Fluctuations of the partial filling factors in competitive RSA from binary mixtures
Arsen V. Subashiev and Serge Luryi
Department of Electrical and Computer Engineering,
State University of New York at Stony Brook, Stony Brook, NY, 11794-2350
Competitive random sequential adsorption on a line from a binary mix of incident particles is
studied using both an analytic recursive approach and Monte Carlo simulations. We find a strong
correlation between the small and the large particle distributions so that while both partial con-
tributions to the fill factor fluctuate widely, the variance of the total fill factor remains relatively
small. The variances of partial contributions themselves are quite different between the smaller and
the larger particles, with the larger particle distribution being more correlated. The disparity in
fluctuations of partial fill factors increases with the particle size ratio. The additional variance in the
partial contribution of smaller particle originates from the fluctuations in the size of gaps between
larger particles. We discuss the implications of our results to semiconductor high-energy gamma
detectors where the detector energy resolution is controlled by correlations in the cascade energy
branching process.
PACS numbers: 02.50.Ey, 05.20.-y, 68.43.-h, 07.85.Nc
I. INTRODUCTION
One dimensional irreversible random sequential ad-
sorption (RSA) has been of interest for several decades.
Its numerous extensions include RSA with particles ex-
panding in the adsorption process [1, 2, 3], two-size par-
ticle adsorption [4, 5, 6, 7], and also RSA with an ar-
bitrary particle-size distribution function [8]. The inter-
est is due to the relevance of this process to a number
of physical phenomena in different fields of application,
such as information processing [9], particle branching in
impact ionization [10] and crack formations in crystals
under external stress [11]. The simplest example of RSA
is the so-called car parking problem (CPP). In the con-
text of CPP, one studies the average number of particles
(“cars”) adsorbed on a long line and the variance of this
number. Equivalently, one is concerned with the distri-
bution function for the size of gaps between the parked
cars (see Refs. [12, 13] for the review).
The problem of competitive RSA from a binary mix-
ture is of special interest because of the non-trivial corre-
∗subashiev@ece.sunysb.edu
lations in both the particle and gap-size distributions, de-
veloped during the deposition. These correlations mani-
fest themselves in the final irreversible state correspond-
ing to the so-called “jamming limit” — when every gap
capable of adsorbing a particle has done so. Numerous
studies, reported in the literature for the binary-mixture
RSA in the jamming limit, addressed the problem of cor-
relations only indirectly, through its manifestation in the
fill factor or the gap distribution. Available results in-
clude binary mixtures with point-like particles [4, 5] and
those with a relatively small particle size ratio, b/a < 2
[6]. Also available are Monte-Carlo studies of the fill
factor and the gap-size distribution for a binary-mixture
deposition with equal abundance of both particles [7].
The present study is concerned with the correlation
between the fluctuations in the number of adsorbed par-
ticles of each kind from a two-size binary mixture, as
well as with their partial contributions to the fill factor.
We present both analytical results and those obtained
by Monte-Carlo simulations for a wide range of binary-
mixture compositions and size ratios.
We are interested in the RSA problem primarily be-
cause of its relevance to the propagation of high-energy γ-
http://arxiv.org/abs/0704.1235v3
mailto:subashiev@ece.sunysb.edu
particles through a semiconductor crystal — with parti-
cle energy branching (PEB) due to cascade multiplication
of secondary electrons and holes [10, 14, 15, 16, 17]. The
correlation of energy distribution between secondaries is
quite similar to that of the gap distribution in the RSA
process [18]. In both cases, the ratio of the variance of
the final number of particles to the average particle num-
ber in the final (jamming) state can be much less than
unity, which is favorable for the detector energy resolu-
tion. This ratio (which would be unity if the particle
number obeyed a Poisson distribution) is called the Fano
factor, Φ [19].
The reported attempts to evaluate Φ employed over-
simplified models of the semiconductor band structure.
In such models, all crystal properties are characterized
by three parameters, namely, the band gap, the phonon
frequency, and the ratio of the rate of phonon emission to
that of impact ionization. The price of this oversimpli-
fication had been that correspondence with experiment
could be achieved only by assuming unphysically large
rates of phonon losses (about 0.5 eV per created e-h pair).
This does not corroborate with the known values for the
ratio of the impact ionization and the phonon emission
probabilities for high-energy electrons in semiconductors.
The model furthermore obscures the role of features in
the band structure and the ionization process that are
specific to a particular semiconductor.
In our earlier work [3], we used an extended RSA model
of particles that expand or shrink upon adsorption. The
shrinking model is relevant to the PEB problem in that it
helps to elucidate such factors as the non-constant den-
sity of states in the semiconductor band and the fact that
due to momentum conservation the ionization threshold
is larger than the actual (bandgap) energy that is lost in
impact ionization.
The recursive technique employed in Ref. [3] allowed
us to assess the accuracy of approximate approaches to
the yield and variance calculations (such as, e.g., the
average-loss approach of Refs. [15, 16]).
In the present work, the RSA model is extended in a
different direction — competitive deposition of different-
size particles from a binary mixture — that is suitable
to simulate the role of multiple channels of pair produc-
tion, owing to the multi-valley nature of semiconductor
bands. We arrive at a number of qualitative conclusions
that should be taken into account in both the interpre-
tation of experimental data and the choice of the crystal
composition and device structure in gamma detectors op-
timized for energy resolution.
The paper is organized as follows. Section II presents
the basic equations of the recursive approach and the an-
alytical results for the fill factor and its variance for the
larger particles. In Sect. III, we analyze the results that
demonstrate high correlation in the particle distribution.
Based on the gained understanding, we formulate in Sec.
IV the implications of our results for the Fano factor of
semiconductor γ detectors. Our conclusions are summa-
rized in Sect. V. Certain analytical results are derived in
the Appendix.
II. PARTIAL CONTRIBUTIONS TO THE FILL
FACTOR AND ITS VARIANCE FOR TWO-SIZE
RSA PROBLEM
We consider the problem of competitive deposition
from a binary mixture of particles with sizes a and b,
whose relative contributions to the total flux on the ad-
sorbing line are q and p = 1 − q, respectively. We shall
use a recursive approach to first study the mean number
of particles na(x) and nb(x), adsorbed on a line of length
x (in the jamming limit), and then the corresponding
variances.
Consider a large enough empty length x > a, b. We
assume that the adsorption is sequential, i.e. only one
particle is adsorbed at a time. The first adsorbed par-
ticle will be of size a with the probability of landing at
any point q(x − a)/(x − l) or of size b with the landing
probability p(x − b)/(x − l). Here l = qa + pb is the
“average” particle size in the binary flux. After the first
particle is adsorbed, it fills a certain interval [y, y+a] (or
[y, y + b]), and leaves two independent segments, whose
combined size is either x− a or x− b. The average num-
bers of a-particles na(y) and na(x− y− a) (or na(y) and
na(x−y−b)) will be subsequently adsorbed in these gaps.
Thus, the recursion relation is of the form
na(x) =
q(x− a)
[1 + na(y) + na(x− a− y)]
p(x− b)
[na(y) + na(x− b− y)] ,
where the first and the second terms (upper and lower
lines) correspond to the cases of the first landed particle
being a particle of sort a or b, respectively. These cases
must be averaged over all possible landing coordinates y
of the first particle in a different way, viz. for a first,
< na(y) >a=
∫ x−a
na(y)dy ,
whereas for b first,
< na(y) >b=
∫ x−b
na(y)dy .
Performing the average and using the symmetry between
left and right segments we obtain, finally:
na(x) =
q(x − a)
∫ x−a
na(y)dy
∫ x−b
na(y)dy. (1)
A similar equation holds for the particles of size b:
nb(x) =
p(x− b)
∫ x−a
nb(y)dy
∫ x−b
nb(y)dy. (2)
With the help of Eqs. (1,2) one can readily derive an
equation for the average total covered length f(x), de-
fined as f(x) = ana(x) + bnb(x), giving
f(x) =
xl − qa2 − pb2
∫ x−a
f(y)dy
∫ x−b
f(y)dy. (3)
Equation (3) agrees with that of Ref. [8] for the total cov-
ered length in RSA from a multi-size mixture. However,
the advantage of Eqs. (1,2) is that they permit studying
the partial contributions to the coverage by each of the
two sorts of particles separately.
Note that the symmetry between the a- and the b- par-
ticles is broken by the initial conditions. To be specific,
let b > a. Then, for b-particles the boundary condition
at small x is simply
nb(x) = 0, 0 ≤ x < b (4)
whereas for a-particles we have
na(x) =
0, 0 ≤ x < a
1, a < x ≤ min(2a, b)
For b > 2a, Eq. (5) should be supplemented with
na(x) = 1 +
∫ x−a
na(y)dy (6)
Eq. (6) accounts for the deposition of smaller particles in
small gaps where the larger particle does not fit. Clearly,
this process is not influenced by the b-particles and does
not involve particle competition.
More refined arguments are needed to derive the sec-
ond moment of the distribution, i.e. the expected value
of the square of the number of particles of a given sort,
ua(x) = En
a(x). It may not be a priori evident that one
can write independent expressions for particles of both
sorts, because parameters a and b not only describe the
particle size but also designate the sort of a particle. In-
deed, we can even have a = b and distinguish the parti-
cles by some other parameter, like “color”. Our approach
should remain valid in this case too. To be rigorous, we
therefore introduce an artificial parameter, the “mass” of
a particle, ma and mb, whose value may depend on the
particle shape and is simply proportional to the particle
length only for a fixed transverse particle size. Hence one
can regard ma and mb as independent parameters.
Consider a total mass M(x) = mana(x) + mbnb(x)
of the particles adsorbed in a line segment x. We first
evaluate recursively the mean square of the total mass
< M2(x) >= 〈[mana(x)+mbnb(x)]
2〉, and then calculate
the second partial derivatives with respect to ma and mb.
Using the landing probabilities of particles to perform the
averaging, we obtain
ua(x) = (x− l)
q(x− a) + 2q
∫ x−a
ua(y)dy
∫ x−b
ua(y)dy + 4q
∫ x−a
na(y)dy
∫ x−a
na(y)na(x− y − a)dy
∫ x−b
na(y)na(x− y − b)dy
Similarly, equation for ub(x) reads
ub(x) = (x− l)
p(x− b) + 2q
∫ x−a
ub(y)dy
∫ x−b
ub(y)dy + 4p
∫ x−b
nb(y)dy
∫ x−a
nb(y)nb(x − y − a)dy
∫ x−b
nb(y)nb(x − y − b)dy
We could have derived Eqs. (1,2) in a similar way,
by first evaluating the total average mass M(x) =
mana(x) +mbnb(x) recursively, and then calculating the
derivatives. For a more general case, when the total mass
is a linear functional M(x) =
mlnl(x)dl on the mass
distribution ml, one would have to use variational deriva-
tives δM(x)/δml. For the case of binary mixtures we
consider, partial derivatives are sufficient.
Similarly, we derive an equation for the correlation
function uc(x) = 〈na(x)nb(x)〉 by calculating a mixed
derivative of < M2(x) > with respect to ma and mb.
For particles uniform in the transverse direction with unit
mass density, both the mass and the length of particles
are identical, which gives a way to check the equations.
An appropriate linear combination of equations for ua,
ub, and uc then gives an equation for the variance of the
total filled length or, equivalently, for the variance of the
wasted length, w(x) = x − f(x). The resulting equation
can also be obtained directly, by applying recursion ar-
guments to the waste. The identical results obtained can
be viewed as an additional proof of Eqs. (7,8).
Note the asymmetry in the 4-th terms of Eqs. (7,8)
that are proportional, respectively, to 4q and 4p. These
terms ensure the correct (linear) asymptotic behavior of
the variance at large x.
An important feature of Eqs. (1,2,7,8) is that in spite
of the competitive character of the deposition of particles
of different sorts, the equations for na, nb and the higher
moments are independent. This is rooted in the fact that
a single deposition step on an empty length x does not
depend on the already adsorbed particle distribution.
Due to the self-averaging nature of the filling length
(and waste length) in the limit x → ∞ the averaged
(hence approximate) recursion equations yield exact re-
� � � � � � ���
�
������
� � �� �� ��
b/a = 2.4
� ��� !"#$%&'() *+,-./012 3456789:; <=>?@ABCD EFGHIJ
Adsorbtion length,  x  
FIG. 1: (color online) Average number of adsorbed particles
na (solid lines) and nb (dashed lines) as functions of the length
x (measured in units of a) of the adsorption interval, assumed
initially empty. The results are obtained by iterating Eqs.
(1,2) with the assumed ratio of the particle size b/a = 2.4
and the varying fraction q of a-particles in the flux.
sults. The recursive technique is in this sense equiva-
lent to the alternative “kinetic” approach to RSA that
is sometimes regarded as a higher-level theory. In the
kinetic approach one considers the rate equation that
describes the sequential deposition of particles with the
particle distribution on a line characterized by a time-
dependent function G(x, t) representing the average den-
sity of gaps whose size is between x and x + dx [2, 5].
It has been ascertained for a number of problems that
both approaches give the same result for the coverage.
Still, each has its own benefits. The kinetic approach
allows studying the temporal variation of a state with
specified particle distribution. The recursive approach,
while simulating a simplified version of the kinetics, al-
lows to study more complex effects, such as variance of
the adsorbed particles of different size.
Evaluation of na(x) and nb(x) is readily done by re-
peated iterations of Eqs. (1,2), going from the small to
progressively larger lengths x. Results of the numerical
recursion are shown in Fig. 1 for a particle size ratio
b/a = 2.4 and varying q.
The noteworthy features of the functions na(x) and
nb(x) are (i) the step-like features at x = a, x = b
(which are replicated with ever smaller amplitudes at
x = na +mb, where n and m are integers), (ii) the dip
in the number of small particles na(x) at x = b, which
increases with p, and (iii) the reduction of nb with in-
creasing q. We also note that for all q the behavior of
both na(x) and nb(x) becomes very close to linear al-
ready at x ≈ 7.
The asymptotic behavior of na(x) and nb(x) at large
x can be obtained by multiplying Eqs. (1,2) by x− l and
taking the derivative with respect to x. The resulting
differential equations are satisfied by linear functions of
the form
na = αa(x+ l)− q, nb = αb(x+ l)− p, (9)
where αa and αb are arbitrary constants. When correctly
chosen (by matching to the recursive solution) these con-
stants become the partial filling factors. After the match-
ing is done, the total filled length in the asymptotic limit
is given by f(x) = θx + (θ − 1)l, where θ = aαa + bαb is
the specific coverage. It is worthwhile to stress that the
value of the asymptotic solutions (9) consists precisely in
that they are asymptotically exact. Hence they provide
a sanity check on any solution we could have obtained by
a numerical recursion up to moderate values of x.
Similarly, Eqs. (7-9) yield the variances at large x,
ua − n
a = µa(x+ l)− qp[1 + (b − a)αa]
2, (10a)
ub − n
b = µb(x+ l)− qp[1− (b− a)αb]
2. (10b)
Again, these solutions are asymptotically exact; they sat-
isfy Eqs. (7,8) with arbitrary values of µa and µb, pro-
vided of course that na(x) and nb(x) are in the correct
asymptotic form (9) with properly chosen [i.e., satisfying
Eqs. (1,2)] coefficients αa and αb. In principle, we could
now follow a procedure similar to above, viz. determine
µa and µb by matching Eqs. (10) against a numerical
recursive solution at some moderate value of x. How-
ever, it would be rather difficult to control the numerical
accuracy in this procedure, because of the difference of
nonlinear functions that enter Eqs. (10), even though
that difference itself behaves linearly with x at large x.
Fortunately, our model admits of an exact solution
based on the use of Laplace transformation (details can
be found in [3] and references therein). Below we present
an exact evaluation of variance for particles of larger size,
while details of similar though lengthier calculations for
smaller particles are presented in Appendix.
Firstly, we need exact solutions of Eqs. (1,2). To ob-
tain these, we substitute x → x + b in Eq. (2) and mul-
tiply it by x − l. Taking the Laplace transformation of
the resulting equation and using the boundary condition
(4), we obtain
+ b− l
ebsNb(s) =
qe(b−a)s + p
Nb(s)
Here Nb(s) is the Laplace transform of nb(x),
Nb(s) =
e−sxnb(x)dx, (12)
Rearranging the terms and multiplying by e−bs, we put
Eq. (11) into the form
N ′b(s)+
qe−as + pe−bs
Nb(s) = −
e−bs. (13)
For p → ∞, the solution of Eq. (13) is, asymptotically,
Nb(s)|s→∞ =
s(b− l)
e−bs, (14)
as follows from the known variation of nb(x) ≈ p(x −
b)/(b− l) at small x− b. Hence we have
Nb(s) =
p exp(−l s)
s2β(s)
e−q(b−a)tβ(t)dt, (15)
where
β(s) = exp
1− q exp(−at)− p exp(−bt)
To find the asymptotic behavior of nb(x) at large x, it
is convenient to use Karamata’s Tauberian theorem for
the asymptotic growth rate of steadily growing functions
(see e.g. [20], p. 37). According to the theorem, the
asymptotics of nb(x) [or na(x) or their variances] can be
readily obtained (by taking the inverse Laplace transfor-
mation) from the Laurent power series expansion of the
Laplace transforms of these functions at small s (see [9]
for the mathematical details of this analysis).
Function Nb(s) is analytic at all s 6= 0 and at s = 0 it
has a second-order pole with the following asymptotic
Nb(s) =
αb,0l − p
+O(s), (17)
where
αb,0 = p
e−q(b−a)sβ(s)ds. (18)
To calculate nb(x) at large x, we take the inverse Laplace
transformation of (17). This gives
nb(x) = αb,0(x + l)− p, (19)
with an exponentially small error term, in line with the
asymptotics given by Eq. (9).
In the limit p = 1, equation (18) duly gives the so-
called jamming filling factor R for the standard RSA,
αb,0(l = 1) ≡ R = 0.74759 · · · (also called the Renyi
constant [21]). In the limit a → 0, Eq. (18) recovers the
results of Refs. [4, 5] for the coverage of a line from a
binary mixture of finite size particles and point defects.
Moreover, Eq. (18) gives the large particle contribution
to the total coverage, obtained in [6, 8] for the range
a < b < 2a. Here we see that this result remains valid
for arbitrary a < b.
Next, we perform similar manipulations with Eq. (8)
and obtain an equation for the Laplace transform of the
variance Ub(s) = L̂[ub(x)], viz.
U ′b(s)+
qe−as + pe−bs
Ub(s) = −
exp(−bs)
Rb(s),
where
Rb(s) = p+ 4psNb(s) + 2s
2N2b (s)
qe(b−a)s + p
, (21)
with Nb(s) defined by Eq. (15). The solution of Eq. (20)
can be written in a form similar to Eq. (15), namely
Ub(s) =
exp(−sl)
s2β(s)
β(t)e−q(b−a)tRb(t)dt. (22)
The integrand in the right-hand side of Eq. (22) is pro-
portional to 1/t2 causing the integral to diverge as 1/s
for s → 0. This is due to the square-law dependence of
u(x) at large x .
To separate the regular part needed for the estima-
tion of variance, we note that at small t one has Rb(t) ∝
αb,0t
−2. Moreover, the series expansion shows that the
difference β(t) exp[−q(b− a)t]Rb(t)− 2α
−2 is regular
at t → 0. Therefore, it is convenient to define an en-
tire function κb(t) = β(t) exp[−q(b−a)t]Rb(t)−2α
In terms of this function, the solution Ub(s) can be ex-
pressed as follows:
Ub(s) =
exp(−sl)
s2β(s)
+ kb,0 −
κb(t)dt
, (23)
where
kb,0 =
κb(t)dt. (24)
To apply Karamata’s Tauberian theorem, we note that
the asymptotic expansion of Ub(s) near its third-order
pole is of the form
Ub(s) =
kb,0 + 2α
kb,0l − κb(0)− qp(b− a)
2α2b,0
. (25)
Taking the inverse Laplace transformation, we find the
asymptotic form of ub(x):
ub(x) = α
2 + (kb,0 + 2α
b,0l)x
+kb,0l − κb(0)− qp(b− a)
2α2b,0, (26)
with an exponentially small error term. Using Eq. (19)
to subtract n2
(x), we find an equation of the form (10b)
with µb = kb,0 + 2pαb,0. The specific variance of the
adsorbed number of b-particles is given by (at x → ∞)
µb = αb,0(1 + 2p) + 2
β(s)sNb(s)e
2pe−bs
+sNb(s)
qe−as + pe−bs
α2b,0
ds. (27)
Integrating by parts the last term and rearranging the
result, we finally obtain
µb = αb,0(1− 2p) + 4p
αb(u)
e−bu (1− qe−au
−pe−bu
du+ 2
β(u)u2
e−luK(u)du, (28)
where
K(u) = qe−au
2(1− qe−au − pe−bu)− (a+ l)u
+pe−bu
2(1− qe−au − pe−bu)− (b+ l)u
αb(u) = αb,0 − p
e−q(b−a)yβ(y)dy. (30)
In the limit of small p → 0, the Fano factor Φ =
µb/αb,0 → 1. In this limit, large particles are distributed
on the line randomly, without correlations. In the oppo-
site limit, p = 1, Eq. (28) reduces to the standard RSA
result, first obtained for a lattice RSA model by Macken-
zie [22]. The numerical value of the Mackenzie constant,
µ0 = 0.0381564 · · · , corresponds to Φ = 0.0510387 · · · ,
see [9]. Expression (28) for the larger particles has the
same structure as the corresponding formula in the stan-
dard RSA model (fixed-size CPP). Due to the exponen-
tial factors in the integrands of Eq. (28), the dependence
of µb on a for a ≪ b is quite weak. The limiting value of
the specific variance for a/b → 0 gives the specific vari-
ance of the fill factor for the case of finite-size particles
(b = 1) mixed with point-size particles,
µb,p = αb,p,0(1− 2p) + 4p
αb,p(u)
e−u (1− e−u) du
b,p(u)
βp(u)u2
{qe−pu (2− 2e−u − u)
+e−(1+p)u [2p (1− e−u)− (1 + p)u]
du, (31)
where αb,p,0 is the fill factor for this case,
αb,p(u) = p
e−qyβp(y)dy, αb,p,0 = αb,p(u = 0)
βp(u) = exp
1− exp(−t)
. (33)
It is worth to note that Eqs. (2,8) and their solutions can
be readily generalized to the case when particles of the
smaller size have an arbitrary distribution in the interval
[a1, a2] so long as a2 ≤ b [23].
The above analytic results for the variance of larger
particles are essentially exact, as will be confirmed in the
next Section by Monte Carlo simulations. For the smaller
particles, the calculations are messier and accurate ana-
lytical results can be obtained only in a certain range of
particle size ratios. Estimations of the variance for small-
size particles are further discussed in the Appendix.
III. DISCUSSION OF THE RESULTS,
COMPARISON WITH MONTE CARLO
MODELING
Here we present the results of numerical calculations
using both the analytical expressions obtained in the pre-
ceding section and Monte Carlo simulations. For large-
size particles the Monte Carlo results are very close to
analytical expressions both for the fill factor and the vari-
ance, so we shall not dwell on their comparison. For
small-size particles, especially in the range 2 < b/a < 8,
analytical calculations are rather unwieldy, so Monte
Carlo simulations become indispensable. Larger size ra-
tios, b/a > 8, lend themselves to an approximate ana-
lytical approach (see Appendix). In this case, we use
the Monte Carlo to estimate its accuracy for the small
particle contribution.
Traditional studies of the generalized RSA via Monte
Carlo simulations follow a temporal sequence of events.
For the case of adsorption on a line of the length x from
a binary mixture, one step of the sequence comprises:
(i) selection of a particle from the mixture according
to the deposition flux ratio (with the probability q of
choosing the small-size (a) particle, and the probability
p = 1− q of selecting a particle of larger size b);
(ii) random choice of a deposition coordinate of particle
center on the line x with formerly deposited particles;
(iii) rejection of the particle if it overlaps by any part
with formerly deposited particles or with the line bor-
ders; otherwise, the particle deposition proceeds with the
formation of two new disconnected adsorption lengths.
This traditional approach has several drawbacks, that
make the modeling very demanding, both in terms of the
computer time and memory allocation.
Firstly, both the filled length in the jamming limit and
the specific fill factor (coverage) depend on the initial
length. Due to the self-averaging property of the coverage
it tends to a unique exact value in the limit x → ∞. To
obtain the accuracy of about 0.1 %, the common strategy
has been to use large initial length values (105b -107b) and
make additional averaging over a set of about NR =100-
1000 different realizations.
Secondly, as time evolves and the jamming limit is ap-
proached, the probability of finding a free gap for parti-
cle deposition becomes greatly reduced, so that the ad-
sorption time tends to infinity. The process is termi-
nated when variations of the adsorbed particle number
are smaller than those required by the desired accuracy.
The recursive analysis of the generalized RSA suggests
a revision of the above scheme. Since the deposition is
random and sequential, it does not depend on the tem-
poral history of the process or the growing number of
rejected particles and their coordinates. Therefore one
step of the sequence can be chosen as follows:
(i) selection of any free deposition length, l1 > a. It is
convenient to choose for l1 the outermost free deposition
length on the left-hand side.
(ii) if l1 < b, then particle of size a is deposited,
otherwise the deposited particle is chosen according to
the landing probability, given by q(l1 − a)/(l1 − l) for
a-particle and p(l1 − b)/(l1 − l) for b-particle, where
l = qa+ pb.
(iii) random choice of a deposition coordinate (taken
as the coordinate of particle’s left end) on the line l1 for a
given particle size, i.e. within the interval l1−a for a-size
or within l1 − b for b-size particle, with the formation of
two new adsorption lengths from the initial length l1.
It is readily seen that although the sequence of deposi-
tion events is different from the actual temporal sequence
of adsorption (the simulated deposition proceeds by se-
quentially filling the left-hand lengths), the statistics of
divisions is identical and therefore so is the final distri-
bution of the gaps, as well as all statistical properties
of the jamming state. Our sequential scheme excludes
deposition of to-be-rejected particles and therefore is in-
comparably faster. Besides, it terminates exactly when
the jamming limit (with no gaps larger than unity) is
achieved. Direct comparison with the traditional Monte
Carlo results, e.g. [5, 7, 8] exhibits total agreement. The
difference in the calculation time is especially evident for
small (close to zero) q: in the time scale of “real” deposi-
tion, the jamming limit will be strongly delayed because
of the rarity of events with small particle chosen. In our
modified approach, all gaps smaller than b are “rapidly”
populated by small-size particles, however small be the
value of q.
The next step of the revision is to exploit the fact
(proven analytically in the preceding section) that in the
jamming limit, the linear dependence on the adsorption
length of both the average filled length and its variance is
exponentially accurate, starting from a reasonably short
length, certainly not exceeding x ≈ 10b. Since this lin-
ear dependence has only two parameters [actually only
one, as the parameter ratio is exactly fixed by analyti-
cal considerations, Eqs. (9,10)], both the coverage and
the variance can be determined with Monte Carlo simu-
lations of short samples.
To be sure, in order to achieve the same accuracy as
that obtained for long samples, the results should be av-
eraged over a sufficient number of realizations NR. This,
however, takes little memory or time. Calculations show
similar accuracy for different x and NR, so long as their
product x×NR is fixed. The results presented below were
KLM NOP QRS TUV WXY Z[\]^_`abcde
fghijklmn
opqrstuvw
  b/a :
Ratio of  small particles (a) in the flux, q
    2;      5
    10;    20
{|} ~�� ��� ��� ��� ���
���������
��������
�� ¡¢£¤¥¦
total
FIG. 2: (color online) Partial contributions of small and large
particles to the total coverage depicted as functions of q, for
different particle-size ratios b/a in the flow. Open points cor-
respond to the limit b/a ≫ 1, as described by the analytical
formulae (32) for b-particles and (A13) for a-particles. For the
total coverage, the open circles to the wasted length product
approximation, Eq. (34).
obtained using a sample of size x = 200a for b/a < 10
and x = 400a for b/a = 20, 40, subsequently averaged
over 10 000 realizations, which appeared to be sufficient
to eliminate any spread of the results in the graphical pre-
sentation (producing an accuracy of better than 0.1%).
The use of small samples is very effective in reducing
the calculation time (with an ordinary PC, high-accuracy
results can be obtained in minutes, compared to days in
the traditional scheme [8]).
Figure 2 shows partial contributions to the coverage as
functions of the fraction q of small particles in the binary
mixture at different ratios of particle size. As q increases,
the coverage with large particles is substituted by that
with small particles, producing some decrease in the to-
tal coverage. In the regions of corresponding parameters,
our results reproduce those of reported analytical calcu-
lations (i.e. for b/a < 2 [5, 8] and for a = 0 [6] for the
large particle contribution) and those obtained by the
Monte Carlo simulations of [5, 7, 8], demonstrating the
ª«¬ ­®¯ °±² ³´µ ¶·¸ ¹º»¼½¾¿ÀÁÂÃ
ÄÅÆÇÈÉÊËÌÍÎÏ
ÐÑÒÓÔÕÖ×ØÙÚÛ
      b/a :
 infinity
Ratio of small particles in a flux, q
FIG. 3: (color online) Variance of the partial coverage by ad-
sorbed b-particles from the binary mixture for different values
of the particle size ratio b/a in the flow.
validity of our revised approach.
It is evident from Fig. 2, that the total coverage in-
creases at smaller q, as can be explained by sequential
deposition of the two kinds of particles. In the regime of
small q, large particles are adsorbed first and their de-
position, unobstructed by small particles, is tight. Sub-
sequently, the small particle fill the gaps between large
particles and this clearly reduces the total wasted length.
The effect of increasing the particle size ratio b/a is
pronounced only for b/a < 10, then it rapidly saturates.
Therefore, for large b/a, say b/a = 20 the coverage by
large particles is very close to that obtained for a model
mixture of point-like and finite-size particles [by formally
letting a = 0 in Eq. (18)]. Such a model, however, has lit-
tle relevance to any practical situation, because it simply
ignores the partial contribution of small particles to the
total coverage. The latter can be described analytically
in the limiting case b/a → ∞, Eq. (A13).
The partial contribution of small particles steadily
grows with the increasing size ratio due to the expand-
ing gaps between the large particles. In the limit q → 0,
the total coverage can be estimated by observing that
the specific wasted length in this case is a simple prod-
uct of the specific lengths wasted in initial deposition of
large particles and subsequent deposition of small par-
ticles, i.e. 1 − θ = (1 − θa)(1 − θb). Since for q = 0
the specific coverage θb = R and since for large size ra-
tios (when the gaps between large particles are large) the
specific coverage θa = R, we have θ = 1−(1−R)
2=0.936,
in agreement with the results reported in the literature
[5, 7]. However, the sequential nature of the deposition
suggests that the entire q dependence of the total θ can
also be approximated by a product of the specific wasted
lengths in the competitive deposition of large particles
q(1 −R) and subsequent deposition of small particles in
the remaining gaps, which gives
θ = 1− (1−R)(1− pR) = R[1 + p(1−R)]. (34)
This product-waste approximation is shown in Fig. 2 by
the open circles.
Next, we concentrate on the specific waste variance and
the Fano factor. We shall discuss the b- and a-particles
separately, since the effects are rather different in na-
ture and also since they have been evaluated by differ-
ent techniques. Results for large particles are obtained
by numerical integration of Eq. (30) and confirmed by
Monte Carlo simulations. Results for a particles are ob-
tained by Monte Carlo stimulations and are accompanied
by analytical expressions in the limit b/a ≫ 1.
Variance, µ̃, of the partial contribution of b-particles
to the total coverage is shown in Fig. 3 for different par-
ticle size ratios. Unlike the particle number variance µ,
the variance of coverage, µ̃ = µb, depends only on the
size ratio b/a and does not directly scale with b. It is
therefore more indicative of the effect of decreasing size
àáâ ãäå æçè éêë ìíî ïðñòóô
õö÷øùú
ûüýþÿ�
����� �� �	
�� 
�������� �� � ����� �
)*+ ,
456789:;
<= > ? @ A B C D E F G
FIG. 4: (color online) The Fano factor for the number of ad-
sorbed b-particles from the binary mixture as functions of q,
for different values of the ratio b/a of particle sizes in the
flow. Dotted line corresponds to a random particle packing
on a lattice with suitable lattice constant (aka monomer ad-
sorption).
of small particles on the fluctuations of the number of
large particles. At q → 0, when the adsorption of large
particles is unconstrained by small particles, the vari-
ance of large particles is minimal and corresponds to the
highly correlated distribution [18] in the standard CPP
problem (one-size RSA). The variance rapidly increases
with q as the small particle deposition destroys the CPP
correlations. The maximum of this effect is shifted to
larger q values for larger b/a. For q approaching unity,
the variance decreases simply due to the decrease of the
average number of adsorbed b-particles.
Correlation effects are more adequately characterized
by the Fano factor Φb, shown in Fig. 4. With the increas-
ing number of competing small particles in the flux, the
Fano factor grows from the smallest value Φ = 0.051 · · · ,
corresponding to the one-size RSA problem, to unity in
the limit q → 1. Small coverage by the large particles in
the latter limit means that they are distributed randomly
on the line, so that Poisson statistics recovers. The most
noticeable effect is a rapid decrease of the Fano factor
0.0 0.2 0.4 0.6 0.8 1.0
Small particle contribution to the flux, q
              b/a:
 1.6;    2
 5;       10
 20;     40
FIG. 5: (color online) The Fano factor for adsorbed a-particles
from the binary mixture for different values of the particle size
ratio b/a in the flow. Open points show the contribution of
fluctuations of the gap sizes between large particles.
with 1− q, manifesting a strong enhancement of the cor-
relation effects in the large particle distribution. These
correlation effects become exhausted only near q ≤ 0.1.
The correlation effects increase with b/a but saturate at
about b/a = 20.
Figure 5 shows the Fano factor for a-particles com-
petitively deposited along with large particles. The re-
sults are strikingly different at all q 6= 1 (when Φa = Φ,
as expected). While the distribution remains correlated
(Φa ≤ 1) for small ratios b/a ≤ 5, at larger b/a one has
Φa > 1, almost for all q, which means that the number
of small particles per unit length is strongly fluctuating.
This is due to the widely fluctuating size of the gaps
available for small particle deposition between large par-
ticles. For large values of b/a and in the entire range
of q, the Fano factor Φa can be approximated in terms
of the fluctuations of the coverage by the large particles,
viz. Φa = (b/a)µb,pR
2/θa, where µb,p is given by Eq.
(31) and θa by Eq. (A13). This approximation, which
neglects fluctuations of the density of adsorbed a parti-
TUV WXY Z[\ ]^_ `ab cde
fghij
klmno
pqrst
��� � ���
� ���� ����
§¨©ª«¬ ­®¯°±
FIG. 6: (color online) Variance of the partial number of ad-
sorbed a- and b-particles and of the total number of adsorbed
particles for b/a = 1.2. Also shown is the fluctuation correla-
tion function fcor.
cles in the gaps, is shown in Fig. 5 by open points. This
contribution is proportional to b/a and for b/a > 10 it is
evidently dominant.
For the particle energy branching process at small
b/a < 2, both the variance of the partial numbers of small
and large particles and the total number variance are of
importance. We shall illustrate this point in the instance
of b/a = 1.2 shown in Fig. 6. We see that at q ≈ 0.5 the
fill factor fluctuations are larger for a particles and some-
what smaller for b particles, but both are pretty large,
compared to the variance of the total number of adsorbed
particles. This is due to the strong anti-correlation in
their distribution, as evidenced by the specific fluctua-
tion correlation function, fcor = x
−1〈δnaδnb〉, also plot-
ted in Fig. 6. We note that fcor < 0, which means that
any excess in the number of a-particles is accompanied
by a downward fluctuation in the number of adsorbed
b-particles. Importantly, the variance and the Fano fac-
tor for the total number of adsorbed particles does not
exceed substantially its value for the single-size RSA.
Note the asymmetry of the curves for a and b parti-
cles, e.g. the variance of large particles goes to zero as
q → 1 whereas that of small particles remains finite even
as q → 0. This is a feature of our model that allows ”infi-
nite” amount of time for the deposition of small particles
in the gaps left after the deposition of large particles is
completed, but not vice versa. Therefore, the deposition
of small particles remains finite even in the limit of q → 0
and the same is true for the a-particle number variance.
Another interesting feature of the a-particle num-
ber variance, already evident from Fig. 5, is its non-
monotonic behavior as function of b/a at small q. This
variation is displayed directly in Fig. 7 that shows the
dependence of the Fano factor on b/a for q=0.05, 0.1
and 0.2 — where its non-monotonic nature is most pro-
nounced. The minimum of the Fano factor is achieved at
b/a ≈ 2. Note that the non-monotonic dependence of the
Fano factor is accompanied by non-monotonic variations
in the dispersion of the gaps between small particles. In
Ref. [7] it was found that for q =0.5 the dispersion is
noticeably reduced at b/a ≈ 1.55. These effects were
interpreted as a manifestation of the so-called “snug fit”
events, i.e. particle deposition in gaps that are just barely
above the unit length a. In contrast, the Fano factor for
b-particles and that for the total number of particles re-
main monotonic everywhere.
IV. SOME CONSEQUENCES FOR THE
ENERGY BRANCHING IN HIGH-ENERGY
PARTICLE DETECTORS
The model of RSA from binary mixtures is relevant
to an important practical problem of particle energy
branching (PEB) where high-energy particle propagates
in an absorbing medium and multiplies producing sec-
ondary electron-hole (e-h) pairs. Multiplication proceeds
²³´ µ¶· ¸¹º »¼½ ¾¿À ÁÂÃ ÄÅÆ ÇÈÉ
q  :     0.05
          0.1
          0.2
Particle size ratio,  b/a
FIG. 7: (color online) The Fano factor for adsorbed a- and
b-particles as functions of the particle size ratio b/a. Also
shown is the Fano factor for the total number of adsorbed
particles.
so long as the particle energy is above the impact ioniza-
tion threshold [15]. The energy distribution of secondary
particles is random to a good approximation.
The affinity between the two problems was fully rec-
ognized already in 1965 by van Roosbroek [17] (see also
[24]). The PEB process can be considered in terms of
a CPP if one identifies the initial particle kinetic energy
with an available parking length and the pair creation en-
ergy with the car size. Similarly, the kinetic energies of
secondary particles can be identified with the new gaps
created after deposition of a particle. Full equivalence
of PEB to CPP further requires that only one of the
secondary particles takes on a significant energy, which
corresponds to binary cascades [25]. Otherwise, one has
to consider a simultaneous random parking of two cars
in one event.
To estimate the particle initial energy in PEB, one
measures the number N of created electron-hole pairs.
Variance of this number, due to the random character of
energy branching and also due to random energy losses
in phonon emission, limits the accuracy of energy mea-
surements. Both the yield N and the e-h pair variance
var(N) = (N −N)2 are proportional to the initial en-
ergy. The ratio of the e-h pair variance to the yield, i.e.
the Fano factor of the PEB process, is a parameter that
quantifies the energy resolution of high-energy particle
detectors.
For semiconductor crystals, the PEB problem has ad-
ditional complications due to the energy dependence of
phonon losses and the energy dependence of the electron
density of states and the impact ionization matrix ele-
ment. Full quantitative analysis of the PEB is possible
only with detailed numerical calculations, which goes far
beyond the scope of the present article.
A common feature of the energy branching process in
semiconductors is the presence of several pair production
channels, associated with the multi-valley energy band
structure of the crystal. In Si, Ge and common A3B5
semiconductors, the e-h pair creation produces electrons
in one of the ellipsoids near the edge of the Brillouin zone,
in 100 (X) or 111 (L) directions. Owing to the difference
in the final densities of states and the matrix elements,
the impact ionization processes associated with X and L
valleys have different but competitive probabilities. Be-
cause of its low density of states, the Γ valley is usually
not competitive, even when it is the lowest valley.
Ultimately, electrons will end up in the lowest energy
valley but when the final electron valley is itself degener-
ate, as in Ge or Si, the resulting electron states may not
be fully equivalent, because of the different collection ki-
netics owing to the crystal anisotropy. This effect may
have important consequences for the observed variance.
For example, in Si diode detectors electrons are created
in 6 degenerate energy valleys that represent ellipsoids
of revolution elongated along (100) and equivalent direc-
tions in k-space. Suppose the diode structure is such that
the current flows along the (100) direction, as it is usually
the case. Electrons from the two valleys along the cur-
rent have a large mass and low mobility. The measured
current is hence dominated by electrons from the 4 val-
leys elongated perpendicular to the current that have a
low mass and high mobility along the current. Since the
choice of equivalent valley in the PEB process is fully ran-
dom, the number of high-mobility electrons will fluctuate
more strongly than the total number of generated carri-
ers. These fluctuations will dominate if the inter-valley
transition rate is low compared to the inverse collection
time. In the opposite limit of high inter-valley transition
rates, this effect will average out as the collected current
will fluctuate in time. The current fluctuation mecha-
nism due to the carrier escape into heavy-mass valleys is
a well-known source of noise in multi-valley semiconduc-
tors [26]). More detailed account for these effects will be
presented elsewhere [23].
Here we shall discuss an opposite situation, that is
common to direct-gap semiconductors, such as GaAs or
InP. In these materials, the lowest (Γ) electron valley has
a very low density of states, compared to that in the
satellite (X and L) valleys. Therefore, the probability of
electron generation in the Γ-valley can be neglected in
first approximation, so that the branching competition
occurs only between the satellite valleys of two different
kinds. Both the density of states and the threshold en-
ergy are different between X and L valleys and we can use
the results of the present study to interpret and predict
the consequences, at least qualitatively.
The binary-mixture RSA model interprets the higher
density of states as higher deposition rate and the higher
threshold as larger particle size. To make our conclu-
0.0 0.2 0.4 0.6 0.8 1.0
b/a = 1.4
Ratio of the small particles in the flux, q 
 Coverage
FIG. 8: (color online) Partial fill factors and the total coverage
for b/a = 1.4 as a function of q. Also shown is the total
number Ntot of adsorbed particles
sions more transparent, let us re-formulate the required
results in terms of a random parking problem with cars
of two sizes. We are now interested only in the numbers
of parked cars and the fluctuations of these numbers.
Several qualitative conclusions can be drawn from our
results:
(i) The total number of parked cars (in the jamming
state) will decrease with increasing fraction of larger
cars in the flow and with the growth of their size. For
b/a = 1.4 the effect is illustrated in Fig. 8 (which can
be viewed as an extension of Fig. 2). It follows from the
fact that adsorption of a large car excludes larger length
for subsequent parking events and thus causes a decrease
of the total fill factor. Note that the decrease in the to-
tal particle number is accompanied by an increase in the
total filled length, as smaller number of cars cover larger
area.
The next two conclusions (ii) and (iii), illustrated in
Fig. 6, are interconnected and will be discussed jointly.
(ii) Variance of the total number of parked cars and the
Fano factor will both grow with the increasing fraction of
larger cars in the flow and with the growth of their size.
(iii) Variance of the separate numbers of parked small
and large cars and their Fano factors are considerably
larger than that of the total number of cars. Therefore,
if for some reason one type of cars is neglected or under-
counted, the registered variance and the Fano factor can
be substantially increased.
These conclusions are connected with the nature of the
car number fluctuations and the strong anti-correlation
between the fluctuations in the number of small and large
cars. Fluctuations in the number of parked cars of one
kind are strongly enhanced by the presence of more or
less randomly distributed cars of the second kind, espe-
cially when cars of the second kind dominate. This leads
to conclusion (iii). However, the two distributions are
anti-correlated (higher number of parked small cars is
accompanied by a smaller number of large cars and vice
versa). The anti-correlation is particularly strong for a
size ratio that is close to unity.
One can imagine a case when the two kinds of cars dif-
fer only in “color”. In this case, Eqs. (18,28) yield αa,0 =
qR, αb,0 = pR, µa = Rqp+q
2µ0 and µb = Rqp+p
2µ0, so
that at large x we have < δnaδnb > /x = −(R − µ0)qp.
Then, the anti-correlation is almost complete: the fluc-
tuations of the total number are much smaller than those
of a given color, but still non-zero. Both the individual-
color number fluctuations and the anti-correlation are
largest at q ≈ 0.5, cf. Fig. 6. The anti-correlation
decreases with increasing size ratio, as reflected in our
conclusion (ii).
To discuss the above conclusions in terms of the PEB
problem, we note that estimation of the initial particle
energy is equivalent in CPP to a measurement of the
unknown length of a parking lot in terms of the total
number of cars that were able to fit into it by random
parking, assuming that the average fill factor for a given
two-size car mixture is known from earlier measurements.
The absolute accuracy of such a measurement depends on
the variance of the fill factor, and the relative accuracy
is determined by the Fano factor. As shown above for a
mixture of cars, the larger disparity of car sizes leads to
the higher fill-factor variance and therefore reduces the
absolute accuracy.
A particle detector measures the total number of sec-
ondary particles of all sorts (but not their total creation
energy, that would be equivalent to the filled length). In
any channel, all secondaries that have sufficient energy
for further branching will do so. Therefore, only those
pair creation energy ratios that leave the channels com-
petitive (i.e. b/a < 2) are relevant to the PEB problem
— otherwise additional energy branching would be pos-
sible.
We conclude that the presence of competing channels
with different energies [e.g. impact ionization with ex-
citation in X and L valleys] will decrease the quantum
yield (the number of secondaries per unit energy of the
primary particle) and enlarge the Fano factor. The at-
tendant loss in energy resolution is not that bad when
the ionization energies associated with different valleys
are not too disparate. For example, in Ge besides the
lowest eight L valleys (EG = 0.66 eV) one has a non-
competitive Γ valley (EΓ = 0.8 eV) and six very compet-
itive Si-like valleys (EG = 0.85 eV). The downgrading of
energy resolution should be more important for crystals
with larger (≈ 2) threshold energy ratio. For example, in
Si one has besides the 6 lowest valleys (EG = 1.12 eV) in
X direction, eight germanium-like L valleys with the gap
EL = 2.0 eV. Their effect on the Fano factor in silicon
may not be negligible.
Finally, reformulating (iii), we stress that any signif-
icant disparity in the collection efficiency between dif-
ferent equivalent valleys will strongly enhance the Fano
factor and downgrade energy resolution. This happens
because any collection disparity breaks the symmetry
between the equivalent valleys and destroys the anti-
correlation, responsible for keeping the total Fano factor
low even when the partial particle numbers associated
with individual valleys exhibit fully random fluctuations.
One possible origin for the asymmetry in the collection
efficiency in semiconductors has been discussed above in
the case of silicon diodes with the electric field in (100)
direction. In germanium diodes all different valleys are
equivalent relative to the (100) direction and the sym-
metry is not broken. It would be broken, however, if one
were to use Ge diodes oriented in (111) direction. This
would lead to a situation similar to Si — with a possi-
ble degradation in the Fano factor. These effects deserve
additional study, both experimental and theoretical.
V. CONCLUSIONS
We have studied a generalized 1-dimensional competi-
tive random sequential adsorption problem from a binary
mixture of particles with varying size ratio. Using a re-
cursive approach, we obtained independent equations for
the number of adsorbed particles of given sort and exact
analytical expressions for the partial filling factors and
variances for the larger particles. For the smaller parti-
cles analytical expressions were obtained in a number of
limiting cases. The results have been confirmed by direct
Monte Carlo simulations. To do so, we have introduced
a modified Monte Carlo procedure that enabled us to
explore a wide range of particle size ratios and particle
fractions in the flux.
A number of qualitative implications have been for-
mulated, relevant to the energy branching problem in
high-energy particle propagation through a semiconduc-
tor crystal. Conclusions made concern the quantum yield
and the energy resolution in semiconductor detectors
made of crystals with several competing channels of im-
pact ionization with different final electronic states.
We have found a very strong anti-correlation effects
which strongly suppress fluctuations of the total particle
number compared to the fluctuations of partial contribu-
tions by particles of a given sort. This effect is particu-
larly evident when one considers the deposition of similar
competing particles, e.g. parking of cars that are differ-
ent only in “color”. It may have dramatic consequences
for semiconductor γ-radiation detectors, if the symmetry
between anti-correlated particles is broken by a biased
collection. This leads to an important conclusion that the
energy resolution of semiconductor detectors is very sen-
sitive to the collection efficiency of competing secondary
particles.
We have also found a very strong correlation effects
that suppress fluctuations of the larger particle number
for all particle ratios. As a result, the Fano factor for
the larger particles is as a rule considerably smaller than
that for the smaller particles. The variance of the cover-
age by the smaller particles strongly increases with the
growth of the particle size ratio b/a. This effect is due
to the fluctuations in the size of gaps between larger par-
ticles that serve as receptacles for small-particle deposi-
tion. For b/a ≥ 5 the small-particle variance exceeds that
for the Poisson distribution in almost the entire range of
particle fractions in the flux onto the adsorbing line.
Acknowledgement. This work was supported by the
New York State Office of Science, Technology and Aca-
demic Research (NYSTAR) through the Center for Ad-
vanced Sensor Technology (Sensor CAT) at Stony Brook.
APPENDIX A: SMALL PARTICLE
CONTRIBUTIONS TO COVERAGE AND
COVERAGE VARIANCE
To calculate the contribution of small particles to the
total coverage at large x, we use Eq. (1) with the initial
boundary conditions (5). With the substitution x → x+b
and using Eq. (6), we rewrite Eq. (1) in the form
(x + b− l)na(x + b) = q(b − a)na(b) + qx
∫ x+b−a
na(y)dy + 2p
na(y)dy (A1)
Equation (A1) is valid for all x ≥ b. Taking Laplace
transformation of na(x) cut at x < b by a step-function
factor, we find that the transform,
Ña(s) =
e−sxna(x)dx, (A2)
satisfies the following equation
+ b− l
ebsÑa(s) =
[1 + (b − a)na(b)s]
e(b−a)s
Ña(s) + J1(s)
Ña(s) + J2(s)
J1(s) =
e−sxna(x)dx, J2(s) =
e−sxna(x)dx
Rearranging the terms, we rewrite it in form
+ l +
qe−as + pe−bs
Ña(s) = −
e−bsRa(s)
where
Ra(s) = q[1+(b−a)na(b)s]+2s
qe(b−a)sJ1(s) + pJ2(s)
The form of Eq. (A5) is similar to Eq. (20) in which,
however Ra should be calculated through J1(s) and
J2(s), using Eqs. (5,6). For the case b < 2a we have
na(b) = 1, and J1(s) = J2(s), while the explicit expres-
sion for J1(s) is easily obtained by substituting na(x) = 1
in Eq. (A4). Solution of Eq. (A5) then enables one to
retrieve the result of Ref [5]. To calculate J1(s) and J2(s)
for b > 2a, it is necessary to use Eq. (6), which describes
RSA of small particles onto a short line x < b. Its ana-
lytical solution and therefore the explicit expressions for
J1(s) and J2(s) can be obtained for the case b/a < 5 us-
ing direct recursion to find na(x) (for one-particle RSA
problem!). The result is rather cumbersome but suitable
for numerical integration.
For the case b/a > 5 one can exploit the exponentially
rapid approach of the solution of Eq. (6) to its asymp-
totic behavior in the limit x ≫ 1 (see e.g. [27] for the
numerical data). This asymptotic solution,
na(x) =
(x+ 1)− 1, (A7)
can be used to calculate J1(s) and then J2(s). To do this,
we multiply Eq. (6) by exp(−sx) and integrate between
0 and b− 1. We obtain an equation for J2(s) of the form
J ′2(s) +
J2(s) = −
e−asI(s)
+s(b− a)e−bs[na(b)− 1] + 2se
−asJ1(s)
, (A8)
where
I(s) =
∫ (b−a)s
dyye−y. (A9)
Solution of Eq. (A8), satisfying the boundary conditions
for na given by Eq. (5), is of the form
J2(s) =
β̃(s)s2
dtβ̃(t)
na(b)(b − a)te
−(b−a)t
+2tJ1(t)−
1− e−(b−a)t
(A10)
β̃(t) = exp
1− ev
. (A11)
The contribution of small particles to the fill factor is
then given by
β(u)Ra(u)du. (A12)
in which β(u) is given by the Eq. (16) and Ra(u) is
defined by Eq. (A6). The obtained solution, though
rather unwieldy, is suitable for numerical integration and
for b/a > 5 it gives the results that agree with Monte
Carlo simulations.
In the limiting case b/a = b/a ≫ 1 it reduces to a more
compact final expression for the contribution to the total
coverage from the small particles
θa = R
due−quβp(u)
q(u− 1)− 2pe−u
(A13)
with βp(u) defined by (33). For q = 1, Eq. (A13) prop-
erly gives θa = R, while for q = 0 one has θa = R(1−R).
The latter expression corresponds to the coverage by
small particles of the gaps between the large particles
left after their initial deposition. For arbitrary q, the
coverage given by Eq. (A13) is depicted in Fig. 2 by the
open squares.
Similar approach can be used to calculate the small
particle coverage variance. However, for b/a > 2 the
equation for the Laplace transform of ua(x) given by Eq.
(7), including all contributions to Na(s), becomes rather
impractical. In the limiting case b/a ≫ 1, when fluctua-
tions of the large particle gaps dominate the variance of
small-particle coverage, one gets a more compact result
shown in Fig. 5.
[1] G. J. Rodgers and Z. Tavassoli, Phys. Lett. A 246, 252
(1998).
[2] D. Boyer, J. Talbot, G. Tarjus, P. Van Tassel, and P.
Viot, Phys. Rev. E 49, 5525 (1994).
[3] A. V. Subashiev and S. Luryi, Phys. Rev. E 75, 011123
(2007).
[4] M. C. Bartelt and V. Privman, Phys. Rev. A 44, R2227-
R2230 (1991).
[5] M. K. Hassan, J. Schmidt, B. Blasius, J. Kurths, Phys.
Rev. E 65, 045103(R) (2002).
[6] M. K. Hassan and J. Kurths, J. Phys. A 34, 7517 (2001).
[7] N. A. M. Araujo, and A. Cadilhe, Phys. Rev. E 73,
051602 (2006).
[8] D. J. Burridge and Y. Mao, Phys. Rev. E 69, 037102
(2004).
[9] E. G. Coffman, Jr., L. Flatto, P. Jelenkovich, and B.
Poonen, Algorithmica 22, 448 (1998).
[10] M. Inoue, Phys. Rev. B 25, 3856 (1982).
[11] P. Calka, A. Mezin, P. Vallois, Stochastic Processes and
their Applications, 115, 983-1016 (2005).
[12] J. Talbot, G. Tarjus, P.R Van Tassel, P. Viot, Colloids
and surfaces A: Physicochemical and Engineering As-
pects B 165, 278 (2000).
[13] V. Privman, Colloids Surf A 165 231-240 (2000).
[14] R. Devanathan, L. R. Corrales, F. Gao, W. J. Weber,
Nuclear Instrum. Methods Phys. Res. A 565, 637-649
(2006).
[15] H. Spieler, Semiconductor Detector Systems, Oxford Uni-
versity Press, 2005.
[16] C. Klein, J. Appl. Phys. 39, 2029 (1968).
[17] W. van Roosbroeck, Phys. Rev. 139, A 1702 (1965).
[18] This correlation originates from the basic fact that a
simple random division of a segment in two parts pro-
duces highly correlated pieces: if one is short the other
is long and vice versa. Energy branching by impact ion-
ization evidently has the similar property, as the sum of
secondary-particle energies is fixed by energy conserva-
tion. This type of correlations was first pointed out by
Ugo Fano in 1947 [19] and bears his name.
[19] U. Fano, Phys. Rev, 72, 26 (1947).
[20] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regu-
lar Variation, Cambridge University Press, Cambridge,
1987.
[21] A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 3 109
(1958); Trans. Math. Stat. Prob. 4, 205 (1963).
[22] J. K. Mackenzie, Journ. Chem. Phuys. 37, 723 (1962).
[23] S. Luryi and A. V. Subashiev, unpublished.
[24] G.D.Alkhazov, A.A. Vorob’ev, A.P Komar, Nucl. Instr.
Meth. 48, 1-12 (1967).
[25] P. E. Nay, Annals of Mathematical Statistics, 33, 702-718
(1962).
[26] Sh. Kogan, Electronic Noise and Fluctuations in Solids,
Cambridge University Press, Cambridge, 1996.
[27] M. Lal and P. Gillard, Math. Computation 28, 562
(1974).

  Competitive random sequential adsorption on a line from a binary mix of
incident particles is studied using both an analytic recursive approach and
Monte Carlo simulations. We find a strong correlation between the small and the
large particle distributions so that while both partial contributions to the
fill factor fluctuate widely, the variance of the total fill factor remains
relatively small. The variances of partial contributions themselves are quite
different between the smaller and the larger particles, with the larger
particle distribution being more correlated. The disparity in fluctuations of
partial fill factors increases with the particle size ratio. The additional
variance in the partial contribution of smaller particle originates from the
fluctuations in the size of gaps between larger particles. We discuss the
implications of our results to semiconductor high-energy gamma detectors where
the detector energy resolution is controlled by correlations in the cascade
energy branching process.

Fluctuations of the partial filling factors in competitive RSA from binary mixtures
Arsen V. Subashiev and Serge Luryi
Department of Electrical and Computer Engineering,
State University of New York at Stony Brook, Stony Brook, NY, 11794-2350
Competitive random sequential adsorption on a line from a binary mix of incident particles is
studied using both an analytic recursive approach and Monte Carlo simulations. We find a strong
correlation between the small and the large particle distributions so that while both partial con-
tributions to the fill factor fluctuate widely, the variance of the total fill factor remains relatively
small. The variances of partial contributions themselves are quite different between the smaller and
the larger particles, with the larger particle distribution being more correlated. The disparity in
fluctuations of partial fill factors increases with the particle size ratio. The additional variance in the
partial contribution of smaller particle originates from the fluctuations in the size of gaps between
larger particles. We discuss the implications of our results to semiconductor high-energy gamma
detectors where the detector energy resolution is controlled by correlations in the cascade energy
branching process.
PACS numbers: 02.50.Ey, 05.20.-y, 68.43.-h, 07.85.Nc
I. INTRODUCTION
One dimensional irreversible random sequential ad-
sorption (RSA) has been of interest for several decades.
Its numerous extensions include RSA with particles ex-
panding in the adsorption process [1, 2, 3], two-size par-
ticle adsorption [4, 5, 6, 7], and also RSA with an ar-
bitrary particle-size distribution function [8]. The inter-
est is due to the relevance of this process to a number
of physical phenomena in different fields of application,
such as information processing [9], particle branching in
impact ionization [10] and crack formations in crystals
under external stress [11]. The simplest example of RSA
is the so-called car parking problem (CPP). In the con-
text of CPP, one studies the average number of particles
(“cars”) adsorbed on a long line and the variance of this
number. Equivalently, one is concerned with the distri-
bution function for the size of gaps between the parked
cars (see Refs. [12, 13] for the review).
The problem of competitive RSA from a binary mix-
ture is of special interest because of the non-trivial corre-
∗subashiev@ece.sunysb.edu
lations in both the particle and gap-size distributions, de-
veloped during the deposition. These correlations mani-
fest themselves in the final irreversible state correspond-
ing to the so-called “jamming limit” — when every gap
capable of adsorbing a particle has done so. Numerous
studies, reported in the literature for the binary-mixture
RSA in the jamming limit, addressed the problem of cor-
relations only indirectly, through its manifestation in the
fill factor or the gap distribution. Available results in-
clude binary mixtures with point-like particles [4, 5] and
those with a relatively small particle size ratio, b/a < 2
[6]. Also available are Monte-Carlo studies of the fill
factor and the gap-size distribution for a binary-mixture
deposition with equal abundance of both particles [7].
The present study is concerned with the correlation
between the fluctuations in the number of adsorbed par-
ticles of each kind from a two-size binary mixture, as
well as with their partial contributions to the fill factor.
We present both analytical results and those obtained
by Monte-Carlo simulations for a wide range of binary-
mixture compositions and size ratios.
We are interested in the RSA problem primarily be-
cause of its relevance to the propagation of high-energy γ-
http://arxiv.org/abs/0704.1235v3
mailto:subashiev@ece.sunysb.edu
particles through a semiconductor crystal — with parti-
cle energy branching (PEB) due to cascade multiplication
of secondary electrons and holes [10, 14, 15, 16, 17]. The
correlation of energy distribution between secondaries is
quite similar to that of the gap distribution in the RSA
process [18]. In both cases, the ratio of the variance of
the final number of particles to the average particle num-
ber in the final (jamming) state can be much less than
unity, which is favorable for the detector energy resolu-
tion. This ratio (which would be unity if the particle
number obeyed a Poisson distribution) is called the Fano
factor, Φ [19].
The reported attempts to evaluate Φ employed over-
simplified models of the semiconductor band structure.
In such models, all crystal properties are characterized
by three parameters, namely, the band gap, the phonon
frequency, and the ratio of the rate of phonon emission to
that of impact ionization. The price of this oversimpli-
fication had been that correspondence with experiment
could be achieved only by assuming unphysically large
rates of phonon losses (about 0.5 eV per created e-h pair).
This does not corroborate with the known values for the
ratio of the impact ionization and the phonon emission
probabilities for high-energy electrons in semiconductors.
The model furthermore obscures the role of features in
the band structure and the ionization process that are
specific to a particular semiconductor.
In our earlier work [3], we used an extended RSA model
of particles that expand or shrink upon adsorption. The
shrinking model is relevant to the PEB problem in that it
helps to elucidate such factors as the non-constant den-
sity of states in the semiconductor band and the fact that
due to momentum conservation the ionization threshold
is larger than the actual (bandgap) energy that is lost in
impact ionization.
The recursive technique employed in Ref. [3] allowed
us to assess the accuracy of approximate approaches to
the yield and variance calculations (such as, e.g., the
average-loss approach of Refs. [15, 16]).
In the present work, the RSA model is extended in a
different direction — competitive deposition of different-
size particles from a binary mixture — that is suitable
to simulate the role of multiple channels of pair produc-
tion, owing to the multi-valley nature of semiconductor
bands. We arrive at a number of qualitative conclusions
that should be taken into account in both the interpre-
tation of experimental data and the choice of the crystal
composition and device structure in gamma detectors op-
timized for energy resolution.
The paper is organized as follows. Section II presents
the basic equations of the recursive approach and the an-
alytical results for the fill factor and its variance for the
larger particles. In Sect. III, we analyze the results that
demonstrate high correlation in the particle distribution.
Based on the gained understanding, we formulate in Sec.
IV the implications of our results for the Fano factor of
semiconductor γ detectors. Our conclusions are summa-
rized in Sect. V. Certain analytical results are derived in
the Appendix.
II. PARTIAL CONTRIBUTIONS TO THE FILL
FACTOR AND ITS VARIANCE FOR TWO-SIZE
RSA PROBLEM
We consider the problem of competitive deposition
from a binary mixture of particles with sizes a and b,
whose relative contributions to the total flux on the ad-
sorbing line are q and p = 1 − q, respectively. We shall
use a recursive approach to first study the mean number
of particles na(x) and nb(x), adsorbed on a line of length
x (in the jamming limit), and then the corresponding
variances.
Consider a large enough empty length x > a, b. We
assume that the adsorption is sequential, i.e. only one
particle is adsorbed at a time. The first adsorbed par-
ticle will be of size a with the probability of landing at
any point q(x − a)/(x − l) or of size b with the landing
probability p(x − b)/(x − l). Here l = qa + pb is the
“average” particle size in the binary flux. After the first
particle is adsorbed, it fills a certain interval [y, y+a] (or
[y, y + b]), and leaves two independent segments, whose
combined size is either x− a or x− b. The average num-
bers of a-particles na(y) and na(x− y− a) (or na(y) and
na(x−y−b)) will be subsequently adsorbed in these gaps.
Thus, the recursion relation is of the form
na(x) =
q(x− a)
[1 + na(y) + na(x− a− y)]
p(x− b)
[na(y) + na(x− b− y)] ,
where the first and the second terms (upper and lower
lines) correspond to the cases of the first landed particle
being a particle of sort a or b, respectively. These cases
must be averaged over all possible landing coordinates y
of the first particle in a different way, viz. for a first,
< na(y) >a=
∫ x−a
na(y)dy ,
whereas for b first,
< na(y) >b=
∫ x−b
na(y)dy .
Performing the average and using the symmetry between
left and right segments we obtain, finally:
na(x) =
q(x − a)
∫ x−a
na(y)dy
∫ x−b
na(y)dy. (1)
A similar equation holds for the particles of size b:
nb(x) =
p(x− b)
∫ x−a
nb(y)dy
∫ x−b
nb(y)dy. (2)
With the help of Eqs. (1,2) one can readily derive an
equation for the average total covered length f(x), de-
fined as f(x) = ana(x) + bnb(x), giving
f(x) =
xl − qa2 − pb2
∫ x−a
f(y)dy
∫ x−b
f(y)dy. (3)
Equation (3) agrees with that of Ref. [8] for the total cov-
ered length in RSA from a multi-size mixture. However,
the advantage of Eqs. (1,2) is that they permit studying
the partial contributions to the coverage by each of the
two sorts of particles separately.
Note that the symmetry between the a- and the b- par-
ticles is broken by the initial conditions. To be specific,
let b > a. Then, for b-particles the boundary condition
at small x is simply
nb(x) = 0, 0 ≤ x < b (4)
whereas for a-particles we have
na(x) =
0, 0 ≤ x < a
1, a < x ≤ min(2a, b)
For b > 2a, Eq. (5) should be supplemented with
na(x) = 1 +
∫ x−a
na(y)dy (6)
Eq. (6) accounts for the deposition of smaller particles in
small gaps where the larger particle does not fit. Clearly,
this process is not influenced by the b-particles and does
not involve particle competition.
More refined arguments are needed to derive the sec-
ond moment of the distribution, i.e. the expected value
of the square of the number of particles of a given sort,
ua(x) = En
a(x). It may not be a priori evident that one
can write independent expressions for particles of both
sorts, because parameters a and b not only describe the
particle size but also designate the sort of a particle. In-
deed, we can even have a = b and distinguish the parti-
cles by some other parameter, like “color”. Our approach
should remain valid in this case too. To be rigorous, we
therefore introduce an artificial parameter, the “mass” of
a particle, ma and mb, whose value may depend on the
particle shape and is simply proportional to the particle
length only for a fixed transverse particle size. Hence one
can regard ma and mb as independent parameters.
Consider a total mass M(x) = mana(x) + mbnb(x)
of the particles adsorbed in a line segment x. We first
evaluate recursively the mean square of the total mass
< M2(x) >= 〈[mana(x)+mbnb(x)]
2〉, and then calculate
the second partial derivatives with respect to ma and mb.
Using the landing probabilities of particles to perform the
averaging, we obtain
ua(x) = (x− l)
q(x− a) + 2q
∫ x−a
ua(y)dy
∫ x−b
ua(y)dy + 4q
∫ x−a
na(y)dy
∫ x−a
na(y)na(x− y − a)dy
∫ x−b
na(y)na(x− y − b)dy
Similarly, equation for ub(x) reads
ub(x) = (x− l)
p(x− b) + 2q
∫ x−a
ub(y)dy
∫ x−b
ub(y)dy + 4p
∫ x−b
nb(y)dy
∫ x−a
nb(y)nb(x − y − a)dy
∫ x−b
nb(y)nb(x − y − b)dy
We could have derived Eqs. (1,2) in a similar way,
by first evaluating the total average mass M(x) =
mana(x) +mbnb(x) recursively, and then calculating the
derivatives. For a more general case, when the total mass
is a linear functional M(x) =
mlnl(x)dl on the mass
distribution ml, one would have to use variational deriva-
tives δM(x)/δml. For the case of binary mixtures we
consider, partial derivatives are sufficient.
Similarly, we derive an equation for the correlation
function uc(x) = 〈na(x)nb(x)〉 by calculating a mixed
derivative of < M2(x) > with respect to ma and mb.
For particles uniform in the transverse direction with unit
mass density, both the mass and the length of particles
are identical, which gives a way to check the equations.
An appropriate linear combination of equations for ua,
ub, and uc then gives an equation for the variance of the
total filled length or, equivalently, for the variance of the
wasted length, w(x) = x − f(x). The resulting equation
can also be obtained directly, by applying recursion ar-
guments to the waste. The identical results obtained can
be viewed as an additional proof of Eqs. (7,8).
Note the asymmetry in the 4-th terms of Eqs. (7,8)
that are proportional, respectively, to 4q and 4p. These
terms ensure the correct (linear) asymptotic behavior of
the variance at large x.
An important feature of Eqs. (1,2,7,8) is that in spite
of the competitive character of the deposition of particles
of different sorts, the equations for na, nb and the higher
moments are independent. This is rooted in the fact that
a single deposition step on an empty length x does not
depend on the already adsorbed particle distribution.
Due to the self-averaging nature of the filling length
(and waste length) in the limit x → ∞ the averaged
(hence approximate) recursion equations yield exact re-
� � � � � � ���
�
������
� � �� �� ��
b/a = 2.4
� ��� !"#$%&'() *+,-./012 3456789:; <=>?@ABCD EFGHIJ
Adsorbtion length,  x  
FIG. 1: (color online) Average number of adsorbed particles
na (solid lines) and nb (dashed lines) as functions of the length
x (measured in units of a) of the adsorption interval, assumed
initially empty. The results are obtained by iterating Eqs.
(1,2) with the assumed ratio of the particle size b/a = 2.4
and the varying fraction q of a-particles in the flux.
sults. The recursive technique is in this sense equiva-
lent to the alternative “kinetic” approach to RSA that
is sometimes regarded as a higher-level theory. In the
kinetic approach one considers the rate equation that
describes the sequential deposition of particles with the
particle distribution on a line characterized by a time-
dependent function G(x, t) representing the average den-
sity of gaps whose size is between x and x + dx [2, 5].
It has been ascertained for a number of problems that
both approaches give the same result for the coverage.
Still, each has its own benefits. The kinetic approach
allows studying the temporal variation of a state with
specified particle distribution. The recursive approach,
while simulating a simplified version of the kinetics, al-
lows to study more complex effects, such as variance of
the adsorbed particles of different size.
Evaluation of na(x) and nb(x) is readily done by re-
peated iterations of Eqs. (1,2), going from the small to
progressively larger lengths x. Results of the numerical
recursion are shown in Fig. 1 for a particle size ratio
b/a = 2.4 and varying q.
The noteworthy features of the functions na(x) and
nb(x) are (i) the step-like features at x = a, x = b
(which are replicated with ever smaller amplitudes at
x = na +mb, where n and m are integers), (ii) the dip
in the number of small particles na(x) at x = b, which
increases with p, and (iii) the reduction of nb with in-
creasing q. We also note that for all q the behavior of
both na(x) and nb(x) becomes very close to linear al-
ready at x ≈ 7.
The asymptotic behavior of na(x) and nb(x) at large
x can be obtained by multiplying Eqs. (1,2) by x− l and
taking the derivative with respect to x. The resulting
differential equations are satisfied by linear functions of
the form
na = αa(x+ l)− q, nb = αb(x+ l)− p, (9)
where αa and αb are arbitrary constants. When correctly
chosen (by matching to the recursive solution) these con-
stants become the partial filling factors. After the match-
ing is done, the total filled length in the asymptotic limit
is given by f(x) = θx + (θ − 1)l, where θ = aαa + bαb is
the specific coverage. It is worthwhile to stress that the
value of the asymptotic solutions (9) consists precisely in
that they are asymptotically exact. Hence they provide
a sanity check on any solution we could have obtained by
a numerical recursion up to moderate values of x.
Similarly, Eqs. (7-9) yield the variances at large x,
ua − n
a = µa(x+ l)− qp[1 + (b − a)αa]
2, (10a)
ub − n
b = µb(x+ l)− qp[1− (b− a)αb]
2. (10b)
Again, these solutions are asymptotically exact; they sat-
isfy Eqs. (7,8) with arbitrary values of µa and µb, pro-
vided of course that na(x) and nb(x) are in the correct
asymptotic form (9) with properly chosen [i.e., satisfying
Eqs. (1,2)] coefficients αa and αb. In principle, we could
now follow a procedure similar to above, viz. determine
µa and µb by matching Eqs. (10) against a numerical
recursive solution at some moderate value of x. How-
ever, it would be rather difficult to control the numerical
accuracy in this procedure, because of the difference of
nonlinear functions that enter Eqs. (10), even though
that difference itself behaves linearly with x at large x.
Fortunately, our model admits of an exact solution
based on the use of Laplace transformation (details can
be found in [3] and references therein). Below we present
an exact evaluation of variance for particles of larger size,
while details of similar though lengthier calculations for
smaller particles are presented in Appendix.
Firstly, we need exact solutions of Eqs. (1,2). To ob-
tain these, we substitute x → x + b in Eq. (2) and mul-
tiply it by x − l. Taking the Laplace transformation of
the resulting equation and using the boundary condition
(4), we obtain
+ b− l
ebsNb(s) =
qe(b−a)s + p
Nb(s)
Here Nb(s) is the Laplace transform of nb(x),
Nb(s) =
e−sxnb(x)dx, (12)
Rearranging the terms and multiplying by e−bs, we put
Eq. (11) into the form
N ′b(s)+
qe−as + pe−bs
Nb(s) = −
e−bs. (13)
For p → ∞, the solution of Eq. (13) is, asymptotically,
Nb(s)|s→∞ =
s(b− l)
e−bs, (14)
as follows from the known variation of nb(x) ≈ p(x −
b)/(b− l) at small x− b. Hence we have
Nb(s) =
p exp(−l s)
s2β(s)
e−q(b−a)tβ(t)dt, (15)
where
β(s) = exp
1− q exp(−at)− p exp(−bt)
To find the asymptotic behavior of nb(x) at large x, it
is convenient to use Karamata’s Tauberian theorem for
the asymptotic growth rate of steadily growing functions
(see e.g. [20], p. 37). According to the theorem, the
asymptotics of nb(x) [or na(x) or their variances] can be
readily obtained (by taking the inverse Laplace transfor-
mation) from the Laurent power series expansion of the
Laplace transforms of these functions at small s (see [9]
for the mathematical details of this analysis).
Function Nb(s) is analytic at all s 6= 0 and at s = 0 it
has a second-order pole with the following asymptotic
Nb(s) =
αb,0l − p
+O(s), (17)
where
αb,0 = p
e−q(b−a)sβ(s)ds. (18)
To calculate nb(x) at large x, we take the inverse Laplace
transformation of (17). This gives
nb(x) = αb,0(x + l)− p, (19)
with an exponentially small error term, in line with the
asymptotics given by Eq. (9).
In the limit p = 1, equation (18) duly gives the so-
called jamming filling factor R for the standard RSA,
αb,0(l = 1) ≡ R = 0.74759 · · · (also called the Renyi
constant [21]). In the limit a → 0, Eq. (18) recovers the
results of Refs. [4, 5] for the coverage of a line from a
binary mixture of finite size particles and point defects.
Moreover, Eq. (18) gives the large particle contribution
to the total coverage, obtained in [6, 8] for the range
a < b < 2a. Here we see that this result remains valid
for arbitrary a < b.
Next, we perform similar manipulations with Eq. (8)
and obtain an equation for the Laplace transform of the
variance Ub(s) = L̂[ub(x)], viz.
U ′b(s)+
qe−as + pe−bs
Ub(s) = −
exp(−bs)
Rb(s),
where
Rb(s) = p+ 4psNb(s) + 2s
2N2b (s)
qe(b−a)s + p
, (21)
with Nb(s) defined by Eq. (15). The solution of Eq. (20)
can be written in a form similar to Eq. (15), namely
Ub(s) =
exp(−sl)
s2β(s)
β(t)e−q(b−a)tRb(t)dt. (22)
The integrand in the right-hand side of Eq. (22) is pro-
portional to 1/t2 causing the integral to diverge as 1/s
for s → 0. This is due to the square-law dependence of
u(x) at large x .
To separate the regular part needed for the estima-
tion of variance, we note that at small t one has Rb(t) ∝
αb,0t
−2. Moreover, the series expansion shows that the
difference β(t) exp[−q(b− a)t]Rb(t)− 2α
−2 is regular
at t → 0. Therefore, it is convenient to define an en-
tire function κb(t) = β(t) exp[−q(b−a)t]Rb(t)−2α
In terms of this function, the solution Ub(s) can be ex-
pressed as follows:
Ub(s) =
exp(−sl)
s2β(s)
+ kb,0 −
κb(t)dt
, (23)
where
kb,0 =
κb(t)dt. (24)
To apply Karamata’s Tauberian theorem, we note that
the asymptotic expansion of Ub(s) near its third-order
pole is of the form
Ub(s) =
kb,0 + 2α
kb,0l − κb(0)− qp(b− a)
2α2b,0
. (25)
Taking the inverse Laplace transformation, we find the
asymptotic form of ub(x):
ub(x) = α
2 + (kb,0 + 2α
b,0l)x
+kb,0l − κb(0)− qp(b− a)
2α2b,0, (26)
with an exponentially small error term. Using Eq. (19)
to subtract n2
(x), we find an equation of the form (10b)
with µb = kb,0 + 2pαb,0. The specific variance of the
adsorbed number of b-particles is given by (at x → ∞)
µb = αb,0(1 + 2p) + 2
β(s)sNb(s)e
2pe−bs
+sNb(s)
qe−as + pe−bs
α2b,0
ds. (27)
Integrating by parts the last term and rearranging the
result, we finally obtain
µb = αb,0(1− 2p) + 4p
αb(u)
e−bu (1− qe−au
−pe−bu
du+ 2
β(u)u2
e−luK(u)du, (28)
where
K(u) = qe−au
2(1− qe−au − pe−bu)− (a+ l)u
+pe−bu
2(1− qe−au − pe−bu)− (b+ l)u
αb(u) = αb,0 − p
e−q(b−a)yβ(y)dy. (30)
In the limit of small p → 0, the Fano factor Φ =
µb/αb,0 → 1. In this limit, large particles are distributed
on the line randomly, without correlations. In the oppo-
site limit, p = 1, Eq. (28) reduces to the standard RSA
result, first obtained for a lattice RSA model by Macken-
zie [22]. The numerical value of the Mackenzie constant,
µ0 = 0.0381564 · · · , corresponds to Φ = 0.0510387 · · · ,
see [9]. Expression (28) for the larger particles has the
same structure as the corresponding formula in the stan-
dard RSA model (fixed-size CPP). Due to the exponen-
tial factors in the integrands of Eq. (28), the dependence
of µb on a for a ≪ b is quite weak. The limiting value of
the specific variance for a/b → 0 gives the specific vari-
ance of the fill factor for the case of finite-size particles
(b = 1) mixed with point-size particles,
µb,p = αb,p,0(1− 2p) + 4p
αb,p(u)
e−u (1− e−u) du
b,p(u)
βp(u)u2
{qe−pu (2− 2e−u − u)
+e−(1+p)u [2p (1− e−u)− (1 + p)u]
du, (31)
where αb,p,0 is the fill factor for this case,
αb,p(u) = p
e−qyβp(y)dy, αb,p,0 = αb,p(u = 0)
βp(u) = exp
1− exp(−t)
. (33)
It is worth to note that Eqs. (2,8) and their solutions can
be readily generalized to the case when particles of the
smaller size have an arbitrary distribution in the interval
[a1, a2] so long as a2 ≤ b [23].
The above analytic results for the variance of larger
particles are essentially exact, as will be confirmed in the
next Section by Monte Carlo simulations. For the smaller
particles, the calculations are messier and accurate ana-
lytical results can be obtained only in a certain range of
particle size ratios. Estimations of the variance for small-
size particles are further discussed in the Appendix.
III. DISCUSSION OF THE RESULTS,
COMPARISON WITH MONTE CARLO
MODELING
Here we present the results of numerical calculations
using both the analytical expressions obtained in the pre-
ceding section and Monte Carlo simulations. For large-
size particles the Monte Carlo results are very close to
analytical expressions both for the fill factor and the vari-
ance, so we shall not dwell on their comparison. For
small-size particles, especially in the range 2 < b/a < 8,
analytical calculations are rather unwieldy, so Monte
Carlo simulations become indispensable. Larger size ra-
tios, b/a > 8, lend themselves to an approximate ana-
lytical approach (see Appendix). In this case, we use
the Monte Carlo to estimate its accuracy for the small
particle contribution.
Traditional studies of the generalized RSA via Monte
Carlo simulations follow a temporal sequence of events.
For the case of adsorption on a line of the length x from
a binary mixture, one step of the sequence comprises:
(i) selection of a particle from the mixture according
to the deposition flux ratio (with the probability q of
choosing the small-size (a) particle, and the probability
p = 1− q of selecting a particle of larger size b);
(ii) random choice of a deposition coordinate of particle
center on the line x with formerly deposited particles;
(iii) rejection of the particle if it overlaps by any part
with formerly deposited particles or with the line bor-
ders; otherwise, the particle deposition proceeds with the
formation of two new disconnected adsorption lengths.
This traditional approach has several drawbacks, that
make the modeling very demanding, both in terms of the
computer time and memory allocation.
Firstly, both the filled length in the jamming limit and
the specific fill factor (coverage) depend on the initial
length. Due to the self-averaging property of the coverage
it tends to a unique exact value in the limit x → ∞. To
obtain the accuracy of about 0.1 %, the common strategy
has been to use large initial length values (105b -107b) and
make additional averaging over a set of about NR =100-
1000 different realizations.
Secondly, as time evolves and the jamming limit is ap-
proached, the probability of finding a free gap for parti-
cle deposition becomes greatly reduced, so that the ad-
sorption time tends to infinity. The process is termi-
nated when variations of the adsorbed particle number
are smaller than those required by the desired accuracy.
The recursive analysis of the generalized RSA suggests
a revision of the above scheme. Since the deposition is
random and sequential, it does not depend on the tem-
poral history of the process or the growing number of
rejected particles and their coordinates. Therefore one
step of the sequence can be chosen as follows:
(i) selection of any free deposition length, l1 > a. It is
convenient to choose for l1 the outermost free deposition
length on the left-hand side.
(ii) if l1 < b, then particle of size a is deposited,
otherwise the deposited particle is chosen according to
the landing probability, given by q(l1 − a)/(l1 − l) for
a-particle and p(l1 − b)/(l1 − l) for b-particle, where
l = qa+ pb.
(iii) random choice of a deposition coordinate (taken
as the coordinate of particle’s left end) on the line l1 for a
given particle size, i.e. within the interval l1−a for a-size
or within l1 − b for b-size particle, with the formation of
two new adsorption lengths from the initial length l1.
It is readily seen that although the sequence of deposi-
tion events is different from the actual temporal sequence
of adsorption (the simulated deposition proceeds by se-
quentially filling the left-hand lengths), the statistics of
divisions is identical and therefore so is the final distri-
bution of the gaps, as well as all statistical properties
of the jamming state. Our sequential scheme excludes
deposition of to-be-rejected particles and therefore is in-
comparably faster. Besides, it terminates exactly when
the jamming limit (with no gaps larger than unity) is
achieved. Direct comparison with the traditional Monte
Carlo results, e.g. [5, 7, 8] exhibits total agreement. The
difference in the calculation time is especially evident for
small (close to zero) q: in the time scale of “real” deposi-
tion, the jamming limit will be strongly delayed because
of the rarity of events with small particle chosen. In our
modified approach, all gaps smaller than b are “rapidly”
populated by small-size particles, however small be the
value of q.
The next step of the revision is to exploit the fact
(proven analytically in the preceding section) that in the
jamming limit, the linear dependence on the adsorption
length of both the average filled length and its variance is
exponentially accurate, starting from a reasonably short
length, certainly not exceeding x ≈ 10b. Since this lin-
ear dependence has only two parameters [actually only
one, as the parameter ratio is exactly fixed by analyti-
cal considerations, Eqs. (9,10)], both the coverage and
the variance can be determined with Monte Carlo simu-
lations of short samples.
To be sure, in order to achieve the same accuracy as
that obtained for long samples, the results should be av-
eraged over a sufficient number of realizations NR. This,
however, takes little memory or time. Calculations show
similar accuracy for different x and NR, so long as their
product x×NR is fixed. The results presented below were
KLM NOP QRS TUV WXY Z[\]^_`abcde
fghijklmn
opqrstuvw
  b/a :
Ratio of  small particles (a) in the flux, q
    2;      5
    10;    20
{|} ~�� ��� ��� ��� ���
���������
��������
�� ¡¢£¤¥¦
total
FIG. 2: (color online) Partial contributions of small and large
particles to the total coverage depicted as functions of q, for
different particle-size ratios b/a in the flow. Open points cor-
respond to the limit b/a ≫ 1, as described by the analytical
formulae (32) for b-particles and (A13) for a-particles. For the
total coverage, the open circles to the wasted length product
approximation, Eq. (34).
obtained using a sample of size x = 200a for b/a < 10
and x = 400a for b/a = 20, 40, subsequently averaged
over 10 000 realizations, which appeared to be sufficient
to eliminate any spread of the results in the graphical pre-
sentation (producing an accuracy of better than 0.1%).
The use of small samples is very effective in reducing
the calculation time (with an ordinary PC, high-accuracy
results can be obtained in minutes, compared to days in
the traditional scheme [8]).
Figure 2 shows partial contributions to the coverage as
functions of the fraction q of small particles in the binary
mixture at different ratios of particle size. As q increases,
the coverage with large particles is substituted by that
with small particles, producing some decrease in the to-
tal coverage. In the regions of corresponding parameters,
our results reproduce those of reported analytical calcu-
lations (i.e. for b/a < 2 [5, 8] and for a = 0 [6] for the
large particle contribution) and those obtained by the
Monte Carlo simulations of [5, 7, 8], demonstrating the
ª«¬ ­®¯ °±² ³´µ ¶·¸ ¹º»¼½¾¿ÀÁÂÃ
ÄÅÆÇÈÉÊËÌÍÎÏ
ÐÑÒÓÔÕÖ×ØÙÚÛ
      b/a :
 infinity
Ratio of small particles in a flux, q
FIG. 3: (color online) Variance of the partial coverage by ad-
sorbed b-particles from the binary mixture for different values
of the particle size ratio b/a in the flow.
validity of our revised approach.
It is evident from Fig. 2, that the total coverage in-
creases at smaller q, as can be explained by sequential
deposition of the two kinds of particles. In the regime of
small q, large particles are adsorbed first and their de-
position, unobstructed by small particles, is tight. Sub-
sequently, the small particle fill the gaps between large
particles and this clearly reduces the total wasted length.
The effect of increasing the particle size ratio b/a is
pronounced only for b/a < 10, then it rapidly saturates.
Therefore, for large b/a, say b/a = 20 the coverage by
large particles is very close to that obtained for a model
mixture of point-like and finite-size particles [by formally
letting a = 0 in Eq. (18)]. Such a model, however, has lit-
tle relevance to any practical situation, because it simply
ignores the partial contribution of small particles to the
total coverage. The latter can be described analytically
in the limiting case b/a → ∞, Eq. (A13).
The partial contribution of small particles steadily
grows with the increasing size ratio due to the expand-
ing gaps between the large particles. In the limit q → 0,
the total coverage can be estimated by observing that
the specific wasted length in this case is a simple prod-
uct of the specific lengths wasted in initial deposition of
large particles and subsequent deposition of small par-
ticles, i.e. 1 − θ = (1 − θa)(1 − θb). Since for q = 0
the specific coverage θb = R and since for large size ra-
tios (when the gaps between large particles are large) the
specific coverage θa = R, we have θ = 1−(1−R)
2=0.936,
in agreement with the results reported in the literature
[5, 7]. However, the sequential nature of the deposition
suggests that the entire q dependence of the total θ can
also be approximated by a product of the specific wasted
lengths in the competitive deposition of large particles
q(1 −R) and subsequent deposition of small particles in
the remaining gaps, which gives
θ = 1− (1−R)(1− pR) = R[1 + p(1−R)]. (34)
This product-waste approximation is shown in Fig. 2 by
the open circles.
Next, we concentrate on the specific waste variance and
the Fano factor. We shall discuss the b- and a-particles
separately, since the effects are rather different in na-
ture and also since they have been evaluated by differ-
ent techniques. Results for large particles are obtained
by numerical integration of Eq. (30) and confirmed by
Monte Carlo simulations. Results for a particles are ob-
tained by Monte Carlo stimulations and are accompanied
by analytical expressions in the limit b/a ≫ 1.
Variance, µ̃, of the partial contribution of b-particles
to the total coverage is shown in Fig. 3 for different par-
ticle size ratios. Unlike the particle number variance µ,
the variance of coverage, µ̃ = µb, depends only on the
size ratio b/a and does not directly scale with b. It is
therefore more indicative of the effect of decreasing size
àáâ ãäå æçè éêë ìíî ïðñòóô
õö÷øùú
ûüýþÿ�
����� �� �	
�� 
�������� �� � ����� �
)*+ ,
456789:;
<= > ? @ A B C D E F G
FIG. 4: (color online) The Fano factor for the number of ad-
sorbed b-particles from the binary mixture as functions of q,
for different values of the ratio b/a of particle sizes in the
flow. Dotted line corresponds to a random particle packing
on a lattice with suitable lattice constant (aka monomer ad-
sorption).
of small particles on the fluctuations of the number of
large particles. At q → 0, when the adsorption of large
particles is unconstrained by small particles, the vari-
ance of large particles is minimal and corresponds to the
highly correlated distribution [18] in the standard CPP
problem (one-size RSA). The variance rapidly increases
with q as the small particle deposition destroys the CPP
correlations. The maximum of this effect is shifted to
larger q values for larger b/a. For q approaching unity,
the variance decreases simply due to the decrease of the
average number of adsorbed b-particles.
Correlation effects are more adequately characterized
by the Fano factor Φb, shown in Fig. 4. With the increas-
ing number of competing small particles in the flux, the
Fano factor grows from the smallest value Φ = 0.051 · · · ,
corresponding to the one-size RSA problem, to unity in
the limit q → 1. Small coverage by the large particles in
the latter limit means that they are distributed randomly
on the line, so that Poisson statistics recovers. The most
noticeable effect is a rapid decrease of the Fano factor
0.0 0.2 0.4 0.6 0.8 1.0
Small particle contribution to the flux, q
              b/a:
 1.6;    2
 5;       10
 20;     40
FIG. 5: (color online) The Fano factor for adsorbed a-particles
from the binary mixture for different values of the particle size
ratio b/a in the flow. Open points show the contribution of
fluctuations of the gap sizes between large particles.
with 1− q, manifesting a strong enhancement of the cor-
relation effects in the large particle distribution. These
correlation effects become exhausted only near q ≤ 0.1.
The correlation effects increase with b/a but saturate at
about b/a = 20.
Figure 5 shows the Fano factor for a-particles com-
petitively deposited along with large particles. The re-
sults are strikingly different at all q 6= 1 (when Φa = Φ,
as expected). While the distribution remains correlated
(Φa ≤ 1) for small ratios b/a ≤ 5, at larger b/a one has
Φa > 1, almost for all q, which means that the number
of small particles per unit length is strongly fluctuating.
This is due to the widely fluctuating size of the gaps
available for small particle deposition between large par-
ticles. For large values of b/a and in the entire range
of q, the Fano factor Φa can be approximated in terms
of the fluctuations of the coverage by the large particles,
viz. Φa = (b/a)µb,pR
2/θa, where µb,p is given by Eq.
(31) and θa by Eq. (A13). This approximation, which
neglects fluctuations of the density of adsorbed a parti-
TUV WXY Z[\ ]^_ `ab cde
fghij
klmno
pqrst
��� � ���
� ���� ����
§¨©ª«¬ ­®¯°±
FIG. 6: (color online) Variance of the partial number of ad-
sorbed a- and b-particles and of the total number of adsorbed
particles for b/a = 1.2. Also shown is the fluctuation correla-
tion function fcor.
cles in the gaps, is shown in Fig. 5 by open points. This
contribution is proportional to b/a and for b/a > 10 it is
evidently dominant.
For the particle energy branching process at small
b/a < 2, both the variance of the partial numbers of small
and large particles and the total number variance are of
importance. We shall illustrate this point in the instance
of b/a = 1.2 shown in Fig. 6. We see that at q ≈ 0.5 the
fill factor fluctuations are larger for a particles and some-
what smaller for b particles, but both are pretty large,
compared to the variance of the total number of adsorbed
particles. This is due to the strong anti-correlation in
their distribution, as evidenced by the specific fluctua-
tion correlation function, fcor = x
−1〈δnaδnb〉, also plot-
ted in Fig. 6. We note that fcor < 0, which means that
any excess in the number of a-particles is accompanied
by a downward fluctuation in the number of adsorbed
b-particles. Importantly, the variance and the Fano fac-
tor for the total number of adsorbed particles does not
exceed substantially its value for the single-size RSA.
Note the asymmetry of the curves for a and b parti-
cles, e.g. the variance of large particles goes to zero as
q → 1 whereas that of small particles remains finite even
as q → 0. This is a feature of our model that allows ”infi-
nite” amount of time for the deposition of small particles
in the gaps left after the deposition of large particles is
completed, but not vice versa. Therefore, the deposition
of small particles remains finite even in the limit of q → 0
and the same is true for the a-particle number variance.
Another interesting feature of the a-particle num-
ber variance, already evident from Fig. 5, is its non-
monotonic behavior as function of b/a at small q. This
variation is displayed directly in Fig. 7 that shows the
dependence of the Fano factor on b/a for q=0.05, 0.1
and 0.2 — where its non-monotonic nature is most pro-
nounced. The minimum of the Fano factor is achieved at
b/a ≈ 2. Note that the non-monotonic dependence of the
Fano factor is accompanied by non-monotonic variations
in the dispersion of the gaps between small particles. In
Ref. [7] it was found that for q =0.5 the dispersion is
noticeably reduced at b/a ≈ 1.55. These effects were
interpreted as a manifestation of the so-called “snug fit”
events, i.e. particle deposition in gaps that are just barely
above the unit length a. In contrast, the Fano factor for
b-particles and that for the total number of particles re-
main monotonic everywhere.
IV. SOME CONSEQUENCES FOR THE
ENERGY BRANCHING IN HIGH-ENERGY
PARTICLE DETECTORS
The model of RSA from binary mixtures is relevant
to an important practical problem of particle energy
branching (PEB) where high-energy particle propagates
in an absorbing medium and multiplies producing sec-
ondary electron-hole (e-h) pairs. Multiplication proceeds
²³´ µ¶· ¸¹º »¼½ ¾¿À ÁÂÃ ÄÅÆ ÇÈÉ
q  :     0.05
          0.1
          0.2
Particle size ratio,  b/a
FIG. 7: (color online) The Fano factor for adsorbed a- and
b-particles as functions of the particle size ratio b/a. Also
shown is the Fano factor for the total number of adsorbed
particles.
so long as the particle energy is above the impact ioniza-
tion threshold [15]. The energy distribution of secondary
particles is random to a good approximation.
The affinity between the two problems was fully rec-
ognized already in 1965 by van Roosbroek [17] (see also
[24]). The PEB process can be considered in terms of
a CPP if one identifies the initial particle kinetic energy
with an available parking length and the pair creation en-
ergy with the car size. Similarly, the kinetic energies of
secondary particles can be identified with the new gaps
created after deposition of a particle. Full equivalence
of PEB to CPP further requires that only one of the
secondary particles takes on a significant energy, which
corresponds to binary cascades [25]. Otherwise, one has
to consider a simultaneous random parking of two cars
in one event.
To estimate the particle initial energy in PEB, one
measures the number N of created electron-hole pairs.
Variance of this number, due to the random character of
energy branching and also due to random energy losses
in phonon emission, limits the accuracy of energy mea-
surements. Both the yield N and the e-h pair variance
var(N) = (N −N)2 are proportional to the initial en-
ergy. The ratio of the e-h pair variance to the yield, i.e.
the Fano factor of the PEB process, is a parameter that
quantifies the energy resolution of high-energy particle
detectors.
For semiconductor crystals, the PEB problem has ad-
ditional complications due to the energy dependence of
phonon losses and the energy dependence of the electron
density of states and the impact ionization matrix ele-
ment. Full quantitative analysis of the PEB is possible
only with detailed numerical calculations, which goes far
beyond the scope of the present article.
A common feature of the energy branching process in
semiconductors is the presence of several pair production
channels, associated with the multi-valley energy band
structure of the crystal. In Si, Ge and common A3B5
semiconductors, the e-h pair creation produces electrons
in one of the ellipsoids near the edge of the Brillouin zone,
in 100 (X) or 111 (L) directions. Owing to the difference
in the final densities of states and the matrix elements,
the impact ionization processes associated with X and L
valleys have different but competitive probabilities. Be-
cause of its low density of states, the Γ valley is usually
not competitive, even when it is the lowest valley.
Ultimately, electrons will end up in the lowest energy
valley but when the final electron valley is itself degener-
ate, as in Ge or Si, the resulting electron states may not
be fully equivalent, because of the different collection ki-
netics owing to the crystal anisotropy. This effect may
have important consequences for the observed variance.
For example, in Si diode detectors electrons are created
in 6 degenerate energy valleys that represent ellipsoids
of revolution elongated along (100) and equivalent direc-
tions in k-space. Suppose the diode structure is such that
the current flows along the (100) direction, as it is usually
the case. Electrons from the two valleys along the cur-
rent have a large mass and low mobility. The measured
current is hence dominated by electrons from the 4 val-
leys elongated perpendicular to the current that have a
low mass and high mobility along the current. Since the
choice of equivalent valley in the PEB process is fully ran-
dom, the number of high-mobility electrons will fluctuate
more strongly than the total number of generated carri-
ers. These fluctuations will dominate if the inter-valley
transition rate is low compared to the inverse collection
time. In the opposite limit of high inter-valley transition
rates, this effect will average out as the collected current
will fluctuate in time. The current fluctuation mecha-
nism due to the carrier escape into heavy-mass valleys is
a well-known source of noise in multi-valley semiconduc-
tors [26]). More detailed account for these effects will be
presented elsewhere [23].
Here we shall discuss an opposite situation, that is
common to direct-gap semiconductors, such as GaAs or
InP. In these materials, the lowest (Γ) electron valley has
a very low density of states, compared to that in the
satellite (X and L) valleys. Therefore, the probability of
electron generation in the Γ-valley can be neglected in
first approximation, so that the branching competition
occurs only between the satellite valleys of two different
kinds. Both the density of states and the threshold en-
ergy are different between X and L valleys and we can use
the results of the present study to interpret and predict
the consequences, at least qualitatively.
The binary-mixture RSA model interprets the higher
density of states as higher deposition rate and the higher
threshold as larger particle size. To make our conclu-
0.0 0.2 0.4 0.6 0.8 1.0
b/a = 1.4
Ratio of the small particles in the flux, q 
 Coverage
FIG. 8: (color online) Partial fill factors and the total coverage
for b/a = 1.4 as a function of q. Also shown is the total
number Ntot of adsorbed particles
sions more transparent, let us re-formulate the required
results in terms of a random parking problem with cars
of two sizes. We are now interested only in the numbers
of parked cars and the fluctuations of these numbers.
Several qualitative conclusions can be drawn from our
results:
(i) The total number of parked cars (in the jamming
state) will decrease with increasing fraction of larger
cars in the flow and with the growth of their size. For
b/a = 1.4 the effect is illustrated in Fig. 8 (which can
be viewed as an extension of Fig. 2). It follows from the
fact that adsorption of a large car excludes larger length
for subsequent parking events and thus causes a decrease
of the total fill factor. Note that the decrease in the to-
tal particle number is accompanied by an increase in the
total filled length, as smaller number of cars cover larger
area.
The next two conclusions (ii) and (iii), illustrated in
Fig. 6, are interconnected and will be discussed jointly.
(ii) Variance of the total number of parked cars and the
Fano factor will both grow with the increasing fraction of
larger cars in the flow and with the growth of their size.
(iii) Variance of the separate numbers of parked small
and large cars and their Fano factors are considerably
larger than that of the total number of cars. Therefore,
if for some reason one type of cars is neglected or under-
counted, the registered variance and the Fano factor can
be substantially increased.
These conclusions are connected with the nature of the
car number fluctuations and the strong anti-correlation
between the fluctuations in the number of small and large
cars. Fluctuations in the number of parked cars of one
kind are strongly enhanced by the presence of more or
less randomly distributed cars of the second kind, espe-
cially when cars of the second kind dominate. This leads
to conclusion (iii). However, the two distributions are
anti-correlated (higher number of parked small cars is
accompanied by a smaller number of large cars and vice
versa). The anti-correlation is particularly strong for a
size ratio that is close to unity.
One can imagine a case when the two kinds of cars dif-
fer only in “color”. In this case, Eqs. (18,28) yield αa,0 =
qR, αb,0 = pR, µa = Rqp+q
2µ0 and µb = Rqp+p
2µ0, so
that at large x we have < δnaδnb > /x = −(R − µ0)qp.
Then, the anti-correlation is almost complete: the fluc-
tuations of the total number are much smaller than those
of a given color, but still non-zero. Both the individual-
color number fluctuations and the anti-correlation are
largest at q ≈ 0.5, cf. Fig. 6. The anti-correlation
decreases with increasing size ratio, as reflected in our
conclusion (ii).
To discuss the above conclusions in terms of the PEB
problem, we note that estimation of the initial particle
energy is equivalent in CPP to a measurement of the
unknown length of a parking lot in terms of the total
number of cars that were able to fit into it by random
parking, assuming that the average fill factor for a given
two-size car mixture is known from earlier measurements.
The absolute accuracy of such a measurement depends on
the variance of the fill factor, and the relative accuracy
is determined by the Fano factor. As shown above for a
mixture of cars, the larger disparity of car sizes leads to
the higher fill-factor variance and therefore reduces the
absolute accuracy.
A particle detector measures the total number of sec-
ondary particles of all sorts (but not their total creation
energy, that would be equivalent to the filled length). In
any channel, all secondaries that have sufficient energy
for further branching will do so. Therefore, only those
pair creation energy ratios that leave the channels com-
petitive (i.e. b/a < 2) are relevant to the PEB problem
— otherwise additional energy branching would be pos-
sible.
We conclude that the presence of competing channels
with different energies [e.g. impact ionization with ex-
citation in X and L valleys] will decrease the quantum
yield (the number of secondaries per unit energy of the
primary particle) and enlarge the Fano factor. The at-
tendant loss in energy resolution is not that bad when
the ionization energies associated with different valleys
are not too disparate. For example, in Ge besides the
lowest eight L valleys (EG = 0.66 eV) one has a non-
competitive Γ valley (EΓ = 0.8 eV) and six very compet-
itive Si-like valleys (EG = 0.85 eV). The downgrading of
energy resolution should be more important for crystals
with larger (≈ 2) threshold energy ratio. For example, in
Si one has besides the 6 lowest valleys (EG = 1.12 eV) in
X direction, eight germanium-like L valleys with the gap
EL = 2.0 eV. Their effect on the Fano factor in silicon
may not be negligible.
Finally, reformulating (iii), we stress that any signif-
icant disparity in the collection efficiency between dif-
ferent equivalent valleys will strongly enhance the Fano
factor and downgrade energy resolution. This happens
because any collection disparity breaks the symmetry
between the equivalent valleys and destroys the anti-
correlation, responsible for keeping the total Fano factor
low even when the partial particle numbers associated
with individual valleys exhibit fully random fluctuations.
One possible origin for the asymmetry in the collection
efficiency in semiconductors has been discussed above in
the case of silicon diodes with the electric field in (100)
direction. In germanium diodes all different valleys are
equivalent relative to the (100) direction and the sym-
metry is not broken. It would be broken, however, if one
were to use Ge diodes oriented in (111) direction. This
would lead to a situation similar to Si — with a possi-
ble degradation in the Fano factor. These effects deserve
additional study, both experimental and theoretical.
V. CONCLUSIONS
We have studied a generalized 1-dimensional competi-
tive random sequential adsorption problem from a binary
mixture of particles with varying size ratio. Using a re-
cursive approach, we obtained independent equations for
the number of adsorbed particles of given sort and exact
analytical expressions for the partial filling factors and
variances for the larger particles. For the smaller parti-
cles analytical expressions were obtained in a number of
limiting cases. The results have been confirmed by direct
Monte Carlo simulations. To do so, we have introduced
a modified Monte Carlo procedure that enabled us to
explore a wide range of particle size ratios and particle
fractions in the flux.
A number of qualitative implications have been for-
mulated, relevant to the energy branching problem in
high-energy particle propagation through a semiconduc-
tor crystal. Conclusions made concern the quantum yield
and the energy resolution in semiconductor detectors
made of crystals with several competing channels of im-
pact ionization with different final electronic states.
We have found a very strong anti-correlation effects
which strongly suppress fluctuations of the total particle
number compared to the fluctuations of partial contribu-
tions by particles of a given sort. This effect is particu-
larly evident when one considers the deposition of similar
competing particles, e.g. parking of cars that are differ-
ent only in “color”. It may have dramatic consequences
for semiconductor γ-radiation detectors, if the symmetry
between anti-correlated particles is broken by a biased
collection. This leads to an important conclusion that the
energy resolution of semiconductor detectors is very sen-
sitive to the collection efficiency of competing secondary
particles.
We have also found a very strong correlation effects
that suppress fluctuations of the larger particle number
for all particle ratios. As a result, the Fano factor for
the larger particles is as a rule considerably smaller than
that for the smaller particles. The variance of the cover-
age by the smaller particles strongly increases with the
growth of the particle size ratio b/a. This effect is due
to the fluctuations in the size of gaps between larger par-
ticles that serve as receptacles for small-particle deposi-
tion. For b/a ≥ 5 the small-particle variance exceeds that
for the Poisson distribution in almost the entire range of
particle fractions in the flux onto the adsorbing line.
Acknowledgement. This work was supported by the
New York State Office of Science, Technology and Aca-
demic Research (NYSTAR) through the Center for Ad-
vanced Sensor Technology (Sensor CAT) at Stony Brook.
APPENDIX A: SMALL PARTICLE
CONTRIBUTIONS TO COVERAGE AND
COVERAGE VARIANCE
To calculate the contribution of small particles to the
total coverage at large x, we use Eq. (1) with the initial
boundary conditions (5). With the substitution x → x+b
and using Eq. (6), we rewrite Eq. (1) in the form
(x + b− l)na(x + b) = q(b − a)na(b) + qx
∫ x+b−a
na(y)dy + 2p
na(y)dy (A1)
Equation (A1) is valid for all x ≥ b. Taking Laplace
transformation of na(x) cut at x < b by a step-function
factor, we find that the transform,
Ña(s) =
e−sxna(x)dx, (A2)
satisfies the following equation
+ b− l
ebsÑa(s) =
[1 + (b − a)na(b)s]
e(b−a)s
Ña(s) + J1(s)
Ña(s) + J2(s)
J1(s) =
e−sxna(x)dx, J2(s) =
e−sxna(x)dx
Rearranging the terms, we rewrite it in form
+ l +
qe−as + pe−bs
Ña(s) = −
e−bsRa(s)
where
Ra(s) = q[1+(b−a)na(b)s]+2s
qe(b−a)sJ1(s) + pJ2(s)
The form of Eq. (A5) is similar to Eq. (20) in which,
however Ra should be calculated through J1(s) and
J2(s), using Eqs. (5,6). For the case b < 2a we have
na(b) = 1, and J1(s) = J2(s), while the explicit expres-
sion for J1(s) is easily obtained by substituting na(x) = 1
in Eq. (A4). Solution of Eq. (A5) then enables one to
retrieve the result of Ref [5]. To calculate J1(s) and J2(s)
for b > 2a, it is necessary to use Eq. (6), which describes
RSA of small particles onto a short line x < b. Its ana-
lytical solution and therefore the explicit expressions for
J1(s) and J2(s) can be obtained for the case b/a < 5 us-
ing direct recursion to find na(x) (for one-particle RSA
problem!). The result is rather cumbersome but suitable
for numerical integration.
For the case b/a > 5 one can exploit the exponentially
rapid approach of the solution of Eq. (6) to its asymp-
totic behavior in the limit x ≫ 1 (see e.g. [27] for the
numerical data). This asymptotic solution,
na(x) =
(x+ 1)− 1, (A7)
can be used to calculate J1(s) and then J2(s). To do this,
we multiply Eq. (6) by exp(−sx) and integrate between
0 and b− 1. We obtain an equation for J2(s) of the form
J ′2(s) +
J2(s) = −
e−asI(s)
+s(b− a)e−bs[na(b)− 1] + 2se
−asJ1(s)
, (A8)
where
I(s) =
∫ (b−a)s
dyye−y. (A9)
Solution of Eq. (A8), satisfying the boundary conditions
for na given by Eq. (5), is of the form
J2(s) =
β̃(s)s2
dtβ̃(t)
na(b)(b − a)te
−(b−a)t
+2tJ1(t)−
1− e−(b−a)t
(A10)
β̃(t) = exp
1− ev
. (A11)
The contribution of small particles to the fill factor is
then given by
β(u)Ra(u)du. (A12)
in which β(u) is given by the Eq. (16) and Ra(u) is
defined by Eq. (A6). The obtained solution, though
rather unwieldy, is suitable for numerical integration and
for b/a > 5 it gives the results that agree with Monte
Carlo simulations.
In the limiting case b/a = b/a ≫ 1 it reduces to a more
compact final expression for the contribution to the total
coverage from the small particles
θa = R
due−quβp(u)
q(u− 1)− 2pe−u
(A13)
with βp(u) defined by (33). For q = 1, Eq. (A13) prop-
erly gives θa = R, while for q = 0 one has θa = R(1−R).
The latter expression corresponds to the coverage by
small particles of the gaps between the large particles
left after their initial deposition. For arbitrary q, the
coverage given by Eq. (A13) is depicted in Fig. 2 by the
open squares.
Similar approach can be used to calculate the small
particle coverage variance. However, for b/a > 2 the
equation for the Laplace transform of ua(x) given by Eq.
(7), including all contributions to Na(s), becomes rather
impractical. In the limiting case b/a ≫ 1, when fluctua-
tions of the large particle gaps dominate the variance of
small-particle coverage, one gets a more compact result
shown in Fig. 5.
[1] G. J. Rodgers and Z. Tavassoli, Phys. Lett. A 246, 252
(1998).
[2] D. Boyer, J. Talbot, G. Tarjus, P. Van Tassel, and P.
Viot, Phys. Rev. E 49, 5525 (1994).
[3] A. V. Subashiev and S. Luryi, Phys. Rev. E 75, 011123
(2007).
[4] M. C. Bartelt and V. Privman, Phys. Rev. A 44, R2227-
R2230 (1991).
[5] M. K. Hassan, J. Schmidt, B. Blasius, J. Kurths, Phys.
Rev. E 65, 045103(R) (2002).
[6] M. K. Hassan and J. Kurths, J. Phys. A 34, 7517 (2001).
[7] N. A. M. Araujo, and A. Cadilhe, Phys. Rev. E 73,
051602 (2006).
[8] D. J. Burridge and Y. Mao, Phys. Rev. E 69, 037102
(2004).
[9] E. G. Coffman, Jr., L. Flatto, P. Jelenkovich, and B.
Poonen, Algorithmica 22, 448 (1998).
[10] M. Inoue, Phys. Rev. B 25, 3856 (1982).
[11] P. Calka, A. Mezin, P. Vallois, Stochastic Processes and
their Applications, 115, 983-1016 (2005).
[12] J. Talbot, G. Tarjus, P.R Van Tassel, P. Viot, Colloids
and surfaces A: Physicochemical and Engineering As-
pects B 165, 278 (2000).
[13] V. Privman, Colloids Surf A 165 231-240 (2000).
[14] R. Devanathan, L. R. Corrales, F. Gao, W. J. Weber,
Nuclear Instrum. Methods Phys. Res. A 565, 637-649
(2006).
[15] H. Spieler, Semiconductor Detector Systems, Oxford Uni-
versity Press, 2005.
[16] C. Klein, J. Appl. Phys. 39, 2029 (1968).
[17] W. van Roosbroeck, Phys. Rev. 139, A 1702 (1965).
[18] This correlation originates from the basic fact that a
simple random division of a segment in two parts pro-
duces highly correlated pieces: if one is short the other
is long and vice versa. Energy branching by impact ion-
ization evidently has the similar property, as the sum of
secondary-particle energies is fixed by energy conserva-
tion. This type of correlations was first pointed out by
Ugo Fano in 1947 [19] and bears his name.
[19] U. Fano, Phys. Rev, 72, 26 (1947).
[20] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regu-
lar Variation, Cambridge University Press, Cambridge,
1987.
[21] A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 3 109
(1958); Trans. Math. Stat. Prob. 4, 205 (1963).
[22] J. K. Mackenzie, Journ. Chem. Phuys. 37, 723 (1962).
[23] S. Luryi and A. V. Subashiev, unpublished.
[24] G.D.Alkhazov, A.A. Vorob’ev, A.P Komar, Nucl. Instr.
Meth. 48, 1-12 (1967).
[25] P. E. Nay, Annals of Mathematical Statistics, 33, 702-718
(1962).
[26] Sh. Kogan, Electronic Noise and Fluctuations in Solids,
Cambridge University Press, Cambridge, 1996.
[27] M. Lal and P. Gillard, Math. Computation 28, 562
(1974).
