Table Of ContentOptimal path of diffusion over the saddle point and fusion of massive nuclei
Chun-Yang Wang,1 Ying Jia,2 and Jing-Dong Bao1,3∗
1Department of Physics, Beijing Normal University, Beijing 100875, China
2College of Science, The Central University for Nationalities, Beijing 100081, China
3Center of Theoretical Nuclear Physics, National Laboratory of Heavy-ion Accelerator, Lanzhou 730000, China
(Dated: February 3, 2008)
Diffusion of a particle passing over the saddle point of a two-dimensional quadratic potential is
8 studied via a set of coupled Langevin equations and the expression for the passing probability is
0 obtained exactly. The passing probability is found to be strongly influenced by the off-diagonal
0 components of inertia and friction tensors. If the system undergoes the optimal path to pass over
2 the saddle point by taking an appropriate direction of initial velocity into account, which departs
fromthepotentialvalleyandhasminimumdissipation,thepassingprobabilityshouldbeenhanced.
n
Application to fusion of massive nuclei, we show that there exists the optimal injecting choice for
a
J thedeformabletargetandprojectilenuclei,namely,theintermediatedeformationbetweenspherical
and extremely deformed ones which enables thefusion probability to reach its maximum.
6
2
PACSnumbers: 24.10.-i,24.60.-k,25.70.Jj,05.20.-y
]
h
c I. INTRODUCTION componentsofinertiaandfrictiontensorsbeforethesys-
e tem arrives firstly at the conditional saddle point.
m
Recently, theoretical calculations for the fusion bar-
- The saddle point passage problem is of great interest rier distribution, accounting for the surface curvature
t
a in various fields of physics, such as collision of molec- correction to the nuclear potential, have been presented
st ular systems, atomic clusters, biomolecules, and so on. by Hinde et al. [12, 13, 14]. The geometrical effect
. The previous studies on this issue were mostly concen- significantly changes the near-barrier fusion cross sec-
t
a trated on a simple diffusive dynamics with single degree tion and the shape of the barrier distribution through
m of freedom, where the Langevin equation with constant
an angle-dependent potential, where the target nucleus
- coefficientscanbeeasilysolvedinthecaseofaquadratic bears quadrupole and bexadecapole deformations while
d potential [1]. However, since many processes obviously
the projectile one is in a sphericalshape. In these calcu-
n
involvemore than one degreeof freedom, which the one-
o lations,thesurfacecurvaturecorrectiontothesphere-to-
dimensional (1D) model does not distinctly hold, high
c spherenuclearpotentialexertsinfluencesuponthefusion
[ dimension at least two dimension (2D) is necessary. A probability through the height of fusion barrier. How-
casein pointwouldbe the fusionreactionofmassivenu- ever,thedynamicalcouplingeffectofvariousdeformative
1
clei, where the fusion is induced by diffusion [2, 3] and
v degreesoffreedomneedstobeaddedfromtheviewpoint
the asymmetrical or the neck degree of freedom of com-
9 of fusion by diffusion [3].
5 pound nuclei needs to be considered [4, 5]. For these The primary purpose of this paper is to study the in-
0 systemswiththecontactpointoftwocollidingnucleibe-
fluence ofcoupling betweentwodegreesoffreedomupon
4 ing veryclose to the conditionalsaddle point, the poten-
the passing probability. In Sec. II, we report the ana-
. tialenergysurface(PES)aroundthesaddlepointcanbe
1 lytical expressionof the saddle-point passing probability
0 approximatedtobeaquadratictype. Underthisapprox-
by solving the 2D coupled Langevin equation with con-
8 imation, Abe et al. [6] obtained an analytical expression
stant coefficients. In Sec. III, we discuss the effects of
0 for the multi-dimensional saddle-point passing probabil-
off-diagonalcomponentsofinertia,frictionandpotential-
: ity. Some authors discussed quantum effect of the fu-
v curvature tensors and then determine the optimal diffu-
i sion probability by using the real time path integral [7]
X sive path. Sec. IV gives an application of this study to
or the quantum transport equation [8, 9], respectively.
the actual fusion process of massive nuclei. A summary
r Boilleyet al. [10]studiedtheinfluenceofinitialdistribu-
a is written in Sec. V.
tion upon the passing probability. Anomalous diffusion
passing over the saddle point of 1D quadratic potential
was also discussed in Ref. [11]. Nevertheless, it is not
completely clear for the dynamical role of non-transport
II. THE PASSING PROBABILITY
degrees of freedom. This might be very important for
the quasi-fission mechanism in the fusion reaction, be-
We consider the directional diffusion of a particle in a
cause the average path of the fusing system in a multi-
2D quadratic PES: U(x ,x )= 1ω x x with i,j =1,2
dimensionalPESshouldbecontrolledbytheoff-diagonal 1 2 2 ij i j
and detω < 0, the motion of the particle is described
ij
by the Langevin equation
∗Correspondingauthor. Electronicmail: [email protected] mijx¨j(t)+βijx˙j(t)+ωijxj(t)=ξi(t) (1)
2
with x (0) = x and x˙ (0) = v , where x < 0
j j0 j j0 10
and v > 0. Here and below the Einstein summation
10
convention is used. The two components of the ran- 2.15 1.0
dthoemirfcoorrcreelaarteioansssuombeeyd ttohebefluGctauuastsiioann-dwihssitipeantoioisnesthaenod- cv022..0150 00P..pass68
rem ξ (t)ξ (t′) =k Tm−1β δ(t t′), where k is the 2.00
BlotzhmiannjconistantBand Tikthkej tem−perature. B 1.95 0.4
Assumingthatx1-axisisthetransportdirection[ω11 < 1.90 0.2
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
0],we write the reduceddistributionfunctionofthe par-
m12 m12
ticle for x while the variables x (t), v (t) and v (t) are
1 2 1 2
integrated out,
2.0 1.0
1
W(x ,t;x ,x ,v ,v ) = 1.8
1 10 20 10 20 √2πσx1((xt)1(t) x1(t) )2 cv0111...246 P00pass..68
exp(cid:18)− 2σ−x21h(t) i (cid:19). 01..80 0.4
(2) 0.6 0.2
-4 -2 0 2 4 -4 -2 0 2 4
Integratingoverx fromzerotoinfinity,wedeterminethe
1
passing probability over the saddle point [x = x = 0]
1 2
as 2.70
1.0
∞ 2.55
0.8
P(t;x10,x20,v10,v20) = Z0 W(x1,t;x10,x20,v10,v20)dx1 cv022..2450 Ppass0.6
= 1erfc hx1(t)i . (3) 2.10 0.4
2 (cid:18)−√2σx1(t)(cid:19) 1.95 0.2
Applying the Laplace transform technique to Eq. (1), 1.80 -4 -2 0 2 4 0.0 -4 -2 0 2 4
we thus get x (t) and its variance σ2 (t) at any time,
1 x1
2 t FIG.1: Thecriticalvelocity(left)andthestationarypassing
x (t) = x (t) + H (t t′)ξ (t′)dt′, (4) probability(right)asfunctionsofvariousoff-diagonalcompo-
1 1 i i
h i Xi=1Z0 − nents m12, β12, and ω12, respectively. The parameters used
t t1 are: m11 =1.5, m22 =2.0, β11 =1.8, β22 =1.2, ω11 =−2.0,
σ2 (t) = dt H (t t ) dt ξ (t )ξ (t ) H (t t ), ω22 =1.5, andθ=0. Theinitialvelocities oftheparticleare
x1 Z0 1 i − 1 Z0 2h i 1 j 2 i j − 2 v0 =4.0,2.2,1.9,1.6 from top tobottom (right).
(5)
where the mean position of the particle along the trans- the particle increases. The critical velocity is defined by
port direction is given by the passing probability being equal to 1. This leads to
2
2 the condition: limt→∞ x1(t) =0. If all the off-diagonal
h i
x (t) = [C (t)x +C (t)v ], (6) components of the three coefficient tensors are not con-
1 i i0 i+2 i0
h i Xi=1 sidered, as well as x20 and v20 are taken to be zero,
the critical velocity is determined from Eq. (6): vc =
which relates to the initial position and velocity. The [F (a)/F (a)]x , where a is the largest positive roo0t of
1 3 10
time-dependent factors in Eq. (6) with exponential P(s)=0. Thisisinfactidenticaltothe one-dimensional
forms according to the residual theorem are C (t) =
iLn−E1[qFsi.(s()4/)P(asn)d] ((i5=) r1ea..d.4H),(tth)e=two−r1e[sFpo(nss)e/Pfu(sni)c]tiaonnds irsespurlot:povr1ct0io=na−lxt1o0t(hpeβf12r1ic+tio4nω1s1tr+enβg1t1h)/.(2m11)[6],which
1 5
H (t)= −1[F (s)/P(s)], where −L1 denotes the inverse We now consider all the off-diagonal components of
2 6
L L three coefficient tensors, namely, the correlations of two
Laplace transform. The expressions of P(s) and F (s)
i
degrees of freedom are taken into account, the critical
(i=1,...,6) are written in the appendix.
velocity can also be determined by limt→∞ x1(t) = 0
h i
and results in
III. THE OPTIMAL DIFFUSIVE PATH
C ( )x +C ( )x
vc = 1 ∞ 10 2 ∞ 20 , (7)
0 −C ( )cosθ+C ( )sinθ
3 4
A. The coupling effect of two degrees of freedom ∞ ∞
where θ denotes the incident angle between the initial
As one has known in the 1D case, the passing proba- velocity and the x -direction, hence v = v cosθ and
1 10 0
bility increases from 0 to 1 when the initial velocity of v =v sinθ.
20 0
3
0.6
=-1.0 = 1.0 v0=4.0
1.0
0ass.5 =-0.5 = 0.5 0s .8 v0=2.2
p s
P a
p
0.4 P 0.6 v0=1.9
0.4 v =1.6
0
0.3
0.2
0.2 0.0
-2 -1 0 1 2
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
(rad)
FIG.2: Thestationarypassingprobabilityasafunctionofthe
off-diagonal component ω12 for various β12. The parameters FIG.3: Thestationarypassingprobabilityasafunctionofthe
used are thesame as those in thefigure 1. incident angle. Here m12 = 0.6, β12 = 0.8, and ω12 = −0.5,
aswellastheotherparametersareusedasthoseinthefigure
1.
B. Determination of the optimal path
Quantities plotted in the figures 1-7 are dimension-
less and k = 1.0 except for the units having been in-
B Where is the optimal incident direction which enables
cluded in the figure caption. In Fig. 1, we plot the
the particle with given initial kinetic energy to have a
critical velocity and the stationary passing probability
larger passing probability? In order to determine this
as functions of the off-diagonal components of three co-
direction, we need to choose a special angle θ which
m
efficient tensors, where one of off-diagonal components
enables the critical velocity to reach its minimum, i.e.,
varies and the other two are fixed to be zero. The
from Eq. (7),
stationary passing probability is calculated by P =
pass
clirmitti→ca∞l v21eelrofcci(cid:2)t−yhixn1c(rte)ai/se(s√2wσitxh1(ti)n)c(cid:3)r.eaItsiinsgshtohwenatbhsaotltuhtee ddvθ0c|θ=θm =0, θm =arctan(cid:18)CC43((∞))(cid:19). (8)
∞
value of m or ω ; while decreases with the increase
12 12 In fact, the largest analytically root of P(s) = 0 domi-
of β . The larger critical velocity of a system needs,
12
| | nantly determines the passing probability. The optimal
the more difficult for the particle is to arrive at the po-
incidentangleθ canthenbeexpressedbytheLangevin
tentialtop. Thisalsoimpliesthatthepassingprobability m
coefficients as
is small when the dissipation along the diffusive path is
largeifthepotentialdifferencesbetweenthesaddlepoint m12(β22a+ω22) m22(β12a+ω12)
θ =arctan − .(9)
and the initial positions are equivalent. As is shown in m (cid:18) m F (a)+m F (a) (cid:19)
11 5 12 6
the figure, the behavior of the passing probability is op-
Using the same parameters as those written in Figs. 1
posite to that of the corresponding critical velocity.
and3,weobtainθ 0.258rad,asisexplicitlyshownin
m
≃
Fig. 3, corresponding to the maximum of the stationary
Figure 2 shows the stationary passing probability in
passing probability.
thepresenceoftwooff-diagonalcomponentsω andβ ,
12 12
We now define γ, ψ and α to be the rotation angle
simultaneously,form =0. Itisseenthatthemaximum
12
of the major axis of the potential-curvature,friction and
ofthe passingprobabilitydoesnotappearinthe vertical
inertia tensors, respectively. They are found to have the
case (ω = 0). In the 2D PES, the particle is usually
12
following expressions:
supposed to travel along the potential valley and then
the steepest decedent direction. Because this is the di- 2ω 2β
12 12
tan2γ = , tan2ψ= ,
rectionwhichfaces asmaller potentialbarrier. However,
ω ω β β
22 11 22 11
it may not be a path with a weaker damping. Under − −
2m
12
the effect of the off-diagonal component of the friction tan2α = . (10)
m m
tensor, the particle is forced to select a better path with 22− 11
bothlowpotentialbarrierandweakfrictiontosurmount As an example, for the case we have studied in Figs. 1
◦ ◦
the saddle point of the potential. and 3, these angles are: γ 7.973 , ψ 34.722 ,
≃ − ≃ −
4
x = 2.0
−5 (a) 1.0 20
x = 1.0
20
0.8 x = 0.5
20
) 2−10
U (x,x1 Ppass0.6 x20= 0.0
−15
0.4 x = -0.5
20
0.2
−20
0.0
2 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
2
0 0
x x (rad)
2 −2 −2 1
FIG. 5: The stationary passing probability as a function of
θ for various x20. Here x10 = −1.0, v0 = 1.9, m12 = 0.6,
β12 = 0.8, and ω12 = −0.5, as well as the other parameters
x2 (b) are thesame as thefigure 1.
competition of these two effects results in the optimal
r
m diffusive path shown in Fig. 4 (b). This phenomenon is
similarto the quasi-stationaryflow passingoverthe bar-
OP
rier in the fission case [15], where the magnitude of the
currentis stronglyinfluenced by the off-diagonalcompo-
x1 nents of inertia and friction tensors.
0
r
Figure 5 shows the dependence of the stationarypass-
ingprobabilityontheincidentangleoftheparticlestart-
ing from various initial positions but with fixed initial
r kinetic energy. It is seen that the passing probability of
the particle starting from a large positive x position is
20
larger than that of starting from both small and nega-
tive x positions.This is because the energy difference
20
between the potential top and the initial position of the
FIG. 4: (color online.) (a) The 2D potential energy sur-
particle is small for the former. Amusingly, we find that
face, where the dotted curve is the saddle ridge line. (b)
the difference between the passing probabilities of two
A schematic illustration of the optimal diffusion path (OP),
where rm, rβ, and rω denote the major axes of the inertia, symmetricalpositions (−1.0,−0.5)and (−1.0,0.5)is ob-
friction, and potential-curvature tensors, respectively. servably large.
For a clearly understanding of the above results, we
plot in Fig. 6 the meandiffusive path of a particle start-
◦ ing fromdifferent initial positions with different incident
and α 33.690 . For comparison, we write the optimal
≃ angles. Here x (t) has been obtained in Eq. (6) and
incidentdirectionofthe the particlewe haveobtainedin 1
h i
◦
the unit of one degree, i.e., θ 14.779 (0.258 rad).
m ≃ 2
In Fig. 4, we plot the two-dimensional quadratic po-
x (t) = [C (t)x +C (t)v ], (11)
2 i+4 i0 i+6 i0
tentialandtheoptimalpathinthex -x planeinawayof h i
1 2 Xi=1
schematic illustration. All the coefficient elements used
here have been written in the figures 1 and 3. It is illus- where all the time-dependent quantities are given in the
tratedthatthedirectionoftheoptimalpathdepartsfrom appendix. The critical velocities are calculated by us-
the x -direction. The effect ofoff-diagonalcomponentof ing Eq. (7): vc = 1.5791 when x = 0.5; vc = 2.2321
1 0 20 0
the inertia tensor makes the average path of the diffu- when x = 0.5, for x = 1.0 and θ = 0. Hence
20 10
− −
sive system turn toward the positive x -axis, while the thestationarypassingprobabilityoftheparticlestarting
2
off-diagonal component of friction leads the mean path from x = 0.5 is larger than that of the particle start-
20
of the particle toward the negative x -axis. Finally, the ing from x = 0.5. In particular, under the present
2 20
−
5
1 circumstance, the diffusive process of the particle with
0.8 −5.2−646.76833 different incident angles shows an interesting behavior.
00..46 − 4−.3.151679333 Bdthiereceacfutoisroemntefhroerciaθnni=tima0lovviseellotaocrigtaeyrpotfohstaihtnieotnphaabrtteioicnflgeθac=lloons0eg.2rt5th8oertxahd1e-,
0.2 −2. saddle point than the latter. Thus the passing probabil-
x
1 0 ity for θ = 0 is larger than that of θ = 0.258 rad at the
beginning. However,the width of the Gaussiandistribu-
−0.2
tion is independent of the incident angle and increases
−0.4
with the increase of time. Although the center position
−0.6 oftheparticle’sdistributionwithθ =0.258radisbehind
−0.8 that of θ = 0, as the time goes on, it will have a lager
share of distribution passed the saddle point. Therefore,
−1
−1 −0.5 0 0.5 1
x the passing probability for a particle with incident angle
2
θ = 0.258 rad is larger than that of θ = 0 in the long
time.
FIG. 6: (color online.) The mean diffusion path of a particle
starting from the initial positions of two kinds at fixed v0 = Time-dependent passing probabilities shown in Figs.
1.9,wherethesolidanddashedlinescorrespondtoθ=0and 7 (a) and 7 (b) are also in complete agreement with the
θ=0.258rad,respectively. HerealltheLangevinparameters above theoretical analysis.
are thesame as thefigures 1 and 5.
IV. APPLICATION TO FUSION OF MASSIVE
NUCLEI
0.4 Wenowapplythepresent2Dsimplifieddiffusivemodel
)
0 (a) toinvestigatethefusionoftwomassivenuclei,whichhas
2
v been described by directional diffusion over the saddle
,
v100.3 point [6]. As a particular example, we calculate the fu-
,0 sion probability of the nearly symmetrical reaction sys-
2
,x00.2 tem100Mo+110Pd[16], whichis plotted as afunction of
1
;x thecenter-of-massenergyEc.m.inFig. 8. Aschematicil-
(t lustrationofthe deformationofthecompoundnucleusis
P 0.1
also shown in this figure. The temperature of the fusing
system is determined by aT2 = E +Q E , where
c.m. B
−
0.0 a=A/10istheenergylevelconstantwithAthenucleon
0 1 2 3 4 5 numberofthecompoundnucleus,Qdenotesthereaction
t Q value and E the barrier height of fission potential.
B
The c,h,α shapeparametrization[17]withtheelon-
{ }
gationc(thehalfofthenuclearlength)andneckvariable
hareused,i,e, x =c,x =h,andthe asymmetricalpa-
1.0 1 2
) rameterαisfixedtobezero. Theinertiaandfrictionten-
0 (b)
v20.8 sors are calculated by the Werner-Wheeler method and
,0 theone-bodydissipativemechanism[18],respectively,all
1
v
, the Langevin coefficients are considered to be constants
200.6
x at the saddle point. The three components of potential-
,
x100.4 curvature tensor are: ω11 = −28.2304, ω22 = 275.4211,
(t; and ω12 = 50.4551 in the unit of MeV; the compo-
P 0.2 βnent=s of60fr1i.c4t9io3n4 tinensthoreβu1n1it=o7f0110.9−92617,Mβe2V2 =sec6;21th.4e42in5-,
12
·
ertia elements m = 102.4081, m = 134.4673, and
0.00 1 2 3 4 5 m12 =110.3783in11the unit of 10−422M2 eVsec2.
·
t It is found a highlighted interesting result from Fig. 8
that there exists the optimal collision shape for projec-
tileandtargetnuclei,whichinducesthemaximumfusion
FIG. 7: (color online.) Time-dependent passing probability
for various x20 and θ. Here x10=−1.0, v0 =1.9, x20=−0.5 probabilityunder the samecenter-of-massenergy. Itcan
in(a)andx20 =0.5in(b),aswellastheLangevinparameters be easily understood from a combining role of the off-
are thesame as thefigures 1 and 5. diagonal components of three dynamical coefficient ten-
sors. This concludes that the fusion probability of mas-
B
C
B
C
B
C
6
B
C
1.0 1.0
m12=
1D
0.8
0.8
m12=
P0fus .6 Pfus0.6 2D
0.4 12=
0.4
c0=1.75, h0=0.2789
0.2 c0=1.50, h0=0.6157
c0=2.09, h0=0.0013 0.2
0.0 c0=2.28, h0=-0.1091
0.0
210 220 230 240 250 260 270 280 210 220 230 240 250 260 270 280
E (MeV) E (MeV)
cm cm
FIG.8: Thefusion probabilityofthereaction 100Mo+110Pd FIG.9: Thefusion probabilityofthereaction 100Mo+110Pd
as a function of the center-of-mass energy for various initial as a function of the center-of-mass energy for the situations
positions. Schematically illustrated as well are the deformed with and without off-diagonal components.
shape of the compound nucleus.
presenceofm leadstothelargestcriticalkineticenergy.
12
sivenucleicanbeenhancedifthetwocollisionheavyions Therefore,wehavearelationforthefusionprobabilities:
arepolarizedtobeellipsoidandthecollisiondirectionbe- P (β =0)>P (ω =0)>P (m =0) ata fixed
fus 12 fus 12 fus 12
tweenthelongandshortaxisofellipsoidisappropriately center-of6-mass energy. 6 6
selected. Forthefusionofdeformedmassivenuclei,there
exists the optimal angle for the incident nucleus to col-
lidewiththetargetone,whichfavorsthefusionofheavy
ions to be accomplished.
Figure9showsthefusionprobabilityof100Mo+110Pd V. SUMMARY
as a function of the center-of-mass energy when the off-
diagonalcomponentsareconsideredpartly. Heretheini- We have studied the diffusion process of a particle
tial position of the fusing system is chosen into the opti- passing over the saddle point of a two-dimensional non-
malone,i.e., c0 =1.75andh0 =0.2789aswellasallthe orthogonal quadratic potential. The expression of the
parametersusedarethesameasthefigure8. Thisfusion passingprobabilityisobtainedanalytically,wherethein-
reactionsystemisregardedasanexampletocomparethe ertia and friction tensors are not diagonal. The optimal
results with and without off-diagonal components in the incident angle of the particle’s initial velocity is deter-
potential surface, friction and mass parameters, which mined. Ourresultshaveshownthattheoptimaldiffusive
is reflected in the result presented in the above section. path,whichdepartsfromthe potentialvalleyinthetwo-
In fact, the case of without all off-diagonal components dimensional potential energy surface, induces the maxi-
is equivalent to the one dimension or without the neck mumsaddle-pointpassingprobability. Thisisduetothe
situation [4]. competition effect between the off-diagonal components
ItisseenfromFig. 9thattheincreaseofthe1Dfusion of inertia, friction and potential-curvature tensors. We
probability curve versus the energy is faster than that haveinvestigatedthefusionprobabilityofmassivenuclei
of 2D case. It has been known that the pervious 1D and compared the results with and without off-diagonal
diffusion model without the neck variable proposed the terms,forinstance,thereactionof100Mo+110Pd. Dueto
fusion probability larger than the experimental data, so the influences of off-diagonal components of inertia and
thepresentcompletelycoupled2Ddiffusionmodelmight friction upon the diffusive path, which are calculated by
be appropriate. Moreover, the results for the presence the c,h,α parametrization, the fusion probability can
{ }
of only one of three off-diagonal components can also be enhanced for an appropriate choice of the collision
be understoodfromthecriticalvelocity(kinetic energy), direction of the deformable target and projectile nuclei.
to see Fig. 1. Namely, the larger the critical kinetic The optimal configurations of colliding nuclei is between
energy of the system is, the less the fusion probability spherical and extremely deformed ones. In further, the
is for the same center-of-mass energy. The nonvanishing presentstudyalsoprovidesusefulinformationinconnec-
β allows the smallest critical kinetic energy and the tion with the synthesis of superheavy elements.
12
7
ACKNOWLEDGEMENTS where detm=m11m12−m212 and detβ =β11β22−β122.
This workwassupportedby the NationalNaturalSci-
ence Foundation of China under Grant Nos. 10674016, Thetime-dependentfactorsintheexpressionof x (t)
2
10747166 and the Specialized Research Foundation for in Eq. (11) read C (t) = L−1[F (s)/P(s)]h(j =i
j j+2
the Doctoral Programof Higher Education under Grant 5, ,8) being resulted from the inverse Laplace trans-
No. 20050027001. for·m··s, as well as
APPENDIX. THE EXPRESSIONS OF hx1(t)i AND
hx2(t)i
The quantities appear in the expression of x (t) are
1
h i F (s) = (m ω m ω )s+β ω β ω ,
7 12 11 11 12 12 11 11 12
P(s) = (detm)s4+(m11β22+m22β11 2m12β12)s3 F (s) = (detm)s3−+(m β +m β − 2m β )s2
− 8 11 22 22 11 12 12
+(detβ+m ω +m ω 2m ω )s2 −
11 22 22 11 12 12 +(m ω m ω +detβ)s+β ω
− 22 11 12 12 22 11
−
+(β ω +β ω 2β ω )s+detω,
11 22 22 11 12 12 β ω ,
− 12 12
F1(s) = (detm)s3+(m11β22+m22β11 2m12β12)s2 F (s) = −(m β m β )s+m ω m ω ,
− 9 12 11 11 12 12 11 11 12
− −
+(detβ+m11ω22 m12ω12)s+β11ω22 β12ω12, F (s) = (detm)s2+(m β m β )s+m ω
− − 10 22 11 12 12 22 11
F (s) = (m ω m ω )s+β ω β ω , −
2 12 22 22 12 12 22 22 12 m ω . (13)
− − 12 12
F (s) = (detm)s2+(m β m β )s+m ω −
3 11 22 12 12 11 22
−
m ω ,
12 12
−
F (s) = (m β m β )s+m ω m ω ,
4 12 22 22 12 12 22 22 12
− −
F (s) = m s2+β s+ω ,
5 22 22 22
F (s) = m s2 β s ω , (12)
6 12 12 12
− − −
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