**8.2. ZrZn**<sup>2</sup>

Experiments for ZrZn2 [13] gives the following values: *Tf* <sup>0</sup> = 28.5 K, *TFS*(0) = 0.29 K, *P*<sup>0</sup> ∼ *Pc* = 21 kbar. The curve *TF*(*P*) ∼ *Tf*(*P*) is almost a straight line, which directly indicates that *n* = 1 is adequate in this case for the description of the *P*-dependence. The slope for *TFS*(*P*) at *P* = 0 is estimated from the condition that its magnitude should not exceed *Tf* 0/*Pc* ≈ 0.014 as we have assumed that is straight one, so as a result we have −0.014 < *ρ* ≤ 0. This ignores the presence of a maximum. The available experimental data for ZrZn2 do not give clear indication whether a maximum at (*Tm*, *Pm*) exists. If such a maximum were at *P* = 0 we would have *ρ*<sup>0</sup> = 0, whereas a maximum with *Tm* ∼ *TFS*(0) and *Pm* � *P*<sup>0</sup> provides us with an estimated range 0 ≤ *ρ*<sup>0</sup> < 0.005. The choice *ρ*<sup>0</sup> = 0 gives *γ*˜ ≈ 0.02 and *γ*˜1 ≈ 0.01, but similar values hold for any |*ρ*0| ≤ 0.003. The multicritical points A and C cannot be distinguished experimentally. Since the experimental accuracy [13] is less than ∼ 25 mK in the high-*P* domain (*P* ∼ 20 − 21 kbar), we suppose that *TC* ∼ 10 mK, which corresponds to *κ* ∼ 10. We employed these parameters to calculate the *T* − *P* diagram using *ρ*<sup>0</sup> = 0 and 0.003. The differences obtained in these two cases are negligible, with both phase diagrams being in excellent agreement with experiment.

Phase diagram of ZrZn2 calculated directly from the free energy (7) for *n* = 1, the above mentioned values of *Ts*, *P*0, *Tf* 0, *κ*, and values of *γ*˜ ≈ 0.2 and *γ*˜1 ≈ 0.1 which ensure *ρ*<sup>0</sup> ≈ 0 is shown in Fig. 2. Note, that the experimental phase diagram [13] of ZrZn2 looks almost exactly as the diagram in Fig. 2, which has been calculated directly from the model (7) without any approximations and simplifying assumptions. The phase diagram in Fig. 2 has the following coordinates of characteristic points: *PA* ∼ *Pc* = 21.42 kbar, *PB* = 20.79 kbar, *PC* = 20.98 kbar, *TA* = *TF*(*Pc*) = *TFS*(*Pc*) = 0 K, *TB* = 0.0495 K, *TC* = 0.0259 K, and *TFS*(0) = 0.285 K.

The low-*T* region is seen in more detail in Fig. 3, where the A, B, C points are shown and the order of the FM-FS phase transition changes from second to first order around the critical end-point C. The *TFS*(*P*) curve, shown by the dotted line in Fig. 3, has a maximum *Tm* = 0.290 K at *P* = 0.18 kbar, which is slightly above *TFS*(0) = 0.285 K. The straight solid line BC in Fig. 3 shows the first order FM-FS phase transition which occurs for *PB* < *P* < *PC*. The solid AC line shows the first order N-FS phase transition and the dashed line stands for the N-FM phase transition of second order.

Although the expanded temperature scale in Fig. 3, the difference [*Tm* − *TFS*(0)] = 5 mK is hard to see. To locate the point *max* exactly at *P* = 0 one must work with values of *γ*˜ and *γ*˜1 of accuracy up to 10−4. So, the location of the *max* for parameters corresponding to ZrZn2 is very sensitive to small variations of *γ*˜ and *γ*˜1 around the values 0.2 and 0.1, respectively. Our initial idea was to present a diagram with *Tm* = *TFS*(0) = 0.29 K and *ρ*<sup>0</sup> = 0, namely, *max* exactly located at *P* = 0, but the final phase diagram slightly departs from this picture because of the mentioned sensitivity of the result on the values of the interaction parameters *γ* and *γ*1. The theoretical phase diagram of ZrZn2 can be deduced in the same way for *ρ*<sup>0</sup> = 0.003 and this yields *Tm* = 0.301 K at *Pm* = 6.915 kbar for initial values of *γ*˜ and *γ*˜1 which differs from *<sup>γ</sup>*˜ <sup>=</sup> <sup>2</sup>*γ*˜1 <sup>=</sup> 0.2 only by numbers of order 10−<sup>3</sup> <sup>−</sup> <sup>10</sup>−<sup>4</sup> [18]. This result confirms the mentioned sensitivity of the location of the maximum *Tm* towards slight variations of the material parameters. Experimental investigations of this low-temperature/low-pressure region with higher accuracy may help in locating this maximum with better precision.

Fig. 4 shows the high-pressure part of the same phase diagram in more details. In this figure the first order phase transitions (solid lines BC and AC) are clearly seen. In fact the line AC is quite flat but not straight as the line BC. The quite interesting topology of the phase diagram of ZrZn2 in the high-pressure domain (*PB* < *P* < *PA*) is not seen in the experimental phase diagram [13] because of the restricted accuracy of the experiment in this range of temperatures and pressures.

These results account well for the main features of the experimental behavior [13], including the claimed change in the order of the FM-FS phase transition at relatively high *P*. Within the present model the N-FM transition is of second order up to *PC* ∼ *Pc*. Moreover, if the experiments are reliable in their indication of a first order N-FM transition at much lower *P* values, the theory can accommodate this by a change of sign of *b <sup>f</sup>* , leading to a new tricritical point located at a distinct *Ptr* < *PC* on the N-FM transition line. Since *TC* > 0 a direct N-FS phase transition of first order is predicted in accord with conclusions from de Haas–van Alphen experiments [44] and some theoretical studies [40]. Such a transition may not occur in other cases where *TC* = 0. In SFT (*n* = 2) the diagram topology remains the same but points B and C are slightly shifted to higher *P* (typically by about 0.01 − −0.001 kbar).
