# Lanthanide contraction

The lanthanide contraction is the greater-than-expected decrease in atomic radii and ionic radii of the elements in the lanthanide series, from left to right. It is caused by the poor shielding effect of nuclear charge by the 4f electrons along with the expected periodic trend of increasing electronegativity and nuclear charge on moving from left to right. About 10% of the lanthanide contraction has been attributed to relativistic effects.
A decrease in atomic radii can be observed across the 4f elements from atomic number 57, lanthanum, to 70, ytterbium. This results in smaller than otherwise expected atomic radii and ionic radii for the subsequent d-block elements starting with 71, lutetium. This effect causes the radii of transition metals of group 5 and 6 to become unusually similar, as the expected increase in radius going down a period is nearly cancelled out by the f-block insertion, and has many other far ranging consequences in post-lanthanide elements.
The decrease in ionic radii (Ln$^{3+}$) is much more uniform compared to decrease in atomic radii.
| Element | Atomic electron configuration (all begin with [Xe]) | Ln$^{3+}$ electron configuration | Ln$^{3+}$ radius (pm) (6-coordinate) |
| La | 5d$^1$6s$^2$ | 4f$^0$ | 103 |
| Ce | 4f$^1$5d$^1$6s$^2$ | 4f$^1$ | 102 |
| Pr | 4f$^3$6s$^2$ | 4f$^2$ | 99 |
| Nd | 4f$^4$6s$^2$ | 4f$^3$ | 98.3 |
| Pm | 4f$^5$6s$^2$ | 4f$^4$ | 97 |
| Sm | 4f$^6$6s$^2$ | 4f$^5$ | 95.8 |
| Eu | 4f$^7$6s$^2$ | 4f$^6$ | 94.7 |
| Gd | 4f$^7$5d$^1$6s$^2$ | 4f$^7$ | 93.8 |
| Tb | 4f$^9$6s$^2$ | 4f$^8$ | 92.3 |
| Dy | 4f$^{10}$6s$^2$ | 4f$^9$ | 91.2 |
| Ho | 4f$^{11}$6s$^2$ | 4f$^{10}$ | 90.1 |
| Er | 4f$^{12}$6s$^2$ | 4f$^{11}$ | 89 |
| Tm | 4f$^{13}$6s$^2$ | 4f$^{12}$ | 88 |
| Yb | 4f$^{14}$6s$^2$ | 4f$^{13}$ | 86.8 |
| Lu | 4f$^{14}$5d$^1$6s$^2$ | 4f$^{14}$ | 86.1 |
The term was coined by the Norwegian geochemist Victor Goldschmidt in his series "Geochemische Verteilungsgesetze der Elemente" (Geochemical distribution laws of the elements).

## Cause

The effect results from poor shielding of nuclear charge (nuclear attractive force on electrons) by 4f electrons; the 6s electrons are drawn towards the nucleus, thus resulting in a smaller atomic radius.
In single-electron atoms, the average separation of an electron from the nucleus is determined by the subshell it belongs to, and decreases with increasing charge on the nucleus; this, in turn, leads to a decrease in atomic radius. In multi-electron atoms, the decrease in radius brought about by an increase in nuclear charge is partially offset by increasing electrostatic repulsion among electrons.
In particular, a "shielding effect" operates: i.e., as electrons are added in outer shells, electrons already present shield the outer electrons from nuclear charge, making them experience a lower effective charge on the nucleus. The shielding effect exerted by the inner electrons decreases in the order s > p > d > f.
Usually, as a particular subshell is filled in a period, the atomic radius decreases. This effect is particularly pronounced in the case of lanthanides, as the 4f subshell which is filled across these elements is not very effective at shielding the outer shell (n=5 and n=6) electrons. Thus the shielding effect is less able to counter the decrease in radius caused by increasing nuclear charge. This leads to "lanthanide contraction". The ionic radius drops from 103 pm for lanthanum(III) to 86.1 pm for lutetium(III).
About 10% of the lanthanide contraction has been attributed to relativistic effects. The lanthanide contraction was experimentally observed in aqueous solutions of lanthanides, including radioactive promethium, through X-ray absorption spectroscopy measurements.

## Effects

The results of the increased attraction of the outer shell electrons across the lanthanide period may be divided into effects on the lanthanide series itself including the decrease in ionic radii, and influences on the following or post-lanthanide elements.

### Properties of the lanthanides

The ionic radii of the lanthanides decrease from 103 pm (La$^{3+}$) to 86 pm (Lu$^{3+}$) in the lanthanide series, electrons are added to the 4f shell. This first f shell is inside the full 5s and 5p shells (as well as the 6s shell in the neutral atom); the 4f shell is well-localized near the atomic nucleus and has little effect on chemical bonding. The decrease in atomic and ionic radii does affect their chemistry, however. Without the lanthanide contraction, a chemical separation of lanthanides would be extremely difficult. However, this contraction makes the chemical separation of period 5 and period 6 transition metals of the same group rather difficult. Even when the mass of an atomic nucleus is the same, a decrease in the atomic volume has a corresponding increase in the density as illustrated by alpha crystals of cerium (at 77 Kelvin) and gamma crystals of cerium (near room temperature) where the atomic volume of the latter is 120.3% of the former and the density of the former is 120.5% of the latter (i.e., 20.696 vs 17.2 and 8.16 vs 6.770, respectively).
As expected, when more mass (protons & neutrons) is packed into a space that is subject to "contraction", the density increases consistently with atomic number for the lanthanides (excluding the atypical 2nd, 7th, and 14th) culminating in the value for the last lanthanide (Lu) being 160% of the first lanthanide (La). Melting points (in Kelvin) also increase consistently across these 12 lanthanides culminating in the value for the last being 161% of the first. This density-melting point association does not depend upon just a comparison between these two lanthanides because the correlation coefficient (Pearson product-moment) for density and melting point for these 12 lanthanides is 0.982 and 0.946 for all 15 lanthanides. There is a general trend of increasing Vickers hardness, Brinell hardness, density and melting point from lanthanum to lutetium (with europium and ytterbium being the most notable exceptions; in the metallic state, they are divalent rather than trivalent). Cerium, along with europium and ytterbium, are atypical when their properties are compared with the other 12 lanthanides as evidenced by the clearly lower values (than either adjacent element) for melting points (lower by >10<43%), Vickers hardness (lower by >32<82%), and densities (lower by >26<33%, when exclude Ce, where the density increases by 10% vs lanthanum). The lower densities for europium and ytterbium (than their adjacent lanthanides) are associated with larger atomic volumes at 148% and 128% of the average volume for the typical 12 lanthanides (i.e., 28.979, 25.067, and 19.629 cm3/mol, respectively).
Because the atomic volume of Yb is 21% more than that of Ce, it is understandable that the density for Ce (the 2nd lanthanide) is 98% of that of ytterbium (the 14th lanthanide) when there is a 24% increase in atomic weight for the latter, and the melting point for Ce (1068 K) is nearly the same as the 1097 K for ytterbium and the 1099 K for europium. These 3 elements are the only lanthanides with melting points below the lowest for the other twelve, which is 1193 K for lanthanum. Because europium has a half-filled 4f subshell, this may account for its atypical values when compared with the data for 12 of the lanthanides. Lutetium is the hardest and densest lanthanide and has the highest melting point at 1925 K, which is the year that Goldschmidt published the terminology "Die Lanthaniden-Kontraktion."
Unlike the m. p. data for the lanthanides (where the values increase consistently when the 2nd, 7th & 14th are excluded), the b. p. temperatures show a repeated pattern at 162% and 165% for the 8th lanthanide relative to the 6th and the 15th relative to the 13th (which ignores the atypical 7th and 14th). The 8th and 15th are among the four lanthanides with one electron in the 5d shell (where the others are the 1st and 2nd) and the b. p. values for these four are +/- 2.6% about 3642 K. See the post-lanthanides section for more comments on the 5d-shell electrons. There is also a repeated b. p. pattern at 66% and 71% for the 6th and 13th lanthanides (relative to the preceding elements) that differ by one electron in the 4f shell, i.e., 5 to 6 and 12 to 13.
| Element | Vickers hardness (MPa) | Brinell hardness (MPa) | Density (g/cm$^3$) | Melting point (K) | Atomic radius (pm) | Boiling point (K) |
| Lanthanum | 491 | 363 | 6.162 | 1193 | 187 | 3737 |
| Cerium | 270 | 412 | 6.770 | 1068 | 181.8 | 3716 |
| Praseodymium | 400 | 481 | 6.77 | 1208 | 182 | 3403 |
| Neodymium | 343 | 265 | 7.01 | 1297 | 181 | 3347 |
| Promethium | ? | ? | 7.26 | 1315 | 183 | 3273 |
| Samarium | 412 | 441 | 7.52 | 1345 | 180 | 2173 |
| Europium | 167 | ? | 5.264 | 1099 | 180 | 1802 |
| Gadolinium | 570 | ? | 7.90 | 1585 | 180 | 3546 |
| Terbium | 863 | 677 | 8.23 | 1629 | 177 | 3396 |
| Dysprosium | 540 | 500 | 8.540 | 1680 | 178 | 2840 |
| Holmium | 481 | 746 | 8.79 | 1734 | 176 | 2873 |
| Erbium | 589 | 814 | 9.066 | 1802 | 176 | 3141 |
| Thulium | 520 | 471 | 9.32 | 1818 | 176 | 2223 |
| Ytterbium | 206 | 343 | 6.90 | 1097 | 176 | 1469 |
| Lutetium | 1160 | 893 | 9.841 | 1925 | 174 | 3675 |

### Magnetism of the lanthanides

It has been shown that lanthanide contraction plays a crucial role in determining the magnetic phase diagram of the heavy rare-earth elements, i.e. those from Gadolinium onwards.

### Influence on the post-lanthanides

The elements following the lanthanides in the periodic table are influenced by the lanthanide contraction. When the first three post-lanthanide elements (Hf, Ta, and W) are combined with the 12 lanthanides, the Pearson correlation coefficient increases from 0.982 to 0.997. On average for the 12 lanthanides, the melting point (on the Kelvin scale) = 1.92x the density (in g/cm^3) while the three elements following the lanthanides have similar values at 188x, 197x, and 192x before the densities continue to increase but the melting points decrease for the next 2 elements followed by both properties decreasing (at different rates) for the next 8 elements. Hafnium is rather unique because not only do density and m. p. temperature change proportionally (relative to lutetium, the last lanthanide) at 135% and 130% but also the b. p. temperature at 133%. The elements with 2, 3, & 4 electrons in the 5d shell (post-lanthanides Hf, Ta, W) have increasing b. p. values such that the b. p. value for W (wolfram, aka tungsten) is 169% of that for the element with one 5d electron (Lu). The high melting point and two other properties of tungsten originates from strong covalent bonds formed between tungsten atoms by the 5d electrons. The elements with 5 to 10 electrons in the 5d shell (Re to Hg) have progressively lower b. p. values such that the element with ten 5d electrons (Hg) has a b. p. value at 52% of tungsten’s (with four 5d electrons).
The radii of the period-6 transition metals are smaller than would be expected if there were no lanthanides, and are in fact very similar to the radii of the period-5 transition metals since the effect of the additional electron shell is almost entirely offset by the lanthanide contraction. For example, the atomic radius of the metal zirconium, Zr (a period-5 transition element), is 155 pm (empirical value) and that of hafnium, Hf (the corresponding period-6 element), is 159 pm. The ionic radius of Zr$^{4+}$ is 84 pm and that of Hf$^{4+}$ is 83 pm. The radii are very similar even though the number of electrons increases from 40 to 72 and the atomic mass increases from 91.22 to 178.49 g/mol. The increase in mass and the unchanged radii lead to a steep increase in density from 6.51 to 13.35 g/cm$^3$.
Zirconium and hafnium, therefore, have very similar chemical behavior, having closely similar radii and electron configurations. Radius-dependent properties such as lattice energies, solvation energies, and stability constants of complexes are also similar. Because of this similarity, hafnium is found only in association with zirconium, which is much more abundant. This also meant that hafnium was discovered as a separate element in 1923, 134 years after zirconium was discovered in 1789. Titanium, on the other hand, is in the same group, but differs enough from those two metals that it is seldom found with them.