Multipole, surface multipole, and magnetic moments

Previous versions of the code worked in the regime of the conserved $y$-simplex, which ensured that all components of all multipole moments were real, cf. Eq. (I-56). Since in the present version the $y$-simplex symmetry was released, we have to carefully define all phase conventions used in calculating matrix elements of multipole operators. This is done by adopting the following convention for multipole moments:

$$Q_{\lambda\mu}(\mbox{{\boldmath {$r$}}}) = a_{\lambda\mu} r^\lambda Y^*_{\lambda\mu}(\theta,\phi),$$ (2)

where $Y_{\lambda\mu}$ are the standard spherical harmonics in the convention of Ref. [8]. Note that the complex-conjugate spherical harmonics enter the adopted definition of multipole moments, i.e., the phase convention corresponds to that used in electrodynamics, see Eq. (4.3) of Ref. [9]. Real factors $a_{\lambda\mu}$= $a_{\lambda,-\mu}$ ensure a traditional normalization of the low-multipolarity moments (for $\lambda$$\leq$2), such that they correspond to moments of simple polynomials of coordinates, see Table 5. For $\lambda$$>$2 we set $a_{\lambda\mu}$=1. We have kept the factor of $\sqrt{3}$ in the definition of $Q_{22}$ in order to conform to the standard definition of the Bohr $\gamma$ deformation [10], i.e., $\tan\gamma$= $\Re\langle Q_{22}\rangle$/ $\langle Q_{20}\rangle$. As usual, moments for negative values of the magnetic quantum number $\mu$ can be obtained from $Q_{\lambda,-\mu}$= $(-1)^\mu Q^*_{\lambda\mu}$. Whenever the multipole moments become complex, the code prints the real parts of non-negative-$\mu$ components and imaginary parts of negative-$\mu$ components.

Table 5: Adopted definitions of the normalization factors $a_{\lambda\mu}$ and the corresponding multipole moments $Q_{\lambda\mu}$.

$\lambda$	$\mu$	$a_{\lambda\mu}$	$Q_{\lambda\mu}$
0	0	$\phantom{-}\sqrt{ 4\pi}$	1
1	0	$\phantom{-}\sqrt{ 4\pi/3}$	$z$
1	1	$-\sqrt{ 8\pi/3}$	$x-iy$
2	0	$\phantom{-}\sqrt{16\pi/5}$	$2z^2-x^2-y^2$
2	1	$-\sqrt{ 8\pi/15}$	$zx-izy$
2	2	$\phantom{-}\sqrt{32\pi/5}$	$\sqrt{3}(x^2-y^2-2ixy)$

Since the adopted normalization factors depend on the magnetic quantum number $\mu$, rotational invariants are not equal to sums of moduli squared of all magnetic components for a given multipole moment. In fact, it is easy to check that for $\lambda$=1 and 2 the invariant combinations are $\vert Q_{10}\vert^2+\vert Q_{11}\vert^2$ and $\vert Q_{20}\vert^2+12\vert Q_{21}\vert^2+\vert Q_{22}\vert^2$.

Apart from calculating multipole moments, the HFODD code version (v2.07f) also calculates the average values of the so-called surface multipole moments,

$$Q^S_{\lambda\mu}(\mbox{{\boldmath {$r$}}}) = a_{\lambda\mu} r^{\lambda+2} Y^*_{\lambda\mu}(\theta,\phi),$$ (3)

which are much more sensitive to matter distribution in the surface region than are the standard multipole moments (2). Moreover, the surface dipole moment is needed to calculate the Schiff moment of a nucleus, see, e.g., Ref. [11]. Note that within the chosen normalization, the surface monopole moment is equal to the radius squared, $Q^S_{00}$=$r^2$. Values of the surface multipole moments can be constrained in the same way as the values of the multipole moments, so the multipole constraint term (I-22) now takes the form

$${\cal E}^{\mbox{\scriptsize {mult}}} = \sum_{\lambda\mu} C_... ...t Q^S_{\lambda\mu}\rangle - \bar Q^S_{\lambda\mu}\right)^2 .$$ (4)

The new version also calculates the average values of the magnetic moment operators [10],

$$\hat{M}_{\lambda\mu} = \left(g_s\hat{\mbox{{\boldmath {$S$}... ...nabla$}}}\left[r^\lambda Y^*_{\lambda\mu}(\theta,\phi)\right],$$ (5)

where $\hat{\mbox{{\boldmath {$S$}}}}$ and $\hat{\mbox{{\boldmath {$L$}}}}$ are the spin and orbital angular momentum operators, respectively, and $g_s$ and $g_l$ are the standard gyromagnetic factors. Constraints on magnetic moments are not yet implemented.

Multipole, surface multipole, and magnetic moments are calculated only for those multipolarities $\lambda$ and magnetic components $\mu$ which are allowed for a given pattern of conserved symmetries, see Ref. [5]. Whenever a shift of the center of mass from the origin of the coordinate frame is allowed, the code also calculates multipole and surface multipole moments in the center-of-mass reference frame. Calculation of the magnetic moments in the center-of-mass reference frame is not yet implemented.

Considering a rotation of the principal axes of the mass distribution with respect to the center-of-mass reference frame, the code determines the Euler angles corresponding to such a rotation, and then calculates all moments and angular momenta in the principal-axes (intrinsic) reference frame. The Euler angles are determined by diagonalizing the Cartesian matrix of the quadrupole tensor, see Ref. [8], and thus finding the orthogonal 3$\times$3 transformation that corresponds to the required rotation. The order of eigenvalues is chosen in such way that in the principal-axes reference frame one has $\langle Q_{yy}\rangle$$\leq$ $\langle Q_{xx}\rangle$$\leq$ $\langle Q_{zz}\rangle$, where $Q_{aa}$=$3x^2_a$$-$$r^2$, or equivalently $\langle y^2\rangle$$\leq$ $\langle x^2\rangle$$\leq$ $\langle z^2\rangle$. Therefore, the obtained Euler angles bring the system to the first sector $0^\circ\leq\gamma\leq60^\circ$ of the standard Bohr $\gamma$ deformation [10].

Note that the principal-axes (intrinsic) reference frame defined in such a way can differ from the original frame even in the case of conserved signature and parity, i.e, when the system has three symmetry planes. Indeed, in such a case, the Euler angles may correspond to a rotation that simply exchanges names and directions of the Cartesian axes. Since this need not be an interesting transformation, the user can manually switch off the printing of results that pertain to the intrinsic frame, by using switch INTRIP, see Sec. 3.6.

Note that the center-of-mass and principle-axes reference frames are determined from the mass dipole and quadrupole moments, respectively, and hence the neutron and proton distributions need not be individually brought to their respective center-of-mass and principal-axes frames. In other words, in the center-of-mass reference frame the neutron and proton dipole moments can be different from zero, and in the principle-axes reference frame the neutron and proton quadrupole moments need not obey conditions $\langle Q_{21}\rangle$=0 and $\Im\langle Q_{22}\rangle$=0. Note also that constraints on the multipole and surface multipole moments are formulated with respect to moments calculated in the original reference frame.