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the magnetocaloric effect ( mce ) is most commonly characterized through the isothermal entropy change @xmath0 and the adiabatic temperature change @xmath1 of a magnetic solid in response to a variation of the applied magnetic field .
an importance of the mce consists in its potential usage for a magnetic refrigeration through the process of adiabatic demagnetization @xcite .
the magnetic materials with the large mce are therefore of technological interest with regard to their cooling capability either at room or low temperatures @xcite .
the large isothermal entropy change were experimentally found for diverse magnetic systems as cobalt - based chain cov@xmath2o@xmath3 @xcite , gadolinium ladder @xcite , oligonuclear spin cluster ln@xmath3mn@xmath4 @xcite or molecular nanomagnets @xcite .
the isothermal entropy change is most commonly obtained either from magnetization data by means of maxwell relation or from an integration of the specific - heat data .    in the present work
we will examine the isothermal entropy change and the adiabatic temperature change of the antiferromagnetic spin-1/2 ising octahedron and dodecahedron , which can be calculated from the exact analytical results reported in our previous study @xcite .
let us consider the spin-1/2 ising octahedron and dodecahedron ( see fig .
[ fig1 ] ) , which are defined trough the following hamiltonian @xmath5 here , @xmath6 denotes the ising spin placed at @xmath7th vertex of a regular polyhedron , the first summation accounts for the antiferromagnetic interaction @xmath8 between adjacent spins , the second summation accounts for the zeeman s energy of individual magnetic moments in the external magnetic field @xmath9 and finally , @xmath10 stands for the total number of spins ( bonds ) of a regular polyhedron .
the magnetization curves and thermodynamic properties of the regular ising polyhedra defined through the hamiltonian ( [ ham ] ) were comprehensively studied in our previous works @xcite . in the present paper
we will focus our attention on magnetocaloric response functions of the ising octahedron and dodecahedron ( see fig .
[ fig1 ] ) , which exhibit a giant magnetocaloric effect close to zero magnetic field due to an abrupt magnetization jump towards the one - third and one - fifth plateau , respectively @xcite .
the isothermal entropy change can be calculated as a difference of the entropy at non - zero and zero magnetic field @xmath11 at the constant temperature .
the molar entropy can be easily obtained from the exact results ( 6)-(10 ) presented in ref .
@xcite for the gibbs free energy @xmath12 according to @xmath13 where @xmath14 denotes avogadro s number .
the adiabatic temperature change @xmath15 can be obtained from the transcendent equation @xmath16 which can be numerically solved by the bisection method . here , @xmath17 ( @xmath18 ) and
@xmath19 ( @xmath20 ) denote the initial ( final ) temperature and magnetic field for a given adiabatic process .
let us proceed to a discussion of temperature dependences of the isothermal entropy change and adiabatic temperature change of the antiferromagnetic spin-1/2 ising octahedron and dodecahedron .
the isothermal entropy change of the ising octahedron is plotted in fig .
[ fig2](a ) for a few different values of the magnetic - field change .
as one can see , the isothermal entropy change of the ising octahedron is negative for arbitrary magnetic - field change independently of temperature .
this result implies a substantial cooling performance of the ising octahedron in thermal contact with the heat bath .
the zero - temperature asymptotic values of the isothermal entropy change are in agreement with the difference of ground - state degeneracy at zero and non - zero magnetic fields .
if the magnetic - field change is lower than the saturation field of the ising octahedron @xmath21 , then , one finds a gradual increase of the working temperature interval with increasing of the magnetic - field change until roughly a midpoint of the one - third magnetization plateau is reached @xcite .
however , the largest isothermal entropy change of the ising octahedron can be found for the magnetic - field change , which is higher than the saturation field .
the maximum value of isothermal entropy change of the ising octahedron for @xmath22 is @xmath23 at low enough and moderate temperatures ( @xmath24 for @xmath25 ) , whereas one observes its gradual decrease at higher temperatures ( @xmath26 for @xmath27 at @xmath25 ) .
the adiabatic temperature change of the ising octahedron linearly increases with temperature up to the threshold temperature , which corresponds to the residual entropy @xmath28 ( @xmath29 is universal gas constant ) of the ising octahedron for a given magnetic - field change , see fig .
[ fig2](b ) .
consequently , the ising octahedron achieves the absolute zero temperature during the adiabatic demagnetization process whenever the initial temperature is selected below the threshold temperature forming a cusp in the relevant temperature dependence .
this result is apparently in contrast with one of the formulations of the third law of thermodynamics @xcite . on the other hand
, there is a rapid decrease in the adiabatic temperature change above the threshold temperature , which indicates a less efficient cooling .
the threshold temperature , which corresponds to the residual entropy for a given magnetic - field change , thus defines the working temperature interval of the ising octahedron that monotonically extends with increasing of the magnetic - field change
.    the antiferromagnetic spin-1/2 ising dodecahedron also shows large mce @xmath30 for most of the magnetic - field changes , see fig . [
fig3](a ) .
the notable exception to this rule represent the field changes approximately equal to the saturation field @xmath31 , for which the inverse mce @xmath32 can be detected in a relatively wide temperature range .
the most efficient cooling through a thermal contact with the ising dodecahedron can be thus achieved just for sufficiently high values of the magnetic - field change @xmath33 , because the maximal value of isothermal entropy change @xmath34 then persists up to moderate temperatures ( @xmath35 for @xmath36 ) .
contrary to this , the isothermal entropy change decreases down to @xmath37 at much higher temperature ( @xmath38 for @xmath36 ) .
the adiabatic temperature change of the ising dodecahedron has a similar behavior as that of the ising octahedron [ see fig.[fig3](b ) ]
. for arbitrary magnetic - field change there exists the threshold temperature corresponding to the residual entropy @xmath39 , under which the ising dodecahedron reaches the absolute zero temperatures during the adiabatic demagnetization .
contrary to the ising octahedron , the adiabatic temperature change of the ising dodecahedron may become negative at moderate temperatures as a result of the inverse mce .
in addition , the adiabatic temperature change is at the same magnetic - field changes less than a half of the adiabatic temperature change of the ising octahedron . in this respect ,
the ising octahedron is more efficient refrigerant than the ising dodecahedron , since it provides a higher gain of the isothermal entropy change and the adiabatic temperature change at the same values of magnetic - field change due to an absence of the inverse mce .
in the present work we have exactly examined two basic magnetocaloric characteristics of the ising octahedron and dodecahedron : the isothermal entropy change and adiabatic change of temperature . it has been demonstrated that the largest isothermal entropy change can be found whenever the field change is greater than the saturation field of the relevant ising spin cluster .
the adiabatic change of temperature confirms an outstanding cooling capability of the ising octahedron and dodecahedron , which may reach during the adiabatic demagnetization absolute zero temperature .
the ising octahedron and dodecahedron can be therefore regarded as promising frustrated spin structures for magnetic refrigeration , whereas the maximum of the adiabatic change of temperature shifts to higher temperatures with increasing of the magnetic - field change .
this work was financially supported by grant nos .
vega 1/0043/16 , apvv-14 - 0073 and vvgs - pf-2016 - 72606 .
the spin-1/2 ising octahedron and dodecahedron with a unique antiferromagnetic interaction display an outstanding magnetization jump at zero magnetic field , which consequently leads to a giant magnetocaloric effect during the adiabatic demagnetization . in the present work we report temperature dependences of two basic magnetocaloric response functions : the isothermal entropy change and the adiabatic change of temperature . 
 it is shown that the ising octahedron and dodecahedron generally exhibit a large negative isothermal entropy change upon increasing of the magnetic field , which serves in evidence of their cooling performance .