Here we present a systematic study of force-free field equation for simple axisymmetric configurations in spherical geometry. The condition of separability of solutions in radial and angular variables leads to two classes of solutions: linear and non
-linear force-free fields. We have studied these linear solutions Chandrasekhar (1956) and extended the non-linear solutions given in Low & Lou (1990) to the irreducible rational form $n= p/q$, which is allowed for all cases of odd $p$ and to cases of $q>p$ for even $p$. We have further calculated their energies and relative helicities for magnetic field configurations in finite and infinite shell geometries. We demonstrate here a method here to be used to fit observed magnetograms as well as to provide good exact input fields for testing other numerical codes used in reconstruction on the non-linear force-free fields.
We present a complete characterization of ferromagnetic system CeIr2B2 using powder x-ray diffraction XRD, magnetic susceptibility chi(T), isothermal magnetization M(H), specific heat C(T), electrical resistivity rho(T,H), and thermoelectric power S(
T) measurements. Furthermore 11B NMR study was performed to probe the magnetism on a microscopic scale. The chi(T), C(T) and rho(T) data confirm bulk ferromagnetic ordering with Tc = 5.1 K. Ce ions in CeIr2B2 are in stable trivalent state. Our low-temperature C(T) data measured down to 0.4 K yield Sommerfeld coefficient gamma = 73(4) mJ/molK2 which is much smaller than the previously reported value of gamma = 180 mJ/molK2 deduced from the specific heat measurement down to 2.5 K. For LaIr2B2 gamma = 6(1) mJ/molK2 which implies the density of states at the Fermi level D(EF) = 2.54 states/(eV f.u.) for both spin directions. The renormalization factor for quasi-particle density of states and hence for quasi-particle mass due to 4f correlations in CeIr2B2 is 12. The Kondo temperature TK ~ 4 K is estimated from the jump in specific heat of CeIr2B2 at Tc. Both C(T) and rho(T) data exhibit gapped-magnon behavior in magnetically ordered state with an energy gap Eg ~ 3.5 K. The rho data as a function of magnetic field H indicate a large negative magnetoresistance (MR) which is highest for T = 5 K.While at 5 K the negative MR keeps on increasing up to 10 T, at 2 K an upturn is observed near H = 3.5 T. On the other hand, the thermoelectric power data have small absolute values (S ~ 7 {mu}V/K) indicating a weak Kondo interaction. A shoulder in S(T) at about 30 K followed by a minimum at ~ 10 K is attributed to crystal electric field (CEF) effects and the onset of magnetic ordering. 11B NMR line broadening provides strong evidence of ferromagnetic correlations below 40 K.
