Weak quadrupole moments

TL;DR

This paper introduces the concept of the weak quadrupole moment in nuclei, linking it to neutron distribution and parity violation, and calculates its effects in various isotopes to aid experimental detection of neutron quadrupoles.

Contribution

It defines the weak quadrupole moment related to neutron distribution and provides calculations for its magnitude in multiple isotopes, enabling new experimental approaches.

Findings

Weak quadrupole moments are related to neutron quadrupole distributions.

Calculations of weak quadrupole moments for various isotopes.

Predicted Lorentz invariance violating energy shifts.

Abstract

Collective effects in deformed atomic nuclei present possible avenues of study on the non-spherical distribution of neutrons and the violation of the local Lorentz invariance. We introduce the weak quadrupole moment of nuclei, related to the quadrupole distribution of the weak charge in the nucleus. The weak quadrupole moment produces tensor weak interaction between the nucleus and electrons and can be observed in atomic and molecular experiments measuring parity nonconservation. The dominating contribution to the weak quadrupole is given by the quadrupole moment of the neutron distribution, therefore, corresponding experiments should allow one to measure the neutron quadrupoles. Using the deformed oscillator model and the Schmidt model we calculate the quadrupole distributions of neutrons, $Q_{n}$, the weak quadrupole moments ,$Q_{W}^{(2)}$, and the Lorentz Innvariance violating energy shifts in $^{9}$Be, $^{21}$Ne , $^{27}$Al, $^{131}$Xe, $^{133}$Cs, $^{151}$Eu, $^{153}$Eu, $^{163}$Dy, $^{167}$Er, $^{173}$Yb, $^{177}$Hf, $^{179}$Hf, $^{181}$Ta, $^{201}$Hg and $^{229}$Th.

Tables

Table 1. Table 1: This shows ∑ 2 ​ n z + 1 2 subscript 𝑛 𝑧 1 \sum 2n_{z}+1 and ∑ N − n z + 1 𝑁 subscript 𝑛 𝑧 1 \sum N-n_{z}+1 for each completely filled N 𝑁 N shell in the Nilsson model.

Filled $N$ Shell	$\sum 2n_z+1=\sum N-n_z+1$
0	2
1	10
2	28
3	60
4	110
5	182

Table 2. Table 2: Nilsson configuration of 173 Yb nucleons: This table shows the nuclear configuration of nucleons in the Nilsson model for 173 Yb generated from the Nilsson plots in Bohr and Mottelson ( 1975 ) . This table shows only partially filled N 𝑁 N shells. All preceding shells are completely filled.

Neutrons in incomplete shells		$\sum {\left(2n_z+1\right)}_n$	$\sum {\left(N-n_z+1\right)}_n$	Protons in incomplete shells		$\sum {\left(2n_z+1\right)}_p$	$\sum {\left(N-n_z+1\right)}_p$
$2f_{7/2}$:	$\pm 1/2\left[530\right]$, $\pm 3/2\left[521\right]$, $\pm 5/2\left[512\right]$	30	24	$1g_{7/2}$:	$\pm 1/2\left[431\right]$, $\pm 3/2\left[422\right]$, $\pm 5/2\left[413\right]$	30	18
$1h_{9/2}$:	$\pm 1/2\left[541\right]$, $\pm 3/2\left[532\right]$, $5/2\left[523\right]$	37	14	$2d_{5/2}$:	$\pm 1/2\left[420\right]$, $\pm 3/2\left[411\right]$	16	14
$3p_{3/2}$:	$\pm 1/2\left[521\right]$	10	8	$2d_{3/2}$:	$\pm 1/2\left[521\right]$	6	8
$1h_{11/2}$:	$\pm 1/2\left[550\right]$, $\pm 3/2\left[541\right]$, $\pm 5/2\left[532\right]$, $\pm 7/2\left[523\right]$, $\pm 9/2\left[514\right]$, $\pm 11/2\left[505\right]$	72	42	$1g_{9/2}$:	$\pm 1/2\left[440\right]$, $\pm 3/2\left[431\right]$, $\pm 5/2\left[422\right]$, $\pm 7/2\left[413\right]$, $\pm 9/2\left[404\right]$	50	30
$1i_{13/2}$:	$\pm 1/2\left[660\right]$, $\pm 3/2\left[651\right]$, $\pm 5/2\left[642\right]$, $\pm 7/2\left[633\right]$	80	20	$1h_{11/2}$:	$\pm 1/2\left[550\right]$, $\pm 3/2\left[541\right]$, $\pm 5/2\left[532\right]$, $\pm 7/2\left[523\right]$	64	20
Total		229	108	Total		166	90

Table 3. Table 3: Sum of proton and neutron Nilsson quantum numbers and deformation parameters for deformed nuclei: This table shows the sum of 2 ​ n z + 1 2 subscript 𝑛 𝑧 1 2n_{z}+1 and N − n z + 1 𝑁 subscript 𝑛 𝑧 1 N-n_{z}+1 for all nucleons in the nuclei.

	Proton (${\Sigma }_p$)		Neutron (${\Sigma }_n$)		$\frac{\overline{\omega }}{{\omega }_z}$
	$2n_z+1$	$N-n_z+1$	$2n_z+1$	$N-n_z+1$
9Be	8	4	9	6	1.68
21Ne	22	14	25	16	1.30
27Al	29	21	30	24	1.12
151Eu	199	177	312	276	1.08
153Eu	241	162	350	272	1.11
163Dy	250	174	389	287	1.12
167Er	260	182	417	302	1.17
173Yb	266	190	439	318	1.18
177Hf	264	202	439	331	1.18
179Hf	264	202	447	341	1.16
181Ta	285	199	468	338	1.08
229Th	362	268	625	502	1.22

Table 4. Table 4: Results for LLIV and quadrupole tensors for different deformed nuclei: (All used Q p subscript 𝑄 𝑝 Q_{p} values have been compiled in Stone ( 2005 ) .) This table shows the proton ( Q p subscript 𝑄 𝑝 Q_{p} ) and neutron ( Q n subscript 𝑄 𝑛 Q_{n} ) electric quadrupole moments, energy shifts due to Lorentz violation ( ⟨ δ ​ H ⟩ ν subscript delimited-⟨⟩ 𝛿 𝐻 𝜈 \left<\delta H\right>_{\nu} ) and the weak quadrupole moments ( Q W ( 2 ) superscript subscript 𝑄 𝑊 2 Q_{W}^{(2)} ) in 9 Be, 21 Ne, 27 Al, 131 Xe, 133 Cs, 163 Dy, 173 Yb, 177 Hf, 179 Hf, 181 Ta, 201 Hg and 229 Th. All quantities are in the lab frame. Nuclei marked with an asterix (*) are near spherical nuclei.

Nuclei	$I_t$	$Q_p$ (barn)	$Q_n$ (barn)	$M_p$ ($m$ MeV)	$M_n$ ($m$ MeV)	(MeV)	(MeV)	$Q_W^{\left(2\right)}$ (barn)
9Be	${\frac{3}{2}}^{-}$	+0.0529(4)	+0.053	-0.14	-5.88	-0.024	-1	-0.05
21Ne	${\frac{3}{2}}^{+}$	+0.103(8)	0.12	3.19	3.36	0.53	0.56	-0.11
27Al	${\frac{5}{2}}^{+}$	+0.150(6)	0.129	17.12	7.26	2.85	1.21	-0.12
131Xe*	${\frac{3}{2}}^{+}$	-0.114	-0.070	-29.4	-18	-4.9	-3	-0.009
133Cs*	${\frac{7}{2}}^{+}$	-0.00355	0.2	-0.9	53	-0.15	9	-0.20
151Eu	${\frac{5}{2}}^{+}$	+0.87(2)	1.4	2.45	8	0.41	1.3	-1.33
153Eu	${\frac{5}{2}}^{+}$	+2.28(9)	2.53	127	81	21	13.5	-2.35
163Dy	${\frac{5}{2}}^{-}$	+2.318(2)	3.3	103.4	115.5	17	19	-3.10
167Er	${\frac{7}{2}}^{+}$	3.57(3)	5.47	90	107	15	18	-5.17
173Yb	${\frac{5}{2}}^{-}$	+2.8(4)	4.5	54	77	9	13	-4.26
177Hf	${\frac{7}{2}}^{-}$	+3.37(3)	5.73	16.35	45	2.7	7.5	-5.44
179Hf	${\frac{9}{2}}^{+}$	+3.79(3)	6.5	35.33	64	6	10.5	-6.17
181Ta	${\frac{7}{2}}^{+}$	+3.17(2)	5.33	188	269.5	31	45	-5.05
201Hg*	${\frac{3}{2}}^{+}$	+0.35	0.53	67.2	102	11.2	17	-0.50
229Th	${\frac{5}{2}}^{+}$	+4.3(9)	6.62	14	-78	2	-13	-6.25

Equations

$$\delta H=\left[-c_{ij}-\frac{1}{2}{c}_{00}{\delta }_{ij}\right]\frac{p_i{p}_j}{m^2},$$

$$H_{Nilsson}$$

$$T^{Lab}=\frac{I\left(2I-1\right)}{\left(I+1\right)\left(2I+3\right)}{T}^{Intrinsic},$$

$$Q_{zz}=Q=2⟨z^2⟩-⟨x^2⟩-⟨y^2⟩.$$

$$m{\omega }_z^2⟨z^2⟩$$

$$q_{i,\nu }$$

$$Q_{\nu }$$

$$=\frac{\hslash }{m}\left[\frac{1}{{\omega }_z}\sum _i{\left(2n_z+1\right)}_{i,\nu }-\frac{1}{{\omega }_{\perp }}\sum _i{\left(N-n_z+1\right)}_{i,\nu }\right].$$

$$<\delta H{>}_{\nu }=\frac{1}{6m}{C}_{0,\nu }^{(2)}\sum _{N=1}^{N_{\nu }}⟨I,I\left|\hat{M}\right|I,I⟩$$

$${\bar{m}}_{i,\nu }=⟨I,I\left|2{\hat{p}}_z^2-{\hat{p}}_x^2-{\hat{p}}_y^2\right|I,I⟩.$$

$$\frac{⟨p_z^2⟩}{2m}$$

$${\bar{m}}_{i,\nu }$$

$$M_{\nu }$$

$$=\hslash m\left[{\omega }_z\sum _i{\left(2n_z+1\right)}_{i,\nu }-{\omega }_{\perp }\sum _i{\left(N-n_z+1\right)}_{i,\nu }\right].$$

$${⟨\delta H⟩}_{\nu }=\frac{M_{\nu }}{6m}{C}_{0,\nu }^{(2)}.$$

$$3$$

$$Q_{\nu }=\frac{41.5\text{ MeV}{\text{fm}}^2}{\hslash \bar{\omega }}[\eta \sum _i{\left(2n_z+1\right)}_{i,\nu }$$

$$-\xi \sum _i{\left(N-n_z+1\right)}_{i,\nu }]$$

$$M_{\nu }=\hslash \bar{\omega }m[\frac{1}{\eta }\sum _i{\left(2n_z+1\right)}_{i,\nu }$$

$$-\frac{1}{\xi }\sum _i{\left(N-n_z+1\right)}_{i,\nu }].$$

$$Q_n^{Lab}$$

$$M_p^{Lab}$$

$$M_n^{Lab}$$

$${⟨\delta H⟩}_p$$

$${⟨\delta H⟩}_n$$

$$\frac{1}{Z}{Q}_{p,def}=\frac{1}{N}{Q}_{n,def}.$$

$$Q_{p,def}=Q_p=0.35\text{barn}$$

$$Q_n^{Lab}$$

$$=-\frac{I-1/2}{I+1}0.009A^{2/3}\text{ barn}.$$

$${⟨\delta H⟩}_{\nu }=\frac{M_{\nu }}{6m}{C}_{0,\nu }^{(2)}=1100A^{-2/3}{Q}_{\nu }{C}_{0,\nu }^{(2)}\text{MeV}.$$

$$H_W=-\frac{G_F}{2\sqrt{2}}{\gamma }_5\left[Z{q}_{w,p}{\rho }_p(r)+N{q}_{w,n}{\rho }_n(r)\right]$$

$$q_{w,p}$$

$$q_{w,n}$$

$$Q_W^{(2)}=q_{w,p}{Q}_p+q_{w,n}{Q}_n.$$

$$H_{WQM}=-\frac{G_F}{2\sqrt{2}}{\gamma }_5{Y}_{20}{\rho }_0\frac{\sqrt{5\pi }{Q}_W^{(2)}}{⟨r^2⟩}$$

Full text

Non-spherical nuclei present a lucrative avenue for studying the existence and magnitude of second order tensor properties due to the collective properties of deformed nuclei. In this work we focus on the quadrupole moment of the nonspherical distribution of neutrons in the nucleus, $Q_{n}$, the weak quadrupole moment (WQM), $Q_{W}^{(2)}$, and the violation of Local Lorentz invariance (LLI) in the nucleon sector. These properties have yet to be experimentally detected though there are constraints for violation of LLI. In this work we will show how these properties are enhanced in deformed nuclei and therefore present a new possibility for measurement or developing further constraints on their existence. In Section I we calculate the the neutron quadrupole moment of the nucleus (NQMN) and local Lorentz invariance violation (LLIV) in deformed nuclei using the Nilsson model. In Section II we discuss NQMN and LLIV in nuclei with a small deformation and in Section III we discuss the NQMN and WQM in parity nonconserving (PNC) effects.

Measuring the NQMN, $Q_{n}$, is a very difficult task as neutrons are electrically neutral particles unlike the electric quadrupole moment of the nucleus which was first observed and measured nearly a century ago by H. Schuler and Th. Schmidt Schmidt and Schuler (1935); Schuler and Schmidt (1935); Casimir (1935) by studying the hyperfine structure of rare earth elements. The NQMN is an important property which will give insight not only into the structure of atomic nuclei but also other dense collections of neutrons such as neutron stars Brown (2000); Furnstahl (2002); Typel and Brown (2001); Reinhard and Nazarewicz (2010) and also the theory of atomic parity nonconservation. For the last two decades there has been increasing interest in understanding the distribution of neutrons compared to protons in atomic nuclei known as the neutron skin. This focus has largely been on the spherical distribution of neutrons (measurement of the root mean square radius of neutrons) and there has been a large amount of experimental effort in measuring the spherical distribution Clark et al. (2003); Trzcińska et al. (2001); Lenske (2009); Abrahamyan et al. (2012).

The PNC effects appear due to mixing of opposite parity states in atoms and molecules by the weak interaction between the nucleus and electrons. The field of PNC in atoms and molecules has been thoroughly reviewed in Refs. Khriplovich (1981); Ginges and Flambaum (2004); Roberts et al. (2015). It was noted in Ref. Sushkov and Flambaum (1978) that the nuclear quadrupole moment induces a tensor PNC weak interaction between the nucleus and electrons (see also Khriplovich (1981); Khriplovich and Pospelov (1991)). In Ref. Flambaum (2016) it was argued that these tensor effects of the weak quadrupole moments are strongly enhanced for deformed nuclei and may get a significant additional enhancement due to mixing of close atomic and molecular levels of opposite parity with a difference of the electron angular momenta $|J_{1}-J_{2}|\leq 2$. These selection rules are similar to that for the effects of the time reversal ($T$) and parity ($P$) violating nuclear magnetic quadrupole moment (MQM). Therefore, nuclei, molecules and molecular levels suggested for the MQM search in Ref. Flambaum et al. (2014), for example, $|\Omega|=1$ doublets in the molecules 177HfF+, 229ThO, 181TaN will also have enhanced effects of the weak quadrupole (see section III). Following our proposal, there has been recent experimental interest in using the weak quadrupole moment to study PNC effects in HfF*+* molecules (used to measure electron electric dipole moment Cairncross et al. (2017)) and parity violation in 173Yb Antypas et al. (2017) and 163Dy Leefer et al. (2014) atoms.

The other second rank tensor property we study is related to violation of Local Lorentz invariance (LLI). The physical property of LLI is fundamental in all disciplines of physics. It states that the laws of physics are invariant under transformations of velocities and orientations (boosts and rotations). However within the study of unification theories the possibility of local Lorentz invariance violation (LLIV) has been suggested and studied in a variety of experiments. These searches have been motivated by suggestions of LLIV in high energy theories such as string theory Kostelecký and Potting (1995); Mavromatos (2007); Pospelov and Shang (2012); Kostelecký and Samuel (1989); Liberati (2013) which occur above the currently unaccessible Plank energy, $M_{P}$. It is expected that signatures of these high energy phenomena will appear heavily suppressed in low energy electroweak experiments ($m_{ew}$) to an order of $m_{ew}/M_{P}\approx\times 10^{-17}$Kostelecký and Potting (1995). This suggests that high energy phenomena such as LLIV can be observed and studied in a low energy regime with exceptionally sensitive measurements in nuclear and atomic systems. The seminal work on the systematic study of LLIV was performed in Colladay and Kostelecký (1998) and Kostelecký and Lane (1999) where a general form of Standard Model Extension (SME) Lagrangian has been presented to include both $CPT$-odd and $CPT$-even LLIV terms to create a unified model.

Attempts to measure possible violations of LLI date back a century with the classic experiment performed by Michelson and Morley Michelson and Morley (1887). Recent attempts to detect LLIV (in the matter sector specifically) are more sophisticated using a variety of techniques such as resonance cavities Müller (2005); Müller et al. (2003); Wolf et al. (2004) and Doppler shift experiments Lane (2005); Saathoff et al. (2003). However, the best current laboratory limits are from clock comparison experiments Prestage et al. (1985); Chupp et al. (1989); Hohensee et al. (2013); Dzuba et al. (2016) in particular the tensor LLIV parameters $c_{ij}$ for the neutron in Smiciklas et al. (2011) using a 21Ne clock and for the proton in Wolf et al. (2006) using 133Cs clocks which was further constrained by 4 orders of magnitude in Flambaum (2016). In Ref. Brown et al. (2016) the $s-d$ shell model calculations of the tensor LLIV effects have been performed in 21Ne, 131Xe and 201Hg nuclei. It was demonstrated that virtual excitations from the nuclear core enhance the nuclear quadrupole moment and suppress the LLIV tensor. The results of the $s-d$ shell model calculations have been supported by the calculations in the Hartree-Fock-Bogoliubov model published in the same paper Brown et al. (2016).

The best laboratory limits on the Michelson-Morley type LLIV tensor in the photon sector has been obtained in Ref. Flambaum and Romalis (2017) using the 21Ne clock data from Smiciklas et al. (2011). The limits on the LLIV parameters obtained using astrophysical measurements and laboratory experiments can be found in Kostelecký and Lane (1999); Kostelecký and Russell (2017). In the present paper we study the LLIV momentum tensor Hamiltonian $\delta H$ Kostelecký and Lane (1999)

[TABLE]

where $m$ is the nucleon mass, to describe the tensor parameter $c_{ij}$ for the LLIV interaction Kostelecký and Lane (1999) in the laboratory frame. In this paper we will show that there exists a collective effect of the momentum tensor in deformed nuclei and show that certain deformed nuclei of experimental interest present a promising option for constraining LLIV even further.

I Deformed Nuclei

In this paper we will use the empirically successful Nilsson model Nilsson (1955); Bohr and Mottelson (1975) (deformed oscillator model) to describe the single particle model states of the constituent nucleons in deformed nuclei and calculate the magnitude of the quadrupole and momentum tensors. The Nilsson Hamiltonian which governs the system is,

[TABLE]

which has the form of an asymmetric 3D oscillator model where we have chosen the $z-$axis as the axis of deformation. Here $m$ is the nucleon mass, $\omega_{z}$ and $\omega_{\perp}$ are the nucleon oscillation frequencies along the $z-$axis and the perpendicular plane respectively. The deformation of the nucleus has the effect of splitting the degeneracy of the nuclear energy levels within the oscillator shell. The greater the deformation the greater the splitting. Each nucleon can be characterised by the set of quantum numbers $\left[Nn_{z}\Lambda\pm\Omega\right]$ Bohr and Mottelson (1975) where $N=n_{z}+n_{x}+n_{y}$ and each $n_{i}$ ($i=x,y,z$) is the principal quantum number in the direction of $i$, $\Lambda$ and $\Omega$ are the nucleon’s orbital and total angular momentum projection on the deformation axis respectivly. Each energy level is doubly degenerate for $\pm\Omega$. The average oscillator frequency is given by $\hbar\bar{\omega}=\hbar/3\left(2\omega_{\perp}+\omega_{z}\right)\approx 45A^{-1/3}-25A^{-2/3}\ \text{MeV}$Brown et al. (2016). Second order tensor properties in deformed nuclei exhibit a collective enhancement compared to vector properties. We indirectly include spin-orbit and angular momentum terms of the Nilsson Hamiltonian through the split level Nilsson energy plots in ref. Bohr and Mottelson (1975).

Consider the magnetic dipole moment which is proportional to the projection of the total angular momentum $\Omega$ to the nuclear axis. In an even-even nucleus all the nucleons are paired and therefore the magnetic dipole moment of the $+\Omega$ state cancels with the $-\Omega$ state resulting in no net magnetic dipole moment of the nucleus. For an odd $A$ nucleus the magnetic dipole moment of the entire nucleus is simply the dipole moment of the odd nucleon. This is the well known Schmidt model of the nucleus. The cases for second order tensor properties of the nucleus are different. For nucleons in the $+\Omega$ and the $-\Omega$ states tensor properties are additive. Therefore there is a collective effect and many nucleons contribute to the tensor properties of the nucleus.

Due to the rotation of the nucleus the tensor properties transform between the body-fixed (intrinsic) frame which rotates and the laboratory frame. A second order tensor observable $T$ between these two frames has the relationshipBohr and Mottelson (1975)

[TABLE]

where $I=I_{z}=\left|\Omega\right|$ is the projection of total nuclear angular momentum (nuclear spin) on the symmetry axis. This expression shows that only in nuclei with spin $I>1/2$ we can detect these second order tensor properties.

I.1 Quadrupole moment of neutron distribution in deformed nuclei

The quadrupole moment tensor along the symmetry axis of the nucleus in cartesian coordinates is given by

[TABLE]

From the virial theorem for bound particle in a harmonic oscillator potential the average kinetic energy $\left<T\right>$ and average potential energy $\left<U\right>$ are equal. Using the well known energy spectrum for the harmonic oscillator $E_{n}=\hbar\omega(n+1/2)$ the average of the square of the position in the $z$ direction is,

[TABLE]

with similar relations for $\left<x^{2}\right>$ and $\left<y^{2}\right>$. Using equations (4), (5) and $n_{x}+n_{y}=N-n_{z}$ the contribution to the quadrupole moment from a single nucleon in the quantum state $\left[Nn_{z}\Lambda\Omega\right]$ is given by

[TABLE]

where $\nu=p,n$ for the $i$th proton or neutron respectively. The total quadrupole moment is the sum of all the respective nucleon quadrupole moment contributions

[TABLE]

I.2 Lorentz invariance violation in deformed nuclei

Taking the expectation value of (1) we find that the LLIV energy shift for a nucleus with $N_{\nu}$ nucleons is given by

[TABLE]

where the $\hat{M}=2\hat{p}_{z}^{2}-\hat{p}_{x}^{2}-\hat{p}_{y}^{2}$ is the momentum tensor operator for $p_{i}p_{j}$ in the SME in cartesian coordinates. We use the standard notion $C_{0,\nu}^{(2)}=c_{xx}+c_{yy}-2c_{zz}$ Kostelecký and Lane (1999). We define the single nucleon LLIV momentum tensor as,

[TABLE]

Using the virial theorem for a harmonic oscillator ($\left<T\right>=\left<U\right>$) and energy spectrum again we have the average square of the momentum given by

[TABLE]

with similar expressions for $x$ and $y$ coordinates.

Using equations (9) and (10) we write the contribution to the LLIV tensor of a nucleon as,

[TABLE]

The total LLIV tensor, $M_{\nu}$, for the nucleus is the sum of all respective nucleons

[TABLE]

This will result in a quadrupole energy shift given by

[TABLE]

I.3 Example calculation of $Q_{\nu}$ and $M_{\nu}$ for the 173Yb nucleus

Rewriting equations (7), (12) and, the relation between the longitudinal and perpendicular frequencies in dimentionless quantities $\eta=\bar{\omega}/\omega_{z}$ and $\xi=\bar{\omega}/\omega_{\perp}$ we have the equations,

[TABLE]

[TABLE]

[TABLE]

The nucleon angular momentum dependence shows up in the order of the split energy branches in the Nilsson plots Nilsson (1955); Bohr and Mottelson (1975). The parameters $\eta$ and $\xi$ can be viewed as deformation parameters of the nuclei. For positive deformation $\eta>1$ (prolate) and for negative deformation $\eta<1$ (oblate). As there is a predominance on nuclear prolate deformations in nature most of the nuclei we consider will be prolate.

To illustrate the process of calculating NQMN and LLIV values we will present the calculation of 173Yb as an example. To begin we find the nuclear configuration of protons and neutrons in the deformed field using the energy level plots presented in Nilsson (1955); Bohr and Mottelson (1975) by filling each non-degenerate $\Omega$ energy branch with two nucleons until all the nucleons have been distributed. To find the correct deformation, $\delta$, from which to fill we use the experimental value of the nuclear spin (which is solely due to the unpaired nucleon). The filling must be done such that the unpaired nucleon is in the correct spin state. The nuclear spin of 173Yb is $5/2^{-}$ due to an unpaired neutron. Using the method described above we find that this is possible with a minimum deformation of $\delta\approx 0.3$. Here we will only present the configuration of the partially filled $N$ shells. The numbers for each full shell is given in Table 1. The incomplete neutron and proton shells in 173Yb are $N=5,6$ and $N=4,5$ respectively. The nucleon configuration of the incomplete shells are given in Table 2.

Summing up the contribution for all the 173Yb filled shells from Table 1 and the partially filled shells from Table 2 the total values for neutrons are $\sum(2n_{z}+1)_{n}=439$, $\sum(N-n_{z}+1)_{n}=318$ and protons $\sum(2n_{z}+1)_{p}=266$, $\sum(N-n_{z}+1)_{p}=190$. Using (3) to transfer the measured value of the quadrupole moment in the lab frame $Q_{p}^{Lab}=4.39$ barn (where 1 barn = 100 fm2 = $10^{-24}$ cm2) to the internal (rotating) frame we have $Q_{p}^{int}=12.66\ \text{barn}$. Using equations (15) and the values for $\sum\left(2n_{z}+1\right)_{p}$, $\sum\left(N-n_{z}+1\right)_{p}$ and $Q^{int}_{p}$ we find the dimensionless deformation parameters $\eta=1.18$ and $\xi=0.93$ for 173Yb. We can find the NQMN and the proton and neutron LLIV tensors using (15) and (16). Finally in the lab frame using (3) we have,

[TABLE]

From (13) we find that the LLIV energy shift of the nucleus is,

[TABLE]

Considering only the contribution of the unpaired neutron in the Schmidt model (see Section II or Refs. Kostelecký and Lane (1999); Flambaum (2016)) gives energy shifts $\left<\delta H\right>_{p}=0\ C^{(2)}_{0,p}\text{MeV}$ and $\left<\delta H\right>_{n}=0.8\ C^{(2)}_{0,n}\text{MeV}$. The collective contribution of paired nucleons in the core gives non zero LLIV energy shifts for both protons and neutrons (in the Schmidt model either the proton or neutron LLIV shift will always be zero) and enhances the LLIV energy shifts by an order of magnitude.

This method can be completed with all the other deformed nuclei, the Nilsson quantum numbers can be found in Table 3 and the quadrupole moment values are presented in Table 4.

To understand the propagation of error in the calculations consider an error of 5% in $Q_{p}$ (experimental value is $Q_{p}=2.80(4)$ barn). Using (7), (12) with the Nilsson numbers for 173Yb results in an error of $\approx 5\%$ for $Q_{n}$ and an error of $\approx 25\%$ for $M_{\nu}$. This large error is not unique to our calculation of $M_{\nu}$ since it involves the subtraction of large numbers. Consider the results from Brown et al. (2016) where a sophisticated $s-d$ model resulted in similar uncertainty for slight variations of an effective charge for the quadrupole moment operator.

II Nuclei with a small deformation

For nuclei with very small deformations the splitting of energy levels with different angular momentum projections is small. In these circumstances the effect of nuclear pairing becomes significant resulting in mixing of the nucleon configurations. Therefore the Nilsson model approach is no longer applicable as it assumes that there is no mixing when counting the nucleon occupation numbers. For these near-spherical nuclei we need to approach the second order tensor properties differently. It is well known that the Schmidt model (valence, $Q_{\nu,val}$) value of nuclear quadrupole is smaller than the true quadrupole value. Therefore we assume that this discrepancy is explained by the quadrupole moment due to a small deformation (deformed, $Q_{\nu,def}$), i.e. the true quadrupole moment is the sum of these two contributions, $Q_{\nu}=Q_{\nu,val}+Q_{\nu,def}$. We also assume that $Q_{\nu,deformed}$ for protons and neutrons is related to the total number of protons and neutrons,

[TABLE]

Then using measured $Q_{p}$ values from Stone (2005) we can find an estimate for the neutron quadrupole moment of slightly deformed nuclei. As an example consider the 201Hg nucleus which has an electric quadrupole moment $Q_{p}=0.35$ barn with a valence neutron. There is no proton Schmidt contribution to $Q_{p}$ meaning the quadrupole moment due to the deformation is

[TABLE]

From Eq. (17) the contribution to $Q_{n}$ due to deformation is $Q_{n,def}=0.53\ \text{barn}$. The Schmidt model contribution of a valence nucleon is given by Bohr and Mottelson (1998); Flambaum (2016)

[TABLE]

The 201Hg nucleus has a valence neutron in the $f_{5/2}$ state with angular projection $I=3/2$. Using 18 the valence contribution is $Q_{n,val}=-0.15\ \text{barn}$. Therefore the total NQMN for 201Hg is $Q_{n}=0.38\ \text{barn}$ where all values are in the labratory frame. As expected this value is larger than $Q_{p}$. For the LLIV tensor we use the method outlined in Flambaum (2016) which relates the LLIV energy shift to the quadrupole moment of the nucleus,

[TABLE]

In this work we use the total quadrupole moment including the valence and deformed contribution discussed above. Similar to the NQMN this will give non zero values for both nucleons unlike the Schmidt model. Using 20 for 201Hg nucleus we have $\left<\delta H\right>_{n}=17\ C_{0,n}^{(2)}\ \text{MeV}$ and $\left<\delta H\right>_{p}=11.2\ C_{0,p}^{(2)}\ \text{MeV}$. Similar calculations can be performed for $Q_{n}$ and $M_{\nu}$ in other slightly deformed nuclei such as 131Xe, 133Cs which are presented in Table 4. In Brown et al. (2016) the LLIV and quadrupole moments were calculated for 21Ne, 131Xe and 201Hg numerically using a self-consistent mean field theory. Our results for 21Ne, where we use the large deformation method, are in a reasonable agreement with Ref. Brown et al. (2016) results for both $Q_{n}$ and $M_{\nu}$ ($Q_{n}=0.097\ \text{barn}$, $M_{p}=2.8\ m\text{ MeV}$ and $M_{n}=4.2\ m\text{ MeV}$). For nuclei 201Hg and 131Xe, where we use the small deformation method, there is a reasonable agreement for $Q_{n}$ and significant differences for $M_{\nu}$. In Ref. Brown et al. (2016) they obtained for 201Hg $Q_{n}=0.584\ \text{barn}$, $M_{p}=-20.5\ m\text{ MeV}$ and $M_{n}=1.5\ m\text{ MeV}$, and for 131Xe their results are $Q_{n}=-0.136\ \text{barn}$, $M_{p}=-4.7\ m\text{ MeV}$ and $M_{n}=5.17\ m\text{ MeV}$.

III Weak quadrupole moments and parity nonconservation in atomic and molecular systems

As previously mentioned above a consequence of studying the NQMN will be a further insight into parity nonconservation (PNC) effects in atomic and molecular systems. PNC has been intensively studied in atomic systems (see e.g. book Khriplovich (1981) and reviews Roberts et al. (2015); Ginges and Flambaum (2004)). The $P$-odd weak nucleon-electron interaction is given by,

[TABLE]

Here $G_{F}$ is the Fermi weak constant, $q_{w,\nu}$ and $\rho_{\nu}(r)$ are the nucleon weak charge and density of protons or neutrons normalised to 1. It is well known that the magnitude of the neutron weak charge is significantly larger than that of the proton. Not including radiative corrections the weak charges are given by

[TABLE]

where $\theta_{W}$ is the Weinberg angle. Previously the interaction (21) treated either the shapes of the proton and neutron densities the same or included a correction due to some neutron skin (see e.g. review Roberts et al. (2015)). The quadrupole moment in the nuclear density produces the tensor weak interaction which is proportional to the weak quadrupole moment (WQM) defined as Flambaum et al. (2017)

[TABLE]

Similar to the weak charge of a nucleus the WQM is dominated by the neutron contribution: $Q_{W}^{(2)}\approx q_{w,n}Q_{n}$ with a small correction due to the proton contribution.

The nuclear WQM induces PNC effects in atomic and molecular systems where the effective single electron PNC Hamiltonian for the nuclear WQM in atomic systems is presented in ref. Flambaum et al. (2017) and is given by,

[TABLE]

where $G_{F}$ is the Fermi weak constant, $\gamma_{5}$ is the standard Dirac matrix, $Y_{20}$ is the spherical harmonic, $\rho_{0}$ is the spherical nucleon density and $\left<r^{2}\right>$ is the mean squared nuclear radius. While calculations of these PNC effects is outside the scope of this work they can be observed in atomic and molecular systems in many ways (see refs. Flambaum et al. (2017); Roberts et al. (2015); Khriplovich (1981); Ginges and Flambaum (2004)) using interference of forbidden electric dipole transition amplitude with M1 ( or E2) amplitude between the states of equal parity Flambaum et al. (2017).

Also the PNC effects of the tensor weak interaction which has different selection rules are strongly enhanced and can be measured in atoms and molecules having close opposite parity energy levels with the difference of the electron angular momenta equal to 2. Corresponding states can be mixed by the tensor weak interaction but not the scalar (proportional to the weak charge) and vector (proportional to the nuclear anapole moment) components. If the difference of the electron angular momenta is 1 the effects of the anapole and the weak quadrupole may be separated due the difference in their contributions to the different hyperfine components of the electromagnetic transitions. Corresponding atomic calculations have been performed in Ref. Flambaum et al. (2017). Measurements of the tensor PNC effects in atoms and molecules will allow one to extract the neutron quadrupole moment of the nuclei.

We present the nuclear weak quadrupole moments $Q_{W}^{(2)}$ in Table 4. As expected heavily deformed nuclei with a large number of neutrons have large WQMs.

IV Conclusion

We presented a versatile semi-empirical method to calculate several second order tensor properties of nuclei which are enhanced in deformed nuclei including the weak quadrupole moment. Though we present values for only a few deformed nuclei the method can be applied to many nuclei of interest. In particular the highly deformed 181Ta, 167Er, 163Dy and 153Eu have large enhancement of tensor properties and are particularly promising candidates for further study. The theoretical results presented should facilitate experimentalists in probing the nuclear structure and fundamental physics, specifically the previously unstudied quadrupole distribution of neutrons and the violations of Lorentz symmetry. Currently such systems are of high interest as they allow the study of physics beyond the standard model with low energy systems.

This enhancement in deformed nuclei can also be applied for other second order tensor observables of interest such as the time-reversal and parity violating magnetic quadrupole moment. Further study of these tensor properties in deformed nuclei could lead to further understanding of fundamental physics.

This work is supported in part by the Australian Research Council and the Gutenberg Fellowship.