Incollection,
Gauss law constraints on Debye-Hückel screening
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
The celebrated concept of static (time-independent) or dynamic
(time-dependent) Debye-Huckel (DH) shielding has been
extensively used by research workers in many
areas of theoretical as well as experimental physics/chemistry. The
present work is aimed at pointing out some serious conceptual
ambiguities present in the standard DH treatment of static
induced screening (caused in an otherwise neutral plasma by an
externally
introduced test charge $q_t$) and to overcome them via a new formulation proposed by us.\\
Ambiguities in standard DH treartment of induced screening\\
(i) The test particle together with the neutral plasma have net
charge $q_t$, all contained in a spherical glass beaker of radius
$R\gg\mu^-1$ and volume V. Since $\phi_D(r)$ gets rapidly damped
for $r>^-1$ hence Gauss law boundary conditions are violated
at the edge, i.e., $\phi_D(r)q_t/R$,\quad
$\phi\acute_D(R)\neq-q_t/R^2$. (ii) The total DH charge of the
system vanishes, i.e., $Q_D\equivd^3 r \rho_D(r) 0$.
(iii) The individual ionic numbers get altered i.e., $n_aD=ınt
d^3 r n_aD(r)=N_a^0- 4n_a^0 q_a q_t/(k T \mu^2)$ which is
awkward since the test charge can only polarize the medium without
creating/destroying ions. (iv)The setting $A_D = 0$, $B_D = q_t$ has
to be done by hand for ensuring the short-distance behaviour as
$r0$. (v)If $q_t$ was positive then the term $q_t
e^-r/r$ can account for the accumulation of the negative
charges around centre but the appearance of induced positive charge
at the surface
remains unexplained.\\
Our formulation of induced screening\\
We invoke the normalized Boltzmann distribution along with the weak
coupling approximation to write the perturbed ionic number density
as
$$n_a(r=N_a^0\ 1-q_a \phi(r)/k Tınt_V d^3s
łeft(1-q_a\phi(s)/k T\right)\\approx$$
$$n_a^0łeft(1-\xi(r)kT\right)\quad$$
equation
equation
$$\xi(r)\equiv\phi(r)-<\phi>\equiv\phi(r)-ınt_Vd^3s \phi(s)/V$$
where $<\phi>$ is the spatial average of the unknown potential
$\phi$. The corresponding charge density reads
equation
\rho(r)=q_t \delta(r)-\mu^2\xi(r)/4\pi
equation
and hence the poisson equation has the solution
equation
\phi(r)=łeft(A e^r+B e^-\mu
r\right)/r+<\phi>\quad;\,\,0<rR.
equation
The three unknown constants $A$, $B$, $<\phi>$ can be found
explicitly by imposing the Gauss law boundary condition on the
potential $\phi(R)= q_t/R$, on the derivative $\phi\acute(R)
=-q_t/R^2$, and normalization requirement $ınt_Vd^3r \xi(r)=0.$\\
Advantages\\
(i) The Gauss law constraints are in-built. (ii) The total charge of
the system becomes correct, i.e.,$Q\equivınt_V d^3r \rho(r)=q_t$.
(iii) The individual ionic numbers are preserved, i.e.,
$N_a\equivınt_Vd^3r n_a(r)=N_a^0$. (iv) The short-distance
behaviour $\phi(r)q_t/r$ as $r0$ is
satisfied automatically. (v) For positive $q_t$ the $Be^-r/r$
term accounts for negative charge accumulation around centre while
the $A e^r/r$ term explains the appearance of induced positive
charge at the surface. (vi) For $r\gg1$ the two approaches agree
(within an additive constant) but for $rłeq1$ they differ
noticeably as seen from figure 1.
