbitrary degeneracy. For our homogenous beam of electron gas moving with
constant speed $v$ one arrives at $S(x,t)=px+f(t)$ in which $p=mv$ is the
average beam momentum and $f(t)$ is an arbitrary function of time. Therefore,
the state function reduces to ${\cal N}(x,t)=\psi(x,t)\exp(ipx/\hbar)$ where
$\psi(x,t)=\sqrt{n(x,t)}\exp[if(t)/\hbar]$.

To this end, let us now consider separable solution ${\cal N}(x,t)={\cal
R}(x){\cal T}(t)$. After decoupling of spatial and temporal parts of first
equation in (5), we find

$\displaystyle\frac{{{\hbar^{2}}}}{{2m}}\frac{{{\partial^{2}}{\cal
R}(x)}}{{\partial{x^{2}}}}+e\phi(x){\cal
R}(x)+\left[{\epsilon-\mu(n,T)}\right]{\cal R}(x)={V_{0}}{\cal R}(x),$ (6a)
$\displaystyle\frac{{\partial^{2}{\phi(x)}}}{{\partial{x^{2}}}}=4\pi e(|{\cal
R}(x)|^{2}-n_{0}),$ (6b) $\displaystyle i\hbar\frac{{\partial{\cal
T}(t)}}{{\partial t}}=\epsilon{\cal T}(t),$ (6c)

where $\epsilon$ is the energy eigenvalue of the electronic system, ${\cal
R}(x)=\sqrt{n(x)}\exp(ipx/\hbar)$ and ${\cal
T}(t)=\sqrt{n(t)}\exp[if(t)/\hbar]$. The equation (6c) immediately leads to
the solution $n(t)=\exp\\{-2i[\epsilon t+f(t)]/\hbar\\}$. Here we want to find
the solution for spatial part, hence, the first two equations in (6) are our
first concern. The linear system for small perturbations is akbquant

$\displaystyle\frac{{{\partial^{2}}\Psi(x)}}{{\partial{x^{2}}}}+\Phi(x)+2E\Psi(x)={U_{0}}\exp(i{k_{d}}x),$
(7a)
$\displaystyle\frac{{{\partial^{2}}\Phi(x)}}{{\partial{x^{2}}}}-\Psi(x)=0,$
(7b)

where $\mu_{0}$ is the equilibrium constant chemical potential at $x=0$ and
$E=(\epsilon-\mu_{0})/2E_{p}$ with $E_{p}=\hbar\omega_{p}$ being the plasmon
energy and $\omega_{p}=\sqrt{4\pi e^{2}n_{0}/m}$ the electron plasma
frequency. We have also used the normalization scheme $\Psi(x)={\cal
R}(x)/{\cal R}(0)$ (${\cal R}(0)=\sqrt{n_{0}}$ is the value of ${\cal R}(x)$
at $x=0$), $\Phi(x)=e\phi/E_{p}$ and $U_{0}=V_{0}/E_{p}$. The space coordinate
$x$ is normalized to the plasmon wavelength $\lambda_{p}=2\pi/k_{p}$ with
$k_{p}=\sqrt{2mE_{p}}/\hbar$ being the characteristic plasmon wavenumber. The
parameter $k_{d}=p/\hbar k_{p}$ is the normalized de Broglie’s matter
wavenumber which is to play a central role in this research. Note that we have
further linearized the potential component leading to the important term
$V_{0}R(0)\exp(ipx/\hbar)$ which plays the role of a driving pseudoforce in
the coupled linear differential system (7).

##  III The Pseudoresonance Effect

In order to elucidate correspondence to the single particle quantum effect, we
consider the particular case of $\Phi=U_{0}=0$. Then, we have in dimensional
form

$\frac{{{\partial^{2}}\Psi(x)}}{{\partial{x^{2}}}}+{k^{2}}\Psi(x)=0,\hskip
8.53581ptk=\frac{{\sqrt{2m(\epsilon-{\mu_{0}})}}}{\hbar}.$ (8)

The solution to this equation is $\Psi(x)=\Psi_{0}\cos(kx)$ with tentative
boundary conditions $\Psi(0)=\Psi_{0}$ and $\Psi^{\prime}(0)=0$, which reduces
to the free particle wavefunction in the limit of dilute gas $\mu_{0}\to 0$
with $\epsilon$ being particle energy. However, the fundamental relation
between the particle momentum $p=mv$ and the wavenumber $k$ has been
hypothesized by de Broglie deb . The later can be more clarified if we
consider the driven pseudoforce equation

$\frac{{{\partial^{2}}\Psi(x)}}{{\partial{x^{2}}}}+{k^{2}}\Psi(x)=k_{0}^{2}\cos(px/\hbar),\hskip
8.53581ptk_{0}=\frac{\sqrt{2mV_{0}}}{\hbar},$ (9)

in which we have taken the real part of the potential energy, for illustration
purpose. The pseudoresonant solution with the boundary conditions
$\Psi(0)=\Psi^{\prime}(0)=0$ reads

$\Psi(x)=\frac{{k_{0}^{2}\left[{\cos(kx)-\cos(px/\hbar)}\right]}}{{{p^{2}}/{\hbar^{2}}-{k^{2}}}}.$
(10)

And here comes the particle-beam momentum into the picture. Note that the
solution (10) has the principal amplitude at the resonant point, $p=\pm\hbar
k$, hence, the de Broglie’s relation. In the dilute beam limit $\mu_{0}\to 0$
one obtains the single particle parabolic energy dispersion
$\epsilon=\hbar^{2}k^{2}/2m$. However, in the limit $p=\pm\hbar k$ the
solution (10) becomes problematic.

Let us now go back to the normalized system (7) with real part of potential
energy

$\displaystyle\frac{{{\partial^{2}}\Psi(x)}}{{\partial{x^{2}}}}+\Phi(x)+2E\Psi(x)=U_{0}\cos(k_{d}x),$
(11a)
$\displaystyle\frac{{{\partial^{2}}\Phi(x)}}{{\partial{x^{2}}}}-\Psi(x)=0,$
(11b)

The system (11) has the following solution for the boundary condition
$\Phi(0)=\Phi_{0}$, $\Psi(0)=\Psi_{0}$ and
$\Phi^{\prime}(0)=\Psi^{\prime}(0)=0$

$\displaystyle\Phi(x)={\Phi_{g}}+\frac{{{U_{0}}\left[{\left({{k_{d}^{2}}-k_{1}^{2}}\right)\cos\left({{k_{2}}x}\right)-\left({{k_{d}^{2}}-k_{2}^{2}}\right)\cos\left({{k_{1}}x}\right)-2\alpha\cos\left({k_{d}x}\right)}\right]}}{{2\alpha\left({{k_{d}^{2}}-k_{1}^{2}}\right)\left({{k_{d}^{2}}-k_{2}^{2}}\right)}},$
(12a)
$\displaystyle\Psi(x)={\Psi_{g}}+\frac{{{U_{0}}\left[{\left({1-{k_{d}^{2}}k_{1}^{2}}\right)\cos\left({{k_{1}}x}\right)-\left({1-{k_{d}^{2}}k_{2}^{2}}\right)\cos\left({{k_{2}}x}\right)-2\alpha{k_{d}^{2}}\cos\left({k_{d}x}\right)}\right]}}{{2\alpha\left({{k_{d}^{2}}-k_{1}^{2}}\right)\left({{k_{d}^{2}}-k_{2}^{2}}\right)}}.$
(12b)

where the general solutions $\Phi_{g}(x)$ and $\Psi_{g}(x)$ are

$\left[{\begin{array}[]{*{20}{c}}{{\Phi_{g}}(x)}\\\
{{\Psi_{g}}(x)}\end{array}}\right]=\frac{1}{{2\alpha}}\left[{\begin{array}[]{*{20}{c}}{{\Psi_{0}}+k_{2}^{2}{\Phi_{0}}}&{-\left({{\Psi_{0}}+k_{1}^{2}{\Phi_{0}}}\right)}\\\
{-\left({{\Phi_{0}}+k_{1}^{2}{\Psi_{0}}}\right)}&{{\Phi_{0}}+k_{2}^{2}{\Psi_{0}}}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{\cos({k_{1}}x)}\\\
{\cos({k_{2}}x)}\end{array}}\right],$ (13)

in which the dual characteristic wavenumbers $k_{1}$ and $k_{2}$ are given as

${k_{1}}=\sqrt{E-\alpha},\hskip 8.53581pt{k_{2}}=\sqrt{E+\alpha},\hskip
8.53581pt\alpha=\sqrt{{E^{2}}-1}.$ (14)

The characteristic wavenumbers satisfy complementarity-like relation
$k_{1}k_{2}=1$. It is interesting to see that solutions (12) are doubly
resonant at $k_{1}=k_{d}$ and $k_{2}=k_{d}$ where the wavenumber $k_{1}$ and
$k_{2}$, respectively, characterize the wave-like and particle-like behavior
of the electron beam propagation. The plasmon wavenumbers lead to a
generalized energy dispersion of $E=(1+k^{4})/2k^{2}$ where $E$ and $k$ are
scaled in terms of $E_{p}$ and $k_{p}$, respectively. This dispersion is
graphically compared to the single particle one in Fig. 1(a). The two wave and
particle branches meet each other at the quantum beating point $k=k_{p}$. The
particle branch of plasmon energy dispersion curve asymptotically approaches
that of the single particle one at very higher energies. There is also an
energy gap below the critical $E_{cr}=E_{p}$ which vanishes at the limit
$\mu_{0}\to 0$.

Figure 1: (a) Energy dipersion curve for plasmon excitations in the absence of
charge screening effect. The dashed curve indicates the standard parabolic
free electron energy dispersion. (b) The plasmon energy dispersion in the
presense of charge screening effect for different values of screening
parameter. The thickness varies according to the value of changed parameter in
plot (b).

Assuming the average energy eigenvalue for electrons in the beam as
$\epsilon=p^{2}/2m$ and using the generalized plasmon energy dispersion, we
arrive at the generalized de Broglie relation
${k_{1,2}}=\sqrt{({\gamma^{2}}-\mu)/2\mp\sqrt{{{\left({{\gamma^{2}}-\mu}\right)}^{2}}/4-1}}$
in which the wave and particle wavenumbers correspond respectively to the
minus and plus signs. Also, $\gamma=v/v_{d}$ with $v_{d}=\sqrt{2E_{p}/m}$
being the plasmon velocity and $\mu=\mu_{0}/2E_{p}$ is the normalized chemical
potential. Note that in the limit $\mu_{0}\ll 2E_{p}$ for dilute beam and
$v\gg v_{p}$ for particle branch we get the de Broglie relation for single
particle as $k_{2}/k_{p}\simeq v/v_{p}$, i.e., $k_{2}\simeq p/\hbar$, where we
have used the definition $v_{p}=\hbar k_{p}/m$. in the same limit from the
wave branch one gets $k_{1}\simeq 0$. Note that for the arbitrary value of the
beam speed one obtains a dual lengthscale de Broglie relation. These
lengscales correspond to single particle and collective excitations in the
beam propagation.

##  IV Charge Screening and Matter Wave Instabilities

Amplitudes of the solutions (12) are nonphysically divergent at resonant
points. The later aspect is shown to be due to the resonant scattering and is
better described by a generalized pseudodamped driven plasmon model in which
the damping term represents the charge screening effect akbdual

$\displaystyle\frac{{{d^{2}}\Psi(x)}}{{d{x^{2}}}}+2\xi\frac{{d\Psi(x)}}{{dx}}+\Phi(x)+2{E_{k}}\Psi(x)=U_{0}\cos(k_{d}x),$
(15a)
$\displaystyle\frac{{{d^{2}}\Phi(x)}}{{d{x^{2}}}}+2\xi\frac{{d\Phi(x)}}{{dx}}-\Psi(x)=0,$
(15b)

where $\xi=K/k_{p}$ with ${\xi^{2}}=({E_{p}}/2)\partial
n/\partial\mu=(1/2\theta){\rm{L}}{{\rm{i}}_{1/2}}\left[{-\exp({\mu}/\theta)}\right]/{\rm{L}}{{\rm{i}}_{3/2}}\left[{-\exp({\mu}/\theta)}\right]$
in which $\theta=T/T_{p}$ ($T_{p}=\hbar\omega_{p}/k_{B}$) and
$\mu=\mu_{0}/2E_{p}$ is the normalized screening parameter. The general
solution to (15) is given as follows

$\displaystyle\Phi(x)={\Phi_{gd}}(x)-\frac{{{{\rm{U}}_{0}}{{\rm{e}}^{-\xi
x}}}}{{2\alpha{\eta_{1}}{\eta_{2}}}}\left[{{\eta_{1}}\left({k_{d}^{2}-k_{2}^{2}}\right)\cos({\beta_{2}}x)-{\eta_{2}}\left({k_{d}^{2}-k_{1}^{2}}\right)\cos({\beta_{1}}x)}\right]+$
(16a) $\displaystyle\frac{{{U_{0}}{{\rm{e}}^{-\xi
x}}}}{{2\alpha{\eta_{1}}{\eta_{2}}}}\left[{\xi{\eta_{1}}\left({k_{d}^{2}+k_{2}^{2}}\right)\frac{{\sin({\beta_{2}}x)}}{{{\beta_{2}}}}-\xi{\eta_{2}}\left({k_{d}^{2}+k_{1}^{2}}\right)\frac{{\sin({\beta_{1}}x)}}{{{\beta_{1}}}}}\right]+\frac{{{U_{0}}}}{{{\eta_{1}}{\eta_{2}}}}\times$
(16b)
$\displaystyle\left\\{{\left[{\left({k_{d}^{2}-k_{1}^{2}}\right)\left({k_{d}^{2}-k_{2}^{2}}\right)-4k_{d}^{2}{\xi^{2}}}\right]\cos({k_{d}}x)+2{k_{d}}\xi\left({k_{1}^{2}+k_{2}^{2}-2k_{d}^{2}}\right)\sin({k_{d}}x)}\right\\},$
(16c) $\displaystyle\Psi(x)={\Psi_{gd}}(x)+\frac{{{U_{0}}{{\rm{e}}^{-\xi
x}}}}{{2\alpha{\eta_{1}}{\eta_{2}}}}\left[{{\eta_{1}}k_{2}^{2}\left({k_{d}^{2}-k_{2}^{2}}\right)\cos({\beta_{2}}x)-{\eta_{2}}k_{1}^{2}\left({k_{d}^{2}-k_{1}^{2}}\right)\cos({\beta_{1}}x)}\right]-$
(16d) $\displaystyle\frac{{{U_{0}}{{\rm{e}}^{-\xi
x}}}}{{2\alpha{\eta_{1}}{\eta_{2}}}}\left[{\xi{\eta_{1}}k_{2}^{2}\left({k_{d}^{2}+k_{2}^{2}}\right)\frac{{\sin({\beta_{2}}x)}}{{{\beta_{2}}}}-\xi{\eta_{2}}k_{1}^{2}\left({k_{d}^{2}+k_{1}^{2}}\right)\frac{{\sin({\beta_{1}}x)}}{{{\beta_{1}}}}}\right]-\frac{{{U_{0}}}}{{{\eta_{1}}{\eta_{2}}}}\times$
(16e)
$\displaystyle\left\\{{\left[{1-\left({k_{1}^{2}+k_{2}^{2}-k_{d}^{2}}\right)\left({k_{d}^{2}+4{\xi^{2}}}\right)}\right]k_{d}^{2}\cos({k_{d}}x)+2{k_{d}}\xi\left({1+k_{d}^{4}-4{\xi^{2}}k_{d}^{2}}\right)\sin({k_{d}}x)}\right\\},$
(16f)

where $\beta_{1}=\sqrt{k_{1}^{2}-\xi^{2}}$,
$\beta_{2}=\sqrt{k_{2}^{2}-\xi^{2}}$,
$\eta_{1}=(k_{d}^{2}-k_{1}^{2})^{2}+4k_{d}^{2}\xi^{2}$ and
$\eta_{2}=(k_{d}^{2}-k_{2}^{2})^{2}+4k_{d}^{2}\xi^{2}$. The amplitude of the
solutions (16) does not diverge but has two strong peaks at the resonant de
Broglie wavenumbers. Some parts of the solution (16) is strongly damped away
from the screening centers and the remaining stable parts may be written in
the following part

$\displaystyle\Phi_{s}(x)=\frac{{{U_{0}}\cos(k_{d}x-\theta_{\phi})}}{{{\eta_{1}}{\eta_{2}}}},\hskip
8.53581pt\theta_{\phi}=\arctan\left[{\frac{{2\xi
k_{d}\left({k_{1}^{2}+k_{2}^{2}-2{k_{d}^{2}}}\right)}}{{\left({{k_{d}^{2}}-k_{1}^{2}}\right)\left({{k_{d}^{2}}-k_{2}^{2}}\right)-4{k_{d}^{2}}{\xi^{2}}}}}\right],$
(17a)
$\displaystyle\Psi_{s}(x)=-\frac{{{U_{0}}\cos(k_{d}x-\theta_{\psi})}}{{{\eta_{1}}{\eta_{2}}}},\hskip
8.53581pt\theta_{\psi}=\arctan\left[{\frac{{2(\xi/k_{d})\left({1+{k_{d}^{4}}-4{\xi^{2}}{k_{d}^{2}}}\right)}}{{1-\left({k_{1}^{2}+k_{2}^{2}-{k_{d}^{2}}}\right)\left({{k_{d}^{2}}+4{\xi^{2}}}\right)}}}\right].$
(17b)

It is clearly remarked that the stable solution propagates with the de Broglie
wavelength $\lambda=h/p$ in space in which $h$ is the planck constant. Also,
the damped solution (16) gives rise to a new energy dispersion relation
$E=[1+(k^{2}+\xi^{2})^{2}]/2(k^{2}+\xi^{2})$. The variation of the new energy
dispersion relation is shown in Fig. 1(b) for various values of the charge
screening parameter. It is remarked that with increase in the value of
screening parameter the wave branch of the dispersion diminishes and at the
critical screening point $\xi=1$ it totaly disappears. Therefore, for a given
value of the plasmon energy this corresponds to only a particle-like de
Broglie wavenumber below a critical plasmon energy,
$E_{cr}=(1+\xi^{4})/2\xi^{2}$. Using the parabolic nonrelativistic energy
relation $\epsilon=p^{2}/2m$ for average beam particles one arrives at the
normalized de Broglie relation

${k_{1,2}}=\sqrt{\frac{{{\gamma^{2}}-\mu}}{2}-{\xi^{2}}\mp\sqrt{\frac{{{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}{4}-1}},$
(18)

where the minus/plus sign correspond to the wave/particle de Broglie
wavenumbers, respectively. The quantum de Brodlie wavenumber, $k_{d}$, has a
value larger than that of the particle branch of plasmon excitation. These two
wavenumbers however become identical as the beam speed approaches infinity.
The normalized relative de Broglie wavenumber difference $\delta
k=(k_{d}-k_{2})/k$ may be written as

$\frac{{\delta
k}}{{{k}}}=1-\sqrt{\frac{{1-\mu/{\gamma^{2}}}}{2}-\frac{{{\xi^{2}}}}{{{\gamma^{2}}}}+\sqrt{\frac{{{{\left({1-\mu/{\gamma^{2}}}\right)}^{2}}}}{4}-\frac{1}{{{\gamma^{4}}}}}}.$
(19)

The equation (18) can be written as $k_{1,2}=\chi_{1,2}(p/\hbar)$ in which the
wave-like and particle-like de Broglie coefficients $\chi_{1,2}$ are given as

${\chi_{1,2}}(\mu,\theta,E_{k})=\sqrt{\frac{1}{2}\left({1-\frac{\mu}{{{E_{K}}}}}\right)-\frac{{{\rm{L}}{{\rm{i}}_{1/2}}\left[{-\exp(\mu/\theta)}\right]}}{{2\theta
E_{k}{\rm{L}}{{\rm{i}}_{3/2}}\left[{-\exp(\mu/\theta)}\right]}}\mp\sqrt{\frac{1}{4}{{\left({1-\frac{\mu}{{{E_{K}}}}}\right)}^{2}}-\frac{1}{{E_{K}^{4}}}}},$
(20)

in which $E_{K}$ is the nonrelativistic normalized kinetic energy of the beam
to the plasmon energy. Note that for relativistic beam $E_{K}$ can not be
simply replaced with the relativistic kinetic energy due to the fact that
nonrelativistic hydrodynamic model has been used in current research. Note
also that in the classical beam limit, $\mu/\theta\ll-1$, we have ${\rm
Li_{\nu}}[-\exp(\mu/\theta)]\approx-\exp(\mu/\theta)$ and $\mu_{0}\simeq 0$.
On the other hand, in the quantum beam limit $\mu/\theta\gg 1$, one has
$\lim_{\mu/\theta\rightarrow\infty}{\rm
Li}_{\nu}[-\exp(\mu/\theta)]=-(\mu/\theta)^{\nu}/\Gamma(\nu+1)$ and
$\mu_{0}\simeq E_{F}$ in which $E_{F}$ is the Fermi energy of the electron
fluid. It is clearly evident that in the classical beam limit, $\mu_{0}\ll
E_{p}$ and $E_{K}\gg E_{p}$, we have $\chi_{1,2}=\\{0,1\\}$, as expected for
the single electron wavenumbers.

Figure 2: (a) Complex wavenumbers of plasmon excitations for plasmon
excitation in a classical dilute electron beam. (b) The beam propagation
stability region in terms of the beam electron density. (c) Effect of charge
screening on the stability of wave-like and particle-like de Broglie
oscillations in an electron beam. (d) Variations in real and imaginary parts
of de Broglie matter waves with changes in the screening parameter.

The complex form of wave-like and particle-like de Broglie wavenumbers may be
written as $k_{1}=k_{r}-ik_{i}$ and $k_{2}=k_{r}+ik_{i}$, the real and
imaginary parts for $\gamma<\sqrt{\mu}$ and $\xi<1$ are given as

$\displaystyle{k_{r}}=\sqrt[\frac{1}{4}]{{{{\left({\frac{{{\gamma^{2}}-\mu}}{2}-{\xi^{2}}}\right)}^{2}}-\left[{\frac{{{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}{4}-1}\right]}}\cos\left\\{{\frac{1}{2}{{\tan}^{-1}}\left[{\frac{{\sqrt{4-{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}}{{2{\xi^{2}}-{{\gamma^{2}}+\mu}}}}\right]}\right\\},$
(21a)
$\displaystyle{k_{i}}=\sqrt[\frac{1}{4}]{{{{\left({\frac{{{\gamma^{2}}-\mu}}{2}-{\xi^{2}}}\right)}^{2}}-\left[{\frac{{{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}{4}-1}\right]}}\sin\left\\{{\frac{1}{2}{{\tan}^{-1}}\left[{\frac{{\sqrt{4-{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}}{{2{\xi^{2}}-{{\gamma^{2}}+\mu}}}}\right]}\right\\}.$
(21b)

Note that in this regime the real and imaginary parts of the wave- and
particle-like branches have the same magnitude but the imaginary parts have
opposite signs. On the other hand for $\sqrt{\mu}<\gamma<\sqrt{\mu+2}$ and
$\xi<1$ with $k_{1}=k_{r}-ik_{i}$ and $k_{2}=k_{r}+ik_{i}$, the real/imaginary
component reads

$\displaystyle{k_{r}}=\sqrt[\frac{1}{4}]{{{{\left({\frac{{{\gamma^{2}}-\mu}}{2}-{\xi^{2}}}\right)}^{2}}-\left[{\frac{{{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}{4}-1}\right]}}\sin\left\\{{\frac{1}{2}{{\tan}^{-1}}\left[{\frac{{\sqrt{4-{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}}{{2{\xi^{2}}-{{\gamma^{2}}+\mu}}}}\right]}\right\\},$
(22a)
$\displaystyle{k_{i}}=-\sqrt[\frac{1}{4}]{{{{\left({\frac{{{\gamma^{2}}-\mu}}{2}-{\xi^{2}}}\right)}^{2}}-\left[{\frac{{{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}{4}-1}\right]}}\cos\left\\{{\frac{1}{2}{{\tan}^{-1}}\left[{\frac{{\sqrt{4-{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}}{{{{\gamma^{2}}+\mu-2{\xi^{2}}}}}}\right]}\right\\}.$
(22b)

Moreover, for $\sqrt{\mu+2}<\gamma<\sqrt{\xi^{2}+\xi^{-2}+\mu}$ and $\xi<1$
both of $k_{1}$ and $k_{2}$ are purely real with the components

$\displaystyle k_{1i}=0,\hskip
8.53581ptk_{1r}=\sqrt{\frac{{{\gamma^{2}}-\mu}}{2}-{\xi^{2}}-\sqrt{\frac{{{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}{4}-1}},$
(23a) $\displaystyle k_{2i}=0,\hskip
8.53581ptk_{2r}=\sqrt{\frac{{{\gamma^{2}}-\mu}}{2}-{\xi^{2}}+\sqrt{\frac{{{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}{4}-1}}.$
(23b)

Finally, for $\gamma>\sqrt{\xi^{2}+\xi^{-2}+\mu}$ and $\xi<1$, the particle-
like branch is purely real and the wave-like branch is purely imaginary the
components of which are given as

$\displaystyle k_{1r}=0,\hskip
8.53581ptk_{1i}=\sqrt{\sqrt{\frac{{{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}{4}-1}-\left(\frac{{{\gamma^{2}}-\mu}}{2}\right)+{\xi^{2}}},$
(24a) $\displaystyle k_{2i}=0,\hskip
8.53581ptk_{2r}=\sqrt{\frac{{{\gamma^{2}}-\mu}}{2}-{\xi^{2}}+\sqrt{\frac{{{{\left({{\gamma^{2}}-\mu}\right)}^{2}}}}{4}-1}}.$
(24b)

However, the case $\xi>1$ is not studied here. Variations of the de Broglie
wavenumbers in terms of normalized beam speed is shown in Fig. 2(a) in the
absence of screening, $\xi=0$ and for classical dilute beam $\mu_{0}\ll
E_{p}$. It is clearly remarked that below a critical velocity
$v_{cr}=\sqrt{2}v_{p}$ the wavenumbers are both complex. Above this critical
beam speed the wavenumbers are real and the particle-like de Broglie
wavenumber increases rapidly with increase in the beam speed while the wave-
like wavenumber decreases slowly. The imaginary component of the wavenumbers
below the critical point indicates spatial damping or growing of the wave
amplitude depending on its sign. It is remarked from Fig. 2(a) (22) that below
the critical beam speed amplitude of the wave-like branch grows while that of
the particle-like damps and this effect enhances as the beam speed decreases
towards zero. The real part of the two branches coincide at the
damping/growing regime and it decreases as the beam speed decreases. The
stable/unstable regimes of electron beam propagation in terms of the beam
density is depicted in Fig. 2(b). It is remarked that the stable beam speed
region rapidly increase as the electron beam number density increases. This
plot is only valid for the classical beam density since we have plotted the
figure assuming $\mu_{0}\ll E_{p}$. A better figure may be produced using the
generalized relation $v_{cr}=v_{p}\sqrt{\mu+2}$ in which $\mu=\mu_{0}/E_{p}$
is the density and temperature dependent normalized equilibrium chemical
potential.

Figure 2(c) shows de Broglie wavenumber components for given value of the
screening parameter $\xi=0.5$ for a classical dilute beam $\mu=0$. It is
remarked that the introduction of charge screening for beam electrons leads to
an upper limit instability of the wave-like branch beyond a critical beam
speed. It is found that both the beam oscillation components are always
spatially stable when the speed varies in the range
$\sqrt{\mu+2}<v/{v_{p}}<\sqrt{\mu+{\xi^{-2}}+{\xi^{2}}}$. Note that when
$\xi=1$ for th ecomplete screening case the stable range completely vanishes.
It is further revealed that the charge screening leads to increase/decrease of
the imaginary/real part of the de Broglie wavenumber in the range
$v>v_{p}\sqrt{\mu+2}$. Note that the screening parameter $\xi$ is a function
of the normalized chemical potential and electron temperature. It is to be
noted that the upper-limit instability of wave-like branch is damping type and
increases with increase in the beam speed. The variation of beam stability is
depicted in Fig. 2(d) in terms of the change in the screening parameter for a
given value of fractional beam speed $v/v_{p}=2$. It is seen that the wave-
like de Broglie branch becomes unstable at lower value of the screening
parameter compared to that of the particle-like branch. Also, the
real/imaginary component of wave-like branch are always lower/higher than that
for the particle branch in the whole range of screening parameter, $\xi$.

Figure 3: The matter wave instability in (a) Aluminium and (b) Silver metals
in the absence of the charge screening effect. Complex wavenumber of de
Broglie’s matter waves for (a) Aluminium and (b) Silver in the presence of the
charge screening effect at the fractional temperature $\theta=0.1$.

Figure 3 shows the real and imaginary parts of matter waves for an electron
beam in Aluminium and Silver metals. Aluminium has the Fermi energy of
$\mu_{0}\simeq 11.7$eV and the plasmon energy of $E_{p}\simeq 15$eV. On the
other hand, these energies for Silver are $\mu_{0}\simeq 5.49$eV and
$E_{p}\simeq 3.76$eV, respectively. As remarked by comparing the plots 3(a)
and 3(b) in the absence of charge screening there is slight difference between
them. In the Silver the unstable range extends to higher value of the
fractional parameter $v/v_{p}$ as compared to the Aluminium due to the
relatively smaller chemical potential of the Silver. Moreover, the magnitude
of imaginary/real part of the de Broglie wavenumber in Silver is slightly
higher/lower compared to that in Aluminium. It is however concluded that, a
beam of electron with a given under critical speed penetrates more distance or
has a longer penetration depth in Aluminium as compared to the Silver metal.
On the other hand, plots 3(c) and 3(d) shows the de Broglie instability in
these metal in the presence of screening which has corresponding parameters
$\xi_{Al}=0.25317$ and $\xi_{Ag}=0.36951$ at the temperature
$\theta=T/T_{p}=0.1$. It is seen that in Silver in $\theta=0.1$ the wave-like
branch of the de Broglie wave damps at much lower fractional speed $v/v_{p}$
value compared to that in Aluminium. It is however confirmed by comparing
these plots that the speed range of stability for wave-like plasmon branch in
Aluminium is greatly higher compared to the Silver metal.

Figure 4: (a) The resonant Bragg scattering by de Broglie’s matter waves for
Aluminium (thin curve) and Silver (thick curve). (b) Effect of charge
screening on the resonant Bragg scattering. The thickness of curves in this
plot indicates incarese in the value of varied screening parameter. (c) Bragg
scattering for Aluminium (thin curve) and Silver (thick curve) in the presence
of charge screening effect at fractional temperature $\theta=0.1$. (d) Effect
of fractional electron fluid temperature $\theta$ on Bragg scattering in
Aluminium.

##  V Electron Beam-Lattice Interaction

For our final purpose we would like to investigate the effect of de Broglie
pseudoresonance on Bragg scattering in a one dimensional periodic density
structure. In order to model such effect we consider the following coupled
pseudoforce system

$\displaystyle\frac{{d^{2}{\Psi(x)}}}{{d{x^{2}}}}+\Phi(x)+2E_{k}\Psi(x)=U_{0}\cos(k_{d}x),$
(25a)
$\displaystyle\frac{{d^{2}{\Phi(x)}}}{{d{x^{2}}}}-{\Psi}(x)=U_{g}\cos(Gx),$
(25b)

in which $U_{g}$ is the lattice potential as normalized to the plasmon energy
and $G=2\pi/a$ is the reciprocal lattice vector and $a$ is lattice constant as
normalized to the plasmon wavelength, $\lambda_{p}$. The general solution this
system may be written as follows

$\displaystyle\Phi(x)={\Phi_{g}}(x)+\frac{{{U_{0}}\cos(k_{d}x)}}{{\left({{k_{d}^{2}}-k_{1}^{2}}\right)\left({{k_{d}^{2}}-k_{2}^{2}}\right)}}-\frac{{{U_{g}}\left({{G^{2}}-k_{1}^{2}-k_{2}^{2}}\right)\cos(Gx)}}{{\left({{G^{2}}-k_{1}^{2}}\right)\left({{G^{2}}-k_{2}^{2}}\right)}}-$
(26a)
$\displaystyle\left[{\frac{{{U_{0}}}}{{\left({{k_{d}^{2}}-k_{1}^{2}}\right)}}+\frac{{{U_{g}}k_{2}^{2}}}{{\left({{G^{2}}-k_{1}^{2}}\right)}}}\right]\frac{{\cos({k_{1}}x)}}{{2\alpha}}-\left[{\frac{{{U_{0}}}}{{\left({{k_{d}^{2}}-k_{2}^{2}}\right)}}+\frac{{{U_{g}}k_{1}^{2}}}{{\left({{G^{2}}-k_{2}^{2}}\right)}}}\right]\frac{{\cos({k_{2}}x)}}{{2\alpha}},$
(26b)
$\displaystyle\Psi(x)={\Psi_{g}}(x)-\frac{{{U_{0}}{k_{d}^{2}}\cos(k_{d}x)}}{{\left({{k_{d}^{2}}-k_{1}^{2}}\right)\left({{k_{d}^{2}}-k_{2}^{2}}\right)}}-\frac{{{U_{g}}\cos(Gx)}}{{\left({{G^{2}}-k_{1}^{2}}\right)\left({{G^{2}}-k_{2}^{2}}\right)}}+$
(26c)
$\displaystyle\left[{\frac{{{U_{0}}k_{1}^{2}}}{{\left({{k_{d}^{2}}-k_{1}^{2}}\right)}}+\frac{{{U_{g}}}}{{\left({{G^{2}}-k_{1}^{2}}\right)}}}\right]\frac{{\cos({k_{1}}x)}}{{2\alpha}}+\left[{\frac{{{U_{0}}k_{2}^{2}}}{{\left({{k_{d}^{2}}-k_{2}^{2}}\right)}}+\frac{{{U_{g}}}}{{\left({{G^{2}}-k_{2}^{2}}\right)}}}\right]\frac{{\cos({k_{2}}x)}}{{2\alpha}}.$
(26d)

The solution constants $\Phi(0)$ and $\Psi(0)$ may be fixed using the periodic
condition of the lattice structure by the following boundary condition

$\displaystyle\Phi(0)=\Phi(a),\hskip
8.53581pt{\left.{\frac{{d\Phi(x)}}{{dx}}}\right|_{x=0}}={\left.{\frac{{d\Phi(x)}}{{dx}}}\right|_{x=a}},$
(27a) $\displaystyle\Psi(0)=\Psi(a),\hskip
8.53581pt{\left.{\frac{{d\Psi(x)}}{{dx}}}\right|_{x=0}}={\left.{\frac{{d\Psi(x)}}{{dx}}}\right|_{x=a}}.$
(27b)

The lattice wavefunction solution is

$\displaystyle\Phi_{n}(x)=\frac{{{U_{0}}\cos(k_{d}x)}}{{\left({{k_{d}^{2}}-k_{1}^{2}}\right)\left({{k_{d}^{2}}-k_{2}^{2}}\right)}}-\frac{{{U_{g}}\left({{n^{2}G^{2}}-k_{1}^{2}-k_{2}^{2}}\right)\cos(nGx)}}{{\left({{n^{2}G^{2}}-k_{1}^{2}}\right)\left({{n^{2}G^{2}}-k_{2}^{2}}\right)}},$
(28a)
$\displaystyle\Psi_{n}(x)=-\frac{{{U_{0}}{k_{d}^{2}}\cos(k_{d}x)}}{{\left({{k_{d}^{2}}-k_{1}^{2}}\right)\left({{k_{d}^{2}}-k_{2}^{2}}\right)}}-\frac{{{U_{g}}\cos(nGx)}}{{\left({{n^{2}G^{2}}-k_{1}^{2}}\right)\left({{n^{2}G^{2}}-k_{2}^{2}}\right)}},$
(28b)

where $n$ is an arbitrary integer indexing the discrete reciprocal lattice
vector, $G$. It is clearly remarked that the solutions (28) has four resonant
elements. These are of course doubly degenerate resonant conditions reduced to
the generalized Bragg condition $k_{d}=nG$ for which the de Broglie’s
wavenumbers of a beam entering the lattice are defined through the conditions
$k_{d}=k_{1}$ and $k_{d}=k_{2}$ discussed previously. It is seen that de
Broglie wavelengths of the beam resonantly interact with the lattice
periodicity through the generalized Bragg condition. The resonant beam speed
components are given through the relation
$\gamma=\sqrt{{{\left({{n^{2}G^{2}}+{\xi^{2}}}\right)}^{-1}}+\left({{n^{2}G^{2}}+{\xi^{2}}+\mu}\right)}$.
Figure 4(a) depicts the variations of the fractional beam speed
$\gamma=v/v_{p}$ in Al and Ag metals in terms of the first ($n=1$) reciprocal
lattice vector in the absence of charge screening. It is remarked that for
beam speed higher than the critical speed $v_{cr}=v_{p}\sqrt{\mu+2}$ two
quantized resonant Bragg scattering corresponding to the wave-like and
particle-like dual lengthscales of the beam exist. For beam speed below this
critical value the Bragg interaction does not take place due to the de Broglie
instabilities. It is further remarked that for very energetic beam the
particle-like branch interacts resonantly with the lattice vector at point
$v/v_{p}\simeq G$ indicated by the dashed line. In Fig. 4(b) depicts the
effect of charge screening on resonant beam-lattice interaction for different
values of the screening parameter. It is seen that there is a cut-off value
for the wave-like Bragg interaction in the presence of the charge screening.
For the critical value of $\xi=1$ the beam-lattice Bragg interaction only
becomes limited to the particle-like channel. Figure 4(c) shows the Bragg
interaction for Al and Ag metals in the presence of charge screening at
fractional temperature $\theta=0.1$. It is evident that the wave-like branch
cut-off speed is relatively higher in Aluminium as compared to Silver. Figure
4(d), on the other hand, reveals the insignificant effect of temperature on
the Bragg interaction pattern.

##  VI Conclusion

The Schrödinger-Poisson system derived from the quantum hydrodynamic model was
reduced to the appropriate coupled driven pseudoforce equation in order to
describe the propagation of an electron beam with arbitrary degree of
degeneracy. The fundamental de Broglie matter wave equation of quantum
mechanics is discovered to be due to resonant interaction of particle-like
plasmonic oscillation branch with the constant electron drift in the beam. A
generalized dual scalelength de Broglie matter wave relation is obtained
through which various wave-particle type instabilities were discussed. We also
investigated the important effects such as the charge screening and periodic
lattice density effect on these new phenomenon. Current investigation may
further elucidate the underlying nature of less understood particle waves as a
fundamental building block of the quantum theory and interaction of such waves
with real material.

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