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A transport-based model of material trends in nonproportionality of scintillators

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A transport based model of material trends in nonproportionality ofscintillators
Qi Li, Joel Q. Grim, R. T. Williams, G. A. Bizarri, and W. W. Moses
Citation: J. Appl. Phys.
109
, 123716 (2011); doi: 10.1063/1.3600070
View online: http://dx.doi.org/10.1063/1.3600070
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v109/i12
Published by the American Institute of Physics.
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A transport-based model of material trends in nonproportionalityof scintillators
Qi Li,
1,a)
Joel Q. Grim,
1
R. T. Williams,
1
G. A. Bizarri,
2
and W. W. Moses
2
1
Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109, USA
2
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
(Received 26 January 2011; accepted 12 May 2011; published online 27 June 2011)Electron-hole pairs created by the passage of a high-energy electron in a scintillator radiationdetector ﬁnd themselves in a very high radial concentration gradient of the primary electron track.Since nonlinear quenching that is generally regarded to be at the root of nonproportional responsedepends on the fourth or sixth power of the track radius in a cylindrical track model, radialdiffusion of charge carriers and excitons on the
10 picosecond duration typical of nonlinear quenching can compete with and thereby modify that quenching. We use a numerical model of transport and nonlinear quenching to examine trends affecting local light yield versus excitationdensity as a function of charge carrier and exciton diffusion coefﬁcients. Four trends are found: (1)nonlinear quenching associated with the universal “roll-off” of local light yield versus
dE
/
dx
is afunction of the lesser of mobilities
l
e
and
l
h
or of
D
EXC
as appropriate, spanning a broad range of scintillators and semiconductor detectors; (2) when
l
e
l
h
, excitons dominate free carriers intransport, the corresponding reduction of scattering by charged defects and optical phononsincreases diffusion out of the track in competition with nonlinear quenching, and a rise inproportionality is expected; (3) when
l
h
l
e
as in halide scintillators with hole self-trapping, thebranching between free carriers and excitons varies strongly along the track, leading to a “hump”in local light yield versus
dE/dx
; (4) anisotropic mobility can promote charge separation alongorthogonal axes and leads to a characteristic shift of the “hump” in halide local light yield. Trends1 and 2 have been combined in a quantitative model of nonlinear local light yield whichis predictive of empirical nonproportionality for a wide range of oxide and semiconductor radiation detector materials where band mass or mobility data are the determinative materialparameters.
V
C
2011 American Institute of Physics
. [doi:10.1063/1.3600070]
I. INTRODUCTION
Proportionality between scintillator light yield and inci-dent gamma-ray energy is a prerequisite for achieving thebest energy resolution consistent with counting statistics in aradiation detector.
1 – 5
Although it has been known for about50 years that scintillator materials have an intrinsic nonpro-portionality of response,
6 – 10
efforts to understand the physi-cal basis of nonproportionality in order to more efﬁcientlydiscover and engineer materials with better gamma resolu-tion have intensiﬁed in the last decade or so.
11 – 18
The moti-vation comes in part from the need for highly discriminatingnuclear material monitoring and some types of security scan-ning, but improved resolution can also beneﬁt medical mo-lecular imaging and particle physics experiments.It would be very useful if one or more material “designrules” for proportionality could be found. What we mean bya material design rule in this context is a predictive relation-ship between one independently measurable material param-eter and a trend of response affecting nonproportionality,valid across a broad range of radiation detector materials.Such predictive trends or rules would be useful of them-selves, but more so because their existence would suggestsimple underlying physical mechanisms that can be tweakedand engineered for improved detector resolution.The sheer number of physical interactions interspersedbetween gamma-ray energy deposition and the detection of scintillator light pulses, as well as the number of variationsof scintillator materials that one can introduce, may makethe existence of one or more simple material design rulesseem unlikely. In fact, the scintillator nonproportionalityproblem has seemed so far to be particularly resistive todeﬁning a single trend that follows from independentlymeasured physical parameters. Payne
et al
.,
11,18
Jaffe,
19
andBizarri
et al
.,
17
among others, have ﬁt empirical parameter-ized models to nonproportionality data for a wide range of materials. The data can be ﬁt with a moderate number of empirically determined parameters. For example,Payne
et al
.
18
have ﬁt electron yield data from the SLYNCI(Scintillator Light Yield Nonproportionality CharacterizationInstrument)
20
experiment for 27 materials using two empiri-cal ﬁtting parameters: in their terms a Birks parameter characterizing how strong the second-order dipole-dipolequenching term is, and a branching fraction
g
e/h
of initialelectron-hole excitations into independent carriers rather than excitons. What is missing so far is the ability to deﬁneor calculate those ﬁtting parameters on the basis of independ-ently measureable properties of the material. Bizarri
et al
.
17
chose ﬁtting parameters to be identiﬁed with a series of
a)
Author to whom correspondence should be addressed. Electronic mail:liq9@wfu.edu.
0021-8979/2011/109(12)/123716/17/$30.00
V
C
2011 American Institute of Physics
109
, 123716-1
JOURNAL OF APPLIED PHYSICS
109
, 123716 (2011)
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radiative and nonradiative rate constants and branchingratios, but the number of such rate and branching parametersis large and so far the independent measurements of them donot exist in a sufﬁciently broad material set to allow a goodtest. Gao
et al
.
21
and Kerisit
et al
.
22,23
have performed MonteCarlo simulations starting from the energy deposition proc-esses. While important to ultimately achieving simulation of the precise light pulse in a given material, the results havenot yet been extended to processes such as thermalized bandtransport of carriers on the
10 ps time scale that we willshow are important with respect to nonlinear quenching.In 2009, we began looking at the effect that electron andhole diffusion occurring in thermal equilibrium within theextreme radial concentration gradient of high-energy elec-tron tracks may have upon nonlinear quenching and thebranching from electron-hole pairs to independent car-riers.
24 – 26
Our interest was provoked partly by the antici-pated extreme sensitivity of high-order nonlinear quenchingto small changes in the track radius given an initial depositedlocal carrier concentration on-axis. Since the carrier densityis inversely proportional to the square of the track radius,second-order dipole-dipole quenching and third-order Auger quenching depend on the inverse fourth and sixth power,respectively, of a cylindrical track radius expanding by diffu-sion.
24 – 26
Even modest diffusion can have controlling inﬂu-ence on nonlinear quenching in such a case, and the extremeconcentration gradient promotes substantial diffusion effectseven on the
10 ps time scale on which nonlinear quenchingtypically occurs.
25,27
The carrier mobilities therefore becomecandidates for physical material parameters that can controlnonlinear quenching and through it, nonproportionality.In Sec. III of this paper, we will present results of a nu-merical model showing how the quenching rates andquenched fractions depend on the carrier mobilities and exci-ton diffusion coefﬁcient as well as upon excitation density(
dE/dx
) along the track. It should be noted that carrier mobil-ity is not a single parameter of the material. Electrons andholes have independent carrier mobilities. We will see thatthe lesser of the two mobilities is an important parameter, asis their ratio. In addition, many important scintillators areanisotropic crystals, so we consider effects of anisotropies inthe carrier mobilities.However, a practical problem for testing the predictionsof our diffusion and quenching model against experiment isthe scarcity of carrier mobility measurements among scintil-lators. Scintillators are, as a class, mostly insulators. This sit-uation has arisen as a result of wanting transparency tovisible and near-ultraviolet activator emissions, and further-more selecting large enough host bandgap to avoid ioniza-tion of activator-trapped charges to either band edge.Although not impossible, the measurement of mobilities ininsulators is challenging, particularly because of the typicallack of ohmic contacts, and so has been performed only in afew inorganic scintillators – notably the alkali halides
28
including CsI
29
and NaI.
30
In contrast, good mobility dataexist for the charge-collecting solid-state radiation detectorssuch as high-purity germanium (HPGe).
31
Therefore in Refs.24 – 26, we used the known mobility data for CsI and Ge with
measured rate constants for dipole-dipole
25 – 27
and Auger recombination
32
to compare our model simulation of nonlin-ear quenching and its dependence on excitation density (
dE/dx
) in these two paradigms of radiation detectors. The agree-ment of the model simulation with experiment was verygood. The model predicts that carrier diffusion is conﬁnedtightly near the track end in CsI:Tl, causing a nonlinear quenched fraction of
60% simulated near the track end.This can be compared with the results of K-dip spectroscopyon the similar scintillator NaI:Tl,
33
which shows
52%quenched at the track-end (
50 eV). In sharp contrast, thehigh mobilities of both carriers in HPGe resulted in fast dif-fusion out of the track core, diluting the carrier density to alevel that terminated Auger decay within 2 femtoseconds,
24 – 26
rendering nonlinear quenching irrelevant for HPGe. Thisis in agreement with the excellent resolution of HPGe.Within this set of two materials representing nearly oppositeextremes of carrier mobility, the modeled effect of diffusionon nonlinear quenching (
nonproportionality) was bothphysically justiﬁed and predictive of experiment.The group of Setyawan, Gaume, Feigelson, and Curtar-olo has investigated the link between carrier mobility(actually band effective masses) and nonproportionality con-currently with our modeling studies.
34
Also faced with thescarcity of measured mobilities or effective masses for mostscintillators, they took the course of calculating electronicband structure for a wide range of scintillators in order todeduce effective masses from the band curvatures. Theyextracted experimental measures of nonproportionality fromthe literature and plotted the parameters versus the ratio
m
h
/m
e
of the calculated (average) band masses for each ma-terial. Excluding most halide materials and also ZnSe:Te, therest of the (largely oxide and two tri-halide) scintillatorswere found to fall on an empirical trend line in Ref. 34. Areason for the group of some halides and ZnSe to fall welloff the primary trend line was suggested generally in termsof a classiﬁcation of “excitonic” versus “non-excitonic”materials. They characterized the alkali halides as beingexcitonic, although Dietrich, Purdy, Murray, and Williams
35
have shown that in NaI:Tl and KI:Tl the majority of scintilla-tion light comes from recombination of independent elec-trons and holes trapped as Tl
0
and Tl
þþ
, respectively,changing what had been earlier assumed in the model of Murray and Meyer.
9
Setyawan
et al
. characterized mostoxides including YAP (YAlO
3
, yttrium aluminum perov-skite) as transporting energy mainly by free carriers. Withthe halide and selenide exceptions noted, Setyawan
et al
.found a signiﬁcant degree of correlation between nonpropor-tionality and the single parameter
m
h
/m
e
coming from calcu-lated band structure.
34
In particular, the materials typiﬁed byYAP and YAG (Y
3
Al
5
O
12
, yttrium aluminum garnet), with
m
h
/m
e
1, peaked up sharply in proportionality compared tothe other oxides. Setyawan
et al
. discussed possible reasonsfor a correlation between nonproportionality and
m
h
/m
e
related to separation of charge carriers with different effec-tive masses, but did not offer a quantitative model.In the remainder of this paper, we will demonstrate thenumerical model basis for trends in scintillator responsedepending on carrier mobilities. These will include the pri-mary correlation of nonlinear quenching (speciﬁcally the
123716-2 Li
et al.
J. Appl. Phys.
109
, 123716 (2011)
Downloaded 07 Feb 2013 to 152.17.132.60. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
amount of yield roll-off versus
dE
/
dx
) with the lesser of holeand electron mobility in a panoramic view. A basis for improved proportionality when
l
h
¼
l
e
will be describedwith the help of numerical simulations. We will also presentthe model basis for ﬁnding that the alkali halides have such auniquely small value of the mobility ratio
l
h
/
l
e
, that the“hump” in electron yield occurs, and furthermore that itshould
improve
the proportionality over what it would bewithout considering the light yield from independent car-riers. Reasons for expecting hole self-trapping to occur gen-erally in the class of halide scintillators and to lead to effectssimilar to alkali halides in the class as a whole will be dis-cussed. Our recent work on modeling the
anisotropy
of mobilities in scintillators will be discussed. Within the classof halide scintillators, the materials with isotropic bandmasses empirically have the poorest proportionality whilethe anisotropic materials have better proportionality, becom-ing quite good in many of them such as SrI
2
:Eu, LaBr
3
:Ce,LaCl
3
:Ce, KLC:Ce (K
2
LaCl
5
:Ce). The model suggests a rea-son. Finally, we will present a quantitative physical model of nonlinear local light yield which is predictive of empiricalproportionality for a wide range of oxide and semiconductor radiation detector materials where band mass or mobilitydata are the determinative material parameters.
II. MODELING METHOD
We use time-step ﬁnite-element analysis to solve for thediffusion and drift currents, electric ﬁeld, and local carrier density in the vicinity around the initial cylindrical distribu-tion of carriers with a radius of about 3 nm produced by theincident electron.
24 – 26
(Varying the initial radius from 2 to 5nm had little effect.) The longitudinal dependence isneglected since the characteristic value for the electron tracklength is generally hundreds of micrometers; while the radialdimension is described in nanometers. The problem cantherefore be solved in a cross-section of the track. We evalu-ate different longitudinal positions along the track by chang-ing the initial carrier density (proportional to
dE/dx
). Theequations used are
~
J
e
ð
~
r
;
t
Þ¼
D
e
r
n
e
ð
~
r
;
t
Þ
l
e
n
e
ð
~
r
;
t
Þ
~
E
ð
~
r
;
t
Þ
;
(1)
@
n
e
ð
~
r
;
t
Þ
@
t
¼r
~
J
e
ð
~
r
;
t
Þ
;
(2)for electrons and an equivalent set of equations for holes.
~
J
e
is the electron number current density (electrons/cm
2
s),
n
e
isthe electron density (electrons/cm
3
),
~
E
is the electric ﬁeld. Inour earlier simulations with isotropic mobilities,
24 – 26
theelectric ﬁeld could be evaluated from Gauss’s law. To handleanisotropic transport in this study, we use the Poissonequation
r
2
u
¼
q
ð
~
r
;
t
Þ
ee
0
:
(3)A fast Poisson solving algorithm was introduced to calculatethe potential and electric ﬁeld at each time step and cell posi-tion, where
q
ð
~
r
;
t
Þ¼
e
½
n
h
ð
~
r
;
t
Þ
n
e
ð
~
r
;
t
Þ
. The Einstein rela-tion
D
¼
l
k
B
T/e gives the diffusion coefﬁcients for electronsand holes in terms of their mobilities
l
e
and
l
h
. The staticdielectric constant
e
is used in the Poisson equation.If the carriers are paired as excitons at concentration
n
ex
,the bimolecular quenching rate due to dipole-dipole Fo¨rster transfer can be included in the simulation through theequation.
@
n
ex
ð
~
r
;
t
Þ
@
t
dipole
dipole
¼
k
2
ð
t
Þ
n
2
ex
ð
~
r
;
t
Þ
;
(4)where
k
2
ð
t
Þ
is the bimolecular quenching rate parameter,
k
2
ð
t
Þ¼
23
p
32
R
3
dd
ﬃﬃﬃﬃﬃ
s
R
p
1
ﬃﬃ
t
p
:
(5)
s
R
is the radiative lifetime of the excited state and
R
dd
is theFo¨rster transfer radius depending on the overlap of emissionand absorption bands.
36 – 38
In each radial cell and time stepof the ﬁnite-element simulation, the loss of carrier pairs todipole-dipole quenching is evaluated from Eq. (4) with
n
ex
taken as the lesser of electron or hole concentration in thatcell, i.e., the local concentration of balanced charges. Excesscharges of one sign in a given cell are considered incapableof having dipole moments to participate in the dipole-dipolequenching. They are counted as excluded from the paired or exciton population
n
ex
. Thus our hypothesis for the pairingof carriers in scintillators is that electrons and holes are cre-ated initially as pairs but they can be “ripped apart” by thedifferences in electron and hole diffusion as time goes for-ward. The question of whether an existing exciton canactually be ripped apart by different carrier diffusion rates isan interesting one that we will discuss in a future publication.We remark that in the ﬁrst place many excitons will be inhigher excited states during the ﬁrst picoseconds and theyshould be easily susceptible to ripping apart. In the secondplace, the strong gradients of electrochemical potential in thetrack region around a hole actually distort exciton groundstates in the crowded region and make them more susceptibleto ripping apart, and in the third place the electrochemicalpotential versus internal coordinates of an exciton in thelandscape of a particle track can lead to tunneling ionization.Pending publication of our study of excitons in the trackenvironment, we already suggest that there are grounds for the computationally convenient hypothesis in this paper thatclose carrier pairs should be regarded as excitons with capa-bility for dipole-dipole quenching until diffusion in a stronggradient rips them apart. The fraction of quenched carriers(
QF
) at time
s
after excitation is evaluated by
QF
¼
Ð
V
Ð
s
0
k
2
ð
t
Þ
n
2
ex
ð
~
r
;
t
Þ
dtdadz
Ð
V
n
ð
~
r
;
0
Þ
dadz
:
(6)In the very high radial concentration gradient of the track,different diffusion rates of electrons and holes can controlwhether carriers pair as excitons or become independent,according to the discussion above. In activated scintillators,it is particularly important to know the independent versuspaired status of carriers at the time of trapping on the
123716-3 Li
et al.
J. Appl. Phys.
109
, 123716 (2011)
Downloaded 07 Feb 2013 to 152.17.132.60. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
activators. As in the game of musical chairs, will the carrierssit on the same or on different activators at the time
s
trap
?The average displacement of an electron at position
~
r
andtime
s
can be evaluated within the diffusion model as
~
d
e
ð
~
r
;
s
Þ¼
ð
s
0
~
J
e
ð
~
r
;
t
Þ
n
e
ð
~
r
;
t
Þ
dt
(7)and similarly for holes. We will write the average relativedisplacement of electrons from holes at a given position andtime
s
as
D
d
¼j
~
d
e
~
d
h
j
. In the case of an activated scintilla-tor, the time of interrogation
s
should be the average time totrapping on the activator. The independent fraction (
IF
) ishence evaluated as
IF
¼
Ð
V
n
ð
~
r
;
s
Þ
min
ð
1
;
D
d
=
s
Þ
dadz
Ð
V
n
ð
~
r
;
s
Þ
dadz
;
(8)where
s
is the average spacing between two nearest activa-tors. In non-activated (excitonic) scintillators, the distance
s
would correspond to a spatial criterion of likely excitonrecombination. However, we have not yet attempted tomodel non-activated excitonic scintillators. We have mod-eled non-activated semiconductor radiation detector materi-als, but in that case the high mobilities of both carriersrender Eq. (8) moot after tens of femtoseconds, typically.Separated and paired charges are subject to differentdominant quenching processes, taken as ﬁrst and secondorder in excitation density, respectively, for the followingreasons. The thermalized transport on which this model isbased is not primarily the slow hopping transport of activa-tor-trapped and self-trapped carriers during the severalmicroseconds duration of a typical scintillator light outputpulse in activated alkali halides. Rather, it is the fast thermal-ized band transport on the
10 picosecond time scale of non-linear quenching. The
10 ps time scale of dipole-dipoleSTE quenching was measured at 2
10
20
e-h/cm
3
excitationdensity in CsI.
25,27
Were it not for fast transport out of thetrack,
24 – 26
a similar time scale would apply for nonlinear quenching in Ge due to its measured Auger rate constant
c
¼
1.1
10
31
cm
6
/s.
32
As illustrated in our modeled com-parison of CsI and Ge, when diffusion of thermalized carrierscan signiﬁcantly dilute the carrier concentration within thetrack core within
10 ps, the nonlinear quenching iscurtailed.In CsI:Tl, as the example we consider in most detail, in-dependent trapped charges undergo de-trapping and re-trap-ping processes until they recombine as Tl
þ
*
in order to yieldluminescence. The primary emphasis of the present model isthe initial few picoseconds because that is where the nonli-nearity of response is mainly determined. One may think of an initial period spent populating activators and traps in adense carrier environment where diffusion and nonlinear quenching are dominant for some picoseconds. Then a slowrecombination of independent trapped charges begins, wherethe quenching is primarily linear because the carrier concen-trations are now lower. Nevertheless, even the linear quench-ing can express effects of the nonlinearities that setup theoccupied trap distributions during the fast phase. The slowrecombination by de-trapping and diffusion of carriers is adifferent recombination path with different time dependenceand perils for quenching by deep trapping than in the alter-nate path taken by electron and hole initially trapped as apair (exciton) on the same thallium. The longer the path over which hopping migration of independent charges must occur,the greater is the chance that one or both will encounter atrap that removes them from the light-emission process dur-ing the scintillation gate width. We incorporate this trappinghazard for migrating free carriers in the model as a linear quenching fraction
k
1
that multiplies the independent carrier fraction
IF
to give the “Independent Nonradiative Fraction”,
INF
:
INF
¼
IF
k
1
:
(9)Since
k
1
was assumed proportional to the migration pathbetween charge-trapping activators, we should expect it tobe proportional to [activator concentration]
1/3
. However,the independent fraction
IF
itself as deﬁned in Eq. (8)depends inversely on the activator spacing
s
, and is thus pro-portional to [activator concentration]
þ
1/3
. Thus to lowestorder,
INF
is independent of activator concentration. How-ever, the dependence on min(1,
D
d/s
) in Eq. (8) leads to mod-erate dependence on activator concentration at highconcentration. We will extend this deﬁnition of
INF
to other activated scintillators in the model to be discussed. Thereader may have noticed that in the present model we handlethe independent carriers in two parts of the calculation insomewhat different ways: for the initial fast diffusion andquenching (picoseconds) we treat it computationally as
j
n
e
-n
h
j
, and for the slow de-trapping diffusion (nanoseconds-microseconds) we treat it via
IF
and
INF
.In the context of this model, we deﬁne “simulated locallight yield” (
SLLY
) as follows:
SLLY
ð
normalized
Þ¼ð
1
QF
Þð
1
INF
Þ
:
(10)It predicts an upper limit of the local light yield as a functionof initial carrier concentration. The most complete set of ma-terial parameters is available for CsI:Tl. The electron mobil-ity in pure CsI has been measured as
l
e
¼
8 cm
2
/Vs at roomtemperature.
29
The static dielectric constant of CsI is 5.65.
39
The trapping time of electrons on Tl
þ
in CsI was measuredas 6 ps.
40
The bimolecular quenching rate in CsI has beenmeasured as
k
2
ð
t
Þ
ﬃﬃ
t
p ¼
2
:
4
10
15
cm
3
s
1/2
.
25,27
Due to thepreviously mentioned lack of mobility data generally in other scintillators, we will in some cases scale mobility valuesfrom calculated band masses, and set missing parametersequal to the CsI values for all materials when attempting toillustrate trends versus mobility alone.
III. MATERIALTRENDS AFFECTINGNONPROPORTIONALITY BASED ON THETRANSPORT/QUENCHING MODELA. Nonlineardipole-dipole and Augerquenchingdependent on min(
l
h
,
l
e
)
Nonlinear quenching processes such as second-order dipole-dipole transfer and third-order Auger recombination
123716-4 Li
et al.
J. Appl. Phys.
109
, 123716 (2011)
Downloaded 07 Feb 2013 to 152.17.132.60. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

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