44
2 Computational Methods
where r 0 ≈ 1.2 fm and A is the mass number.
Of course, it is only an approximation, there is not a sharp cut off with a finite
density of nucleus inside and zero density outside.
Furthermore, some nuclei do not have a spherical symmetry. The deviation may
be pointed out by the value of the quadrupole moment Q of the nucleus.
The rms radius is often measured by scattering of electrons. Another method is
to use optical isotopic shifts. When the s electrons are inside the nucleus, they are
submitted to a potential different from the Coulomb potential. This potential depends
on the volume of the nucleus, i.e., its radius. It is also possible to use muonic atoms
where an electron is replaced by a muon. As the muon is more massive than the
electron, the Bohr orbits are closer to nucleus permitting more precise measurements.
The K α X-ray isotope shift may also be used. The rms radii and the radii changes in
isotopic sequences are tabulated in Angeli and Marinova (2013) where the methods
of measurements are also given.
See also Sect. 3.9.3 where the effect of the nuclear size on the structure is discussed.
2.19.3 The Product of Two Gaussians is Another Gaussian
Assume that the function g a = e
−α a r
2
a is on atom A and function g b = e
−α b r
2
b on
atom B.
We choose the axis Ox such that A and B lie on it.
We may write
g a = e
−α a [(x−xa)
2 +y
2 +z
2 ]
(2.56)
and the same for g b. Then
g a g b = e
−(α α x
2
a +α b x
2
b ) e
2(α a x a +α b x b )x e
−(α a +α b )r
2
(2.57)
If we choose the origin O such that
x a
x b
= −
α b
α a
we obtain another Gaussian
g a g b = ce
−(α a +α b )r
2 with the coefficient c = e
−(α α x
2
a +α b x
2
b )
(2.58)
2.19.4 Relativistic Correction Due to the Dependence
of the Electron Mass on Velocity
For a Hg atom (Z = 80), the velocity of the 1 s electrons is 58% of the speed of light,
its mass is then m = 1.23m 0 , m 0 being the rest mass.
The nonrelativistic energy is
2 Computational Methods
where r 0 ≈ 1.2 fm and A is the mass number.
Of course, it is only an approximation, there is not a sharp cut off with a finite
density of nucleus inside and zero density outside.
Furthermore, some nuclei do not have a spherical symmetry. The deviation may
be pointed out by the value of the quadrupole moment Q of the nucleus.
The rms radius is often measured by scattering of electrons. Another method is
to use optical isotopic shifts. When the s electrons are inside the nucleus, they are
submitted to a potential different from the Coulomb potential. This potential depends
on the volume of the nucleus, i.e., its radius. It is also possible to use muonic atoms
where an electron is replaced by a muon. As the muon is more massive than the
electron, the Bohr orbits are closer to nucleus permitting more precise measurements.
The K α X-ray isotope shift may also be used. The rms radii and the radii changes in
isotopic sequences are tabulated in Angeli and Marinova (2013) where the methods
of measurements are also given.
See also Sect. 3.9.3 where the effect of the nuclear size on the structure is discussed.
2.19.3 The Product of Two Gaussians is Another Gaussian
Assume that the function g a = e
−α a r
2
a is on atom A and function g b = e
−α b r
2
b on
atom B.
We choose the axis Ox such that A and B lie on it.
We may write
g a = e
−α a [(x−xa)
2 +y
2 +z
2 ]
(2.56)
and the same for g b. Then
g a g b = e
−(α α x
2
a +α b x
2
b ) e
2(α a x a +α b x b )x e
−(α a +α b )r
2
(2.57)
If we choose the origin O such that
x a
x b
= −
α b
α a
we obtain another Gaussian
g a g b = ce
−(α a +α b )r
2 with the coefficient c = e
−(α α x
2
a +α b x
2
b )
(2.58)
2.19.4 Relativistic Correction Due to the Dependence
of the Electron Mass on Velocity
For a Hg atom (Z = 80), the velocity of the 1 s electrons is 58% of the speed of light,
its mass is then m = 1.23m 0 , m 0 being the rest mass.
The nonrelativistic energy is