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Protons Range in some superconductor Materials

Wafaa.N.Jassim , Rashid. A. Kadhum   

Physics Department College for Girls Kufa 54001,Iraq

Corresponding authors : [email protected]

                                    [email protected]


In this research, a theoretical study for calculation of the range of protons with energy (2-10MeV) when passing in the superconductor  media (Ti, Mo, Sn and  W). The range of proton were calculated using A. K. CHAUBEY and H. V. GUPTA equation , SRIM 2012 program , PASTR program ,  and then a semi empirical equation we got for calculating least square method  . we get a good agreement among these results. As well as that we calculate  the maximum difference between the semi empirical calculation and with all results we found using the statistical test ( kstest2)  in Matlab program .

Keyword: Range, stopping power,  heavy charged, superconductor, proton.


The information of the features of the transmission and absorption of low, intermediary and high energy proton in elemental materials is of great importance for the experimental methods in nuclear and atomic physics. It is also useful in thoughtful the various interactions of these particles with matter .The knowledge of the ranges of this particle in matter has useful applications for the study of biological effects, radiation damage dosage 'rates, structure analysis of solid target by Rutherford backscattering spectroscopy and energy dissipation at a variety of depths of an absorber. It has also useful applications in the design of detection systems, radiation technology, semi-conductor detectors, shielding and choosing the exact thickness of the target [1,2]. If an ion beam penetrates through matter it loses energy due to collisions with electrons (electronic stopping) and target nuclei (nuclear stopping) [3]. The first (classical) calculation of the energy loss of energetic particles was made by Bohr, while the first quantum mechanical treatment was done by Bethe. This latter theory of stopping power is particularly accurate when the projectile's velocity is sufficiently high[2]. The range of a charged particle is the space it travels before coming to rest. The common of the stopping power gives the distance traveled per unit energy loss. Therefore, the range R(T) of a particle of kinetic energy T is the integral of this quantity down to zero energy[4,5,6]:

R(T)='_0^T''(-dE/dx)^(-1) dE'..(1)'

which for heavy charged projectiles is nearly the same as the mean range R, i.e., the average traversed absorber thickness, because heavy ions are very little scattered and travel almost on a straight line [5]. There have been several theoretical and experimental studies of variation of range of protons with energy in several materials. These studies have guide to the development of empirical relations exact for the material under inquiry and within the energy range used in the experiment. Let us have a look at the proton range relations for air. In air the range of protons having energy Ep can be computed from [6]:

R_p^air [m]=[E_p/9.3]^1.8                   for Ep<200 MeV'..(2)

Ranges of protons as well as other charged particles such as a alpha particles and deuterons given energy in absorber elements of atomic number    Z > 10 in units of absorber mass thickness can be calculated directly by comparison to the calculated range of the same charged particles of the same energy in air according to the following formula described by Friedlander et al. (1964)[7,8]:

where Rz is the range of the charged particle in mass thickness units    mgcm-2, Rair is the range of the charged particle in air in the same mass thickness units, Z is the atomic number of the absorber element, E is the particle energy in MeV, and M is the mass number of the particle (i.e.,1 for protons, 2 for deuterons, and 4 for  alpha particles).The formula provided by Eq. (11) is applicable to charged particles over a wide range of energies(approximately over the range (0.1'1000MeV) and for absorber elements of Z> 10. For lighter absorber elements the term 0.90+0.0275Z is replaced by the value 1.00 with the exception of hydrogen and helium, where the value of 0.30 and 0.82 are used, respectively (Friedlander et al., 1964)[7,8] .For two heavy charged particles at the same initial speed '', the ratio of their ranges is simply[4]:


where M1 and M2 are the rest masses  for projectile and target respectively and Z1 and Z2 are the charges. If particle number 2 is a proton (M2 = 1 and Z2 = 1), then we can write for the range R of the other particle (mass M1= m_p proton masses and charge Z1 = Z) .


The range of a charged particle is computed by numerical integration of the stopping power. The range R in the nonstop slowing down approximation (csda) is given as[10] :

R='_(E_min)^(E_max)''(-dE/''dx)^(-1 ) dE+R(E_min ) ' '..(7)

where R(Emin) is the calculated range at energy Emin and the stopping power of protons is :

-dE/''dX=a/A E^(-b)  Z^(c logE+d  )   '..       (8)

Which represented  empirical relation for the stopping power of protons have arrived by A. K. CHAUBEY and H. V. GUPTA[1,10].The appropriate values of the constants a, b, c, and d are a = 915.0, b = 0.85,

 c = 0.145, d = 0.635.'', A and Z denote the density, atomic weight and atomic number of the stopping material while E is the kinetic energy of the particle in MeV/amu. where R( Emin) is the measured range at energy Emin which is added to the integral equation (7) and treated as a constant for a particular particle and material. Substituting eq. (8) into eq. (7) and converting energy units from MeV to MeV/amu we get,

R_p='_(E_1)^E''m_p [a/A  E^(-b) Z^('  '^(clogE+d) ) ]^(-1) dE+R_1 '(E_1)  '..(9)

 After integration and putting in the values of the constants we get :

R_p=m_p [A/(915''1.85Z^0.635(1-0.145logZ/1.85)  )]''[E^1.85 Z^(-0.145logE)-E_1^1.85 Z^(-0.145logE_1 ) ]+R_(1 ) (E_1 )  '..(10)

The R_(1 ) (E_1 ) is the experimental range of the proton at energy E_1 which slightly differs from the calculated range at the same energy E_1. Therefore, the second term in the second bracket of eq. (10) can be combined with R_(1 ) (E_1 ) and we may define a correction term Fp to the range in a specific medium as:

F_p=R_(1 ) (E_1 )-m_p G_p E_1^1.85 Z^(-0.145logE_1 ) '..(11)

G_p=[A/(915''1.85Z^0.635(1-0.145logZ/1.85)  )] '..(12)

Therefore, eq. (10) reduces to

R_p=m_p G_p E^1.85 Z^(-0.145logE)+F_p  '..(13)

This equation gives the ranges of protons in gm/cm2 in solid medium in the energy region 0.7 to 12.0 MeV/amu[1,10].

2-Results and discussion :

Superconducting materials will have a significant role in advancing industrial and scientific applications with major benefits in various sectors including energy, environment, and healthcare [11]. By using the equation (13) we calculate the range of protons with energy (2-10MeV) when passing in the superconductor  media (Titanium, Molybdnum , Tin and  Tinjasten), for this calculation we have taken Emin as 1 MeV. We have arrived at the following semi empirical relation for the range of protons for that four superconductor  media by using least square method  :

R=aE+b '..(14)

a=(10'''ER_srim-'''E''R_srim '')/(10''T^2 -(''T)^2 )  ''(15)

b=('''E^2 '''R_srim-''E '''TR_srim ''')/(10''E^2 -(''E)^2 )  '..(16)


Energy Function Element Constant

2-10MeV R=aE+b Ti a = 20.2258

 b = -26.5584

Mo a = 25.4094

 b = -31.4339

Sn a = 27.5067

 b = -33.6185

W a = 33.4659

 b = -38.6250

Table(1) the equation which represent the Range  of protons in (Ti, Mo, Sn and W)

We programming this equation depended on Matlab program . As well as we using the SRIM2012 program to calculate the range of protons in this superconductor  mediums[12] and we compared our results with the results obtained from the STAR program groups . STAR program groups include three different stopping power and range calculation programs: ESTAR for electrons, PSTAR for protons and ASTAR for alphas. These programs were developed at the NIST[13]. The figures (1- 4) are plots of the Range versus the incident proton energy from (2 -10Mev) for the elements Ti, Mo, Sn and  W by using Matlab Language. These figures represented comparison among the range calculated from equation(13), the corresponding values obtained from SRIM-2012 program , the corresponding values obtained from PASTR program and the range calculated from equation(14) for the  same elements. From figures(1,2,3) we  note the semi empirical formula agree with all results we achieved compare with it.To show The maximum difference between the curves we use the statistical test k ( kstest2Two-sample test) by using Matlab program. The maximum difference between the present semi empirical results(equations (14)) and all results for figure (1,2,3,4) which for Ti , Mo,Si and W gave   k = 0.1111.

Fig.1Range of proton versus energy in Titanium with others workers value

Fig.2 Range of proton versus energy in Molybdenum with others workers value

Fig.3 Range of proton versus energy in Tin with others workers value        

Fig.4 Range of proton versus energy in Tinjasten with others workers value

3- Conclusions:

From this research we get semi empirical equation to calculate the range of protons  in some superconductor mediums depend on least square method. The results of this equation were agree with the results which calculate by SRIM-2012 program and PASTR program. By using the test statistic k  ( kstest2) we get the present  semi empirical results  have good agreement with the values of all results we compare with it  .


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[8] M.M. Al- Qysi," Radio Chemistry", University of Baghdad,  p(111-115),  (1986)

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[10] K. Chaubey and H. V. Gupta, '' New empirical relations for stopping  power and range of charged particles '', revue de physique appliqu''e, Vol. 12, p (321-329) , (1977) .

[11] Z. Melhem, "In the future of energy, superconducting materials matter and will make a material difference" Materials UK Preliminary Review, p(2-3), 2011.

[12] see:

[13] see:

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