Insights into nature of magnetization plateaus of a nickel complex [Ni4(CO3)2(aetpy)8](ClO4)4 from a spin-1 Heisenberg diamond cluster

Magnetic and magnetocaloric properties of a spin-1 Heisenberg diamond cluster with two different coupling constants are investigated with the help of an exact diagonalization based on the Kambe's method, which employs a local conservation of composite spins formed by spin-1 entities located in opposite corners of a diamond spin cluster. It is shown that the spin-1 Heisenberg diamond cluster exhibits several intriguing quantum ground states, which are manifested in low-temperature magnetization curves as intermediate plateaus at 1/4, 1/2 and 3/4 of the saturation magnetization. Besides, the spin-1 Heisenberg diamond cluster may also exhibit an enhanced magnetocaloric effect, which may be relevant for a low-temperature refrigeration achieved through the adiabatic demagnetization. It is evidenced that the spin-1 Heisenberg diamond cluster with the antiferromagnetic coupling constants J1/kB = 41.4 K and J2/kB = 9.2 K satisfactorily reproduces a low-temperature magnetization curve recorded for the tetranuclear nickel complex [Ni4(CO3)2(aetpy)8](ClO4)4 (aetpy = 2-aminoethyl-pyridine) including a size and position of intermediate plateaus detected at 1/2 and 3/4 of the saturation magnetization. A microscopic nature of fractional magnetization plateaus observed experimentally is clarified and interpreted in terms of valence-bond crystal with either a single or double valence bond. It is suggested that this frustrated magnetic molecule can provide a prospective cryogenic coolant with the maximal isothermal entropy change - Delta S = 10.6 J/(K.kg) in a temperature range below 2.3 K.


I. INTRODUCTION
Molecular-based magnetic materials have attracted a considerable research interest over the past few decades, because they provide perspective building blocks for a development of new generation of nanoscale devices with a broad application potential 1-4 . Small magnetic molecules composed a few exchange-coupled spin centers might for instance serve for the rational design of high-density storage devices 5 and various spintronic devices [6][7][8] . Another intriguing feature of a special class of molecular magnetic materials with an extremely slow magnetic relaxation, which are commonly referred to as single-molecule magnets, is their possible implementation for developing novel platform for a quantum computation and quantum information processing [9][10][11][12][13][14][15] .
Appearance of plateaus in low-temperature magnetization curves of molecular magnetic materials at rational values of the magnetization represents other fascinating topical issue of current research interest, which can be experimentally easily validated due to a recent development of high-field facilities [16][17][18][19][20][21][22][23] . The magnetization plateaus often bear evidence of unconventional quantum states of matter theoretically predicted by the respective quantum Heisenberg spin models (see Ref. 24 and references cited therein). It should be pointed out, however, that the underlying mechanism for formation of intermediate magnetization plateau does not necessarily need to be of a purely 'quantum' origin, but it may sometimes have 'classical' character. The 'classical' plateau is a simple adiabatic continuation of a commensurate classical spin state realized in the Ising limit that is of course being subject to a quantum reduction of the local magnetization caused by quantum fluctuations, while the purely 'quantum' plateau relates to a massive quantum spin state with an energy gap that does not have any classical counterpart [24][25][26][27][28] .
Naturally, the most comprehensively understood are nowadays rational magnetization plateaus of the simplest molecular materials, which consist of well isolated magnetic molecules involving just a few spin centers coupled through antiferromagnetic exchange interactions. High-field measurements performed at sufficiently low temperatures have for instance evidenced presence of intermediate magnetization plateau(s) for the dinuclear nickel complex {Ni 2 } as an experimental realization of the spin-1 Heisenberg dimer [29][30][31] , the dinuclear nickel-copper complex {NiCu} as an experimental realization of the mixed spin-(1,1/2) Heisenberg dimer 32 , the trinuclear copper {Cu 3 } and nickel {Ni 3 } complexes as experimental realizations of the spin-1/2 and spin-1 Heisenberg triangles [33][34][35] , the oligonuclear compound {Mo 12 Ni 4 } as an experimental realization of the spin-1 Heisenberg tetrahedron [36][37][38][39] , the pentanuclear copper complex {Cu 5 } as an experimental realization of the spin-1/2 Heisenberg hourglass cluster 40,41 , the hexanuclear vanadium A full energy spectrum can be obtained from Eq. (3) after considering all available combinations of the quantum spin numbers S 12 = 0, 1, 2 and S 34 = 0, 1, 2 together with the composition rules for the total spin angular momentum S T = |S 12 − S 34 |, |S 12 − S 34 |+ 1, · · · , S 12 + S 34 and its z-component S z T = −S T , −S T + 1, ..., S T according to the Kambe coupling scheme 61,62 . For completeness, all energy eigenvalues assigned to allowed combinations of the quantum spin numbers S T , S 12 , S 34 and S z T are listed in Tab. I. At this stage, it is quite straightforward to obtain from the full energy spectrum quoted in Tab. I an exact result for the partition function of the spin-1 Heisenberg diamond cluster Z = Tr e −βĤ = 81 i=1 e −βEi with β = 1/(k B T ) (k B is Boltzmann's constant, T is the absolute temperature), which is explicitly given by the following lengthy expression: The magnetization per one spin can be subsequently obtained from the associated Gibbs free energy G = −k B T ln Z by making use of the following formula: whereas the expression Z h ≡ ∂Z/∂(βh) is defined as follows:

Energy
ST S12 S34 S z The magnetic molar entropy of the spin-1 Heisenberg diamond cluster can be similarly obtained from the exact result (4) for the partition function according to the formula: where N A and R stand for Avogadro's and universal gas constant, respectively. It should be mentioned that the final formula for a temperature derivative of the partition function is too lengthy in order to write it down here explicitly.

III. THEORETICAL RESULTS
In this part, we will proceed to a comprehensive analysis of the most interesting results for the ground state, magnetization curves and magnetocaloric properties of the spin-1 Heisenberg diamond cluster. The ground-state phase diagram of the spin-1 Heisenberg diamond cluster is displayed in Fig. 2 in the J 2 /|J 1 | − h/|J 1 | plane for two particular cases, which differ from one another in antiferromagnetic (J 1 > 0) vs. ferromagnetic (J 1 < 0) character of the coupling constant along a shorter diagonal of the diamond spin cluster. One finds by inspection eight different ground states unambiguously given by the eigenvectors |S T = S z T , S 12 , S 34 , which are classified through a set of the quantum spin numbers determining the total spin and its z-component being equal S T = S z T within all ground states, as well as, two composite spins S 12 and S 34 formed by spin-1 entities from opposite corners of the diamond spin cluster. Within the framework of the Kambe's coupling scheme 61,62 , it is convenient to express first the relevant ground states as a linear combination over a tensor product of eigenvectors of two considered spin pairs |S T , S 12 , S 34 = i a i |S 12 , S z 12 ⊗ |S 34 , S z 34 before writing them more explicitly as a linear combination over spin states of the usual Ising basis |S T , S 12 , S 34 = i b i |S z 1 , S z 2 , S z 3 , S z 4 . The exact formulas for the eigenvectors |S 12 , S z 12 and |S 34 , S z 34 of the spin-1 Heisenberg dimers are not quoted here explicitly, because they can be found in our preceding work 30 .

FIG. 2:
The ground-state phase diagram of the spin-1 Heisenberg diamond cluster in the J2/|J1| − h/|J1| plane for two particular cases with: (a) the antiferromagnetic interaction J1 > 0; (b) the ferromagnetic interaction J1 < 0. The eigenvectors |ST = S z T , S12, S34 are specified according to the quantum spin numbers determining the total spin and its z-component ST = S z T , as well as, two composite spins S12 and S34 formed by spin-1 entities from opposite corners of a diamond spin cluster.
Typical isothermal magnetization curves of the spin-1 Heisenberg diamond cluster are plotted in Fig. 4 for the antiferromagnetic interaction J 1 > 0 and a few selected values of the interaction ratio J 2 /J 1 in order to provide an independent check of all possible magnetization profiles and field-driven phase transitions. It should be emphasized that the magnetization curves calculated at the lowest temperature k B T /J 1 = 0.01 are strongly reminiscent of zero-temperature magnetization curves with discontinuous jumps of the magnetization, which take place at the aforementioned critical magnetic fields in agreement with the ground-state phase diagram shown in Fig. 2(a). Note furthermore that the rising temperature causes just a gradual melting of the relevant magnetization curves. The first particular case, which is shown in Fig. 4(a) for the interaction ratio J 2 /J 1 = −1.25 with the dominant ferromagnetic interaction along the sides of a diamond spin cluster, illustrates a smooth magnetization curve without any intermediate plateau. The second particular case with the weaker ferromagnetic interaction J 2 /J 1 = −0.75 shows an abrupt rise of the magnetization in vicinity of zero magnetic field, which is subsequently followed by the intermediate 3/4-plateau ending up just at the saturation field [see Fig. 4(b)]. It is noteworthy that the intermediate 3/4-plateau as well as a steep rise of the magnetization close to the saturation field is gradually smeared out upon increasing of temperature. The magnetization curves with a steep rise of the magnetization followed by the intermediate 1/2-and 3/4-plateaus is depicted in Fig. 4(c) for the specific value of the interaction ratio J 2 /J 1 = 0.25. The magnetization curves of the spin-1 Heisenberg diamond cluster displayed in Fig. 4(d) for the higher value of the interaction ratio J 2 /J 1 = 0.5 indicate presence of the intermediate 1/4-, 1/2-and 3/4-plateaus, which follow-up the initial abrupt rise of the magnetization observable near zero magnetic field. It should be stressed, moreover, that the most narrow 1/4-plateau becomes already indiscernible at relatively low temperature k B T /J 1 ≈ 0.1 due to its tiny energy gap. The magnetization curves of the spin-1 Heisenberg diamond cluster for the last two values of the interaction ratio J 2 /J 1 = 0.75 and 1.25, which are plotted in Fig. 4 for formation of the magnetization plateaus is preserved just for zero plateau, while the microscopic nature of all other magnetization plateaus is completely different as evidenced by the ground-state phase diagram shown in Fig. 2(a). The isothermal entropy change of the spin-1 Heisenberg diamond cluster invoked by the change of magnetic field ∆h = h i − h f is plotted in Fig. 5 as a function of temperature for four different values of the interaction ratio J 2 /J 1 , whereas h i = 0 stands for the initial magnetic field and h f = 0 is the final magnetic field during the isothermal demagnetization. Within the proposed notation the conventional MCE occurs for positive values of the isothermal entropy change −∆S m = S m (h f = 0) − S m (h i = 0) > 0, while the inverse MCE is manifested through its negative values −∆S m < 0. It should be pointed out, moreover, that the zero-temperature asymptotic value of the molar entropy change −∆S m = R ln Ω 0 can be simply related to a degeneracy Ω 0 of the zero-field ground state whenever the magnetic-field change does not coincide with any critical magnetic field ∆h = h c,n . In the reverse case ∆h = h c,n the molar entropy change converges in the zero-temperature limit to the smaller asymptotic value −∆S m = R(ln Ω 0 −ln 2) due to a two-fold degeneracy of two coexistent ground states at a critical magnetic field h c,n . The temperature dependences of the molar entropy change of the spin-1 Heisenberg diamond cluster is shown in Fig. 5(a) and (b) for a few different values of the magnetic-field change and the fixed value of the interaction ratio J 2 /J 1 = 0.25, which is consistent with presence of the valence-bond-crystal ground state |2, 0, 2 in the zero-field limit. It is worthwhile to remark that the singlet state of the near-distant spin pair emergent within the ground state |2, 0, 2 effectively decouples all spin correlations of two further-distant spins. Owing to this fact, the further-distant spins behave at zero magnetic field as free paramagnetic entities and the respective degeneracy of the zero-field ground-state is Ω 0 = 9. It can be seen from Fig. 5(a) and (b) that the molar entropy change actually tends to the specific value −∆S m = R ln 9 ≈ 18.3 J.K −1 .mol −1 for all magnetic-field changes except those being equal to the critical magnetic fields ∆h/J 1 = 1.5 and 2.5. In this latter case, the molar entropy change acquires in zero-temperature limit smaller asymptotic value −∆S m = R(ln 9 − ln 2) ≈ 12.5 J.K −1 .mol −1 in accordance with the previous argumentation [see the curves for ∆h/J 1 = 1.5 and 2.5 in Fig. 5(a) and (b)]. Although the isothermal entropy change generally diminishes upon increasing of temperature, it is quite evident from Fig. 5(a) and (b) that the reverse may be true in a range of moderate temperatures whenever the magnetic-field change is chosen sufficiently close to one of the critical magnetic fields.
The isothermal entropy changes of the spin-1 Heisenberg diamond cluster are depicted in Fig. 5(c) and (d) for relatively small and moderate changes of the magnetic field by assuming the interaction ratio J 2 /J 1 = 0.5 supporting another zero-field ground state |1, 1, 2 . It should be pointed out that the conventional MCE with −∆S m > 0 occurs for any magnetic-field change quite similarly as in the previous case. In spite of this qualitative similarity, the molar entropy change converges in zero-temperature limit to completely different asymptotic values on account of a triply degenerate (Ω 0 = 3) ground state |1, 1, 2 realized in zero-field limit. As a matter of fact, it is obvious from Fig.  5(c) and (d) that the molar entropy change reaches either the asymptotic value −∆S m = R ln 3 ≈ 9.1 J.K −1 .mol −1 or −∆S m = R(ln 3 − ln 2) ≈ 3.4 J.K −1 .mol −1 depending on whether or not the magnetic-field change coincides with the critical magnetic field, whereas the latter smaller value of −∆S m applies only if the magnetic-field change corresponds to one of three critical magnetic fields ∆h/J 1 = 0.5, 2.0 or 3.0. Under these specific conditions, the isothermal entropy change starts from this lower asymptotic value, then it increases with rising temperature to its local maximum before it finally tends to zero upon further increase of temperature. The most interesting temperature dependences of the isothermal entropy change can be found when the magnetic-field change is selected slightly below or above the critical magnetic fields [e.g. ∆h/J 1 = 0.4 or 0.6 in Fig. 5(c)], because the molar entropy change then starts from its higher zero-temperature asymptotic limit, then it shows a rapid decline to a local minimum subsequently followed by a continuous rise to a local maximum upon increasing of temperature before it finally decays to zero in the high-temperature region. If the magnetic-field change is sufficiently far from the critical magnetic fields one either finds a monotonic temperature decline of the isothermal entropy change upon increasing of temperature [see curve for ∆h/J 1 = 0.2 in Fig. 5(c)] or one recovers a nonmonotonic temperature dependence with a single round maximum emerging at some moderate temperature [see the curves for ∆h/J 1 = 1.0 and 1.5 in Fig. 5(c) or ∆h/J 1 = 4.0 in Fig.  5(d)].
The completely different magnetocaloric features of the spin-1 Heisenberg diamond cluster can be traced back from temperature variations of the isothermal entropy change, which are shown in Fig. 5(e)-(h) for two selected values of the interaction ratio J 2 /J 1 = 0.75 and 1.25. The common feature of these two particular cases is that the zerofield ground state is the non-degenerate singlet state |0, 2, 2 , which is responsible for existence of zero magnetization plateau in the respective low-temperature magnetization curves [see Fig. 4(e) and (f)]. in the consequence of that, the molar entropy change asymptotically tends in zero-temperature limit either to zero or to the specific value −∆S m = −R ln 2 ≈ −5.8 J.K −1 .mol −1 depending on whether the magnetic-field change differs or equals to the critical magnetic fields, respectively. It can be seen from Fig. 5(g) and (h) that the spin-1 Heisenberg diamond cluster with the interaction ratio J 2 /J 1 = 1.25 exhibits the inverse MCE with −∆S m < 0 for most of the magneticfield changes in a relatively wide range of temperatures. The exception to this rule are just the isothermal entropy changes, which are induced by sufficiently large change of the magnetic field exceeding the saturation field [see the curve ∆h/J 1 = 6.0 in Fig. 5(h)]. Contrary to this, the spin-1 Heisenberg diamond cluster with the interaction ratio J 2 /J 1 = 0.75 shows an outstanding crossover between the inverse and conventional MCE. While the inverse MCE with −∆S m < 0 prevails at lower temperatures and magnetic-field changes, the conventional MCE with −∆S m > 0 dominates at higher temperatures and magnetic-field changes [see Fig. 5(e)-(f)].
Last but not least, let us examine the adiabatic change of temperature as another basic magnetocaloric property of the spin-1 Heisenberg diamond cluster. For this purpose, density plots of the molar entropy are displayed in Fig. 6(a)-(d) in the magnetic field versus temperature plane for four selected values of the interaction ratio J 2 /J 1 , which have been previously used in order to demonstrate a diversity of the magnetization profiles. It should be emphasized that black contour lines shown in Fig. 6(a)-(d) correspond to isentropy lines, from which one can easily deduce adiabatic changes of temperature achieved upon lowering of the external magnetic field. It is quite evident from Fig. 6(a)-(d) that the most notable changes of temperature occur in vicinity of all critical magnetic fields, whereas a sudden drop (rise) in temperature occurs during the adiabatic demagnetization slightly above (below) critical magnetic field. Hence, it follows that the abrupt magnetization jump manifest itself during the adiabatic demagnetization as a critical fan spread over a respective critical magnetic field. Two critical fans can be accordingly observed in Fig. 6(a), three critical fans are visible in Fig. 6(b) and four critical fans appear in Fig. 6(c) and (d). It can be seen from Fig. 6(a)-(d) that most of isentropes converge to some nonzero temperature as the external magnetic field gradually vanishes. More specifically, all isentropes of the spin-1 Heisenberg diamond cluster with the interaction ratio J 2 /J 1 = 0.75 or 1.25 acquire nonzero temperature as the external magnetic field goes to zero [see Fig. 6(c)-(d)]. This observation can be related with presence of zero-field singlet ground state |0, 2, 2 , which is responsible for zero magnetization plateau. On the other hand, the spin-1 Heisenberg diamond cluster with the interaction ratio J 2 /J 1 = 0.25 or 0.5 may exhibit during the adiabatic demagnetization a sizable drop of temperature down to ultra-low temperatures due to absence of zero magnetization plateau 27 . To achieve this intriguing magnetocaloric feature, the molar entropy should be fixed to a smaller value than the entropy corresponding to a degeneracy of the respective zero-field ground state, i.e. S m < R ln 9 ≈ 18.3 J.K −1 .mol −1 for the zero-field ground state |2, 0, 2 emergent for J 2 /J 1 = 0.25 or S m < R ln 3 ≈ 9.1 J.K −1 .mol −1 for the zero-field ground state |1, 1, 2 emergent for J 2 /J 1 = 0.5, respectively. These findings could be of particular importance when the molecular compound {Ni 4 } would be used for refrigeration at ultra-low temperatures.

IV. THEORETICAL MODELING OF TETRANUCLEAR NICKEL COMPLEX {NI4}
In this part, we will interpret available experimental data for the magnetization and susceptibility of the tetranuclear nickel complex {Ni 4 } 59,60 , which can be theoretically modeled by the spin-1 Heisenberg diamond cluster given by the Hamiltonian (1). It actually follows from Fig. 7 that the magnetic core of the tetranuclear coordination compound {Ni 4 } constitutes a 'butterfly tetrameric' unit composed of four exchange-coupled Ni 2+ ions, which is formally identical with the magnetic structure of the spin-1 Heisenberg diamond cluster schematically illustrated in Fig. 1. High-field magnetization data of the nickel complex {Ni 4 } recorded in pulsed magnetic fields up to approximately 68 T at the sufficiently low temperature 1.3 K are presented in Fig. 8(a) together with the respective theoretical fit based on the spin-1 Heisenberg diamond cluster. It is evident from Fig. 8(a) that the measured magnetization data bear evidence of two wide intermediate plateaus roughly at 1.11 and 1.65 µ B per Ni 2+ ion, which are consistent with 1/2-and 3/4-plateaus when the total magnetization is scaled with respect to its saturation value and the appropriate value of the gyromagnetic factor g = 2.2 of Ni 2+ ions is considered. The abrupt magnetization jumps detected at the critical magnetic fields B c,1 ≈ 40.5 T and B c,2 ≈ 68.5 T clearly delimit a width of these intermediate magnetization plateaus. The distinct magnetization profile with a sole presence of the intermediate 1/2-and 3/4-plateaus enables a simple estimation of the relevant coupling constants. First, it has been argued by the ground-state analysis that the intermediate 1/2-and 3/4-plateaus emerge in a zero-temperature magnetization curve as the only magnetization plateaus just if the interaction ratio falls into the range J 2 /J 1 ∈ (−1/2, 1/3). Second, one may take advantage of the fact that the width of 3/4-plateau ∆B 3/4 = B c,2 − B c,1 is independent of the interaction ratio J 2 /J 1 in contrast with the width of 1/2-plateau ∆B 1/2 = B c,1 . Hence, the relative width of two magnetization plateaus δ r = ∆B 3/4 : ∆B 1/2 = 28 T : 40.5 T ≈ 0.69 observed in experiment can be straightforwardly exploited for an unambiguous determination of a relative strength of the coupling constants: Once determined, the absolute values of the coupling constants J 1 and J 2 can be easily calculated for instance from the first critical field B c,1 = 40.5 T when taking into account knowledge of the interaction ratio (16): In accordance with this argumentation, the spin-1 Heisenberg diamond cluster with the coupling constants J 1 /k B = 41.4 K, J 2 /k B = 9.2 K and the gyromagnetic factor g = 2.2 indeed satisfactorily reproduces the high-field magnetization data of the butterfly-tetramer compound {Ni 4 } as convincingly evidenced by the respective theoretical fit, which is shown in Fig. 8(a)  the coupling constants J 1 and J 2 does not significantly improve a theoretical fit of these experimental data. It has been found in Ref. 60 that the significant improvement of the theoretical fit can be achieved only when considering a weak ferromagnetic exchange coupling J 3 /k B = −0.66 K between the further-distant spins S 3 and S 4 , which allows a steeper uprise of the magnetization in a low-field range. A consideration of the exchange coupling between the further-distant spins S 3 and S 4 is however beyond the scope of the present article. Next, we will employ the coupling constants (17) ascribed to the coordination compound {Ni 4 } for a theoretical interpretation of a temperature dependence of the susceptibility times temperature (χT ) product. To this end, the available experimental data for the χT product of the tetranuclear nickel complex {Ni 4 } are confronted in Fig. 9(a) with the respective theoretical prediction based on the spin-1 Heisenberg diamond cluster by assuming the model parameters (17) previously extracted from the fitting procedure of the high-field magnetization data. Although a theoretical curve qualitatively captures all essential features for temperature variations of the χT product including a local minimum experimentally observed around 14 K, the good quantitative accordance between the experimental and theoretical data is found just in a relatively narrow range of temperatures T ∈ (25, 80) K while outside of this temperature range the theoretical data generally underestimate the experimental ones. We have therefore adapted the optimization technique based on a hill-climbing procedure in order to find the best fitting set for the χT data. This procedure provided for the tetranuclear nickel compound {Ni 4 } described by the spin-1 Heisenberg diamond cluster another fitting set of the model parameters J 1 /k B = 54.3 K, J 2 /k B = 13.9 K and g = 2.31, which not only qualitatively but also quantitatively captures the experimental data in a full range of temperatures as exemplified in Fig. 9(b). While the rise of the gyromagnetic factor by a few percent (cca. 5 %) could be attributed to a substantial temperature difference within the magnetization and susceptibility measurements, the relatively large discrepancy in assessment of both coupling constants clearly indicates an oversimplified nature of the spin-1 Heisenberg diamondcluster model given by the Hamiltonian (1). It is quite reasonable to conjecture from nearly isotropic character of the magnetization curves measured along two orthogonal crystallographic axes 60 that the axial and/or rhombic zero-fieldsplitting parameters acting on Ni 2+ ions are presumably negligible and hence, the discrepancies in the magnetization and susceptibility data could be resolved when taking into consideration the biquadratic interaction and/or the pair exchange interaction between the further-distant spins S 3 and S 4 . Last but not least, the best fitting set (17) extracted for the spin-1 Heisenberg diamond-cluster model from the highfield magnetization curve of the tetranuclear nickel complex {Ni 4 } will be used for making a theoretical prediction of its basic magnetocaloric properties not reported experimentally hitherto. More specifically, we will investigate in detail temperature variations of the isothermal magnetic entropy change as well as field-induced changes of temperature during the adiabatic demagnetization. It is evident from Fig. 10(a) that the isothermal mass entropy change of the nickel compound {Ni 4 } gradually diminishes from its maximum value −∆S M ≈ 10.6 J.K −1 .kg −1 upon increasing of temperature whenever the magnetic-field change is sufficiently small ∆B < 15 T. On assumption that the magneticfield change is set ∆B = 7 T the molecular compound {Ni 4 } provides an efficient refrigerant below 2.3 K with the enhanced MCE −∆S M > 10 J.K −1 .kg −1 . It should be stressed that a subtle rise of the isothermal entropy change −∆S M can be detected for the higher magnetic-field changes [e.g. see the curve for ∆B = 20 T in Fig. 10(a)], which is however of very limited applicability for the cooling technologies.
On the other hand, the density plot of the magnetic mass entropy in the magnetic field versus temperature plane is displayed in Fig. 10(b) with the aim to elucidate a parameter space suitable for cooling purposes. The relevant contour lines with constant magnetic entropy bring insight into magnetic-field driven changes of temperature during the process of adiabatic demagnetization. A considerable drop and rise of temperature apparently occurs in the isentropes near the critical magnetic fields, which correspond to the magnetic-field-driven magnetization jumps. If the magnetic entropy is set sufficiently close to the particular value S M ≈ 10.6 J.K −1 .kg −1 , moreover, the adiabatic demagnetization should cause a sizable drop of temperature of the molecular complex {Ni 4 } with up to −∆T ≈ 10 K achieved due to the magnetic-field change ∆B = 7 T.

V. CONCLUSIONS
In the present article we have investigated in detail magnetic and magnetocaloric properties of the spin-1 Heisenberg diamond cluster with two different coupling constants through an exact diagonalization based on the Kambe's method, which takes advantage of a local conservation of composite spins formed by spin-1 entities located in opposite corners of a diamond spin cluster. It has been verified that the spin-1 Heisenberg diamond cluster exhibits several intriguing quantum ground states, which come to light in low-temperature magnetization curves as intermediate 1/4-, 1/2-or 3/4-plateau depending on a specific choice of the interaction ratio and the magnetic field. We have demonstrated a substantial diversity of the magnetization curves, which may exhibit different magnetization profiles with either a single 3/4-plateau, a sequence of two consecutive 1/2-and 3/4-plateaus, three consecutive 1/4-, 1/2-and 3/4plateaus, four consecutive 0-, 1/4-, 1/2-and 3/4-plateaus or is completely free of any plateau. In addition, the spin-1 Heisenberg diamond cluster may also exhibit the enhanced MCE, which may be relevant for a low-temperature refrigeration achieved through the adiabatic demagnetization on assumption that a relative strength of the coupling constants J 2 /J 1 ∈ (−1, 2/3) is consistent with absence of the zero magnetization plateau.
It has been evidenced that the spin-1 Heisenberg diamond cluster with the antiferromagnetic coupling constants J 1 /k B = 41.4 K, J 2 /k B = 9.2 K and the gyromagnetic factor g = 2.2 satisfactorily captures low-temperature magnetization curves recorded for the tetranuclear nickel complex {Ni 4 } including a size and position of the intermediate 1/2and 3/4-plateaus 60 . Moreover, it turns out that the fractional magnetization plateaus observed experimentally bear evidence of two remarkable valence-bond-crystal ground states with either a single or double valence bond between the near-distant spin-1 Ni 2+ ions. It has been also suggested that the molecular compound {Ni 4 } may provide a prospective cryogenic coolant with the maximal isothermal entropy change −∆S M = 10.6 J.K −1 .kg −1 suitable for a low-temperature refrigeration below 2.3 K.