New Formula of Nuclear Force Md
International Journal of Scientific & Engineering Research Volume 2, Issue 4, April-2011 1 ISSN 2229-5518 New Formula of Nuclear Force Md. Kamal Uddin
Abstract -It is well established that the forces between nucleons are transmitted by meson. The quantitative explanation of nuclear forces in terms of meson theory was extremely tentative & in complete but this theory supplies a valuable point of view . it is fairly certain now that the nucleons within nuclear matter are in a state made rather different from their free condition by the proximity of other nucleons charge independence of nuclear forces demand the existence of neutral meson as amongst the same type of nucleolus (P-P) or (N-N). this force demand the same spin & orbital angular momentum. The exchange interaction in produced by only a neutral meson. The involving mesons without electric charge, that it gives exchanges forces between proton & Neutron & also therefore maintains charge in dependence character. It is evident for the nature of the products that neutral mesons decay by strong & weak interaction both. It means that neutral mesons constituents responsible for the electromagnetic interaction. Dramatically neutral mesons plays important role for electromagnetic & nuclear force both. Index Terms - Restmass energy,Mesons,photons,protons,neutrons,velocity of light,Differentiation
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1. INTRODUCTION
T is well established that the forces application of quantum mechanics to Ibetween nucleons are transmitted by nuclear phenomena gives us confidence in meson. The quantitative explanation of general use of quantum mechanics for the nuclear forces in terms of meson theory description of the force between heavy was extremely tentative & incomplete, but particles in nuclei. this theory supplies a valuable point of Where dm0c2 = either rest mass energy of Δ0 view. Yukawa first pointed out that mesons(For neutral theory),or rest mass nuclear force can be explained by assuming energy of Δ+, Δ-& Δ0mesons( for symmetrical theory) that particle of mass about 200 times the dm0 =either mass of Δ0 mesons or mass of electron mass(mesons) exist & can be Δ+ , Δ-& Δ0 mesons emitted & absorbed by nuclear m0 = mass of nucleons particles(neutrons & protons) with such an m0cdc = rest mass energy of nucleons assumption a force between nuclear dr= Range of nuclear force, which can be particles of right range & right calculated from differentiation of Nuclear shape(rapid decrease at large distances is radius.( The force between two nucleons is now obtaining. attractive for distance r(radius) greater Now we have the rest mass energy = m0 c2 than dr (range) & is repulsive Differentiating with respect to r (Inner radius otherwise).This strongly suggests & well at which nuclear force comes into play) proved that to some degree of d(m0c2) approximation the total isotopic spin T is a dr constant of the motion & is conserved in = c2dm0 + m0d(c2) = c2dm0 + all processes, at least with a high dr dr dr probability. mo d(c2) dc = c2dm0 + 2m0c.dc dc= The average velocity of neutron & dc dr dr dr proton. A large velocity is used in nuclear . This force is short range, attractive & along disintegration. the line joining the two particles (central force).(The wide success of this first c = Velocity of light
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2 = multiplicity of interacting particles interact only with neutral mesons( particles is given by (2T+1), the isotopic spin neutral theory) or(2) that they interact has no such meaning for leptons or a gamma equally strongly with neutral, positive & rays negative mesons(symmetrical theory). It is 1 = either multiplicity of obvious that the part of the force which Δ0 mesons or Δ+, Δ- & Δ0 mesons (evidence of does not depend on the spin of the involving of mesons (all type)) nuclear does not fulfill any useful Where T = Vector sum of function in the theory. The force between isotopic spin of proton & proton, neutron& proton & neutron are result from the neutron, neutron & proton The success of transfer of positive meson from the former these applications supplies additional to the latter or a negative meson in the support for the hypothesis of the charge opposite direction. So there vector sum of independence & charge symmetry of component of isotopic spin of these particle nuclear force. As the nuclear interactions do must be zero. not extend to very large distances beyond The charges on charged the nuclear radius & this character is mesons must be equal in their magnitude. useful to solve the problem. The Full charge independence for any system in which the number of neutrons equals the Charge independence of nuclear forces numbers of protons, this formula give the demand the existence of Δ0 meson as amongst evidence the charge symmetry, merely the same type of nucleons (p-p) or (N-N).This means that the neutron-neutron & proton force demand the same spin & orbital proton interaction are equal but says angular momentum. Positive pions are not nothing about the relations of neutron able to surmount the nuclear coulomb proton interaction to others. Nuclear forces barrier & there fore undergo spontaneous are symmetrical in neutrons & protons. i.e. decay while negative pions are captured the force between two protons are the by nuclei. The exchange of a pion is thus same as those between two neutrons. This equivalent to charge exchange. we can think identity refers to the magnitude as well as of nucleons as exchanging their space & the spin dependence of the forces. spin co-ordinates In the neutral theory, therefore neutron & protons are completely Now we can see the following reaction equivalent & indistinguishable as far as the P+ P ----- P+P+Δ0 associated meson fields are concerned. Such P+N -----P+N+Δ+ particle decay into two gamma rays. These N +P----- N+ P + Δ- gamma rays are Δ0 – rest systems are emitted These reactions are soon as Y + P -- in opposite direction & therefore spin Δ0 must P + Δ0 be Zero as the spin of photon is unity. It is The capture of photons can effect evident from the nature of the products the production of mesons by an that neutral mesons decay by the electromagnetic interaction, decay electromagnetic interaction while charged electromagnetically since these processes pions decay by strong & weak interaction involve no change of strangeness. both. It means that neutral mesons P + P --- P + P + Δ0 constituents responsible for the P + N ----- P + N + Δ0 electromagnetic interaction. We know that It is found that only two assumptions neutron & proton can change into one are in agreement with theoretical & another by meson capture. Protons &neutron experimental facts, notably the equality of can transform into each other by capture of the forces between two like & two unlike positive & negative pion respectively, or get nuclear particles in the singlet state. These transform into the same particle through assumption are either(1) that nuclear neutral meson interaction. During these
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transformation either an emission or an stable than hydrogen atom.Becouse binding absorption of meson is essential. The energy of helium atom is larger than that attraction between any nucleons can arise of hydrogen atom, so in this process large from the transfer of a neutral meson from energy is released.( here the negative pion one nucleon to the other. If the meson has significant role to produce nuclear were assumed to be charged ( positive or interaction. While charge pion main tain the negative) the resulting force between character of proton.) Now, as the hydrogen nuclear particles turned out to be of the nucleus converted into helium nucleus, exchange type which had been successful These are happen when resonances of in the interpretation in nuclear physics. The nucleons is in excited state. The scattering mesons must obey Bose statistics because cross- section can be interpreted by they are emitted in the transformation of a assuming that in the strong interactions the neutron into a proton( or vise versa) both total isospin is converted as well as the obey Fermi statistics. third component. The total isotopic spin of the system is 3/2.The ratio of radius & Important point-- range is about 2:1. It shows that those 1. The deuteron does posses nuclei has maximum number of nucleons measurable properties which might serve as are most stable than less mass number a guide in the search for the correct nuclei. nuclear interaction. The mass number of deuteron A is very minimum. From the 2*.Also, Strongly, The findings we must regard the deuteron as velocity of light depends loosely bound. The deuteron consist of two upon range of the nuclear particle roughly equal mass M, so that the force. The velocity of light reduced mass of the system is 1/2M.The equal like photons, lack deuteron has spin T=1, the neutron & mass & force carrying protons spins might be a parallel particle of other forces like combination. The magnetic moment of strong force. Because range deuteron will, therefore be sum of is variable, then the velocity magnetic moment of proton & neutron. of light must be variable. According to conclusion, As the range As, velocity of light= Range becomes larger in deuteron nucleus & they of nuclear force(distance becomes more unstable. travel by meson)/ Life time *When light nuclei of hydrogen atom of resonances. In this comes within the range of nuclear force relation we can see that the they can fuse together to form helium velocity of light must be nucleus. In this process (fusion) the range is variable. It is clear that not effected. However the force is twice of the fundamental particles the hydrogen’s nuclear force & so on. are not wholly Further the energy requires to bring independent, The neutron nucleons inside the range of force is twice observed to change of the rest energy of hydrogen nucleus. In spontaneously into a other word, we can say that the minimum proton. Neutron decay energy requires to form a helium nucleus is takes on the average some twice of the hydrogen rest mass energy. thousands seconds for free The mass of He atom (alpha-particle) equal neutron, whereas within a to the four times of the hydrogen atom, so nucleus the characteristic that the nuclear force of helium nucleus is time between nucleon- four times stronger than that of hydrogen nucleon collisions is 10-24 atom. It means that the He atom is more seconds. For a satisfactory
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picture it is often enough to depths are not identical but depends upon think of the nucleus as a the state of their actual binding in the grouping of protons & potential well. The range also depends neutrons interaction, with upon mass number A & binding energy. the appearance or we know that the atomic mass number A disappearance of is approximately equal to a twice the photons. One atomic number Z. For the light & should be noted that this intermediate nuclei. It shows that light relation hold only inside nuclei prefer to add nucleons is n-p pair. the nucleus. Out side the i.e there is a strong interaction between nucleus the evident is to neutrons & protons. The range of nuclear be contrary. It is a fact no force depends on the mass number A & body(even mesons or the velocity of light depends on the range, gamma rays ) can have so it is obviously thought that the spin & velocity greater than the velocity of light depends on the mass velocity of light. From this number A of the nucleus & spin is zero formula we can find the or an integer for A even & is an odd half nuclear force acts between integral for A odd. The total rest mass the pair of nucleons & energy also depends on the mass number does not influenced by the A. For increasing of rest energy, we must presence of neighboring increase the mass number A. obviously, the nucleons. It is necessary rest mass energy must be depends on the that any one particle must radial distance. brings the velocity of light. . This is purely a quantum mechanical We know that the nuclear effect. If the mass number A increases the force is short ranged. Out range decreases, & the force are stronger. side of the range it is This binding energy displays saturation repulsive. effect. This property of the nuclear force can be explain in term of exchange nature *Range of nuclear force :- To show that the of nuclear force. It should be noted that range of force is related to the mass of nucleons attract each other strongly only if exchanged particle, assume that the they are in same orbital state. This formula Δ0-meson is contained virtually in a proton. prove the pauli hypothesis. This formula If this virtual particle travels with the usually attributed to the effect of higher- velocity light as might be expected for a order interactions in which two or more field particle, then greatest distance the mesons are simultaneously transmitted meson could travel in this time also known between the nucleons. as range of the pion exchange force. 3. It would seem that in a nucleus 4.The velocity of light depends on the consisting of the many nucleons the binding wavelength of it constituents, If the energy per nucleon should increase with particles has longer wavelength then the the increase of the mass number A . In range decreases & therefore force is reality evidence is to contrary, the binding stronger. we can find the effective range of energy per nucleons decreases with nuclear force in terms of the Compton wave increasing mass number A, The binding length of pi—meson. We know that energies of the different nucleon placed at different(variable) constituents (color various depths are not identical but depend particles) has different wavelength, so it is upon the states of their actual binding in obviously thought the velocity light must the potential well. The binding energies of be variable. Evidently, if meson interacted the different nucleon placed at various with nucleons strongly enough to be
IJSER © 2011 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 2, Issue 4, April-2011 5 ISSN 2229-5518 responsible for the nuclear forces. Its mean meson comes to close together, the can free path within the nucleus should be neutralize each other then the force about the same as is that of a nucleons. between neutron & proton come into play. Then one question must be arise, How & So obviously we can say that only neutral why the velocity of light vary in free path meson plays important role in charge or in vacuum. So far this problem is independent nuclear force. The concern, the velocity of light influenced by mesons(positive & negative) can be its internal matter, which must have absorbed by the nucleus of an element or different value. However it may be it may be combine with the another meson possible that the velocity of light equal at then the sum of the masses of these all ranges within the nucleus. . The forces mesons converted into energy. This process responsible for binding the individual is called annihilation of matter. Before this particle inside the nucleus must therefore process one positive meson & one negative be exceptionally strong.If the particles has meson unite to make neutral particles motion then the material body has physical called k0 meson. The process of construction significance otherwise not. It means the & destruction has proved very help full is force between elementary particles depends considering the origin of universe. The on the velocity of the body as well as neutral K-- mesons is a stable particle but mass of body. It should be remarked that stability last for a small time. Its half life the particles travels with velocity of light is of the order of micro seconds. This are not a conservable quantities. In particle is an essential constituent of the quantum theory, every field must be nucleons of all elements.We know that quantized. These quanta produce a field, neutral—meson decay into two photons & which is responsible for different forces. never into three photons. It is clear that the neutral pions has been produced by bombarding hydrogen & deuteron with high energy photons. Gamma rays has sufficient energy to maintain energy of 5.The emission of a charged meson will be nucleons then the nucleons produce the accompanied by a change of charge of the neutral mesons.One can speak of the meson emitting nuclear particle, Thus a neutron field associated with a proton(or a netron) can only emits a negative meson or absorb because the nature (charge) of the nuclear a positive meson & will thereby be particle does not change by emitting or transformed into proton. When we consider absorbing a neutral meson.It has developed the emission of one meson by a nuclear a theory of nuclear forces in which neutral particle & reabsorbtion by another.It is way the equality of the forces between like obvious that in this way no force will be & unlike nuclear particles.The theory obtain between two nuclear particle of involving charged meson only giving no same kind i.e. two neutrons or two forces between two like nuclear particles. It protons. For the same kind particles, there is obvious that the negative meson & is neutral meson is responsible for positive meson gave the symmetrical force interaction.This solution would make the between protons & neutrons & the interact interaction coused by neutral meson alone, equally strongly with these meson. An Since for unlike particles the charged alternative way of explaining this equality mesons given an additional contribution, is to assume interaction with neutral meson while for like particles they do not, the only. Then the charge of the nuclear total interaction will not be the same3 for particles( whether it is a proton or a like & unlike particles in S-first state.This neutron) becomes entirely irrelevant & the will lead to forces between a neutron & a equality of forces follows immediately. This proton.The negative & positive charge alternative is discussed in present paper.
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6.According to the pauli principle only of neutral mesons move only a few atomic two neutrons & two protons will found in diameters before they decay( so that it the same orbital state. Therefore it is influenced few neighbors nucleons)& thus possible to find four nucleons strongly are not affected by the matter through bound or Alpha particle structure, also which they pass & thus nuclear force work confirm by binding energy curve. The properly. It should be also noted that in whole extraordinary stability of the alpha shows universe there is only mass will be conserved that the most stable nuclei are those in and energy will be destroyed, then the mass which number of nucleons & photons are will not change into energy. equal. We can find it from this formula. It *It is enough to think that S- mesons which is obviously thought that the full charge form a nuclear cloud around the individual independence for any system in which the number of neutrons equal to the number nucleons & are in a virtual state get their of protons. From conclusion, we get, requisite rest mass energy from the Number of photons = number of nucleons= incident particle & are released from the 2(number of neutral mesons).The discovery nuclear binding potential. Nuclear binding of the neutral meson & the fact that charge potential compensate the rest mass energy. independence is now consistent with all It produce enough energy to maintain the nuclear data, confirm fully the use of the rest mass energy for production of mesons. symmetric meson theory, containing Since the rest mass energy of S - mesons positive, negative &neutral mesons about 275 Me, the threshold energy for a described by three way wave functions. gamma rays to produce the rest mass With the form of Yukawa potential for energy of these particles should be high. scalar mesons, it is easy to see that the pi- But, if protons projectiles are used to meson can not be scalar. This theory prove produce mesons, it requires a large this argument. threshold as a particle with mass retains some energy in the collision. It should be Change of Law-- remarked that the binding potential is Since there is no requirement for the independent of spin & range of the conservation of Pions so there is no particle, when they compensate the rest conservation law in rest mass energy & energy. The energy require to pull out the even in the universe. This formula show that nucleons from the nucleus is less than half there is no meaning of the word’ constant. of rest mass energy. The slow motion There is no conservation law controlling neutron play this role. Similarly if the the total number of K—aons or meson. The nucleus brings (from binding potential) energy of formation of mesons comes from sufficient energy for the existing of nuclear binding potential ( which has the energy to force. It maintains stability. In order to formation of meson for a long time), but approach particle to within short range or when these potential has not enough closer the energy of the approaching energy, the production of pions end & particle should be very high. nuclear force does not exist.
*The life time of any radioactive substances Since there is no depends on the total number of pions limitation of formation of production and another particles production. Mesons even in strong pions are commonly formed in the decay interaction. This is due to of k—oans, hyperons & resonant states. It high energy photons ( J - should be noted that pions are formed only at rays ) then this cyclic chain high energy.Becouse of their short life time should be possible
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S0o J rays J+d o d+ S0 This reaction shows that the kinetic energy as well as potential energy of nucleon in the nucleus will be over and above of the rest mass energy. In these phenomena the total charge of fundamental particles are conserved. It is reasonable to assume that the nuclear force between two protons has the same characteristic as that between neutron & proton. The argument about short range forces involves both proton- proton & neutron - proton forces. The main difference between proton & neutron seems to be the electric charge, & the nuclear force apparently does not arise from charge. We assume there fore that the potential between two protons is confined within some short range as before, although the value of range need not necessarily be the same.
Finally-- some of the peculiarities of nuclear forces can enumerated as follows— (a) short range character (b) Large strength, the nuclear potential energies are quite large. (c) Exchange character & saturable nature. (d) Dependence on spin (e) charge independence& charge symmetry.
References—1. Elementry nuclear theory.H.A.Bathe 2.Nuclear physics, Roy & Nigam 3.Nuclear physics, Srivastva
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