# The Possibility of Measuring Nuclear Shapes by Using Spectral Lines of Muonic Ions

## Abstract

**:**

## 1. Introduction

## 2. Analytical Results

_{e}= 1, unless specified to the contrary. The ellipsoid has the size 2a in the equatorial plane and the size 2c along the axis of symmetry. The potential outside the nucleus is usually approximated in the following way

_{outside}(r, cosθ) = − (Z/r)[1 + β(a/r)

^{2}P

_{2}(cosθ)],

_{2}(cosθ) = (3 cos

^{2}θ − 1)/2

^{2}),

^{2}− a

^{2})/5

^{(2)}

_{nLM}= (2βa

^{2}Z

^{4}μ

^{3}/n

^{3})[3M

^{2}− L(L + 1)]/[L(L + 1)(2L − 1)(2L + 1)(2L + 3)],

^{(2)}

_{n00}= 2βa

^{2}Z

^{4}μ

^{3}/(3n

^{3}).

^{(2)}

_{nLM}from Equation (5) over the M-sublevels, it vanishes. Therefore, the average of M energy correction of the second order is only E

^{(2)}

_{n00}given by Equation (6).

_{outside}= 2βa

^{2}Z

^{4}μ

^{3}/(3n

^{3}) = 4(c

^{2}− a

^{2}) Z

^{4}μ

^{3}/(15n

^{3}),

_{inside}

^{sphere}(r) = (Z/R)[r

^{2}/(2R

^{2}) − 3/2].

_{inside}

^{sphere}(r) = U

_{inside}

^{sphere}(r) − (−Z/r),

_{inside}

^{sphere}= 2Z

^{4}μ

^{3}R

^{2}/(5n

^{3}),

_{inside}(r) = (Z/R)[ρ

^{2}/(2a

^{2}) + z

^{2}/(2c

^{2}) − 3/2],

^{1/3}= a

^{2/3}c

^{1/3}.

^{2}c/3 to the volume of the sphere 4πR

^{3}/3 of the radius R. In spherical polar coordinates, Equation (11) can be rewritten as follows:

_{inside}(r) = (Z/R)[r

^{2}sin

^{2}θ/(2a

^{2}) + r

^{2}cos

^{2}θ/(2c

^{2}) − 3/2],

_{inside}(r) = U

_{inside}(r) − (−Z/r),

_{inside}= [2Z

^{4}μ

^{3}R

^{4}/(15n

^{3})](2/a

^{2}+ 1/c

^{2}).

_{tot}= [2Z

^{4}μ

^{3}R

^{2}/(15n

^{3})][2(c/a)

^{2/3}− 2(a/c)

^{2/3}+ 2(c/a)

^{4/3}+ (a/c)

^{4/3}],

_{tot}/ΔE

_{inside}

^{sphere}= [2(c/a)

^{2/3}− 2(a/c)

^{2/3}+ 2(c/a)

^{4/3}+ (a/c)

^{4/3}]/3.

^{2}− 1]/5

^{1/2}.

_{tot}/ΔE

_{inside}

^{sphere}= [2(1 + 5β/2)1/3 − 2(1 + 5β/2)−1/3 + 2(1 + 5β/2)2/3 + (1 + 5β/2)−2/3]/3.

_{tot}presented in Equation (16), then the shift S of the Lyman lines in the frequency scale is

_{tot}(n = 1) = −(2Z

^{4}μ

^{3}R

^{2}/15)[2(1 + 5β/2)

^{1/3}−2(1 + 5β/2)

^{−1/3}+ 2(1 + 5β/2)

^{2/3}+ (1 + 5β/2)

^{−2/3}],

_{tot}(n = 1) is positive for any β (see Figure 1), then S is always negative for any β, so the shift of the Lyman lines in the wavelength scale is always red for any β.

^{1/3}, it is possible to use Equation (21) (or Figure 1) to determine the experimental value of the parameter β and thus the experimental value of the nuclear quadrupole moment D connected to β. For the latter purpose, the expression for D in Equation (4) can be rewritten as follows;

^{2}/5)[(c/a)

^{4/3}− (a/c)

^{2/3}] = (2ZR

^{2}/5)[(1 + 5β/2)

^{2/3}− (1 + 5β/2)

^{−1/3}].

^{4}μ

^{3}R

^{2}/15 = 4.2 × 10

^{−4}Z

^{4}[R(fermi)]

^{2}.

^{–1}.

_{0}(a.u.) = 2.6 × 10

^{−5}A

^{1/3}(A being the atomic number) to the characteristic size of the muonic cloud r

_{0}(a.u.) = 1/(μZ) (in the ground state) is relatively small:

_{0}/r

_{0}= 5.5 × 10

^{−3}ZA

^{1/3}< 1.

_{0}/r

_{0}= (6.9−7.9) × 10

^{−3}Z

^{4/3}< 1,

_{max}~ 40.

_{0}/r

_{0}) can be taken into account in future publications.

## 3. Comparison with Competing Effects and Numerical Examples

_{inside}

^{sphere}(as demonstrated in Section 2, with ΔE

_{inside}

^{sphere}being presented in Equation (10)), it is sufficient to show that the energy shift due to the spherical nucleus significantly exceeds the competing effects. One of these effects is the fine structure splitting:

_{fs}(n) = E(n, j = 3/2) − E(n, j = 1/2) = α

^{2}μZ

^{4}/(4n

^{3}), (n > 1); ΔE

_{fs}(1) = 0,

_{inside}

^{sphere}(n) with ΔE

_{fs}(n) for n > 1, we get

_{inside}

^{sphere}(n)/ΔE

_{fs}(n) = (8/5)(μR/α)

^{2}> 1

_{0})]

^{2}, where R

_{0}(a.u.) = 2.6 × 10

^{−5}A

^{1/3}, so that the relative correction for muonic ions is ~1/A

^{2/3}and thus is insignificant for A >> 1.

^{4}. By scaling the natural width data from the NIST database to muonic ions, we find the following:

^{−6}Z

^{4}, ΔE

_{inside}

^{sphere}(n = 2)/γ = 1.3 × 10A

^{2/3},

^{−7}Z

^{4}, ΔE

_{inside}

^{sphere}(n = 3)/γ = 3.2 × 10

^{2}A

^{2/3},

^{−8}Z

^{4}, ΔE

_{inside}

^{sphere}(n = 4)/γ = 6.0 × 10

^{2}A

^{2/3},

_{0}(n), where

_{0}(n) = (μZ

^{2}/2)(1 − 1/n

^{2})

_{spher}= −2Z

^{4}μ

^{3}R

^{2}/5,

_{spher}/ω

_{0}(n) = − 4Z

^{2}μ

^{2}R

^{2}/[5(1 − 1/n

^{2})] = −2.4 × 10

^{−5}Z

^{2}A

^{2/3}.

_{0}(n) > S

_{min}/ω

_{0}(n) ~ −2.4 × 10

^{−5}Z

^{2}A

^{2/3}.

_{min}/ω

_{0}(n) for the muonic Lyman lines for several nuclei that are listed, e.g., in [5] as known to be oblate (β < 0).

^{20}

_{6}C

_{14}(oblate in the nuclear ground state):

_{min}/ω

_{0}(2) ~ −0.008 (for Ly-α), S

_{min}/ω

_{0}(3) ~ −0.007 (for Ly-β), S

_{min}/ω

_{0}(4) ~ −0.007 (for Ly-γ)

^{42}

_{14}Si

_{28}(oblate in the nuclear ground state):

_{min}/ω

_{0}(2) ~ −0.07 (for Ly-α), S

_{min}/ω

_{0}(3) ~ −0.06 (for Ly-β), S

_{min}/ω

_{0}(4) ~ −0.06 (for Ly-γ)

^{12}

_{6}C

_{6}(oblate in the 1st excited 2

^{+}nuclear state):

_{min}/ω

_{0}(2) ~ −0.006 (for Ly-α), S

_{min}/ω

_{0}(3) ~ −0.005 (for Ly-β), S

_{min}/ω

_{0}(4) ~ −0.005 (for Ly-γ)

^{28}

_{14}Si

_{14}(oblate in the 1st excited 2

^{+}nuclear state):

_{min}/ω

_{0}(2) ~ −0.06 (for Ly-α), S

_{min}/ω

_{0}(3) ~ −0.05 (for Ly-β), S

_{min}/ω

_{0}(4) ~ −0.05 (for Ly-γ)

^{34}

_{16}S

_{18}(oblate in the 1st excited 2

^{+}nuclear state):

_{min}/ω

_{0}(2) ~ −0.08 (for Ly-α), S

_{min}/ω

_{0}(3) ~ −0.07 (for Ly-β), S

_{min}/ω

_{0}(4) ~ −0.07 (for Ly-γ)

^{36}

_{18}Ar

_{18}(oblate in the 1st excited 2

^{+}nuclear state):

_{min}/ω

_{0}(2) ~ −0.11 (for Ly-α), S

_{min}/ω

_{0}(3) ~ −0.09 (for Ly-β), S

_{min}/ω

_{0}(4) ~ −0.09 (for Ly-γ)

## 4. Conclusions

## Conflicts of Interest

## References

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1 | We note that the result for L = 0 was obtained in [9] by considering a quasi-Coulomb potential −Z/r ^{1 − ε}, where ε << 1, and then taking the limit of ε = 0. This method allowed us to remove the uncertainty that would arise if one used the Coulomb potential for calculating the energy correction for L = 0. We also note that Equation (6) can be obtained from Equation (5), first by setting M = 0 in Equation (5), then by cancelling out L(L + 1) in the numerator and denominator, and then setting L = 0. |

**Figure 1.**Ratio of the energy shift due to the ellipsoid-shaped nucleus to the energy shift due to the corresponding spherical nucleus versus the parameter β, characterizing the degree of non-sphericity of the nucleus.

**Figure 2.**The magnified part of the dependence from Figure 1 around the minimum of this dependence, which is equal to 0.454 and corresponds to β = −0.322.

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**MDPI and ACS Style**

Oks, E.
The Possibility of Measuring Nuclear Shapes by Using Spectral Lines of Muonic Ions. *Atoms* **2018**, *6*, 14.
https://doi.org/10.3390/atoms6020014

**AMA Style**

Oks E.
The Possibility of Measuring Nuclear Shapes by Using Spectral Lines of Muonic Ions. *Atoms*. 2018; 6(2):14.
https://doi.org/10.3390/atoms6020014

**Chicago/Turabian Style**

Oks, Eugene.
2018. "The Possibility of Measuring Nuclear Shapes by Using Spectral Lines of Muonic Ions" *Atoms* 6, no. 2: 14.
https://doi.org/10.3390/atoms6020014