Calculation of Deuteron Magnetic Dipole Moment Using Constituent Quarks for All Possible,

In the quark like model, deuteron is considered as a system of six quarks clustering into two baryons out of several possible baryon states. The deuteron wave function is written in terms of all possible baryon combinations and the quark constituent of each baryon. In addition to (p, n) pair the ( , ) ++ − ∆ ∆ , 0 ( , ) + ∆ ∆ , ( , ) n + ∆ , 0 ( , ) p ∆ pairs are also considered. The expectation value of the deuteron magnetic dipole moment is calculated and the value 0.8554514 D N μ μ = is obtained. This finding is in a better agreement with the experimental value as compared to the value given in the shell model. PACS Number: 21. 65. Qr. 12. 38. Mh


Introduction
According to the present quark model, the hadrons are made of quarks with three generations and six flavors. Each quark has its own spin, charge, bare mass and different effective masses in baryons and mesons [1]. In the shell model, the magnetic dipole moment of the deuteron D µ is calculated in the ground state ( 0) l = , by using the proton and neutron spin g-factors namely, 5.5856912  [3][4][5]. In order to reduce the difference between the measured and calculated values, it was assumed that the deuteron ground state is a combination of l=0 and l=2 states [3].To improve the calculated value of magnetic dipole moment of deuteron, the spin-orbit coupling is also considered in the nuclear Shell model [6][7][8]. By using the same spin g-factor for the proton and neutron, the magnetic dipole moments of other nuclei were also determined and resulted in a larger difference between the measured and calculated values. In order to achieve a better match, it was assumed that the spin g-factors of proton and neutron are different from the bound proton and neutron and their approximation relation is, 0.6 bound free s s g g ≅ [9].
In this work a different approach is chosen and the deuteron constituent quarks are considered. In this quark model, the constituent quarks are not completely bounded and continuously participate in the process of baryon creation and annihilation. The two baryons inside deuteron do not have to be only proton and neutron, they can be one of these pairs, and ( , ) p n . Although the masses of the Delta baryons are greater than the nucleon mass, but by taking into account the Heisenberg uncertainty principle between energy and time, it is possible for deuteron to be made of two Delta baryons for very short time. It is noticeable that the Based upon this quark model of nuclei, all magic numbers are determined [11]. In this model, the nuclear binding energy is also given in a simpler expression [12].

Determination of Deuteron Wave Function
Based upon the assumption that deuteron is made of 3 up and 3 down quarks, clustering continuously into a pair of baryons, the deuteron wave function is obtained. The In Eq. (1), the square of each coefficient represents the probability of each baryon pair formation. These coefficients are determined later. Each term of Eq. (1) consists of four independent parts namely, spin, flavor, color and space, therefore [1,10]: The total wave function is anti-symmetric (under exchange of two baryons in nuclei or two quarks in baryon). Although the functional dependency of the ground state of the space part is not known but it is symmetric due to the fact that l=0 for all quarks. The three colors are generators of SU(3) color symmetry and three colors together makes one decuplet and two octet and one color singlet namely, Naturally made particles are colorless and color singlet and in SU(3) color singlet is anti-symmetric [13]. Therefore each term of the deuteron wave function is written in terms of baryon wave functions as follow: Where, 1 b and 2 b are wave functions of the two baryon states which form the deuteron.
The total wave function of the fermions is anti-symmetric. The spin part is obtained from Clebsch -Gordan's method as follow [14]: All members of the spin 3/2 state are symmetric and for 1/2 state are partially anti-symmetric and change sign under exchange of two particles. There are also the following combinations, made from particles 1 and 3: which are not independent and can be constructed from the two previous combinations namely, 13 12 23 | | | > = > + > (9) The flavor part of the baryons, ( ) flavor ψ is similar to the spin part due to the fact that these baryons are made of 2 kinds of quarks namely, u and d and belong to the SU(2) group. In fact by substituting ↑ spin for u and ↓ spin for d, the flavor part is studied. As stated before, the color function is anti-symmetric for baryons and the space function is symmetric, The same is for the other pairs involved. The total numbers of possible states are counted and then those states with spin other than one are eliminated due to the fact that, deuteron has spin 1. The obtained results are summarized in table 1. As a result of the precise counting of the number of states, the coefficients introduced in Eq. (1) are determined as follow: For example, based upon the constituent quark's spin, the first term in Eq. (13) is expanded as follow: 3  3 3  3  3  3 3  1  1  1  10  2 2  2 2  2 2  2 2  2   3  3  3 3  3 3  3  1  1  1  10  2 2  2 2  2 2  2 2  2   3  3  3  4  1  1  1  10  2 2  2 In Eq. (14) the coefficients before each terms are the Clebsch-Gordon coefficients. By considering the constituent quark's flavor in addition to the spin, the first term of the total deuteron wave function, which is given in Eq. (14), is expanded and given in the appendix. For other baryon pairs, the same procedure is considered for the wave function of deuteron and the total wave function is obtained.

Determination of the Deuteron Magnetic Dipole Moment
In the absence of the orbital motion, the magnetic dipole moment of deuteron is simply the vector sum of the six constituent quarks namely, 1 2 3 4 5 6 µ µ µ µ µ µ µ The terms of this equation depend upon the flavor of quarks (since up and down quarks have different magnetic dipole moments) and also depend upon the spin direction of six quarks.
The magnetic dipole moment of spin 1/2 particle with mass m and charge q is given [1,3] as q S mc µ = The deuteron wave function is given in section 2. The first term of this wave function is And the expectation value for the magnetic dipole moment of this first term is,