Magic Number

Magic number nuclei

It has been observed that nuclei having either the number of protons Z or number of neutrons N = A-Z equal to the numbers 2, 8, 20, 50, 82, and 126 have a very high stability ass compared to their neighbors. These numbers are called magic numbers. Similarly, nuclei with 14,18 and 40 nucleons are slightly less stable but are more stable than the rest. These numbers are called semi-magic numbers.

Magic number on shell model

The shell model explains the existence of magic numbers. The ground state of the nucleus is represented as (1s1/22 ). It has radial quantum number n=1; orbital angular momentum quantum number i=0 and total angular momentum quantum number j = l+s = 1/2 . Since the spin quantum number s=1/2. The maximum number of nucleons it contain is (2j+1) =(\(\frac 12 \times 2\)+1) =2.

The first shell closure occurs at nucleon number two which is, therefore, the magic number.The second shell has two subshells represented as (1P3/4)4 and (1P1/2)2. It has radial quantum number l=1 and total angular momentum quantum number $$ j= l+s = 1+ \frac 12 = \frac 32 \; and l- \frac 12 = \frac12 $$

The subshell 1P3/2 can contain maximum number of nucleons$$ =2 \times \frac 32 +1 =4 $$

and \text{the sub shell(1P1/2) can contain maximum number of nucleons $$=2 \times \frac 12 +1 =2$$

Thus the second shell closure takes place at total nucleons number 2+6 =8 Which is a magic number.

The third shell has 3-sub shell (1d5/3)6,(1d3/2)4and (2S1/2)2

$$ for 1d_{5/2}, n=1 l=2.j =2+\frac 12 =\frac 52$$

$$ \text{ It can cnontain maximum nucleons}=2.\frac 52 +1=6$$

$$for\; 1d_{3/2},n=1, l=2 j= 2- \frac 12 =\frac 32 $$

$$\text{it can contains maximum nucleons} =2.\frac 32+1 =4$$

$$For \; 2S_{1/2} n=2, l=0, j= 0+ \frac 12 =\frac 12 $$

$$\text{it can contains maximum nucleons} =2.\frac 12+1 =2$$

Thus third shell closure takes place at total total nucleons $$=2+6+12 =20$$

$$\text{Which is magic number.}$$

The first shell of fourth shell is 1f7/2 .

$$It \;has \;n=1, l=3, j=l+1/2, = 1+\frac 32=\frac 72.$$

$$\text{it can contains maximum nucleons} =2.\frac 72+1 =8$$

Combined with the 20 nucleons of the first, third shells, there are a total nuber of nucleons is 28 ,which is a semi magic number In the case of doubly magic nuclei, both Z and N are even -even nuclei, the single particle model agular moment cancel each other to giive the total anglar momentum j=o.Hence the spin of the ground state of even of even_even nuclei is zero.

Also for ground state of even-even nuclei, the orbital angular momentum 'l' values are zero and hence parity

$$= (-1 )^l =+(i.e. even)$$

According to the shell model, for Z and N equal to magic numbers, the protons, and neutrons from a closed shell giving rise to the complete spherical symmetry to the nucleus and hence the quadrupole moment is very low. i.e. nearly zero.

for eg let us consider take$$ _{82}Pb^{208}(N=126, Z=82) for; _{82}Pb^{208}, j=0 (i.e. spin) $$

$$parity \;= (-1)^r=(-1)^0 = +(even)$$

Experimental Evidence.

(i) Stability

A very important experimental evidence in favor of magic number nuclei comes from the study of stables nuclides. It has been experimentally observed that \(_2He^4\) with Z=N=2 and \(_8O^{16}\) with Z=N=8 are highly stable. A graph between binding energy per nucleon and mass number shows sudden increase in binding energy per nucleons for

$$_{20}Ca^{40}(Z=20,N=20)$$

$$_{30}Sr^{88}(Z=38,N=50)$$

$$_{50}Sn^{120}(Z=50,N=70)$$

$$_{58}Ce^{140}(Z=58,N=82)$$

$$_{82}Pb^{282}(Z=82,N=126)$$

(ii) Relative abundance

It has been observed from experimental data that nuclei having a number of nuclei as the magic number are in abundance as compared to other nuclei in nature. The most abundant nuclei are

$$_{8}O^{16}(Z=8,N=8)$$

$$_{20}Ca^{40}(Z=20,N=20)$$

$$_{30}Sr^{88}(Z=38,N=50)$$

$$_{50}Sn^{120}(Z=50,N=70)$$

$$_{58}Ce^{140}(Z=58,N=82)$$

$$_{82}Pb^{282}(Z=82,N=126)$$.

(iii) Number of isotopes

The nuclides whose proton number Z corresponds to a magic number have the highest number of isotopes. for example$$_{20}Ca^{40}(Z=20,N=20)$$ has six isotopes. Whereas the average number of isotopes in that region is about two. Similarly$$_{50}Sn^{120}(Z=50,N=70)$$ has 10 isotopes. Whereas the average number in this region is about 4.

(v) End product of radioactive series

The stable end products of the natural radioactive series is$$_{82}Pb^{282}(Z=82,N=126)$$. It has Z=82 and N=126 both is magic number.

(vi) Energy of the emitted \(\beta\) particles

The energy of the emitted \(\beta\) particles is very high when the number of protons or neutrons in the daughter nucleus is equal to the magic number.

(vii) Neutron capture cross section

For nuclei with the number of neutrons equal to the magic number and the number of protons Z= 50 and Z= 82, the neutrons capture cross-section is particularly low as compared to their neighbors.

(viii) Electric quadrupole moment.

The electric quadrupole moment of nuclei with protons and neutrons equal to magic number is zero.These nuclei are, therefore, spherically symmetric because electric quadrupole moment measures the departure of nuclear charge distribution from spherical symmetry. Nuclei in the neighborhood of magic number nuclei have a certain value of quadrupole moment.i.e these have a distorted shape.

(ix) The first Excitation energy

The first Excitation energy of magic number nuclei is very high.

When the number of neutrons or protons exceeds the magic number by one, the binding energy of the extra nucleons is much less than the average value.

Significance;-

The significance of magic number lies in the fact that the corresponding number of protons and/or neutrons give high stability to the nucleons.The presence of magic number leads to the development of the shell model of the nucleus, by analogy with the closed shells and subshells in the case of atoms.There is a compelling similarity of stability between magic number nuclei and the inert gas atoms.Corresponding to magic number 2, 8, 20, 50, 82 and 126, the first ,second, third etc. shell closures in nuclei occurs at numbers of protons or neutrons in the nucleus equal to 2, 8, 20, 50, 82 and 126.