Research Article The Effect of Nuclear Elastic Scattering ...downloads.hindawi.com/journals/ahep/2014/739491.pdfResearch Article The Effect of Nuclear Elastic Scattering on Temperature

Research ArticleThe Effect of Nuclear Elastic Scattering on TemperatureEquilibration Rate of Ions in Fusion Plasma

M Mahdavi1 R Azadifar1 and T Koohrokhi2

1 Physics Department University of Mazandaran PO Box 47415-416 Babolsar Iran2 Physics Department Faculty of Sciences Golestan University Shahid Beheshti Street PO Box 155 Gorgan Iran

Correspondence should be addressed to M Mahdavi mmahdaviumzacir

Received 30 May 2014 Accepted 27 October 2014 Published 17 November 2014

Academic Editor Luis A Anchordoqui

Copyright copy 2014 M Mahdavi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

A plasma with two different particle types and at different temperatures has been considered so that each type of ion withMaxwell-Boltzmann distribution function is in temperature equilibrium with itself Using the extracted nuclear elastic scattering differentialcross-section from experimental data solving the Boltzmann equation and also taking into account themobility of the backgroundparticles temperature equilibration rate between two different ions in a fusion plasma is calculatedThe results show that at highertemperature differences effect of nuclear elastic scattering is more important in calculating the temperature equilibration rate Theobtained expressions have general form so that they are applicable to each type of particle for background (119887) and each type forprojectile (119901) In this paper for example an equimolar Deuterium-Hydrogen plasma with density 119899 = 5 times 10

25 cmminus3 is chosen inwhich the deuteron is the background particle with temperature (also electron temperature) 119879

119887= 1 keV (usual conditions for a

fusion plasma at the ignition instant) and the proton is the projectile with temperature 119879119901gt 119879119887 These calculations particularly are

very important for ion fast ignition in inertial confinement fusion concept

1 Introduction

In a fusion plasma because of interactions collisions andgain and loss powers we encounter an unstable plasma [12] The thermal equilibrium between charged particles iscrucial for understanding the overall energy balance in afusion plasma where the ignition and burn of the plasmaare strongly temperature dependent [3] Since a plasma iscomposed of charged particles they are affected by Coulombforce in the plasma Coulomb force is a long-range force andthus may also be involved in the short-range collisions andthe long-range collective effects The problem of temperatureequilibration was studied first by Landau [4 5] and Spitzer(LS) [6]TheLSmodel is applicable to dilute hot fully ionizedplasmas where the collisions are weak and binary Recentlythere have been several theoretical and computational workswhich aim to study temperature equilibration [7ndash12] Amongthe theoretical studies Brown Preston and Singleton (BPS)by solving the Boltzmann equation for short-range collisionsand the Lenard-Balescu equation for long-range collective

effects and also considering the quantum corrections haveachieved more accurate results [9] In a hot and densefusion plasma in addition to the Coulomb interactions thenuclear elastic scattering (NES) is able to role via short-rangeinteractions (hard collisions) The effect of the NES on thestopping power of a charged particle has been investigatedin previous studies [13ndash15] It has been indicated that thestopping power due to NES is greater at the higher projectileenergies This fact motivated us to study the effect of NESon thermal equilibration rate for a fusion plasma A fusionplasma is formed from charged particles at high temperaturesin which they interact with each other to achieve balanceIn this paper for the simplest case a plasma with twodifferent ions at different temperatures has been consideredso that each type of ionwithMaxwell-Boltzmann distributionfunction is in temperature equilibrium with itself Herefor example proton at higher temperature is considered asprojectile and deuteron at lower temperature is assumed asbackground By solving the Boltzmann equation and takinginto account the thermal motion of plasma ions the thermal

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 739491 7 pageshttpdxdoiorg1011552014739491

2 Advances in High Energy Physics

equilibration rate due to NES is calculated The results arecompared with Coulomb calculations and the conditions inwhich each contribution is dominated are discussed

2 Elastic Collision

Consider a projectile with mass 119898119901and velocity k

119901interact-

ing with a background particle with mass 119898119887and velocity k

119887

in laboratory frame (Vectors are written in bold type andnonbold letters indicate quantity magnitude) In an elasticcollision V

119888= (119898

119901k119901+ 119898119887k119887)119872119901119887

is the velocity of thecenter of mass (CM) and relative velocity of the system isk119901119887

= k119901minus k119887 After the collision the projectile and target

velocities consider k1015840119901and k1015840

119887 respectively Therefore k1015840

119901119887=

k1015840119901minus k1015840119887is the relative velocity of two particles after collision

These quantities are related to each other by

k119901= V119888+

119898119887

119872119901119887

k119901119887

k1015840119901= V119888+

119898119887

119872119901119887

k1015840119901119887

(1)

where 119872119901119887

= 119898119901+ 119898119887is the total mass The conservation

laws for linear momentum and kinetic energy in the CMsystem indicate that the relative speeds of the particles donot change before and after collision as a result of an elasticcollision |k

119901119887| = |k1015840

119901119887| = V

119901119887 As a result the magnitude

of the relative velocities remains unchanged and only theangle Θ (scattering angle in CM system) between k

119901119887and

k1015840119901119887

is changed Since the magnitude and direction of CMvelocities (before and after a collision) will not change V

119888

can be selected as the reference Using the above relationsbetween V

119888and k119901119887

velocities the kinetic energy of the pro-jectile is changed as follows

1198641015840

119901minus 119864119901=1

2119898119901(V10158402119901minus V2119901) = 119898

119901119887V119888sdot (k1015840119901119887

minus k119901119887) (2)

where119898119901119887

= 119898119901119898119887119872119901119887is reducedmass of the projectile and

target system If 120593 is the angle betweenV119888and k119901119887 according

to scattering angle in CM system Θ then angle between V119888

and k1015840119901119887

is (120593 minus Θ) The change in kinetic energy of the pro-jectile is equal to

1198641015840

119901minus 119864119901= 119898119901119887119881119888V119901119887(cos120593 (cosΘ minus 1) + sin120593 sinΘ) (3)

Since the scattering in the CM frame is axially symmetricabout the relative speed k

119901119887 transverse components average

is zero in the scattering process and so the second term onthe right of this equation will be removed

3 Temperature Equilibration Rate due toCoulomb Interaction

The problem of temperature equilibration was first addressedby Landau and Spitzer who used the Fokker-Planck equation

to derive an equilibration rate for one ion species temperature119879119901 given another ion species temperature 119879

119887[4ndash6]

119889120576LS119901119887

119889119905= 119862

LS119901119887(119879119901minus 119879119887) (4)

where 119862LS119901119887

is defined as

119862LS119901119887

= minus8

3radic2120587

radic1198981199011198981198871198852

1199011198852

119887119899119901119899119887lnΛLS119901119887

(119898119901119879119887+ 119898119887119879119901)32

(5)

where 119898119901119887

are the ion species masses 119885119901119887

are the ion char-ges and 119899

119901119887are the number densities The LS Coulomb log-

arithm lnΛLS119901119887

is defined as follows

lnΛLS119901119887

= 23 minus ln[

[

119885119901119885119887(119898119901+ 119898119887)

119898119901119879119887+ 119898119887119879119901

(1198991199011198852

119901

119879119901

+1198991198871198852

119887

119879119887

)

12

]

]

(6)

Recently there have been several theoretical and compu-tational works which aim to study temperature equilibration[7ndash9] The computational studies have all been done withclassical molecular dynamics and have focused on temper-ature equilibration in a multi-eV hydrogen plasma [10ndash12]The results of these studies showed that deviations fromLS approach are to be expected even for moderate valuesof lnΛLS

119901119887 Among the theoretical works Brown Preston

and Singleton (BPS) produced an analytic calculation forCoulomb energy exchange processes for a fusion plasma[9 16 17] These precise calculations are accurate to leadingand next-to-leading order in the plasma coupling parameterand to all orders for two-body quantum scattering within theplasma In general the energy density exchange rate betweenthe two charged particles 119901 and 119887 in a plasma can be writtenas follows

119889120576Coul119901119887

119889119905= 119862

Coul119901119887

(119879119887minus 119879119901) (7)

where 119862Coul119901119887

is called the rate coefficient and it can be writtenas a sum of three terms

119862Coul119901119887

= (119862119862

119901119887119878+ 119862119862

119901119887119877) + 119862119876

119901119887 (8)

where the first two arise from classical short and long distancephysics and the latter term arises from short distance two-body quantum diffraction The first term in (8) the short-distance classical scattering contribution reads

119862119862

119901119887119878= minus1198962

1199011198962

119887

(119898119901120573119901119898119887120573119887)12

(119898119901119879119901+ 119898119887119879119887)32

(1

2120587)

32

times [ln1198851199011198851198871198902119870

41198981199011198871198812119901119887

+ 2120574]

(9)

Advances in High Energy Physics 3

where 120574 ≃ 057721 is Euler constant The second termin (8) the long-distance dielectric term that accounts forcollective effects in the plasma is given by

119862119862

119901119887119877=1198962

1199011198962

119887

2120587(119898119901120573119901

2120587)

12

(119898119887120573119887

2120587)

12

times int

infin

minusinfin

119889VV2 exp minus12(119898119901120573119901+ 119898119887120573119887) V2

times119894

2120587

119865 (V)120588total (V)

ln 119865 (V)1198702

(10)

Here 119870 is an arbitrary wave number so that the total resultdoes not depend upon 119870 However sometimes choosing 119870

to be a suitable multiple of the Debye wave number of theplasma simplifies the formula We write the inverse temper-ature of the projectile as 120573

119901= 119879minus1

119901and the plasma species 119887

as 120573119887= 119879minus1

119887 which we measure in energy units Debye wave

number 119896119887of this species is defined by

1198962

119887= 4120587120573

1198871198852

119887119899119887 (11)

where 119899119887is the number density of species 119887 The total Debye

wave number 119896119863is defined by

1198962

119863= sum

119887

1198962

119887 (12)

The function 119865(119906) is related to the leading-order plasmadielectric susceptibility in which it may be expressed in thedispersion form

119865 (119906) = minusint

infin

minusinfin

119889V120588total (V)119906 minus V + 119894120578

(13)

where the limit 120578 rarr 0+ is understood The spectral weight

120588total(V) is defined by

120588total (V) = sum

119887

120588119887(V) (14)

where

120588119887(V) = 119896

2

119887Vradic

120573119887119898119887

2120587exp(minus1

2120573119887119898119887V2) (15)

According to the BPS calculations the general case isobtained by adding a quantum correction which is relatedto the short distance two-body quantum diffraction and tothe classical result Therefore the third term in (8) 119862119876

119901119887is a

coefficient which is calculated from the quantum correctionand is defined as follows

119862119876

119901119887= minus

1

21198962

1199011198962

119887

(119898119901120573119901119898119887120573119887)12

(119898119901119879119901+ 119898119887119879119887)32

(1

2120587)

32

int

infin

0

119889120577 exp(minus1205772)

times [Re120595(1 + 119894120578119901119887

12057712) minus ln

120578119901119887

12057712]

(16)

the strength of the quantum effects associated with the scat-tering of two plasma species 119901 and 119887 is characterized by thedimensionless parameter

120578119901119887

=119885119901119885119887

ℏ119881119901119887

(17)

where the square of the thermal velocity in this expression isdefined by

1198812

119901119887=

119879119901

119898119901

+119879119887

119898119887

(18)

and120595(119911) is the logarithmic derivative of the gamma function

4 Temperature Equilibrium due to NuclearElastic Scattering

Since nuclear force is a short-range force it can only beinvolved in short-range collisions and therefore does notcontribute to long-range cumulative effects of plasma Boltz-mann equations are described generally as short-rangeencounters The equation for the phase-space density 119891

119901(P119901)

of species 119901 is [18]

[120597

120597119905+ k119901sdot nabla]119891

119901(r p119901 119905) = sum

119887

119870119901119887(r p119901 119905) (19)

We suppress the common space and time coordinates r 119905 andwrite the collision term involving species 119887 in the followingform

119870119901119887(r p119901 119905)

= int1198893p1015840119887

(2120587ℏ)3

1198893p1015840119901

(2120587ℏ)3

1198893p119887

(2120587ℏ)3

10038161003816100381610038161003816119879 (119882 119902

2)10038161003816100381610038161003816

2

(2120587ℏ)31205753

times (p1015840119887+ p1015840119901minus p119887minus p119901) (2120587ℏ) 120575

times (1

2119898119887V10158402119887+1

2119898119886V10158402119901minus1

2119898119887V2119887minus1

2119898119886V2119901)

times [119891119887(p1015840119887) 119891119901(p1015840119901) minus 119891119887(p119887) 119891119901(p119901)]

(20)

This relation describes the scattering of the particles ofmasses119898119901and 119898

119887and the scattering from the initial momenta

p119901= 119898119901k119901 p119887= 119898119887k119887to the final momenta p1015840

119901= 119898119901k1015840119901

p1015840119887= 119898119887k1015840119887with the scattering amplitude 119879(119882 119902

2) depending

on the center-of-mass energy 119882 and the squared momen-tum transfer 1199022 It is convenient to employ this quantum-mechanical notation for several reasons It explicitly displaysthe complete kinematical character of a scattering processincluding the detailed balance symmetry Furthermore itshows that the collision term (19) vanishes when all theparticles are in thermal equilibriumwith the generic densities119891(p) sim exp[minus120573119898V22] because of the conservation of energyenforced by the delta function In case we have a spatialuniformity (homogeneity) include the gradient removed


from (19) and thus the rate of change of energy is obtainedas follows

119889120576119901119887

119889119905= int

1198893p119901

(2120587ℏ)3

1199012

119901

2119898119901

120597119891119901(p119901 119905)

120597119905

= int1198893p119901

(2120587ℏ)3

1199012

119901

2119898119901

119870119901119887(p119901)

(21)

Placement of relation (20) in (21) and rate of change of energydensity of species 119901 in collisions with plasma species 119887will bewritten as follows

119889120576119901119887

119889119905

= int1198893p1015840119887

(2120587ℏ)3

1198893p1015840119901

(2120587ℏ)3

1198893p119887

(2120587ℏ)3

1198893p119901

(2120587ℏ)3(

11990110158402

119901

2119898119901

minus1199012

119901

2119898119901

)

times |119879|2(2120587ℏ)

3119891119887(P119887) 119891119901(P119901) 120575(3)


times (1

2119898119887V10158402119887+1

2119898119901V10158402119901minus1

2119898119887V2119887minus1

2119898119901V2119901)

(22)

Now in general the cross section for the scattering of par-ticles 119901 and 119887 into a restricted momentum interval Δ is givenby

V119901119887intΔ

119889Ω(119889120590119901119887

119889Ω)

= int1198893p1015840119887

(2120587ℏ)3

1198893p1015840119901

(2120587ℏ)3

10038161003816100381610038161003816119879 (119882 119902

2)10038161003816100381610038161003816

2

(2120587ℎ)31205753


times (1

2119898119887V10158402119887+1

2119898119901V10158402119901minus1

2119898119887V2119887minus1

2119898119901V2119901)

(23)

We can write (23) in terms of the speed of the center of massV119888and relative speed k

119901119887 We use the conversion 119889k

119901119889k119887=

119895(k119901 k119887V119888 k119901119887)119889V119888119889k119901119887 where (119895(k

119901k119887V119888k119901119887) = 1) is the

Jacobian Using these definitions we find that

119889120576119901119887

119889119905= int

1198893p119887

(2120587ℏ)3

1198893p119901

(2120587ℏ)3119891119887(P119887) 119891119901(P119901) V119901119887

times int119889Ω(119889120590119901119887

119889Ω) [1198641015840

119901minus 119864119901]

(24)

Equation (24) represents the rate of change of energy densityof projectile particles 119901 in collisions with plasma species 119887Now consider the particles distribution functions 119901 and 119887119891119901(p119901) and 119891

119887(p119887) where

119891119895(p119895) = 119899119895(2120587ℏ2120573119895

119898119895

)

32

exp(minus12120573119895119898119895V2119895) (25)

is the Maxwellian distribution function Then temperatureEquilibration rate can be calculated as follows

119889120576NI119901119887

119889119905= 119862

NI119901119887(119879119887minus 119879119901) (26)

where the coefficient 119862NI119901119887

is given by

119862NI119901119887

= minus11989911990111989911988712058712

272

(119898119901119898119887)12

119872119901119887

(119898119901119879119901+ 119898119887119879119887)52

times int

infin

0

1198642

119888119889119864119888exp(minus

119872119901119887

119898119901119879119887+ 119898119887119879119901

119864119888)

times 119868NI(119864119862)

(27)

where the integral

119868NI(119864119862) = int

120587

0

(119889120590

NI119901119887

119889Ω) (cosΘ minus 1) sinΘ119889Θ (28)

and 119864119888

= 12119898119901119887V2119901119887

is the total energy of the projectileand the target is in the center of mass system Extractingof NES differential cross-section from experimental values(119889120590

NI119901119887119889Ω) rate of temperature equilibration is archived It

is seen that (26) when the two types of ion temperatureare equal 119889120576NI

119901119887119889119905 = 0 which indicates the temperature

equilibrium

5 Results and Discussion

Now we will consider the thermal equilibration rate betweenthe proton and deuteron in an equimolar plasma withdensity 119899 = 5 times 10

25 cmminus3 The deuteron is consideredas the background at temperature 119879

119887= 1 keV and the

proton is the projectile at temperature 119879119901 greater than the

background temperature (119879119901

gt 119879119887) It is assumed that

each type of particle is in equilibrium with itself so that theequilibrium is described byMaxwell-Boltzmann distributionfunction The density and temperatures are chosen so thatthe their values are close to initial conditions of an inertialconfinement fusion fuel at ignition moment The thermalequilibration rate due to Coulomb interactions is obtainedfrom (5) and (7) for LS and BPS relations respectively(Figure 1) The portions of singular term (see (9)) regularterm (see (10)) and quantum term (see (16)) of BPS relationare shown separately It is seen that the values obtainedfrom the BPS calculations would give lower values thanLS calculations However the most important point is thatboth equilibration rates are further reduction at the highertemperature differences This event is predictable of coursebecause the Coulomb differential cross section (Rutherfordrsquosdifferential cross section) is proportionalwith the total energyin center of mass system (119864

119862) as 119889120590Coul

119889Ω prop 119864minus2

119862 The total

energy in the center of mass system increases with increasingthe temperature difference between the two ions and thus


260

240

220

200

180

160

120

140

100

60

80

40

20

0minus20

0 5 10 15 20 25 30 35 40 45 50 55 60

(Tb minus Tp) (keV)LSBPSBPS-R

BPS-SBPS-Q

(d120576 pbdt)

n(k

eV p

sminus1 )

Figure 1 The thermal equilibration rate due to Coulomb interac-tions is obtained from (4) and (7) for LS and BPS relations respec-tively

106

0067MeV

105

133MeV333MeV

104

1066MeV4320MeV100MeV

103

102

101

100

0 20 40 60 80 100 120 140 160 180

Θ (deg)

(d120590

exp dΩ)(d120590

Cou

l dΩ)

Figure 2 The ratio of the experimental differential cross-section(119889exp119901119887

120590119889Ω) to the Coulomb differential cross-section (119889Coul119901119887

120590119889Ω)is plotted versus the scattering angle in the center of mass system(Θ) for different energies 119864

119862 for proton-deuteron elastic scattering

the rate of Coulomb interaction decreases due to reductionof its cross-section

In Figure 2 the ratio of the experimental differentialcross-section (119889exp

119901119887120590119889Ω) to the Coulomb differential cross-

section (119889Coul119901119887

120590119889Ω) is plotted versus the scattering anglein the center of mass system (Θ) for different energies forproton-deuteron elastic scattering The labeled energies in

008

007

006

005

004

003

002

001

000

minus0010 1 2 3 4 5 6 7 8 9 10 11 12

EC (MeV)

INI (E

C)

Figure 3The integral 119868NI(119864119862) (see (28)) versus the total energies in

the center of mass coordinate 119864119862

the figure are the total energy in the center of mass system(119864119862) As can be seen this ratio is different for different

energies and also is associated with fluctuations However ingeneral this ratio is increased more by increasing the energyand this means that the experimental differential cross-section ismore deviated from the Coulomb differential cross-section Deviation increase reflects the growth of the contri-bution of NES differential cross-section (119889NI

119901119887120590119889Ω) at higher

energies (the superscript NI denotes the sum of nuclearand interference terms that for simplicity are called nuclearelastic scattering (NES))

By extracting the NES differential cross-section fromexperimental data (119889NI

119901119887120590119889Ω = 119889

exp119901119887

120590119889Ω minus 119889Coul119901119887

120590119889Ω)the integral 119868NI(119864

119862) (see (28)) can be calculated (Figure 3)

The integral 119868NI(119864119862) is submitted in (27) and the thermal

equilibration rate due toNES can be obtained Figure 4 showsthe thermal equilibration rate due to NES (NI) and Coulombinteraction (LS and BPS) and also the total term that isobtained from sum of nuclear and Coulomb contributions(NI +BPS) It is seen that the thermal equilibration rate due toCoulomb interaction has the main contribution in the lowertemperature differences With increasing temperature theCoulomb interaction contribution decreases and increases inthe share of NES Thus at temperature differences Δ119879 ge

3278MeV the NES is the dominant contribution to thethermal equilibration rate

The rapid development of laser-accelerated ion beamsis developing the ion fast ignition (IFI) scheme in inertialconfinement fusion concept [19] In the IFI scheme anintense laser pulse accelerates an ion beam to multi-MeVtemperatures The accelerated ion beam deposits its energyin the ignition region of fuel pellet that is named ldquohot spotrdquoand provides the required energy for ignition [20 21] Forthis reason the diver ion beam is named ignitor The poweris deposited through an ion beam energy exchange withthe background plasma particles with few-keV temperatures


70

65

60

55

50

45

40

35

30

25

20

15

10

5 3278 (MeV)

20 22 24 26 28 30 32 34 36 38 40

LSBPS

NIBPS + NI

(Tb minus Tp) (MeV)

0

(d120576 pbdt)

n(k

eV p

sminus1 )

Figure 4 The thermal equilibration rate due to NES (NI) andCoulomb interaction (LS and BPS) and also the total term that isobtained from the sum of nuclear and Coulomb contributions (NI +BPS)

In the energy exchange process some of the ion beam energyis delivered to the plasma ions and another part of energy isdelivered to the plasma electronsThe electrons are the sourceof power loss in a fusion plasma that it is better to be kept atlow temperature At the high temperature difference (multi-MeV) between ion beam and background plasma particlesthe rate of energy exchange via Coulomb interaction isreduced and in turn the rate of energy exchange via NES isincreased The NES causes the ion temperature to be furtherincreased resulting in reduced power loss

The laser-accelerated ion beams can be produced withdifferent distribution functions such as exponential Max-wellian and quasimonoenergy distribution functions [2223] Furthermore some of ion beams such as the deuteronbeam can be on their way to interact with the backgroundions [24 25] Therefore research in this area requires moredetailed calculations that will be covered in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] JD Lindl Inertial Confinement Fusion SpringerNewYorkNYUSA 1998

[2] J D Lindl P Amendt R L Berger et al ldquoThe physics basisfor ignition using indirect-drive targets on theNational IgnitionFacilityrdquo Physics of Plasmas vol 11 no 2 pp 339ndash491 2004

[3] S Atzeni and J Meyer-ter-VehnlThe Physics of Inertial FusionOxford University Press Oxford UK 2004

[4] L D Landau ldquoDie kinetische Gleichung fur den Fall Coulomb-scherWechselwirkungrdquo in Physikalische Zeitschrift der Sowjetu-nion vol 10 pp 154ndash164 1936

[5] L D Landau ldquoKinetic equation for the case of Coulomb inter-actionrdquo Zhurnal Eksperimentalrsquonoi i Teoreticheskoi Fiziki vol 7p 203 1958

[6] L Spitzer Jr Physics of Fully Ionized Gases Interscience NewYork NY USA 2nd edition 1992

[7] M W C Dharma-Wardana and F Perrot ldquoEnergy relaxationand the quasiequation of state of a dense two-temperature non-equilibrium plasmardquo Physical Review E vol 58 no 3 pp 3705ndash3718 1998

[8] D O Gericke M S Murillo and M Schlanges ldquoDense plasmatemperature equilibration in the binary collision approxima-tionrdquo Physical Review EmdashStatistical Nonlinear and Soft MatterPhysics vol 65 no 3 Article ID 036418 2002

[9] L S Brown D L Preston and R L Singleton Jr ldquoChargedparticlemotion in a highly ionized plasmardquo Physics Reports vol410 no 4 pp 237ndash333 2005

[10] J N Glosli F R Graziani RMMore et al ldquoMolecular dynam-ics simulations of temperature equilibration in dense hydrogenrdquoPhysical Review E vol 78 no 2 Article ID 025401(R) 2008

[11] G Dimonte and J Daligault ldquoMolecular-dynamics simulationsof electron-ion temperature relaxation in a classical Coulombplasmardquo Physical Review Letters vol 101 no 13 Article ID135001 2008

[12] B Jeon M Foster J Colgan et al ldquoEnergy relaxation rates indense hydrogen plasmasrdquo Physical Review EmdashStatistical Non-linear and Soft Matter Physics vol 78 no 3 Article ID 0364032008

[13] MMahdavi and T Koohrokhi ldquoNuclear elastic scattering effecton stopping power of charged particles in high-temperaturemediardquoModern Physics Letters A vol 26 no 21 pp 1561ndash15702011

[14] M Mahdavi and T Koohrokhi ldquoEnergy deposition of multi-MeV protons in compressed targets of fast-ignition inertial con-finement fusionrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 85 no 1 Article ID 016405 2012

[15] M Mahdavi T Koohrokhi and R Azadifar ldquoThe interactionof quasi-monoenergetic protons with pre-compressed inertialfusion fuelsrdquo Physics of Plasmas vol 19 no 8 Article ID 0827072012

[16] L S Brown andR L Singleton ldquoTemperature equilibration ratewith Fermi-Dirac statisticsrdquo Physical Review E vol 76 no 6Article ID 066404 2007

[17] R L Singleton ldquoCharged particle stopping power effects onignition some results from an exact calculationrdquo Physics of Plas-mas vol 15 no 5 Article ID 056302 2008

[18] D Kremp M Schlanges and W-D Kraeft Quantum Statisticsof Nonideal Plasma vol 25 of Atomic Optical and PlasmaPhysics Springer Berlin Germany 2005

[19] J C Fernandez B J Albright F N Beg et al ldquoFast ignition withlaser-driven proton and ion beamsrdquoNuclear Fusion vol 54 no5 Article ID 054006 2014

[20] M Tabak J Hammer M E Glinsky et al ldquoIgnition and highgain with ultrapowerful lasersrdquo Physics of Plasmas vol 1 no 5pp 1626ndash1634 1994

[21] M Roth T E Cowan M H Key et al ldquoFast ignition by intenselaser-accelerated proton beamsrdquoPhysical Review Letters vol 86no 3 pp 436ndash439 2001


[22] BMHegelich B J Albright J Cobble et al ldquoLaser accelerationof quasi-monoenergetic MeV ion beamsrdquo Nature vol 439 no7075 pp 441ndash444 2006

[23] M Roth D Jung K Falk et al ldquoBright laser-driven neutronsource based on the relativistic transparency of solidsrdquo PhysicalReview Letters vol 110 no 4 Article ID 044802 2013

[24] D-X Liu W Hong L-Q Shan S-C Wu and Y-Q Gu ldquoFastignition by a laser-accelerated deuteron beamrdquo Plasma Physicsand Controlled Fusion vol 53 no 3 Article ID 035022 2011

[25] X Yang G H Miley K A Flippo and H Hora ldquoEnergyenhancement for deuteron beam fast ignition of a precom-pressed inertial confinement fusion targetrdquo Physics of Plasmasvol 18 no 3 Article ID 032703 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


equilibration rate due to NES is calculated The results arecompared with Coulomb calculations and the conditions inwhich each contribution is dominated are discussed

2 Elastic Collision

Consider a projectile with mass 119898119901and velocity k

119901interact-

ing with a background particle with mass 119898119887and velocity k

119887

in laboratory frame (Vectors are written in bold type andnonbold letters indicate quantity magnitude) In an elasticcollision V

119888= (119898

119901k119901+ 119898119887k119887)119872119901119887

is the velocity of thecenter of mass (CM) and relative velocity of the system isk119901119887

= k119901minus k119887 After the collision the projectile and target

velocities consider k1015840119901and k1015840

119887 respectively Therefore k1015840

119901119887=

k1015840119901minus k1015840119887is the relative velocity of two particles after collision

These quantities are related to each other by

k119901= V119888+

119898119887

119872119901119887

k119901119887

k1015840119901= V119888+

119898119887

119872119901119887

k1015840119901119887

(1)

where 119872119901119887

= 119898119901+ 119898119887is the total mass The conservation

laws for linear momentum and kinetic energy in the CMsystem indicate that the relative speeds of the particles donot change before and after collision as a result of an elasticcollision |k

119901119887| = |k1015840

119901119887| = V

119901119887 As a result the magnitude

of the relative velocities remains unchanged and only theangle Θ (scattering angle in CM system) between k

119901119887and

k1015840119901119887

is changed Since the magnitude and direction of CMvelocities (before and after a collision) will not change V

119888

can be selected as the reference Using the above relationsbetween V

119888and k119901119887

velocities the kinetic energy of the pro-jectile is changed as follows

1198641015840

119901minus 119864119901=1

2119898119901(V10158402119901minus V2119901) = 119898

119901119887V119888sdot (k1015840119901119887

minus k119901119887) (2)

where119898119901119887

= 119898119901119898119887119872119901119887is reducedmass of the projectile and

target system If 120593 is the angle betweenV119888and k119901119887 according

to scattering angle in CM system Θ then angle between V119888

and k1015840119901119887

is (120593 minus Θ) The change in kinetic energy of the pro-jectile is equal to

1198641015840

119901minus 119864119901= 119898119901119887119881119888V119901119887(cos120593 (cosΘ minus 1) + sin120593 sinΘ) (3)

Since the scattering in the CM frame is axially symmetricabout the relative speed k

119901119887 transverse components average

is zero in the scattering process and so the second term onthe right of this equation will be removed

3 Temperature Equilibration Rate due toCoulomb Interaction

The problem of temperature equilibration was first addressedby Landau and Spitzer who used the Fokker-Planck equation

to derive an equilibration rate for one ion species temperature119879119901 given another ion species temperature 119879

119887[4ndash6]

119889120576LS119901119887

119889119905= 119862

LS119901119887(119879119901minus 119879119887) (4)

where 119862LS119901119887

is defined as

119862LS119901119887

= minus8

3radic2120587

radic1198981199011198981198871198852

1199011198852

119887119899119901119899119887lnΛLS119901119887

(119898119901119879119887+ 119898119887119879119901)32

(5)

where 119898119901119887

are the ion species masses 119885119901119887

are the ion char-ges and 119899

119901119887are the number densities The LS Coulomb log-

arithm lnΛLS119901119887

is defined as follows

lnΛLS119901119887

= 23 minus ln[

[

119885119901119885119887(119898119901+ 119898119887)

119898119901119879119887+ 119898119887119879119901

(1198991199011198852

119901

119879119901

+1198991198871198852

119887

119879119887

)

12

]

]

(6)

Recently there have been several theoretical and compu-tational works which aim to study temperature equilibration[7ndash9] The computational studies have all been done withclassical molecular dynamics and have focused on temper-ature equilibration in a multi-eV hydrogen plasma [10ndash12]The results of these studies showed that deviations fromLS approach are to be expected even for moderate valuesof lnΛLS

119901119887 Among the theoretical works Brown Preston

and Singleton (BPS) produced an analytic calculation forCoulomb energy exchange processes for a fusion plasma[9 16 17] These precise calculations are accurate to leadingand next-to-leading order in the plasma coupling parameterand to all orders for two-body quantum scattering within theplasma In general the energy density exchange rate betweenthe two charged particles 119901 and 119887 in a plasma can be writtenas follows

119889120576Coul119901119887

119889119905= 119862

Coul119901119887

(119879119887minus 119879119901) (7)

where 119862Coul119901119887

is called the rate coefficient and it can be writtenas a sum of three terms

119862Coul119901119887

= (119862119862

119901119887119878+ 119862119862

119901119887119877) + 119862119876

119901119887 (8)

where the first two arise from classical short and long distancephysics and the latter term arises from short distance two-body quantum diffraction The first term in (8) the short-distance classical scattering contribution reads

119862119862

119901119887119878= minus1198962

1199011198962

119887

(119898119901120573119901119898119887120573119887)12

(119898119901119879119901+ 119898119887119879119887)32

(1

2120587)

32

times [ln1198851199011198851198871198902119870

41198981199011198871198812119901119887

+ 2120574]

(9)



119862119862

119901119887119877=1198962

1199011198962

119887

2120587(119898119901120573119901

2120587)

12

(119898119887120573119887

2120587)

12

times int

infin

minusinfin

119889VV2 exp minus12(119898119901120573119901+ 119898119887120573119887) V2

times119894

2120587

119865 (V)120588total (V)

ln 119865 (V)1198702

(10)



119901= 119879minus1


as 120573119887= 119879minus1



1198962

119887= 4120587120573

1198871198852

119887119899119887 (11)



1198962

119863= sum

119887

1198962

119887 (12)


119865 (119906) = minusint

infin

minusinfin

119889V120588total (V)119906 minus V + 119894120578

(13)




119887

120588119887(V) (14)

where

120588119887(V) = 119896

2

119887Vradic

120573119887119898119887

2120587exp(minus1

2120573119887119898119887V2) (15)


119901119887is a


119862119876

119901119887= minus

1

21198962

1199011198962

119887

(119898119901120573119901119898119887120573119887)12

(119898119901119879119901+ 119898119887119879119887)32

(1

2120587)

32

int

infin

0

119889120577 exp(minus1205772)

times [Re120595(1 + 119894120578119901119887

12057712) minus ln

120578119901119887

12057712]

(16)


120578119901119887

=119885119901119885119887

ℏ119881119901119887

(17)


1198812

119901119887=

119879119901

119898119901

+119879119887

119898119887

(18)




119901(P119901)


[120597

120597119905+ k119901sdot nabla]119891

119901(r p119901 119905) = sum

119887

119870119901119887(r p119901 119905) (19)


119870119901119887(r p119901 119905)

= int1198893p1015840119887

(2120587ℏ)3

1198893p1015840119901

(2120587ℏ)3

1198893p119887

(2120587ℏ)3

10038161003816100381610038161003816119879 (119882 119902

2)10038161003816100381610038161003816

2

(2120587ℏ)31205753


times (1

2119898119887V10158402119887+1

2119898119886V10158402119901minus1

2119898119887V2119887minus1

2119898119886V2119901)

times [119891119887(p1015840119887) 119891119901(p1015840119901) minus 119891119887(p119887) 119891119901(p119901)]

(20)




119901= 119898119901k1015840119901


2) depending




119889120576119901119887

119889119905= int

1198893p119901

(2120587ℏ)3

1199012

119901

2119898119901

120597119891119901(p119901 119905)

120597119905

= int1198893p119901

(2120587ℏ)3

1199012

119901

2119898119901

119870119901119887(p119901)

(21)


119889120576119901119887

119889119905

= int1198893p1015840119887

(2120587ℏ)3

1198893p1015840119901

(2120587ℏ)3

1198893p119887

(2120587ℏ)3

1198893p119901

(2120587ℏ)3(

11990110158402

119901

2119898119901

minus1199012

119901

2119898119901

)

times |119879|2(2120587ℏ)

3119891119887(P119887) 119891119901(P119901) 120575(3)


times (1

2119898119887V10158402119887+1

2119898119901V10158402119901minus1

2119898119887V2119887minus1

2119898119901V2119901)

(22)


V119901119887intΔ

119889Ω(119889120590119901119887

119889Ω)

= int1198893p1015840119887

(2120587ℏ)3

1198893p1015840119901

(2120587ℏ)3

10038161003816100381610038161003816119879 (119882 119902

2)10038161003816100381610038161003816

2

(2120587ℎ)31205753


times (1

2119898119887V10158402119887+1

2119898119901V10158402119901minus1

2119898119887V2119887minus1

2119898119901V2119901)

(23)



119901119889k119887=

119895(k119901 k119887V119888 k119901119887)119889V119888119889k119901119887 where (119895(k

119901k119887V119888k119901119887) = 1) is the


119889120576119901119887

119889119905= int

1198893p119887

(2120587ℏ)3

1198893p119901

(2120587ℏ)3119891119887(P119887) 119891119901(P119901) V119901119887

times int119889Ω(119889120590119901119887

119889Ω) [1198641015840

119901minus 119864119901]

(24)


119887(p119887) where

119891119895(p119895) = 119899119895(2120587ℏ2120573119895

119898119895

)

32

exp(minus12120573119895119898119895V2119895) (25)


119889120576NI119901119887

119889119905= 119862

NI119901119887(119879119887minus 119879119901) (26)


is given by

119862NI119901119887

= minus11989911990111989911988712058712

272

(119898119901119898119887)12

119872119901119887

(119898119901119879119901+ 119898119887119879119887)52

times int

infin

0

1198642

119888119889119864119888exp(minus

119872119901119887

119898119901119879119887+ 119898119887119879119901

119864119888)

times 119868NI(119864119862)

(27)

where the integral

119868NI(119864119862) = int

120587

0

(119889120590

NI119901119887


and 119864119888

= 12119898119901119887V2119901119887





equilibrium









119862) as 119889120590Coul


119862 The total



260

240

220

200

180

160

120

140

100

60

80

40

20

0minus20

0 5 10 15 20 25 30 35 40 45 50 55 60


BPS-SBPS-Q

(d120576 pbdt)

n(k

eV p

sminus1 )


106

0067MeV

105

133MeV333MeV

104


103

102

101

100

0 20 40 60 80 100 120 140 160 180

Θ (deg)

(d120590

exp dΩ)(d120590

Cou

l dΩ)








section (119889Coul119901119887


008

007

006

005

004

003

002

001

000

minus0010 1 2 3 4 5 6 7 8 9 10 11 12

EC (MeV)

INI (E

C)





119901119887120590119889Ω) at higher



119901119887120590119889Ω = 119889

exp119901119887

120590119889Ω minus 119889Coul119901119887

120590119889Ω)the integral 119868NI(119864







70

65

60

55

50

45

40

35

30

25

20

15

10

5 3278 (MeV)

20 22 24 26 28 30 32 34 36 38 40

LSBPS

NIBPS + NI

(Tb minus Tp) (MeV)

0

(d120576 pbdt)

n(k

eV p

sminus1 )






References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119862119862

119901119887119877=1198962

1199011198962

119887

2120587(119898119901120573119901

2120587)

12

(119898119887120573119887

2120587)

12

times int

infin

minusinfin

119889VV2 exp minus12(119898119901120573119901+ 119898119887120573119887) V2

times119894

2120587

119865 (V)120588total (V)

ln 119865 (V)1198702

(10)



119901= 119879minus1


as 120573119887= 119879minus1



1198962

119887= 4120587120573

1198871198852

119887119899119887 (11)



1198962

119863= sum

119887

1198962

119887 (12)


119865 (119906) = minusint

infin

minusinfin

119889V120588total (V)119906 minus V + 119894120578

(13)




119887

120588119887(V) (14)

where

120588119887(V) = 119896

2

119887Vradic

120573119887119898119887

2120587exp(minus1

2120573119887119898119887V2) (15)


119901119887is a


119862119876

119901119887= minus

1

21198962

1199011198962

119887

(119898119901120573119901119898119887120573119887)12

(119898119901119879119901+ 119898119887119879119887)32

(1

2120587)

32

int

infin

0

119889120577 exp(minus1205772)

times [Re120595(1 + 119894120578119901119887

12057712) minus ln

120578119901119887

12057712]

(16)


120578119901119887

=119885119901119885119887

ℏ119881119901119887

(17)


1198812

119901119887=

119879119901

119898119901

+119879119887

119898119887

(18)




119901(P119901)


[120597

120597119905+ k119901sdot nabla]119891

119901(r p119901 119905) = sum

119887

119870119901119887(r p119901 119905) (19)


119870119901119887(r p119901 119905)

= int1198893p1015840119887

(2120587ℏ)3

1198893p1015840119901

(2120587ℏ)3

1198893p119887

(2120587ℏ)3

10038161003816100381610038161003816119879 (119882 119902

2)10038161003816100381610038161003816

2

(2120587ℏ)31205753


times (1

2119898119887V10158402119887+1

2119898119886V10158402119901minus1

2119898119887V2119887minus1

2119898119886V2119901)

times [119891119887(p1015840119887) 119891119901(p1015840119901) minus 119891119887(p119887) 119891119901(p119901)]

(20)




119901= 119898119901k1015840119901


2) depending




119889120576119901119887

119889119905= int

1198893p119901

(2120587ℏ)3

1199012

119901

2119898119901

120597119891119901(p119901 119905)

120597119905

= int1198893p119901

(2120587ℏ)3

1199012

119901

2119898119901

119870119901119887(p119901)

(21)


119889120576119901119887

119889119905

= int1198893p1015840119887

(2120587ℏ)3

1198893p1015840119901

(2120587ℏ)3

1198893p119887

(2120587ℏ)3

1198893p119901

(2120587ℏ)3(

11990110158402

119901

2119898119901

minus1199012

119901

2119898119901

)

times |119879|2(2120587ℏ)

3119891119887(P119887) 119891119901(P119901) 120575(3)


times (1

2119898119887V10158402119887+1

2119898119901V10158402119901minus1

2119898119887V2119887minus1

2119898119901V2119901)

(22)


V119901119887intΔ

119889Ω(119889120590119901119887

119889Ω)

= int1198893p1015840119887

(2120587ℏ)3

1198893p1015840119901

(2120587ℏ)3

10038161003816100381610038161003816119879 (119882 119902

2)10038161003816100381610038161003816

2

(2120587ℎ)31205753


times (1

2119898119887V10158402119887+1

2119898119901V10158402119901minus1

2119898119887V2119887minus1

2119898119901V2119901)

(23)



119901119889k119887=

119895(k119901 k119887V119888 k119901119887)119889V119888119889k119901119887 where (119895(k

119901k119887V119888k119901119887) = 1) is the


119889120576119901119887

119889119905= int

1198893p119887

(2120587ℏ)3

1198893p119901

(2120587ℏ)3119891119887(P119887) 119891119901(P119901) V119901119887

times int119889Ω(119889120590119901119887

119889Ω) [1198641015840

119901minus 119864119901]

(24)


119887(p119887) where

119891119895(p119895) = 119899119895(2120587ℏ2120573119895

119898119895

)

32

exp(minus12120573119895119898119895V2119895) (25)


119889120576NI119901119887

119889119905= 119862

NI119901119887(119879119887minus 119879119901) (26)


is given by

119862NI119901119887

= minus11989911990111989911988712058712

272

(119898119901119898119887)12

119872119901119887

(119898119901119879119901+ 119898119887119879119887)52

times int

infin

0

1198642

119888119889119864119888exp(minus

119872119901119887

119898119901119879119887+ 119898119887119879119901

119864119888)

times 119868NI(119864119862)

(27)

where the integral

119868NI(119864119862) = int

120587

0

(119889120590

NI119901119887


and 119864119888

= 12119898119901119887V2119901119887





equilibrium









119862) as 119889120590Coul


119862 The total



260

240

220

200

180

160

120

140

100

60

80

40

20

0minus20

0 5 10 15 20 25 30 35 40 45 50 55 60


BPS-SBPS-Q

(d120576 pbdt)

n(k

eV p

sminus1 )


106

0067MeV

105

133MeV333MeV

104


103

102

101

100

0 20 40 60 80 100 120 140 160 180

Θ (deg)

(d120590

exp dΩ)(d120590

Cou

l dΩ)








section (119889Coul119901119887


008

007

006

005

004

003

002

001

000

minus0010 1 2 3 4 5 6 7 8 9 10 11 12

EC (MeV)

INI (E

C)





119901119887120590119889Ω) at higher



119901119887120590119889Ω = 119889

exp119901119887

120590119889Ω minus 119889Coul119901119887

120590119889Ω)the integral 119868NI(119864







70

65

60

55

50

45

40

35

30

25

20

15

10

5 3278 (MeV)

20 22 24 26 28 30 32 34 36 38 40

LSBPS

NIBPS + NI

(Tb minus Tp) (MeV)

0

(d120576 pbdt)

n(k

eV p

sminus1 )






References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119889120576119901119887

119889119905= int

1198893p119901

(2120587ℏ)3

1199012

119901

2119898119901

120597119891119901(p119901 119905)

120597119905

= int1198893p119901

(2120587ℏ)3

1199012

119901

2119898119901

119870119901119887(p119901)

(21)


119889120576119901119887

119889119905

= int1198893p1015840119887

(2120587ℏ)3

1198893p1015840119901

(2120587ℏ)3

1198893p119887

(2120587ℏ)3

1198893p119901

(2120587ℏ)3(

11990110158402

119901

2119898119901

minus1199012

119901

2119898119901

)

times |119879|2(2120587ℏ)

3119891119887(P119887) 119891119901(P119901) 120575(3)


times (1

2119898119887V10158402119887+1

2119898119901V10158402119901minus1

2119898119887V2119887minus1

2119898119901V2119901)

(22)


V119901119887intΔ

119889Ω(119889120590119901119887

119889Ω)

= int1198893p1015840119887

(2120587ℏ)3

1198893p1015840119901

(2120587ℏ)3

10038161003816100381610038161003816119879 (119882 119902

2)10038161003816100381610038161003816

2

(2120587ℎ)31205753


times (1

2119898119887V10158402119887+1

2119898119901V10158402119901minus1

2119898119887V2119887minus1

2119898119901V2119901)

(23)



119901119889k119887=

119895(k119901 k119887V119888 k119901119887)119889V119888119889k119901119887 where (119895(k

119901k119887V119888k119901119887) = 1) is the


119889120576119901119887

119889119905= int

1198893p119887

(2120587ℏ)3

1198893p119901

(2120587ℏ)3119891119887(P119887) 119891119901(P119901) V119901119887

times int119889Ω(119889120590119901119887

119889Ω) [1198641015840

119901minus 119864119901]

(24)


119887(p119887) where

119891119895(p119895) = 119899119895(2120587ℏ2120573119895

119898119895

)

32

exp(minus12120573119895119898119895V2119895) (25)


119889120576NI119901119887

119889119905= 119862

NI119901119887(119879119887minus 119879119901) (26)


is given by

119862NI119901119887

= minus11989911990111989911988712058712

272

(119898119901119898119887)12

119872119901119887

(119898119901119879119901+ 119898119887119879119887)52

times int

infin

0

1198642

119888119889119864119888exp(minus

119872119901119887

119898119901119879119887+ 119898119887119879119901

119864119888)

times 119868NI(119864119862)

(27)

where the integral

119868NI(119864119862) = int

120587

0

(119889120590

NI119901119887


and 119864119888

= 12119898119901119887V2119901119887





equilibrium









119862) as 119889120590Coul


119862 The total



260

240

220

200

180

160

120

140

100

60

80

40

20

0minus20

0 5 10 15 20 25 30 35 40 45 50 55 60


BPS-SBPS-Q

(d120576 pbdt)

n(k

eV p

sminus1 )


106

0067MeV

105

133MeV333MeV

104


103

102

101

100

0 20 40 60 80 100 120 140 160 180

Θ (deg)

(d120590

exp dΩ)(d120590

Cou

l dΩ)








section (119889Coul119901119887


008

007

006

005

004

003

002

001

000

minus0010 1 2 3 4 5 6 7 8 9 10 11 12

EC (MeV)

INI (E

C)





119901119887120590119889Ω) at higher



119901119887120590119889Ω = 119889

exp119901119887

120590119889Ω minus 119889Coul119901119887

120590119889Ω)the integral 119868NI(119864







70

65

60

55

50

45

40

35

30

25

20

15

10

5 3278 (MeV)

20 22 24 26 28 30 32 34 36 38 40

LSBPS

NIBPS + NI

(Tb minus Tp) (MeV)

0

(d120576 pbdt)

n(k

eV p

sminus1 )






References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




260

240

220

200

180

160

120

140

100

60

80

40

20

0minus20

0 5 10 15 20 25 30 35 40 45 50 55 60


BPS-SBPS-Q

(d120576 pbdt)

n(k

eV p

sminus1 )


106

0067MeV

105

133MeV333MeV

104


103

102

101

100

0 20 40 60 80 100 120 140 160 180

Θ (deg)

(d120590

exp dΩ)(d120590

Cou

l dΩ)








section (119889Coul119901119887


008

007

006

005

004

003

002

001

000

minus0010 1 2 3 4 5 6 7 8 9 10 11 12

EC (MeV)

INI (E

C)





119901119887120590119889Ω) at higher



119901119887120590119889Ω = 119889

exp119901119887

120590119889Ω minus 119889Coul119901119887

120590119889Ω)the integral 119868NI(119864







70

65

60

55

50

45

40

35

30

25

20

15

10

5 3278 (MeV)

20 22 24 26 28 30 32 34 36 38 40

LSBPS

NIBPS + NI

(Tb minus Tp) (MeV)

0

(d120576 pbdt)

n(k

eV p

sminus1 )






References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




70

65

60

55

50

45

40

35

30

25

20

15

10

5 3278 (MeV)

20 22 24 26 28 30 32 34 36 38 40

LSBPS

NIBPS + NI

(Tb minus Tp) (MeV)

0

(d120576 pbdt)

n(k

eV p

sminus1 )






References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics













FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article The Effect of Nuclear Elastic Scattering ...downloads.hindawi.com/journals/ahep/2014/739491.pdfResearch Article The Effect of Nuclear Elastic Scattering on Temperature