# Pressure-Dependent Crystal Radii

## Abstract

**:**

^{2−}, Cl

^{−}, and Br

^{−}are also reported. It is shown that all examined cation radii vary linearly with pressure. Cation radii obey strict correlations between ionic compressibilities and reference 0 GPa radii, thus reducing previous empirical rules of the influence of valence, ion size, and coordination to a simple formula. Both cation and anion radii are functions of nuclear charge number and a screening function which for anions varies with pressure, and for cations is pressure-independent. The pressure derivative of cation radii and of the anion radii at high pressure depends on electronegativity with power −1.76.

## 1. Introduction

_{2}SiO

_{4}, three-quarters of the Earth’s mantle is oxygen as constituent chemical species, and within this approximation the O

^{2−}anion dominates the increase of the electronic inner energy induced in Earth material by compression. Between 0 and 136 GPa (the pressure of the core mantle boundary), the contraction of the crystal radius of O

^{2−}in four- to six-fold coordination by Mg and Si is from 1.28 or 1.26·10

^{−12}m, respectively to ~1.2210

^{−12}m (see Section 3), hence: 4/3π Δr

^{3}· 1.36·10

^{11}N/m

^{2}= 2.50·10

^{−19}J/at = 1.56 eV/at (of O

^{2−}). Differences between energy levels of bonding and anti-bonding electron states of ions in the crystal field of oxide anions are of comparable magnitude. Hence the pressure-induced increase in electronic energy is within a range that allows for formation of bond states different from those that we know from ambient pressure. This type of change defines proper high-pressure minerals [4,5].

^{2−}and found that these cation radii are not significantly dependent on individual structures. Hence, the concept of crystal radii appears to be valid over pressures of 10–100 GPa, at least for those ions. Further, within uncertainties, all three cations exhibited linear dependence on pressure. The O anion was found to exhibit initially a marked nonlinear compression converging towards weaker linear compression, consistent with the observations by Prewitt and Downs [6] that anions compress more strongly than cations. However, the potential coordination dependence of the O

^{2−}anion radius was not examined in [13]. Tschauner and Ma [4] reported radii of K, Mg, Ca, Al, Si for different cation coordinations and explicitly considered the effect of coordination on the oxide anion radius. Tschauner and Ma [4] found that K, Mg, Ca, Al, Si exhibit linear contraction over the examined pressure intervals within uncertainties but with the exception of fourfold-coordinated Si. The heavier cations K and Ca are more compressible than the lighter ions Mg, Al, and Si. In addition, it was observed that the higher the cation valence, the lesser the pressure effect, again consistent with [6], whereas the available data did not support the general trend for the pressure dependence of crystal radii with bond coordination that was proposed in [6].

^{−}and Br

^{−}are assessed for six- and eightfold coordination in order to obtain comparative sets of cation radii from oxides and halogenides for alkaline elements (Table 1). It is found that cation crystal radii are rather independent of the difference in anion electron affinity and on individual structures within small differences. Furthermore, within uncertainties, all crystal radii contract linearly with pressure. The range of pressure dependencies is similar to that reported by Gibbs et al. [10] for Ca, Si, and La. Based on the present augmented set of pressure dependencies, general correlations of ionic compressibility and nuclear charge number of the radii and their pressure derivatives of the radii and electronegativity are established, and these correlations encompass the rules previously stated by Prewitt and Downs [6].

## 2. Methods

- (1)
- For given valence and bond coordination the assessed radii should be independent of the structures, therefore;
- (2)
- Radii should be assessed through different crystal structures, if available. Wherever possible cation radii are assessed through structures with different anions
- (3)
- Radii derived from structures with ions on general positions are given preference.

_{2}because the effect of the proton appears to be within the scatter of the data for this difficult cation see Section 3.2). Ambient-pressure radii were not fixed in the fits of radii compression, thus allowing for a comparison of the interpolated 1 bar radii with literature values that were taken from [16]. Unless stated differently, all data are for 300 K. In order to minimize the number of superscripts in text and figures, valences are specified only for multivalent ions and in case of ambiguity. For convenience, radii are given in Å, unless stated otherwise. The term coordination is used here in the sense of bond coordination. To avoid lengthy wording but also conflicts with the reference style, coordination is given as Roman numbers in square brackets. All radii presented in this study are crystal radii. Henceforth, the term radii shall be used for crystal radii and ionic radii shall be mentioned explicitly as such.

## 3. Discussion

#### 3.1. Anions

^{2−}, Cl

^{−}, and Br

^{−,}exhibit compressibilities that are nonlinear at low pressure but approach linear compression asymptotically with increasing pressures. This behaviour can be cast into the functional form r = const·P

^{−m}(Table 1). In [13], an empirical equation was presented for the pressure effect on the O

^{2−}anion in sixfold coordination. The equation was obtained from interatomic distances of Mg-O and Si-O in binary and ternary compounds with emphasis on high-pressure minerals. As a starting point for pressure-dependent crystal radii of O

^{2−}[VI], the Baader radii of Si and O in compressed silica by Du and Tse [18] were used. The Baader radii at ambient pressure were corrected to match the crystal radii of Mg and Si in octahedral bond configuration. Then, the pressure dependence of O

^{2−}[VI]—as computed in [18]—was corrected to yield a consistent set of structure-independent crystal radii for Mg and Si. This resulted in a relation r(O

^{2−}) = 1.269·P

^{−0.0176}, where P in GPa and r is the anion crystal radius in Å (Table 1). In the present paper, the set of oxide phases is much expanded (Table 2). The augmented set of crystal data requires a better assessment of the effect of O-anion coordination by cations (Table 1) than previously. Initially, only the constant term was varied to match the 1 bar radius of O

^{2−}in each coordination. However, it was found that cation radii for some structures (eskolaite, ABO

_{3}-perovskites) exhibited slightly nonlinear or non-monotonous pressure dependencies. It was found that these nonlinearities vanish if the power of the O-anion compression s modified for coordinations less than six (Table 1) and that the resulting cation radii re structure-independent within uncertainties as defined by the variance of cation radii obtained from different phases.

^{2−}[VI], then corrected as to obtain equal radii of Na[VI,VIII] and K[VI,VIII] from both chlorides and bromides and to match K[VI] from K-O bond distances. In a second step, the obtained Cl

^{−}and Br

^{−}radii were tested with CsCl, and CsBr [22]: correct anion radii should give equal radii of Cs[VIII] for both salts. This was found to be the case.

**Table 1.**Power and constant prefactors for r(anion) = r

_{0}·P

^{m}(P in GPa). The physical meaning of the functional form of r(anion, P) is discussed in Section 4.3.

Ion | M | r_{0} [Å/GPa^{m}] |
---|---|---|

O^{2−} [II,III] | −0.0040(2) | 1.210(5) |

O^{2−} [IV] | −0.0045(5) | 1.238(3) |

O^{2−} [VI] | −0.0176(2) | 1.269(2) |

Cl^{−} [VI] | - | 1.62 − 0.02·P * |

Cl^{−} [VIII] | −0.07(1) | 1.66(1) |

Br^{−} [VIII] | −0.078(3) | 1.85(2) |

#### 3.2. Alkaline and Alkaline Earths Elements

^{2−}coordination in bridgmanite. With this correction, Mg[X] exhibits a contraction that intersect the crystal radius of Mg in postperovskite-type MgSiO

_{3}at 116 and 120 GPa [23,24]. All alkaline earths compress linearly within uncertainties over the examined pressure range. Radii of Be[IV] were obtained from compression studies of bromellite (BeO, [25]). Compression studies on other Be-silicates and of chrysoberyl [26] exhibit much scatter around the values obtained from bromellite. Hence, the third criterion in Methods was applied here: to give preference to radii obtained from interatomic distances in structures without internal degrees of freedom in case of discrepancies with more complicated structures. Hence, only radii from bromellite were used to define the pressure dependence of Be[IV]. It is noted that Ca[VIII] from the high-pressure CsCl-type CaO phase interpolates to the ambient pressure radius Ca[VII] (based on Shannon 1976 [16]) and that Ca[X] in compressed synthetic davemaoite [27] interpolates to Ca[IX] at ambient pressure (Table 2). This observation has been made already by Tschauner and Ma [4] and an interpretation is presented here in the Results.

#### 3.3. Rare Earths

#### 3.4. Al, Cr, Fe

^{3+}[VI] were obtained from a single compression study of eskolaite [36]. Compression -is linear.

^{2+}and Fe

^{3+}(and higher oxidation states) are affected not only by coordination and valence but also by the valence electron spin state [16]. Moreover, valence electronic states of Fe are often mixed because of charge transfer between different sites. Mixed states along certain bond directions also affect bond distances and coordination. Charge transfer and spin state may change with pressure. Charge transfer blurs the bond coordination (and accounts for the large variety of pure iron oxides phases of different stoichiometry that occur at high pressure). Consequently, the large number of compression studies and structure analyses of compounds of Fe

_{x}O

_{y}yields a large scatter of apparent radii which cannot be corrected for the electronic effects mentioned above without ad hoc assumptions about spin states and charge transfer. Instead, only single crystal compression data for wuestite [37] were used to obtain radii for Fe

^{2+}in the high spin (HS) state to 50 GPa. No robust data of Fe

^{3+}could be obtained because not even hematite keeps a fixed spin state over a sufficiently large range of pressure (because of the Morin transition).

#### 3.5. Si, Ge, Ti

^{4+}in [VI]-coordination was studied in [13]. Tschauner and Ma [4] provide an augmented set of data for Si[VI] and for Si[IV]. Pressure-dependent radii of Ti

^{4+}were obtained from compression data of rutile, perovskite, and geikielite (see Table 2). Compression data of ilmenite in the literature show much variance, probably as result of a minor component of hematite in the examined specimens, and are not used here. Both Si [VI] and Ti

^{4+}[VI] exhibit linear compression (see Figure 4).

_{3}[42].

**Table 2.**Pressure dependencies of crystal radii r, interpolated ambient pressure radii r

_{0}, reference crystal radii r

_{cryst}at 1 bar from reference [15], R

^{2}of the fitted pressure dependencies, and references to the studies whose structure data were used to obtain interatomic distances.

Element, [Coordination] | −dr/dP [Å/GPa] | r_{o} [Å] | r_{cryst} [Å] (Shannon 1976) | R^{2} | Ref. |
---|---|---|---|---|---|

Li[IV] | 0.006(1) | 0.713(1) | 0.73 | 0.90 | [43] |

Na[VI] | 0.0110(2) | 1.198(3) | 1.16 | 0.85 | [21] |

Na[VII] | 0.008(4) | 1.28(3) | 1.26 | 0.32 | [44] |

Na[VIII] | 0.00152(4) | 1.335(3) | 1.32 | 0.99 | [21] |

K[VI] | 0.0030(3) | 1.528(1) | 1.52 | 0.98 | [19,20,45] |

K[VII] * | 0.0030(1) | 1.584(3) | 1.6 | 0.99 | [20] |

K[VIII] | 0.0049(1) | 1.642(2) | 1.65 | 0.99 | [20] |

K[VIII′] ** | 0.0014(1) | 1.53(1) | 1.52[VI] | 0.97 | [20] |

K[IX] | 0.0066(24) | 1.68(1) | 1.69 | 0.63 | [46] |

Cs[VI] * | 0.0026(5) | 1.79(4) | 1.81 | 0.98 | [22] |

Cs[VIII] | 0.0053(2) | 1.88(1) | 1.88 | 0.99 | [22] |

Cs[XI] | 0.0057(2) | 1.994(3) | 1.99 | 0.99 | [47] |

Be[IV] | 0.0010(2) | 0.414(1) | 0.41 | 0.76 | [25] |

Mg[VI] | 0.0014(2) | 0.84(1) | 0.86 | 0.76 | [4,13,48,49] |

Mg[X] | 0.0024(1) | 1.11(3) | - | 0.99 | [4,23,24,50,51,52] |

Ca[VI] | 0.0024(1) | 1.143(4) | 1.14 | 0.99 | [53] |

Ca[VII,VIII *] | 0.0016(1) | 1.19(1) | 1.2[VII] | 0.97 | [53] |

Ca[VIII] | 0.0034(2) | 1.28(1) | 1.26 | 0.98 | [53] |

Ca[IX,X] | 0.0020(2) | 1.319(8) | 1.32[IX] | 0.94 | [26,54,55] |

Sr[VI] | 0.0033(6) | 1.27(1) | 1.32 | 0.77 | [56,57,58,59] |

Ba[VI,VII *] | 0.0025(2) | 1.512(4) | 1.52[VII] | 0.88 | [60,61,62,63,64,65] |

Ba[VIII] | 0.0025(1) | 1.569(4) | 1.56 | 0.98 | [60,61,62,63,64,65] |

Sc[X] | 0.00141(2) | 1.032(1) | - | 0.99 | [29] |

Y[X] | 0.0025(6) | 1.176(3) | 1.215[IX] | 0.71 | [28] |

La[XI,XII] | 0.0030(2) | 1.463(1) | 1.4[X], 1.5[XII] | 0.97 | [30] |

Pr^{3+} [XI,XII] | 0.0024(1) | 1.446(1) | - | 0.98 | [31] |

Gd[X] | 0.0017(3) | 1.332(2) | 1.33[X] | 0.86 | [32] |

Ti^{4+}[VI] | 0.0016(2) | 0.746(8) | 0.745 | 0.95 | [42,54,65] |

Cr^{3+}[VI] | 0.00144(2) | 0.752(1) | 0.755 | 0.99 | [36] |

Fe^{2+}[VI,HS] | 0.00202(3) | 0.907(1) | 0.92 | 0.99 | [37] |

Al[VI] | 0.00117(2) | 0.677(1) | 0.675 | 0.99 | [26,28,29,30,31,32,33,34,35] |

Ge[VI] | 0.00085(7) | 0.665(1) | 0.67 | 0.97 | [42] |

Si[IV] | 0.00069(8) | 0.408(2) | 0.4 | 0.88 | [48,49,66,67,68,69] |

Si[VI] | 0.0009(1) | 0.565(2) | 0.54 | 0.79 | [4,13,23,24,27,42,50,51,52,70,71,72,73,74] |

## 4. Results

#### 4.1. Pressure-Induced Structure Changes and Pressure-Dependent Radii

_{3}in a correlation of the cation ratios rA/rB or of the ratio rB/rO versus the tolerance factor t = (rA + rO)/[√2·(rB + rO)] [3], the minerals enstatite, MgSiO

_{3}, and wollastonite, CaSiO

_{3}, plot far outside the range of perovskite structures [5]. Hence, this discrepancy in combination with the occurrence of high-pressure polymorphs such as perovskite-type Mg and Ca silicates was suggested to serve as an indicator for marked pressure-induced changes in the chemical bonds between the constituent atomic species [5]. If this concept is correct, the pressure-induced changes of cat- and anion radii are expected to shift the high-pressure polymorphs into the field of tolerance of their structures. This is the case, indeed, and it is illustrated here for perovskite-type oxides as a particularly large and well examined class of materials [3].

_{3}and CaSiO

_{3}just into the perovskite field of oxides in a simple correlation of rA and rB (see [5]). However, this correlation neglects the influence of the oxide anion. Li et al. [3] have shown that at ambient pressure, a correlation between the tolerance factor t and the octahedral factor o = rB/rO provides a much better means of predicting perovskite structures. With the pressure-dependent radii from Table 2, the octahedral factor o for bridgmanite and for davemaoite ranges between 0.444 and 0.456 and t is below 0.919 and 1.00, respectively. This relation of o and t holds to about 100 GPa where o drops below the lower bound of the perovskite field. This range of pressure is close to the transformation of bridgmanite to the postperovskite phase [23,24,]. However, it is noted that CaSiO

_{3}remains in the perovskite structure despite o < 0.4. For octahedrally coordinated Si and O, and tenfold coordinated Mg, the parameters t and o remain within the perovskite field even at ambient pressure. This observation tentatively explains the metastability of bridgmanite: if t and o are far outside this range, a spontaneous collapse of the structure is expected. This is the case for the stable electronic configurations that correspond to the radii of Si[IV] and Mg[VI]. Hence, the transition from the metastable to the stable valence electron configuration in decompressed bridgmanite is sterically hindered (see Section 9 in Grochala et al. [8]) but the energetic barrier is low. Davemaoite, with a t around unity, is close to the upper limit in t of the perovskite field and may need chemical substitution to survive at low pressures [75]. FeTiO

_{3}remains at the border between the fields of ilmenite and perovskite structures, crossing into the latter around 7–10 GPa, which is slightly below the 10–12 GPa of the phase boundary interpolated to 300 K. These borderline values of o and t are consistent with the existence of a metastable LiNbO

_{3}-type phase of FeTiO

_{3}: wangdaodeite [4]. However, with O

^{2−}in fourfold coordination, the octahedral factor o for ilmenite and wangdaodeite drops below that of liuite around 6.5 GPa and this is a potential indicator for the stabilization of the perovskite- over the ilmenite- and LiNbO

_{3}-type structures above that pressure. The fact that above 9–10 GPa, and along with further increase in pressure, o and t do not shift further into the perovskite field, is consistent with the observed breakdown of this phase to cottunite-type TiO

_{2}and FeO at high pressure [5]: the perovskite structure does not gain in stability with increasing pressure and is replaced by simple oxides once a denser arrangement of Ti in a titania phase becomes energetically favourable. Thermal contributions as well as reaction kinetics influence the actual pressure where breakdown of liuite is observed. Charge transfer and changes in the spin state of Fe are expected to influence the effective radius of iron but appear to have no decisive effect in this particular case since the pressure-dependent parameters o and t are overall consistent with the observed phase transformations in FeTiO

_{3}.

#### 4.2. General Considerations: Cations

^{−4}to 7·10

^{−3}Å/GPa extends further but is overall comparable to the range of values previously assessed for Ca, Y, La, and Al (9·10

^{−4}to 5·10

^{−3}[10], and for those elements the present study and the earlier work agree within uncertainties). However, the conclusion by those authors that crystal and ionic radii may be considered as basically incompressible within the range of pressures in Earth cannot be supported here in that generality. For instance, the radii of Cs[VIII,XI] and K[VIII,IX] decrease by ~30% between 0 and 100 GPa (Table 2, Figure 2). Moreover, the differences in radii contraction extend over one order of magnitude and such large relative differences, although across overall small values of compression, have potential impact on element partitioning over the range of the Earth mantle. Generally, the pressure dependence of crystal radii follows a simple systematics that extends from the more compressible alkaline cations and the anions to rather incompressible ions like Si, Ge, and Mg. The correlation between dr/dP and r

_{0}, the crystal radius of the ions at reference ambient conditions, is shown in Figure 5 (left panel).

^{−1.75}= dr/dP also deviate from the linear correlation dr/dp = 0.00329(15)·r − 0.0010(2) (Equation (2)). The deviations indicate coordinations of ions whose electronegativity is smaller (Li[IV], Na[VII], K[IX]) or higher than those along the general trend and accordingly higher or smaller dr/dP. Note that each of the deviating ions occurs in a coordination which falls onto the correlation between dr/dP and r and χ, respectively. Thus, it is not the chemical species but specific coordinations of ions that exhibit excess compressibility (see Section 4.1 and Section 4.2). The three data points for χ > 15 eV represent the high-pressure dr/dP of the anions O, Cl, Br (see Section 4.3). The ions represented with filled symbols in Figure 5 give a linear correlation.

^{2}= 0.96. However, a good number of ions have been excluded from this fit (hollow symbols, a fit for all data gives a slope of 0.0029(4) and the same constant 0.0010(5) but with R

^{2}= 0.66) and this requires an explanation.

^{2}= 0.66) marked variations that are due to specific reasons for each ion or may better be replaced by upper and lower bounds.

_{i}e

_{i}/n (sum running from i = 1 to n), n

_{i}the occupation of the ith energy level e

_{i}and n the total number of electrons [76]. Ambient pressure electronegativities by Rahm et al. [76] are used here. The same elements that exhibit a very strong linear correlation between dr/dP and r

_{0}also show a very strong correlation (R

^{2}= 0.98) with χ:

^{2}= 0.98. The merit of this separate fitting is discussed in Section 4.4. First, the overall linear correlation of the main trend in Equation (1) is discussed. The ionic compressibility is defined here as (r

_{0})

^{−3}(dr

^{3}/dP)

_{T}and with dr

^{3}/dP = 3r

^{2}dr/dP, and with substitution of Equation (1) into this formula one obtains:

^{3}/dP)

_{T}= 3(A· r

_{0}

^{3}− B· r

_{0}

^{2})

⇒ (r

_{0})

^{−3}(dr

^{3}/dP)

_{T}= 3(A − B/r

_{0})

^{3}/dP converges to zero for vanishing radii. B/r is a second order correction term and explains the minor deviation of the very small ions Be[IV] and Si[IV] from the main trend. Constant 3A has the dimension of a compressibility with value 9.8(1)·10

^{−12}m

^{2}/N = 9.8(1)·10

^{−3}/GPa, which quantifies the change of ionic compressibility with increasing radius; that is, the change in compressibility with addition of further outer electrons for given principal quantum number L (because the radii vary periodically as a function of L and Z, see for instance [14]). This brings back the issue of the correlation between dr/dP and electronegativity and the causes of the deviations of ions in some coordinations from the rather strict general trends of Equations (1) and (2).

_{B})

^{−3}(dr

^{3}dP)

_{T}with r

_{B}the Bohr radius. β is given in 1/GPa and this somewhat unusual measure of ion compression describes the compressibility of the ions as multi-electron systems relative to the reference volume of a single electron, as defined by the Bohr radius r

_{B}, whereas the proper ionic compressibility (r

_{0})

^{−3}(dr

^{3}dP)

_{T}is invariant of Z or any other atomic parameter within uncertainties (Equation (4)). β is illustrative in showing the relation between ion compression and nuclear charge number Z. The relation between dr/dP and Z shows an equivalent systematics. It is given below in Section 4.4.

^{4+}[IV] has been obtained from the bond distances of diamond under compression [80]: considering that a spherical average of valence electron distributions is not expected to be a good match for the highly covalent bond of diamond, the extremely low compressibility of this element and monatomic material is captured quite accurately in the relation between the nominal ionic compressibility of C[IV] and Z = 6 (Figure 6). Then, the L = 3 ions Al[VI], Si[IV], Si[VI] are expected to be highly incompressible and this is observed, indeed. Similarly, for L = 4, Ge[VI] is the least compressible of the examined ions in this row, etc. Moreover, from Figure 6 one can conclude that the cations of transition group VIIIb elements, the lanthanides, and the actinides should exhibit similar, low-ionic compressibilities. This conclusion is also consistent with experimental observation. Although it is not possible to extract good cation-oxide distances from X-ray diffraction data from high-pressure experiments on compounds of elements with such high form factors, it is known that compounds of L = 5 and 6 group VIIIb elements are quite incompressible which, in part, is explained by low cationic compressibility for simple compounds of these elements [81,82,83].

_{0}= C/(Z − S) and dr/dP or ionic compressibility β, respectively, explains why pressure-induced changes in outer electron states are correlated to changes in bond coordination rather than inducing an isostructural, continuous evolution of radii and their compressibilities: C and S are functions of electron states and undergo quantized changes [77,78,79]. Upon such changes, bond coordination either increases, thereby reducing repulsion, or remains equal, if consistent with Equation (5). Hence, Equation (5) in combination with (2) can be used as a boundary condition for assessing pressure-induced reconstructive transitions. Furthermore, the pressure dependence of the radii is actually not equal to but smaller than C/(Z − S) by the compressibility factor A (Equations (1), (4) and (5)). Factor A therefore represents the relative change of C and S upon compression for a given Z, valence, and coordination, which together with C and S define the radii [77,78,79]. It is, thus, not surprising the radii r and their dr/dP exhibit some variation around the linear correlation of Equation (1), in consequence of particular settings for S. The effect is seen, for instance, for the dr/dP of Fe

^{2+}[VI] which, in high-spin configuration, is somewhat higher than predicted (Figure 6). More generally, radii and dr/dP for a given chemical species but in different coordination are shifted such that in high coordination the distribution of the outer electrons over a more extensive set of configurations reduces repulsion and allows for higher compressibility relative to Equation (1), and coordinations that enhance repulsion give lower dr/dP relative to (1). These variations are equivalent to the positive deviations from the power relation between electronegativity and dr/dP (Figure 6), and Equation (2) can be placed into context with the electron screening functions through dr/dP = A C/(Z − S) − B = const·χ

^{−1.76}. It is important to note that for each chemical species and valence, there is a least one coordination where the radius and dr/dP fall onto the linear correlation of Equation (1) and the power law dr/dP ~ 1/χ

^{1.76}(Equation (2)); thus, these two relations represent the principal trend of ionic compression behaviour.

#### 4.3. General Considerations: Anions

^{−m}of the pressure-dependent anions O

^{2−}, Cl

^{−}, and Br

^{−}is to be explained in an equivalent fashion: for these anions, dr/dP = −m·r′/P

^{(1+m)}= C′/{Z − S(P)

^{(1+m)}} and the main difference to the cations is in the continuous pressure- dependence of S. In fact, the initial linear anionic compressibilities of O

^{2−}, Cl

^{−}, and Br

^{−}fall -onto the same correlation of β and Z as the cations (Figure 6, hollow symbols). A quite remarkable aspect in this consistency of cat- and anion compression behaviour is that the initial linear anionic compressibilities of the anions for each row L fall onto the extension of the correlation for L + 1 rather than for L, which means that, within the approximation of the crystal radius concept, the initial compression of the anions reflects the full octet state that is formally assigned to them in inorganic chemistry. At large P, the anion radii approach a linear contraction regime like the cations and this change in anion compression regime has been proposed to serve as for a principal distinction of intermediate- and high-pressure phases [5]. The asymptotic high-pressure values also fall into the correlation of β and Z (hollow symbols in Figure 7); however, they are at the tail of the correlations for L rather than on the slopes of the correlation for L + 1, close to the singularities of Z − S. From a purely formal point of view, this observation implies that within this high-pressure regime O

^{2−}[VI], Cl

^{−}[VIII], and Br

^{−}[VIII] have shifted off the octet state. This tentative interpretation is consistent with the proposition by Prewitt and Downs [6] that increasing pressure increases covalency of the bonds. Furthermore, it is consistent with the general pressure-induced reduction of electronegativity [84], which also reduces the difference in electronegativity between cat- and anion, and is therefore in agreement with a reduced electron density gradient along the bond vectors.

#### 4.4. Prediction of Ionic Radii Compression

**Figure 7.**Correlation of pressure dependencies of ionic radii dr/dP in 1/GPa as function of nuclear charge number. Filled diamonds: predicted dr/dP; hollow squares: observed dr/dP. (

**Left panel**): For rows L = 2 to 4, (

**right panel**): for rows L = 5 to 7. For given Z, the different values of dr/dP represent different coordinations. Only Li[IV] deviates markedly and beyond uncertainties from the predicted relation.

## 5. Summary

_{0})

^{−3}(dr

^{3}dP)

_{T}is nearly invariantly 9.8(1)·10

^{−12}m

^{2}/N = 9.8(1)·10

^{−3}/GPa, which quantifies the change of ionic compressibility with increasing radius, that itself is defined through the addition of further outer electrons for a given principal quantum number L, along with increasing Z, and modified by the screening function S (see Section 4.2 and Section 4.3). Therefore, cation compression does generally not result in continuous changes of valence electronic states but in monotonous linear contraction. Changes in valence electron states appear strictly correlated to changes of bond coordination and are constrained to discrete changes in states that reduce repulsion of the outer electrons. Commonly, changes in bond coordination are achieved through reconstructive phase transitions. For cations, the screening function is not pressure-dependent within uncertainties over the examined pressure range. For anions, the screening function scales with a low power of pressure such that with increasing pressure a linear compression behaviour is approached that is then equivalent to the compression behaviour of the cations. The transition between the initial high compressibility of anions to asymptotic linear compression behaviour at high pressure appears to a be a continuous electron transition from a nearly perfect valence electron octet state towards less localized outer electron states, and this appears to define a fundamental difference in anion and cation compression. The more similar compression behaviour at sufficiently high pressure is consistent with the reduced difference in electronegativity between cat- and anion and more shared outer electron states. This behaviour is initially and to first order dominated by the high compressibility of the anions, but in second degree, pressure-induced coordination changes of the cations also modify their electronegativity based on the power-law relation dr/dP ~ 1/χ

^{1.76}that has been established in this paper. Positive and negative excess compressibility of ions in some coordinations is explained as result of lower or higher electron repulsion relative to the coordination-independent electronegativity. These deviations can be quantified in terms of a correction to the electronegativity. The relations between radii, ionic compressibility, electronegativity, and nuclear charge number appear to be general and, thus, allow for predicting pressure dependence of radii for most ions within narrow limits.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Goldschmidt, V.M. The principles of distribution of chemical elements in minerals and rocks. The seventh Hugo Muller Lecture, delivered before the Chemical Society on 17 March 1937. J. Chem. Soc.
**1937**, 655–673. [Google Scholar] [CrossRef] - Manjon, F.J.; Errandonea, D.; Lopez-Solano, J.; Rodriguez, P.; Radescu, S.; Mujica, A.; Munoz, A.; Garro, N.; Pellicer-Porres, J.; Segura, A.; et al. Crystal stability and pressure-induced phase transitions in scheelite AWO(4) (A = Ca, Sr, Ba, Pb, Eu) binary oxides. II: Towards a systematic understanding. Phys. Stat. Sol. B
**2007**, 244, 295–302. [Google Scholar] [CrossRef] - Li, Z.; Yang, M.J.; Park, J.S.; Wei, S.H.; Berry, J.J.; Zhu, K. Stabilizing Perovskite Structures by Tuning Tolerance Factor: Formation of Formamidinium and Cesium Lead Iodide Solid-State Alloys. Chem. Mat.
**2016**, 28, 284–292. [Google Scholar] [CrossRef] - Tschauner, O.; Ma, C. Discovering High-Pressure and High-Temperature Minerals. In Celebrating the International Year of Mineralogy; Bindi, L., Cruciani, G., Eds.; Springer: Berlin/Heidelberg, Germany, 2023; pp. 169–206. [Google Scholar] [CrossRef]
- Tschauner, O. High-pressure minerals. Am. Min.
**2019**, 104, 1701–1731. [Google Scholar] [CrossRef] - Prewitt, C.T.; Downs, R.T. High-pressure crystal chemistry. In Ultrahigh-Pressure Mineralogy: Physics and Chemistry of the Earth’s Deep Interior; Hemley, R.J., Ed.; Mineralogical Society of America: Washington, DC, USA, 1998; Volume 37, pp. 283–317. [Google Scholar]
- Shannon, R.D.; Prewitt, C.T. Coordination and volume changes accompanying high-pressure phase transformations of oxides. Mat. Res. Bul.
**1969**, 4, 57–59. [Google Scholar] [CrossRef] - Grochala, W.; Hoffmann, R.; Feng, J.; Ashcroft, N.W. The Chemical Imagination at Work in Very Tight Places. Angew. Chem. Int. Ed.
**2007**, 46, 3620–3642. [Google Scholar] [CrossRef] - Gibbs, G.V.; Ross, N.L.; Cox, D.F.; Rosso, K.M.; Iversen, B.B.; Spackman, M.A. Bonded radii and the contraction of the electron density of the oxygen atom by bonded interactions. J. Phys. Chem. A
**2013**, 117, 1632–1640. [Google Scholar] [CrossRef] - Gibbs, G.V.; Cox, D.F.; Ross, N.L. The incompressibility of atoms at high pressures. Am. Min.
**2020**, 105, 1761–1768. [Google Scholar] [CrossRef] - Cammi, R.; Rahm, M.; Hoffmann, R.; Ashcroft, N.W. Varying Electronic Configurations in Compressed Atoms: From the Role of the Spatial Extension of Atomic Orbitals to the Change of Electronic Configuration as an Isobaric Transformation. J. Chem. Theory Comput.
**2020**, 16, 5047–5056. [Google Scholar] [CrossRef] - Rahm, M.; Ångqvist, A.; Rahm, J.M.; Erhart, P.; Cammi, R. Non-Bonded Radii of the Atoms Under Compression. ChemPhys. Chem.
**2020**, 21, 2441–2453. [Google Scholar] [CrossRef] - Tschauner, O. An observation related to the pressure dependence of ionic radii. Geosciences
**2022**, 12, 246. [Google Scholar] [CrossRef] - Pauling, L. The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry; Cornell University Press: New York, NY, USA, 1960; p. 644. [Google Scholar]
- Shannon, R.D.; Prewitt, C.T. Effective ionic radii in oxides and fluorides. Acta Crystallogr.
**1969**, 25, 925–946. [Google Scholar] [CrossRef] - Shannon, R.D. Revised effective ionic-radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Cryst. A
**1976**, 32, 751–767. [Google Scholar] [CrossRef] - Morrison, S.M.; Hazen, R.M. An evolutionary system of mineralogy, Part IV: Planetesimal differentiation and impact mineralization (4566 to 4560 Ma). Am. Min.
**2021**, 106, 730–761. [Google Scholar] [CrossRef] - Du, X.P.; Tse, J.S. Oxygen Packing Fraction and the Structure of Silicon and Germanium Oxide Glasses. J. Phys. Chem. B
**2017**, 121, 10726–10732. [Google Scholar] [CrossRef] - Zhang, J.-M.; Ko, J.-D.; Hazen, R.M.; Prewitt, C.T. High-pressure crystal chemistry of KAlSi3O8 hollandite. Am. Min.
**1993**, 78, 493–499. [Google Scholar] - Dewaele, A.; Belonoshko, A.B.; Garbarino, G.; Occelli, F.; Bouvier, P.; Hanfland, M.; Mezouar, M. High-pressure high-temperature equation of state of KCl and KBr. Phys. Rev. B
**2012**, 85, 214105. [Google Scholar] [CrossRef] - Dewaele, A. Equations of State of Simple Solids (Including Pb, NaCl and LiF) Compressed in Helium or Neon in the Mbar Range. Minerals
**2019**, 9, 684. [Google Scholar] [CrossRef] - Dewaele, A. Compression of CsCl and CsBr in the megabar range. High. Pressure Res.
**2020**, 40, 402–410. [Google Scholar] [CrossRef] - Murakami, M.; Hirose, K.; Kawamura, K.; Sata, N.; Ohishi, Y. Post-perovskite phase transition in MgSiO
_{3}. Science**2004**, 304, 855–858. [Google Scholar] [CrossRef] - Ono, S.; Kikegawa, T.; Ohishi, Y. Equation of state of CaIrO
_{3}-type MgSiO_{3}up to 144 GPa. Am. Min.**2006**, 91, 475–478. [Google Scholar] [CrossRef] - Hazen, R.M.; Finger, L.W. High-pressure and high-temperature crystal-chemistry of beryllium-oxide. J. All Phys.
**1986**, 59, 3728–3733. [Google Scholar] [CrossRef] - Au, Y.; Hazen, R.M. Polyhedral modeling of the elastic properites of corundum and chrysoberyl. Geophys. Res. Lett.
**1985**, 12, 725–728. [Google Scholar] [CrossRef] - Chen, H.; Shim, S.-H.; Leinenweber, K.; Prakapenka, V.; Meng, Y.; Prescher, C. Crystal structure of CaSiO
_{3}perovskite at 28–62 GPa and 300 K under quasi-hydrostatic stress conditions. Am. Min.**2021**, 103, 462–468. [Google Scholar] [CrossRef] - Ross, N.L.; Zhao, J.; Angel, R.J. High-pressure single-crystal X-ray diffraction study of YAlO3 perovskite. J. Sol. Stat. Chem.
**2004**, 177, 1276–1284. [Google Scholar] [CrossRef] - Ross, N. L High pressure study of Sc Al O3 perovskite. Phys. Chem. Min.
**1998**, 25, 597–602. [Google Scholar] [CrossRef] - Zhao, J.; Ross, N.L.; Angel, R.J. Polyhedral control of the rhombohedral to cubic phase transition in LaAlO
_{3}perovskite. J. Phys. Cond. Matt.**2004**, 16, 8763–8773. [Google Scholar] [CrossRef] - Zhao, J.; Ross, N.L.; Angel, R.J.; Carpenter, M.A.; Howard, C.J.; Pawlak, D.A.; Lukasiewicz, T. High-pressure crystallography of rhombohedral Pr Al O3 perovskite. J. Phys. Cond.Matt.
**2009**, 21, 235403. [Google Scholar] [CrossRef] [PubMed] - Ross, N.L.; Zhao, J.; Angel, R.J. High-presure structural behavior of GdAlO
_{3}and GdFeO_{3}perovskites. J. Sol. Stat. Chem.**2009**, 177, 3768–3775. [Google Scholar] [CrossRef] - Finger, L.W.; Hazen, R.M. Crystal structure and compression of ruby to 46 kbar. J. Appl. Phys.
**1978**, 49, 5823–5826. [Google Scholar] [CrossRef] - Kim-Zajonz, J.; Werner, S.; Schulz, H. High pressure single crystal X-ray diffraction study on ruby up to 31 GPa. Z. Krist.
**1999**, 214, 331–336. [Google Scholar] [CrossRef] - Lin, J.; Degtyareva, O.; Prewitt, C.T.; Dera, P.; Sata, N.; Gregoryanz, E.; Mao, H.-k.; Hemley, R.J. Crystal structure of a high-pressure/high-temperature phase of alumina by in situ X-ray diffraction. Nat. Mat.
**2004**, 3, 389–393. [Google Scholar] [CrossRef] [PubMed] - Kantor, A.; Kantor, I.; Merlini, M.; Glazyrin, K.; Prescher, C.; Hanfland, M.; Dubrovinsky, L. High-pressure structural studies of eskolaite by means of single-crystal X-ray diffraction. Am. Min.
**2012**, 97, 1764–1770. [Google Scholar] [CrossRef] - Mao, H.-k.; Shu, J.; Fei, Y.; Hu, J.; Hemley, R.J. The wüstite enigma. Phys. Earth Planet. Int.
**1996**, 96, 135–145. [Google Scholar] [CrossRef] - Yamanaka, T.; Kyono, A.; Nakamoto, Y.; Kharlamova, S.; Struzhkin, V.V.; Gramsch, S.A.; Mao, H.-K.; Hemley, R.J. New structure of high-pressure body-centered orthorhombic Fe
_{2}SiO_{4}. Am. Min.**2015**, 100, 1736–1743. [Google Scholar] [CrossRef] - Jacobsen, S.D.; Demouchy, S.; Frost, D.J.; Boffa Ballaran, T.; Kung, J. A systematic study of OH in hydrous wadsleyite from polarized FTIR spectroscopy and single-crystal X-ray diffraction: Oxygen sites for hydrogen storage in earths interior. Am. Min.
**2005**, 90, 61–70. [Google Scholar] [CrossRef] - Hazen, R.M.; Yang, H.X. Effects of cation substitution and order-disorder on P-V-T equations of state of cubic spinels. Am. Min.
**1999**, 84, 1956–1960. [Google Scholar] [CrossRef] - Ma, C.; Tschauner, O.; Beckett, J.R.; Liu, Y.; Rossman, G.R.; Sinogeikin, S.V.; Smith, J.S.; Taylor, L.A. Ahrensite, gamma-Fe
_{2}SiO_{4}, a new shock-metamorphic mineral from the tissint meteorite: Im-plications for the tissint shock event on Mars. Geochim. Cosmochim. Acta**2016**, 184, 240–256. [Google Scholar] [CrossRef] - Yamanaka, T.; Komatsu, Y.; Sugahara, M.; Nagai, T. Structure change of MgSiO
_{3}, MgGeO_{3}, and MgTiO_{3}ilmenites under compression. Am. Min.**2005**, 90, 1301–1307. [Google Scholar] [CrossRef] - Ross, N.L.; Zhao, J.; Slebodnick, C.; Spencer, E.C.; Chakoumakos, B.C. Petalite under pressure: Elastic behavior and phase stability. Am. Min.
**2015**, 100, 714–721. [Google Scholar] [CrossRef] - Benusa, M.D.; Angel, R.J.; Ross, N.L. Compression of albite, NaAlSi
_{3}O_{8}. Am. Min.**2005**, 90, 1115–1120. [Google Scholar] [CrossRef] - Petitgirard, S. Density and structural changes of silicate glasses under high pressure. High Pressure Res.
**2017**, 37, 200–213. [Google Scholar] [CrossRef] - Gatta, G.D.; Angel, R.J.; Zhao, J.; Alvaro, M.; Rotiroti, N.; Carpenter, M.A. Phase stability, elastic behavior, and pressure-induced structural evolution of kalsilite: A ceramic material and high-T/high-P mineral. Am. Min.
**2011**, 96, 1363–1372. [Google Scholar] [CrossRef] - Gatta, G.D.; Lotti, P.; Comboni, D.; Merlini, M.; Vignola, P.; Liermann, H.P. High-pressure behavior of (Cs, K) Al4Be5B11O28 (londonite): A single-crystal synchrotron diffraction study up to 26 GPa. J. Am. Ceram. Soc.
**2017**, 100, 4893–4901. [Google Scholar] [CrossRef] - Lazarz, J.D.; Dera, P.; Hu, Y.; Meng, Y.; Bina, C.R.; Jacobsen, S.D. High-pressure phase transitions of clinoenstatite. Am. Min.
**2019**, 104, 897–904. [Google Scholar] [CrossRef] - Finkelstein, G.J.; Dera, P.K.; Jahn, S.; Oganov, A.R.; Holl, C.M.; Meng, Y.; Duffy, T.S. Phase transitions and equation of state of forsterite to 90 GPa from single-crystal X-ray diffraction and molecular modeling. Am. Min.
**2014**, 99, 35–43. [Google Scholar] [CrossRef] - Ross, N.L.; Hazen, R.M. High-pressure crystal chemistry of Mg Si O3 perovskite. Phys. Chem. Min.
**1990**, 17, 228–237. [Google Scholar] [CrossRef] - Kudoh, Y.; Ito, E.; Takeda, H. Effect of pressure on the crystal structure of perovskite type Mg Si O3. Phys. Chem. Min.
**1987**, 14, 350–354. [Google Scholar] [CrossRef] - Sugahara, M.; Yoshiasa, A.; Komatsu, Y.; Yamanaka, T.; Bolfan Casanova, N.; Nakatsuka, A.; Sasaki, S.; Tanaka, M. Reinvestigation of the MgSiO
_{3}perovskite structure at high pressure. Am. Min.**2006**, 91, 533–536. [Google Scholar] [CrossRef] - Richet, P.; Mao, H.-K.; Bell, P.M. Static compression and equation of state of CaO to 1.35 Mbar. J. Geophys. Res.
**1988**, 93, 15279–15288. [Google Scholar] [CrossRef] - Zhao, J.; Ross, N.L.; Wang, D.; Angel, R.J. High-pressure crystal structure of elastically isotropic CaTiO
_{3}perovskite under hydrostatic and non-hydrostatic conditions. J. Phys. Cond. Matt.**2011**, 23, 455401. [Google Scholar] [CrossRef] [PubMed] - Milani, S.; Comboni, D.; Lotti, P.; Fumagalli, P.; Ziberna, L.; Maurice, J.; Hanfland, M.; Merlini, M. Crystal structure evolution of CaSiO
_{3}polymorphs at earth’s mantle pressures. Minerals**2021**, 11, 652. [Google Scholar] [CrossRef] - Xiao, W.; Tan, D.; Zhou, W.; Liu, J.; Xu, J. Cubic perovskite polymorph of strontium metasilicate at high pressures. Am. Min.
**2013**, 98, 2096–2104. [Google Scholar] [CrossRef] - Loridant, S.; Lucazeau, G.; Le Bihan, T. A high-pressure Raman and X-ray diffraction study of the perovskite SrCeO
_{3}. J. Phys. Chem. Sol.**2002**, 63, 1983–1992. [Google Scholar] [CrossRef] - Knight, K.S.; Marshall, W.G.; Bonanos, N.; Francis, D.J. Pressure dependence of the crystal structure of SrCeO
_{3}perovskite. J. All Comp.**2005**, 394, 131–137. [Google Scholar] [CrossRef] - Errandonea, D.; Kumar, R.S.; Ma, X.; Tu, C. High-pressure X-ray diffraction of SrMoO
_{4}and pressure-induced structural changes. J. Sol. Stat. Chem.**2008**, 181, 355–364. [Google Scholar] [CrossRef] - Crichton, W.; Merlini, M.; Hanfland, M.; Mueller, H. The crystal structure of barite, Ba(SO
_{4}), at high pressure. Am. Min.**2011**, 96, 364–367. [Google Scholar] [CrossRef] - Santamaria-Perez, D.; Chulia-Jordan, R. Compression of mineral barite, BaSO4: A structural study. High. Pressure Res.
**2012**, 32, 81–88. [Google Scholar] [CrossRef] - Errandonea, D.; Pellicer-Porres, J.; Manjón, F.J.; Segura, A.; Ferrer-Roca, C.; Kumar, R.S.; Tschauner, O.; López-Solano, J.; Rodríguez-Hernández, P.; Radescu, S.; et al. Determination of the high-presure crystal structure of BaWO4 and PbWO4. Phys. Rev. B Cond. Matt. Mat. Phys.
**2006**, 73, 224103. [Google Scholar] [CrossRef] - Yusa, H.; Sata, N.; Ohishi, Y. Rhombohedral(9R) and hexagonal(6H) perovskites in barium silicates under high pressure. Am. Min.
**2007**, 92, 648–654. [Google Scholar] [CrossRef] - Friedrich, A.; Kunz, M.; Miletich, R.; Pattison, P. High-pressure behavior of Ba(OH)
_{2}: Phase transitions and bulk modulus. Phys. Rev. B Cond. Matt. Mat. Phys.**2002**, 66, 214103. [Google Scholar] [CrossRef] - Hayward, S.A.; Redfern, S.A.T.; Stone, H.J.; Tucker, M.G.; Whittle, K.R.; Marshall, W.G. Phase transitions in BaTiO
_{3}: A high-pressure neutron diffraction study. Z. Krist.**2005**, 220, 735–739. [Google Scholar] [CrossRef] - Hazen, R.M.; Finger, L.W.; Hemley, R.J.; Mao, H.K. High-pressure crystal chemistry and amorphization of alpha quartz. Solid State Commun.
**1989**, 72, 507–511. [Google Scholar] [CrossRef] - Levien, L.; Prewitt, C.T.; Weidner, D.J. Structure and elastic properties of quartz at pressure. Am. Min.
**1980**, 65, 920–930. [Google Scholar] - Angel, R.J.; Shaw, C.S.J.; Gibbs, G.V. Compression mechanisms of coesite. Phys. Chem. Min.
**2003**, 30, 167–176. [Google Scholar] [CrossRef] - Kudoh, Y.; Takeda, H. Single crystal X-ray diffraction study on the bond compressibility of fayalite, Fe
_{2}SiO_{4}and rutile, TiO_{2}under high pressure. Phys. B + C Phys. Cond. Matt.**1986**, 139, 333–336. [Google Scholar] [CrossRef] - Andrault, D.; Angel, R.J.; Mosenfelder, J.L.; Le Bihan, T. Equation of state of stishovite to lower mantle pressures. Am. Min.
**2003**, 88, 301–307. [Google Scholar] [CrossRef] - Zhang, L.; Popov, D.; Meng, Y.; Wang, J.; Ji, C.; Li, B.; Mao, H.K. In-situ crystal structure determination of seifertite SiO2 at 129 GPa: Studying a minor phase near Earth’s core-mantle boundary. Am. Min.
**2016**, 101, 231–234. [Google Scholar] [CrossRef] - Ross, N.L.; Shu, J.-F.; Hazen, R.M.; Gasparik, T. High-pressure crystal chemistry of stishovite. Am. Min.
**1990**, 75, 739–747. [Google Scholar] - Yamanaka, T.; Fukuda, T.; Tsuchiya, J. Bonding character of SiO
_{2}stishovite under high pressures up to 30 GPa. Phys. Chem. Min.**2002**, 29, 633–641. [Google Scholar] [CrossRef] - Sugiyama, M.; Endo, S.; Koto, K. The crystal structure of stishovite under pressure up to 6 GPa. Miner. J.
**1987**, 13, 455–466. [Google Scholar] [CrossRef] - Tschauner, O.; Huang, S.; Yang, S.; Humayun, M.; Liu, W.; Gilbert Corder, S.N.; Bechtel, H.A.; Tischler, J.; Rossman, G.R. Discovery of davemaoite, CaSiO
_{3}-perovskite, as a mineral from the lower mantle. Science**2021**, 374, 891–894. [Google Scholar] [CrossRef] - Rahm, M.; Zeng, T.; Hoffmann, R. Electronegativity Seen as the Ground-State Average Valence Electron Binding Energy. J. Am. Chem. Soc.
**2019**, 141, 342–351. [Google Scholar] [CrossRef] [PubMed] - Pauling, L. The theoretical prediction of the physical properties of many-electron atoms and ions. Proc. Roy. Acad. Ser. A
**1927**, 114, 181–211. [Google Scholar] - Slater, J.C. Atomic Shielding Constants. Phys. Rev.
**1930**, 36, 57–64. [Google Scholar] [CrossRef] - Clementi, E.; Raimondi, D.L. Atomic Screening Constants from SCF Functions. J. Chem. Phys.
**1963**, 38, 2686–2689. [Google Scholar] [CrossRef] - Gillet, P.; Fiquet, G.; Daniel, I.; Reynard, B.; Hanfland, M. Equations of state of 12C and 13C diamond. Phys. Rev. B
**1999**, 60, 14660–14664. [Google Scholar] [CrossRef] - Crowhurst, J.C.; Goncharov, A.F.; Sadigh, B.; Evans, C.L.; Morrall, P.G.; Ferreira, J.L.; Nelson, A.J. Synthesis and characterization of the nitrides of platinum and iridium. Science
**2006**, 311, 1275–1278. [Google Scholar] [CrossRef] - Haines, J.; Leger, J. Phase-transitions in ruthenium dioxide up to 40 GPa—Mechanism for the rutile-to-fluorite phase-transformatin and a model for the high-pressure behavior of stishoivte SiO
_{2}. Phys. Rev. B**1993**, 48, 13344–13350. [Google Scholar] [CrossRef] - Tschauner, O.; Kiefer, B.; Tetard, F.; Tait, K.; Bourguille, J.; Zerr, A.; Dera, P.; McDowell, A.; Knight, J.; Clark, C. Elastic moduli and hardness of highly incompressible platinum perpnictide PtAs2. Appl. Phys. Lett.
**2013**, 103, 101901. [Google Scholar] [CrossRef] - Rahm, M.; Cammi, R.; Ashcroft, N.W.; Hoffmann, R. Squeezing All Elements in the Periodic Table: Electron Configuration and Electronegativity of the Atoms under Compression. J. Am. Chem. Soc.
**2019**, 141, 10253–10271. [Google Scholar] [CrossRef]

**Figure 1.**Pressure-dependent crystal radii of the examined alkaline earth ions. (

**a**) Be, Mg, Ca. (

**b**) Sr and Ba. Coordination is given in square brackets. For all ions, compression is negative and linear within uncertainties. Lines show the least-square fits of the compression (see Table 2).

**Figure 2.**Crystal radii of Li, Na, K, and Cs, bond coordinations given in square brackets. Filled symbols represent data from cation-oxide distances, hollow symbols = radii from cation-chloride and –bromide distances, respectively. Fitted pressure dependences are shows as lines and are given in Table 2. There is a change in compressibility of K[VIII] between 60 and 70 GPa such that the high-pressure regime interpolates to K[VI] at 0 GPa (see Section 4). The intersections of K[IX] and K[VIII] around 30–40 GPa and of K[VI] and Na[VIII] around 110–130 GPa are noteworthy.

**Figure 3.**(

**Left**): Radii of rare earth elements. (

**Right**): Radii of Al, Fe

^{2+}, and Cr

^{3+}, all three in sixfold coordination. Fitted pressure dependences are shows as lines and are given in Table 2.

**Figure 4.**Crystal radii of the tetravalent cations Si, Ge, and Ti. Coordinations given in square brackets. Lines represent the fitted pressure dependencies.

**Figure 5.**Overview and systematics of the pressure dependencies of the examined ions: correlation between −dr/dP in 1/ GPa and ambient pressure crystal radii r

_{0}in Å. (

**Left side**): dr/dP as function of r

_{0}. Fit through filed symbol data, hollow symbols are for the following ions: Li[IV], Na[VI], Na[VII] Na[VIII], Cs[VI], Ca[VIII], [X], Ba[VI], [VIII], La, Pr, Gd. Filled symbols are for all other ions that represent the ‘mail trend’ (as explained in Section 4.2). (

**Right side**): dr/dP as function of electronegativity χ in eV, at 0 GPa [76]. The fits are for the data that are represented as filled symbols; the same is true for the left panel.

**Figure 6.**Ionic compressibility as a function of the nuclear charge number Z. The correlation is equivalent to that of the ionic radii and Z (see [14,77]) and determines the pressure-induced changes in the outer electron configurations of the ions. Compressibilites of cations are given in Table 2 shown as black squares; initial and asymptotic high-pressure compressibilities of the anions are shown as hollow diamonds (see Section 4.3).

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Tschauner, O.
Pressure-Dependent Crystal Radii. *Solids* **2023**, *4*, 235-253.
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Tschauner O.
Pressure-Dependent Crystal Radii. *Solids*. 2023; 4(3):235-253.
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2023. "Pressure-Dependent Crystal Radii" *Solids* 4, no. 3: 235-253.
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