Average values

The electric and magnetic moments are defined as

   

$$Q_{\lambda\mu} \langle\Psi \vert\hat{Q}_{\lambda\mu}\vert\Psi\rangle = \int q_{\lambda\mu}(\mbox{{\boldmath {$r$ }}})\,d^3\mbox{{\boldmath {$r$ }}} ,$$ = (82)

$$M_{\lambda\mu} \langle\Psi \vert\hat{M}_{\lambda\mu}\vert\Psi\rangle = \int m_{\lambda\mu}(\mbox{{\boldmath {$r$ }}})\,d^3\mbox{{\boldmath {$r$ }}} ,$$ = (83)

where $\vert\Psi\rangle$ is a many-body mean-field state, and $q_{\lambda\mu}(\mbox{{\boldmath {$r$ }}})$ and $m_{\lambda\mu}(\mbox{{\boldmath {$r$ }}})$are the corresponding moment densities:

   

$$q_{\lambda\mu}(\mbox{{\boldmath {$r$ }}}) e\rho(\mbox{{\boldmath {$r$ }}})Q_{\lambda\mu}(\mbox{{\boldmath {$r$ }}}) ,$$ = (84)

$$m_{\lambda\mu}(\mbox{{\boldmath {$r$ }}}) \mu_N\sum_{k=x,y,z}\left( g_s{s_k}\nabla_k Q_{\lambda\mu}(\mbox{{\boldmath {$r$ }}})\right.$$ =

$$\left. {\textstyle\frac{2}{\lambda+1}} g_l{j_k}(\mbox{{\boldmath ... ...x{{\boldmath {$\nabla$ }}}Q_{\lambda\mu}(\mbox{{\boldmath {$r$ }}}))_k\right) ,$$ - (85)

and e, g_s, and g_l are the elementary charge, and the spin and orbital gyromagnetic factors, respectively [1]. In definitions (84), multipole functions[21] (solid harmonics) have the standard form: $Q_{\lambda\mu}(\mbox{{\boldmath {$r$ }}})$= $r^\lambda Y_{\lambda\mu}(\theta,\phi)$.

Similarly, the mean value of the k-component of total angular momentum (in units of $\hbar$) reads

$$I_k = \langle\Psi \vert\hat{I}_k\vert\Psi\rangle = \int \le... ...\mbox{{\boldmath {$r$ }}})\right) d^3\mbox{{\boldmath {$r$ }}}$$ (86)

We may now combine symmetry properties of densities $\rho$, $\mbox{{\boldmath {$s$ }}}$, and $\mbox{{\boldmath {$j$ }}}$, Table 3, with those of multipole operators, Table 5, to obtain symmetry conditions obeyed by the electric and magnetic moments, and by the average angular momenta, for given conserved symmetries of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ groups. In doing so, we have to remember that since the electric multipole operators are time-even, the corresponding electric moments depend only on the time-even component of the density matrix, as given in Eq. (84). This is so irrespective of whether the time reversal is, or is not a conserved symmetry, or whether the system contains even or odd number of fermions. Therefore, the time reversal does not impose any condition on the electric multipole moments. On the other hand, with the time-reversal symmetry conserved, which may occur only for even systems, all magnetic moments and average angular momenta must vanish, because they depend only on the time-odd component of the density matrix, Eqs. (85) and (86).

 

Table 6: Conditions fulfilled by the electric and magnetic multipole moments, $Q_{\lambda\mu}$ and $M_{\lambda\mu}$, for conserved D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators. Where applicable, the upper part of the Table gives expressions in terms of changed signs of magnetic components, and the lower part gives equivalent expressions in terms of the complex conjugation.

k

$\hat{R}_{k}$

$\hat{R}^T_{k}$

$\hat{S}_{k}$

$\hat{S}^T_{k}$

x

$Q_{\lambda\mu}$

=

$(-1)^{\lambda}$

$Q_{\lambda,-\mu}$

$Q_{\lambda\mu}$

=

$(-1)^{\lambda}$

$Q_{\lambda,-\mu}$

$Q_{\lambda\mu}$

=

$Q_{\lambda,-\mu}$

$Q_{\lambda\mu}$

=

$Q_{\lambda,-\mu}$

x

$M_{\lambda\mu}$

=

$(-1)^{\lambda}$

$M_{\lambda,-\mu}$

$M_{\lambda\mu}$

=

$-(-1)^{\lambda}$

$M_{\lambda,-\mu}$

$M_{\lambda\mu}$

=

-

$M_{\lambda,-\mu}$

$M_{\lambda\mu}$

=

$M_{\lambda,-\mu}$

y

$Q_{\lambda\mu}$

=

$(-1)^{\lambda-\mu}$

$Q_{\lambda,-\mu}$

$Q_{\lambda\mu}$

=

$(-1)^{\lambda-\mu}$

$Q_{\lambda,-\mu}$

$Q_{\lambda\mu}$

=

$(-1)^{-\mu}$

$Q_{\lambda,-\mu}$

$Q_{\lambda\mu}$

=

$(-1)^{-\mu}$

$Q_{\lambda,-\mu}$

y

$M_{\lambda\mu}$

=

$(-1)^{\lambda-\mu}$

$M_{\lambda,-\mu}$

$M_{\lambda\mu}$

=

$-(-1)^{\lambda-\mu}$

$M_{\lambda,-\mu}$

$M_{\lambda\mu}$

=

$-(-1)^{-\mu}$

$M_{\lambda,-\mu}$

$M_{\lambda\mu}$

=

$(-1)^{-\mu}$

$M_{\lambda,-\mu}$

z

$Q_{\lambda\mu}$

=

$(-1)^{\mu}$

$Q_{\lambda\mu}$

$Q_{\lambda\mu}$

=

$(-1)^{\mu}$

$Q_{\lambda\mu}$

$Q_{\lambda\mu}$

=

$(-1)^{\lambda+\mu}$

$Q_{\lambda\mu}$

$Q_{\lambda\mu}$

=

$(-1)^{\lambda+\mu}$

$Q_{\lambda\mu}$

z

$M_{\lambda\mu}$

=

$(-1)^{\mu}$

$M_{\lambda\mu}$

$M_{\lambda\mu}$

=

$-(-1)^{\mu}$

$M_{\lambda\mu}$

$M_{\lambda\mu}$

=

$-(-1)^{\lambda+\mu}$

$M_{\lambda\mu}$

$M_{\lambda\mu}$

=

$(-1)^{\lambda+\mu}$

$M_{\lambda\mu}$

x

$Q_{\lambda\mu}$

=

$(-1)^{\lambda+\mu}$

$Q_{\lambda \mu}^*$

$Q_{\lambda\mu}$

=

$(-1)^{\lambda+\mu}$

$Q_{\lambda \mu}^*$

$Q_{\lambda\mu}$

=

$(-1)^{\mu}$

$Q_{\lambda \mu}^*$

$Q_{\lambda\mu}$

=

$(-1)^{\mu}$

$Q_{\lambda \mu}^*$

x

$M_{\lambda\mu}$

=

$(-1)^{\lambda+\mu}$

$M_{\lambda \mu}^*$

$M_{\lambda\mu}$

=

$-(-1)^{\lambda+\mu}$

$M_{\lambda \mu}^*$

$M_{\lambda\mu}$

=

$-(-1)^{\mu}$

$M_{\lambda \mu}^*$

$M_{\lambda\mu}$

=

$(-1)^{\mu}$

$M_{\lambda \mu}^*$

y

$Q_{\lambda\mu}$

=

$(-1)^{\lambda}$

$Q_{\ ambda \mu}^*$

$Q_{\lambda\mu}$

=

$(-1)^{\lambda}$

$Q_{\lambda \mu}^*$

$Q_{\lambda\mu}$

=

$Q_{\lambda \mu}^*$

$Q_{\lambda\mu}$

=

$Q_{\lambda \mu}^*$

y

$M_{\lambda\mu}$

=

$(-1)^{\lambda}$

$M_{\lambda \mu}^*$

$M_{\lambda\mu}$

=

$-(-1)^{\lambda}$

$M_{\lambda \mu}^*$

$M_{\lambda\mu}$

=

-

$M_{\lambda \mu}^*$

$M_{\lambda\mu}$

=

$M_{\lambda \mu}^*$

z

$Q_{\lambda\mu}$

=

$(-1)^{\mu}$

$Q_{\lambda\mu}$

$Q_{\lambda\mu}$

=

$(-1)^{\mu}$

$Q_{\lambda\mu}$

$Q_{\lambda\mu}$

=

$(-1)^{\lambda+\mu}$

$Q_{\lambda\mu}$

$Q_{\lambda\mu}$

=

$(-1)^{\lambda+\mu}$

$Q_{\lambda\mu}$

z

$M_{\lambda\mu}$

=

$(-1)^{\mu}$

$M_{\lambda\mu}$

$M_{\lambda\mu}$

=

$-(-1)^{\mu}$

$M_{\lambda\mu}$

$M_{\lambda\mu}$

=

$-(-1)^{\lambda+\mu}$

$M_{\lambda\mu}$

$M_{\lambda\mu}$

=

$(-1)^{\lambda+\mu}$

$M_{\lambda\mu}$

For the conserved parity, one obtains the standard conditions:

   

$$Q_{\lambda\mu} (-1)^\lambda Q_{\lambda\mu} ,$$ = (87)

$$M_{\lambda\mu} -(-1)^\lambda M_{\lambda\mu} ,$$ = (88)

i.e., odd electric and even magnetic moments must vanish. Similar symmetry properties with respect to other symmetries of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ groups are collected in Table 6.

Within the standard phase convention of Eq. (80), only a conservation of the y-T-simplex symmetry, $\hat{S}^T_{y}$, enforces the reality of all multipole electric and magnetic moments. In such a case, the lower part of Table 6 gives at a glance all the multipole moments which must vanish whenever any other symmetry is additionally conserved; these are those for which the phase factors are negative. On the other hand, a conservation of the x-T-simplex symmetry, $\hat{S}^T_{x}$, enforces the equality of negative and positive magnetic components. In this case, a conservation of any additional symmetry puts to zero the multipole moments with negative phase factors appearing in the upper part of the Table. Of course, numerous other combinations of conserved symmetries can be considered, for example, a conservation of the y-simplex symmetry, $\hat{S}_{y}$, gives real electric moments and imaginary magnetic moments.

Since conditions listed in Table 6 depend only on the parity of $\lambda$ and on the parity of $\mu$, and since condition (80) allows us to consider only non-negative values of $\mu$, one has only six types of the symmetry properties of multipole moments with respect to the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators. These six types are listed in Table 7 for electric and magnetic moments. Column denoted by the identity operator $\hat{E}$ gives the properties resulting solely from condition (80), while the remaining columns give properties of moments when one of the non-identity D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators is conserved.

In the same Table we also give symmetry properties of the Cartesian components of the average angular momenta I_k (86). Although the symmetry properties of the angular momentum are identical to those of the dipole magnetic moment, explicit values shown for its Cartesian components allow for a simple visualization of a direction taken by the angular-momentum vector when various D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators are conserved. In particular, one can see that a conservation of any of the signature or simplex operators for a given axis enforces the angular-momentum direction along that axis, while a conservation of any T-signature or T-simplex operators allows for a tilted angular momentum in the plane perpendicular to the given axis, cf. Ref.[23]. On the other hand, none of these operators may be conserved if the angular momentum is to be tilted beyond any of the x-y, y-z, and z-x planes. Note, however, that the above tilting conditions pertain to the reference frame, and not to the principal axes of the mass distribution. An appropriate choice of the reference frame, as discussed below, has to be performed in order to relate the conserved D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators to the direction of I_k with respect to the mass principal axes.

 

Table 7: Properties of electric multipole moments $Q_{\lambda\mu}$, magnetic multipole moments $M_{\lambda\mu}$, and average angular momenta, I_k for conserved D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators. Symbols C, R, I, or 0 denote values which can be, in general, complex, real, imaginary, or zero, respectively.

$\hat{E}$

$\hat{T}$

$\hat{P}$

$\hat{P}^T$

$\hat{R}_{x}$

$\hat{R}^T_{x}$

$\hat{S}_{x}$

$\hat{S}^T_{x}$

$\hat{R}_{y}$

$\hat{R}^T_{y}$

$\hat{S}_{y}$

$\hat{S}^T_{y}$

$\hat{R}_{z}$

$\hat{R}^T_{z}$

$\hat{S}_{z}$

$\hat{S}^T_{z}$

Q10,Q30,Q50$\ldots$

R

R

0

0

0

0

R

R

0

0

R

R

R

R

0

0

Q11,Q31,Q33$\ldots$

C

C

0

0

R

R

I

I

I

I

R

R

0

0

C

C

Q20,Q40,Q60$\ldots$

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

Q21,Q41,Q43$\ldots$

C

C

C

C

I

I

I

I

R

R

R

R

0

0

0

0

Q22,Q42,Q44$\ldots$

C

C

C

C

R

R

R

R

R

R

R

R

C

C

C

C

Q32,Q52,Q54$\ldots$

C

C

0

0

I

I

R

R

I

I

R

R

C

C

0

0

M10,M30,M50$\ldots$

R

0

R

0

0

R

0

R

0

R

0

R

R

0

R

0

M11,M31,M33$\ldots$

C

0

C

0

R

I

R

I

I

R

I

R

0

C

0

C

M20,M40,M60$\ldots$

R

0

0

R

R

0

0

R

R

0

0

R

R

0

0

R

M21,M41,M43$\ldots$

C

0

0

C

I

R

R

I

R

I

I

R

0

C

C

0

M22,M42,M44$\ldots$

C

0

0

C

R

I

I

R

R

I

I

R

C

0

0

C

M32,M52,M54$\ldots$

C

0

C

0

I

R

R

R

I

R

I

R

C

0

C

0

Ix

R

0

R

0

R

0

R

0

0

R

0

R

0

R

0

R

Iy

R

0

R

0

0

R

0

R

R

0

R

0

0

R

0

R

Iz

R

0

R

0

0

R

0

R

0

R

0

R

R

0

R

0

Independently of any D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ symmetry breaking, the reference frame in the space coordinates can be chosen in such way that some of the moments have simple forms. For example, a shift of the reference frame can bring all electric dipole moments to zero (this corresponds to using the center-of-mass system of reference), i.e.,

$$Q_{10}={\Re}Q_{11}={\Im}Q_{11}=0.$$ (89)

Similarly, a suitable rotation of the reference frame can bring the electric quadrupole moments $Q_{2\mu}$ to the principal axes, where

$${\Re}Q_{21}={\Im}Q_{21}={\Im}Q_{22}=0.$$ (90)

On the other hand, for some conserved symmetries, these conditions can be automatically satisfied. For example, conservation of the D $_{\mbox{\rm\scriptsize {2h}}}$ group (i.e., simultaneous invariance with respect to operators $\hat{P}$, $\hat{R}_{x}$, and $\hat{R}_{y}$) ensures that the center-of-mass (89) and principal-axes (90) conditions are automatically satisfied, see Table 7. Therefore, the breaking of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ symmetries may have non-trivial physical consequences only for higher electric moments; starting from Q₃₀, if the parity is broken, or starting from Q₄₁, for example, if the parity is conserved. In other words, the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ symmetry breaking will not lead to new classes of low-multipolarity shapes. Nevertheless, such symmetry breaking will immediately be reflected in values of magnetic moments, whenever the time reversal is broken too.