Structure variations within RSi2 and R2TSi3 silicides. Part I. Structure overview

Nentwich, M.; Zschornak, M.; Sonntag, M.; Gumeniuk, R.; Gemming, S.; Leisegang, T.; Meyer, D. C.

doi:10.1107/S2052520620001043

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STRUCTURAL SCIENCE
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ISSN: 2052-5206

Volume 76| Part 2| April 2020| Pages 177-200

https://doi.org/10.1107/S2052520620001043

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Structure variations within RSi₂ and R₂TSi₃ silicides. Part I. Structure overview

M. Nentwich,^a ^*‡ M. Zschornak,^a ‡ M. Sonntag,^a R. Gumeniuk,^a S. Gemming,^b,^c T. Leisegang ^a,^d and D. C. Meyer ^a

^aInstitute for Experimental Physics, Technical University Bergakademie Freiberg, 09596 Freiberg, Germany, ^bInstitute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany, ^cInstitute of Physics, Technische Universität Chemnitz, 09107 Chemnitz, Germany, and ^dSamara Center for Theoretical Materials Science, Samara National Research University, 443086 Samara, Russia
^*Correspondence e-mail: Melanie.Nentwich@physik.tu-freiberg.de

Edited by J. Lipkowski, Polish Academy of Sciences, Poland (Received 19 December 2019; accepted 26 January 2020; online 12 March 2020)

Here, structural parameters of various structure reports on RSi₂ and R₂TSi₃ compounds [where R is an alkaline earth metal, a rare earth metal (i.e. an element of the Sc group or a lathanide), or an actinide and T is a transition metal] are summarized. The parameters comprising composition, lattice parameters a and c, ratio c/a, formula unit per unit cell and structure type are tabulated. The relationships between the underlying structure types are presented within a group–subgroup scheme (Bärnighausen diagram). Additionally, unexpectedly missing compounds within the R₂TSi₃ compounds were examined with density functional theory and compounds that are promising candidates for synthesis are listed. Furthermore, a correlation was detected between the orthorhombic AlB₂-like lattices of, for example, Ca₂AgSi₃ and the divalence of R and the monovalence of T. Finally, a potential tetragonal structure with ordered Si/T sites is proposed.

Keywords: silicide; lanthanide; ordering phenomena; structure prediction; DFT.

1. Introduction

The rare earth disilicides RSi₂ have been the subject of numerous studies in the past few decades mainly due to their exciting magnetic properties, such as magnetic ordering phenomena (Wang et al., 2019 ; Pan et al., 2013 ; Kotsanidis et al., 1990 ; Li et al., 1998a , 2002a , 2013 ; Bazela et al., 2003 ; Inosov et al., 2009 ), especially ferromagnetic ordering (Majumdar et al., 1998 , 1999b ; Li et al., 1999 , 2002a,b , 2003 , 2013; Frontzek et al., 2004 ), their spin-glass-like behavior (Li et al., 1998a, 1999, 2002b, 2003; Kimura et al., 1999 ; Szytuła et al., 1999 , 2000 ; Paulose et al., 2003 ; Lu et al., 2013 ) and Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions (Li et al., 2002b; Inosov et al., 2009; Tang et al., 2010a ,b ; Lu et al., 2013), which have been studied since the early 1980s. In the middle of the 20th century, ternary compounds of composition U₂TSi₃ (with a transition metal T substituting one in four Si atoms) were a central research subject due to the emerging use of U-containing compounds in the military and the energy sector. Some of the formed structures are considered as prototypes for further R₂TSi₃ compounds.

As it has been widely discussed in the literature (Hoffmann & Pöttgen, 2001 ; Pan et al., 2013; Peter & Kanatzidis, 2012 ), the RSi₂ and R₂TSi₃ compounds crystallize with the hexagonal AlB₂ and the tetragonal ThSi₂ type and derivative structure types (Hoffmann & Pöttgen, 2001). Some of the disilicides are polymorphic (Perri et al., 1959b ; Brown & Norreys, 1961 ; Mayer et al., 1967 ), meaning that they crystallize in two or more different phases (International Union for Crystallography, 2017 ). This reflects in the now obsolete structure-type names α-USi₂ and α-ThSi₂ for tetragonal ThSi₂ as well as β-USi₂ and β-ThSi₂ for hexagonal AlB₂ (Evers et al., 1980 ; Yashima et al., 1982a ,b ,c ; Yashima & Satoh, 1982 ; Lejay et al., 1983 ; Evers et al., 1983 ; Weigel et al., 1984 ; Sato et al., 1984 ; Zhong et al., 1985 ; Chevalier et al., 1986 ; Dhar et al., 1987 ).

The relationship between the large variety of the derivatives from AlB₂ and ThSi₂ aristotypes can be nicely explained within the group–subgroup scheme, also known as Bärnighausen formalism (Bärnighausen, 1980 ). The AlB₂ structure is one of the simplest inorganic structure types. It has hexagonal space group P6/mmm (No. 191) and its unit cell incorporates only the two Wyckoff sites 1a and 2d (Hofmann & Jäniche, 1935 ) occupied by one R atom on the Al site and two Si atoms on the B site, forming a two-dimensional Si network, similar to graphite. The unit cell of the ThSi₂ structure also has only two occupied Wyckoff positions (4a and 8e), but the Si sublattice forms a more complex 3D network (Brauer & Mittius, 1942 ).

Nowadays, 46 structure types derived from AlB₂ (Hoffmann & Pöttgen, 2001) and four from ThSi₂ are known. They include binary and ternary intermetallic compounds with compositions RX₂, RT₂, RTX or R₂TX₃, where X is an element of the third or fourth group.

In this work, we systematize the occurrence of RSi₂ and R₂TSi₃ compounds, where R = alkaline earth metal, lanthanide, actinide or member of the Sc group and T is a transition metal. We present 12 different structure types of these compounds derived from the AlB₂ type. Six of these structure types have not been considered by Hoffmann & Pöttgen (2001). Additionally, we present three further structure types based on the tetragonal ThSi₂ type. One of these types is purely hypothetical and considers the possibility of ordered Si/T positions in ThSi₂-like structures. Furthermore, we order all structure reports for RSi₂ and R₂TSi₃ compounds according to their R and T elements within an R–T grid. After analyzing all element combinations, we choose nine promising compounds not found in the literature and perform DFT calculations to evaluate the probability of a successful synthesis. We discuss peculiarities of the distribution of structure types among the RSi₂ and R₂TSi₃ compounds, based on a mapping of symmetries on the R–T grid with corresponding symbols.

2. Methods

To gain a comprehensive overview of RSi₂ and R₂TSi₃ compounds, we performed an extensive literature search by scanning the ICSD, SciFinder and Reaxys databases for all possible element combinations for T within the Cr to Zn groups and R within the Sc group, the alkaline earth metal, the lanthanides and the actinides. Only experiments at ambient conditions were considered. Additionally, we did not consider data sets if they were too incomplete, i.e. missing lattice parameters or an insufficient description of the symmetry. Additionally, we did not take incommensurately modulated structures into account, because these modulations mainly arise for nonstoichiometric disilicides within this family of compounds and because the descriptions do not conform with those of conventional symmetry. Please refer to Leisegang (2010 ), Kubata et al. (2005 ) and Dshemuchadse (2008 ) for further information. However, commensurable modulations are interpreted as superstructures.

Table 1 contains the tabulated data of the composition of the compounds as well as their structure parameters, i.e. lattice parameters a and c, ratios c/a, formula units per unit cell, and structure type. These data were used without further refinement. The compounds, discussed within this article, are more than solid solutions as most of them exhibit ordered structures and, therefore, have distinct structure types compared to similar stoichiometries. Within this article, only the formula units and the deviation of the compounds within the range of R and T elements is of interest. Part II (Nentwich et al., 2020 ) will discuss and compare other parameters.

Table 1
Alphabetically sorted list of RSi₂ and R₂TSi₃ compounds and their crystal data

R is an element of the alkaline earth metals, the scandium group, or the lanthanide or actinide series. T is a transition metal, Al or Si; thus a disilicide. The supercell can be identified by the formula units per unit cell. Lines written in blue indicate data sets not used for Fig. 9.

R	T	a (Å)	b (Å)	c (Å)	c/a	Formula units	Structure type	Thermal treatment	Reference	ICSD number
Am	Si	4.0190		13.6880	3.4058	4	ThSi₂	–	Weigel et al. (1977 )
		4.0150		13.7330	3.4204	4	ThSi₂	–	Weigel et al. (1984)	43816
Ba	Ag	8.6130	14.9270	19.6390	2.2802	16	Ba₄Li₂Si₆	550°C, 1.5 days	Cardoso Gil et al. (1999)	410520
Ca	Ag	8.3150	8.6460	14.3910	1.7307	8	Ca₂AgSi₃	550°C, 1.5 days	Cardoso Gil et al. (1999)	410522
	Ni	3.9880		4.3460	1.0898	1	AlB₂	–	Bodak & Gladyshevskii (1968 )	20300
	Si	4.2830		13.5200	3.1567	4	ThSi₂	–	Evers et al. (1977a)	1453
		4.2830		13.5200	3.1567	4	ThSi₂	–	Evers et al. (1978b)
		4.2832		13.5420	3.1617	4	ThSi₂	–	McWhan et al. (1967 )	87392
		4.2830		13.5300	3.1590	4	ThSi₂	–	Nakano & Yamanaka (1994 )
Ce	Au	4.2220		14.3750	3.4048	4	t	750°C, 14 days	Gordon et al. (1997)
		8.2840		8.7010	1.0503	8	h	750°C, 14 days	Gordon et al. (1997)
		8.3060		8.6870	1.0459	8	Er₂RhSi₃ (190/194)	Floating zone	Majumdar et al. (2000)
	Co	4.0440		4.1940	1.0371	1	AlB₂	–	Bodak & Gladyshevskii (1985)	52846
		8.1040		4.1970	0.5179	4	Ce₂CoSi₃/U₂RuSi₃	750°C, 14 days	Gordon et al. (1997)	83895
		8.1100		4.2200	0.5203	4	Ce₂CoSi₃/U₂RuSi₃	750°C, 7 days	Majumdar et al. (1999a)
		8.1130		4.2190	0.5200	4	Ce₂CoSi₃/U₂RuSi₃	Floating zone	Majumdar et al. (2000)
		8.0890		8.4020	1.0387	8	Er₂RhSi₃ (190/194)	800°C, 5 days	Patil et al. (2008 )
	Cu	4.0600		4.2800	1.0542	1	AlB₂	–	Bodak & Gladyshevskii (1985)
		4.0770		4.3140	1.0581	1	AlB₂	–	Gladyshevskii & Bodak (1965 )	20303
		4.0590		4.2940	1.0579	1	AlB₂	–	Hwang et al. (1996 )
		4.0580		4.2960	1.0586	1	AlB₂	850°C, 7 days	Lu et al. (2013)
		8.0920		4.2060	0.5198	4	Ce₂CoSi₃/U₂RuSi₃	850°C, 7 days	Lu et al. (2013)
		4.1360		4.2370	1.0244	1	AlB₂	–	Raman (1967)
		4.0650		4.3020	1.0583	1	AlB₂	–	Raman (1967)
		4.0640		4.3040	1.0591	1	AlB₂	800°C, 7 days	Yubuta et al. (2009 )
		8.1280		8.6080	1.0591	8	Er₂RhSi₃ (190/194)	800°C, 7 days	Yubuta et al. (2009)
	Fe	4.0680		4.1400	1.0177	1	AlB₂	–	Gladyshevskii & Bodak (1965)	20304
		4.0620		4.2120	1.0369	1	h	750°C, 14 days	Gordon et al. (1997)
	Ir	8.2120		4.2374	0.5160	4	Ce₂CoSi₃/U₂RuSi₃	–	Szlawska & Kaczorowski (2011)
	Ni	4.0390		4.2870	1.0614	1	AlB₂	–	Bodak & Gladyshevskii (1985)	621652
		4.0480		4.2910	1.0600	1	AlB₂	–	Dhar et al. (1994 )	658279
		4.0430		4.3020	1.0641	1	AlB₂	–	Gladyshevskii & Bodak (1965)	20302
		4.0406		4.2801	1.0593	1	h	750°C, 14 days	Gordon et al. (1997)
		4.0610		4.1490	1.0217	1	AlB₂	–	Raman (1967)
		4.0710		4.2020	1.0322	1	AlB₂	–	Raman (1967)
		4.0485		4.2887	1.0593	1	AlB₂	800°C, 7 days	Rojas et al. 2010 )
		4.0450		4.2830	1.0588	1	AlB₂	–	Szlawska & Kaczorowski (2012)	187100
	Pd	8.2631		17.1320	2.0733	16	h	750°C, 14 days	Gordon et al. (1997)
		8.2330		8.5650	1.0403	8	Er₂RhSi₃ (190/194)	750°C, 7 days	Mallik & Sampathkumaran (1996)
		4.1215		4.2723	1.0366	1	AlB₂	750°C, 5 days	Szytuła et al. (1999)
	Pt	8.2500		4.3320	0.5251	4	Ce₂CoSi₃/U₂RuSi₃	750°C, 14 days	Majumdar et al. (2001)
	Rh	8.2100		8.4100	1.0244	8	Er₂RhSi₃	800°C, 4 days	Chevalier et al. (1984)	621958
		8.2310		8.4391	1.0253	8	Er₂RhSi₃	–	Kase et al. (2009 )
		8.3270		8.5160	1.0227	8	Er₂RhSi₃ ( $[P\overline{6}2c]$ )	730°C, 4 days	Leciejewicz et al. (1995 )
		8.2370		8.4450	1.0253	8	Er₂RhSi₃ (190/194)	800°C, 5 days	Patil et al. (2008)
		8.2300		8.4400	1.0255	8	Er₂RhSi₃ (190/194)	800°C, 5 days	Sengupta et al. (2003 )
		8.2240		4.2261	0.5139	4	Ce₂CoSi₃/U₂RuSi₃	–	Szlawska et al. (2009)	164827
		8.2620		8.4390	1.0214	8	Er₂RhSi₃ ( $[P\overline{6}2c]$ )	800°C, 54 days	Szytuła et al. (1993)	106425
	Si	4.1900		13.9300	3.3246	4	ThSi₂	–	Benesovsky et al. (1966 )
		4.2700		13.8800	3.2506	4	ThSi₂	–	Binder (1960)
		4.1415		13.7816	3.3277	4	ThSi₂	–	Brauer & Haag (1950)	622204
		4.1560		13.8400	3.3301	4	ThSi₂	–	Brauer & Haag (1952)	25664
		4.1760		13.8480	3.3161	4	ThSi₂-like	1100°C, 14 days	Dhar et al. (1987)
		4.1910		13.8890	3.3140	4	ThSi₂-defect	1100°C, 14 days	Dhar et al. (1987)
		4.1940		13.9300	3.3214	4	ThSi₂	–	Dijkman et al. (1982 )	622206
		4.1900		13.9300	3.3246	4	ThSi₂-defect or Nd□_xSi_2−x	800°C, 1 day	Houssay et al. (1989 )
		4.1900		13.8800	3.3126	4	ThSi₂	–	Lahiouel et al. (1986 )	622197
		4.2700		13.8800	3.2506	4	ThSi₂	–	Lawrence et al. (1984 )	622190
		4.1900		13.9400	3.3270	4	ThSi₂	450°C, 0.5 days	Mayer et al. (1967)	622153
		4.1800		13.8900	3.3230	4	ThSi₂	–	Mayer & Eshdat (1968)
		4.1700		13.8200	3.3141	4	ThSi₂-defect	950°C, 7 days	Murashita et al. (1991 )
		4.1900		13.9200	3.3222	4	ThSi₂	950°C, 7 days	Murashita et al. (1991)
		4.2700		13.8800	3.2506	4	t	–	Perri et al. (1959b)
		4.1900		13.9200	3.3222	4	ThSi₂	–	Pierre et al. (1988)
		4.1500		13.8700	3.3422	4	ThSi₂	1000°C, 4 days	Raman & Steinfink (1967)
		4.1920		13.9030	3.3166	4	ThSi₂	–	Ruggiero & Olcese (1964 )	622138
		4.1780		13.8500	3.3150	4	ThSi₂-defect	1000°C, 3 days	Shaheen & Schilling (1987 )
		4.1880	4.1180	13.8800	3.3142	4	Nd□ _xSi_2−x	1000°C, 3 days	Shaheen & Schilling (1987)	622192
		4.1910		13.9490	3.3283	4	ThSi₂	1000°C, 3 days	Shaheen & Schilling (1987)	622192
		4.1890		13.8920	3.3163	4	ThSi₂	–	Weitzer et al. (1991 )	622175
		4.1840		13.8560	3.3117	4	ThSi₂	–	Yashima et al. (1982c)
		4.1600		13.9000	3.3413	4	ThSi₂	–	Zachariasen (1949 )	31642
Cm	Si	3.9630		13.7200	3.4620	4	ThSi₂	–	Weigel & Marquart (1983 )
Dy	Ni	3.9700		4.0130	1.0108	1	AlB₂	–	Mayer & Felner (1973b)	53369
	Pd	8.1110		8.0550	0.9931	8	h	–	Kotsanidis et al. (1990)
		4.0620		4.0310	0.9924	1	AlB₂	750°C, 10 days	Li et al. (2003)
		4.0620		4.0310	0.9924	1	AlB₂	750°C, 10 days	Nimori & Li (2006 )
		4.0612		4.0334	0.9932	1	AlB₂	750°C, 5 days	Szytuła et al. (1999)
	Pt	8.1000		8.2000	1.0123	8	Er₂RhSi₃ ( $[P\overline{6}2c]$ )	900°C, 23 days	Li et al. (2013)
	Rh	8.0970		7.8230	0.9662	8	Er₂RhSi₃	800°C, 4 days	Chevalier et al. (1984)	630163
	Si	4.0400	3.9500	13.3300	3.2995	4	GdSi₂	–	Binder (1960)
		3.8300		4.1100	1.0731	1	AlB₂	–	Gladyshevskii (1963 )	20248
		3.8310		4.1210	1.0757	1	AlB₂	700°C, 3 days	Iandelli et al. (1979)	630294
		3.8285	6.6312	4.1230	1.0769	2	Er₃□Si₅	1000°C, 10 days	Ji et al. (2004 )
		6.6338		4.1200	0.6211	3	Yb₃□Si₅	–	Knapp & Picraux (1985 )
		3.8310	6.6355	4.1210	1.0757	2	Er₃□Si₅	–	Koleshko et al. (1986 )	53382
		3.8300		4.1200	1.0757	1	AlB₂	450°C, 0.5 days	Mayer et al. (1967)	103369
		4.0450	3.9350	13.3190	3.2927	4	GdSi₂	–	Mayer & Eshdat (1968)	630287
		4.0300	3.9300	13.3200	3.3052	4	GdSi₂	–	Mayer & Eshdat (1968)
		4.0300	3.9310	13.3200	3.3052	4	GdSi₂	–	Mayer & Felner (1973b)
		3.9739		13.6760	3.4415	4	ThSi₂	–	Nesper et al. (1979 )	630314
		4.0400	3.9500	13.3400	3.3020	4	GdSi₂	–	Perri et al. (1959b)	630297
		4.0300		13.3800	3.3201	4	ThSi₂	–	Perri et al. (1959b)	150663
		4.0400	3.9500	13.3300	3.2995	4	GdSi₂	–	Perri et al. (1959a)	630297
		4.0380	3.9370	13.3100	3.2962	4	GdSi₂	–	Pierre et al. (1988)
Er	Cu	3.9670		13.7300	3.4611	4	ThSi₂	–	Raman (1967)	627257
	Ni	3.9600		3.9860	1.0066	1	AlB₂	–	Mayer & Felner (1973b)	53404
	Pd	4.0640		3.9910	0.9820	1	h	Floating zone	Frontzek (2009)
		8.0920		7.9250	0.9794	8	h	–	Kotsanidis et al. (1990)
		4.0427		3.9794	0.9843	1	AlB₂	750°C, 5 days	Szytuła et al. (1999)
	Rh	8.0780		8.7480	1.0829	8	Er₂RhSi₃	800°C, 4 days	Bazela et al. (2003)	97376
		8.0780		7.7480	0.9591	8	Er₂RhSi₃ ( $[P\overline{6}2c]$ )	800°C, 4 days	Bazela et al. (2003)	97375
		8.0360		7.7120	0.9597	8	Er₂RhSi₃ ( $[P\overline{6}2c]$ )	800°C, 4 days	Chevalier et al. (1984)	53413
		8.1130		7.7556	0.9559	8	Er₂RhSi₃	800°C, 14 days	Gladyshevskii et al. (1992)	300248
	Si	3.7930	6.5697	4.0820	1.0762	2	Er₃□Si₅	–	Auffret et al. (1990)
		3.7990		4.0890	1.0763	1	AlB₂	–	Gladyshevskii (1963)	20250
		3.7980		4.0880	1.0764	1	AlB₂	700°C, 3 days	Iandelli et al. (1979)	631146
		3.7990	6.5801	4.0895	1.0765	2	Er₃□Si₅	1000°C, 10 days	Ji et al. (2004)
		6.5818		4.0900	0.6214	3	Yb₃□Si₅	–	Knapp & Picraux (1985)
		3.7990	6.5801	4.0900	1.0766	2	Er₃□Si₅	–	Koleshko et al. (1986)	631159
		3.7800		4.0900	1.0820	1	AlB₂	–	Mayer et al. (1962)	631151
		3.7800		4.0800	1.0794	1	AlB₂	450°C, 0.5 days	Mayer et al. (1967)	631140
		3.7850		4.0800	1.0779	1	AlB₂	700°C, 2 days	Mayer & Felner (1972)	631144
		3.8000		4.0900	1.0763	1	AlB₂	–	Mayer & Felner (1973b)	631153
		3.9370		13.6160	3.4585	4	ThSi₂	–	Nesper et al. (1979)	631164
		3.7920		4.0830	1.0767	1	AlB₂	–	Pierre et al. (1988)	631150
		3.8000		4.0900	1.0763	1	AlB₂	–	Sekizawa & Yasukouchi (1966 )	631155
		6.5783		8.1760	1.2429	6	Tb₃□Si₅	700°C, 0 days	Tsai et al. (2005 )
Eu	Ag	8.4200	14.8580	17.8640	2.1216	16	Ba₄Li₂Si₆	900°C, 3 days	Cardoso Gil et al. (1999)	410521
		4.1500		4.5150	1.0880	1	AlB₂	–	Mayer & Felner (1973a)	58453
		8.3060	9.0369	14.3770	1.7309	8	Ca₂AgSi₃	800°C, 5 days	Sarkar et al. (2013)	250524
	Co	4.0460		4.5000	1.1122	1	AlB₂	–	Mayer & Felner (1973a)	102379
	Cu	4.0762		4.4895	1.1014	1	AlB₂-like	Floating zone	Cao et al. (2010 , 2011 )
		8.1890		8.9760	1.0961	8	Er₂RhSi₃ (190/194)	800°C,	Majumdar et al. (1998)
		4.0950		4.4880	1.0960	1	AlB₂	–	Majumdar et al. (1999b)
		4.0800		4.4660	1.0946	1	AlB₂	–	Mayer & Felner (1973a)	53255
	Ni	4.0340		4.4960	1.1145	1	AlB₂	–	Mayer & Felner (1973a)	53436
	Pd	8.3188		4.3588	0.5240	4	Ce₂CoSi₃/U₂RuSi₃	750°C, 7 days	Rodewald et al. (2003 )	391246
	Si	4.2900		13.3300	3.1072	4	ThSi₂	–	Binder (1960)	631674
		4.3040		13.6500	3.1715	4	ThSi₂	–	Evers et al. (1977a)	1454
		4.3030		13.6600	3.1745	4	ThSi₂	–	Evers et al. (1983)
		4.0520		4.4820	1.1061	1	AlB₂	–	Nesper et al. (1979)	103436
		4.2970		13.7040	3.1892	4	ThSi₂	–	Nesper et al. (1979)	631683
		4.2900		13.6600	3.1841	4	t	–	Perri et al. (1959b)
Gd	Pd	4.0790		4.0980	1.0047	1	h	Floating zone	Frontzek (2009)
		8.1580		8.1180	0.9951	8	h	750°C, 5 days	Kotsanidis et al. (1990)
	Pt	8.1390		8.3030	1.0201	8	Er₂RhSi₃ (190/194)	750°C, 14 days	Majumdar et al. (2001)
	Rh	8.1120		7.9760	0.9832	8	Er₂RhSi₃	800°C, 4 days	Chevalier et al. (1984)	636281
		8.1120		7.9760	0.9832	8	Er₂RhSi₃	–	Mulder et al. (1998 )
	Si	4.0920	4.0130	13.4370	3.2837	4	Nd□Si_2−x	800°C, 1 day	Auffret et al. (1991 )
		4.0900	4.0100	13.4400	3.2861	4	GdSi₂	–	Binder (1960)
		4.0200	4.1000	13.4300	3.3408	4	Nd□_xSi_2−x	800°C, 1 day	Houssay et al. (1989)	636419
		3.8770		4.1720	1.0761	1	AlB₂	700°C, 3 days	Iandelli et al. (1979)	636432
		6.7204		4.1700	0.6205	3	Yb₃□Si₅	–	Knapp & Picraux (1985)
		3.8770	6.7152	4.1720	1.0761	2	Er₃□Si₅	–	Koleshko et al. (1986)	53633
		3.8700		4.1700	1.0775	1	AlB₂	450°C, 0.5 days	Mayer et al. (1967)	636421
		4.0800	4.0100	13.4200	3.2892	4	GdSi₂	–	Mayer & Eshdat (1968)
		3.8690	6.7013	4.1820	1.0809	2	Er₃□Si₅	800°C, 14 days	Mulder et al. (1994 )	658032
		3.8525		4.1470	1.0764	1	AlB₂	–	Nesper et al. (1979)	636450
		4.0438		13.8020	3.4131	4	ThSi₂	–	Nesper et al. (1979)	636452
		4.1000	4.0100	13.6100	3.3195	4	o	–	Perri et al. (1959b)	150661
		4.0900	4.0100	13.4400	3.2861	4	GdSi₂	–	Perri et al. (1959a)
		4.0900	4.0100	13.4400	3.2861	4	GdSi₂	–	Pierre et al. (1988)	636434
		4.0930	4.0090	13.4400	3.2837	4	Nd□_xSi_2−x	–	Pierre et al. (1990)
		4.0800	3.9960	13.4100	3.2868	4	GdSi₂	1000°C, 4 days	Raman & Steinfink (1967)
		4.0900	4.0100	13.4200	3.2812	4	GdSi₂	–	Sekizawa & Yasukouchi (1966)	636440
Ho	Pd	8.1520		32.1680	3.9460	32	h	Floating zone	Frontzek (2009)
		8.1010		7.9960	0.9870	8	h	750°C, 5 days	Kotsanidis et al. (1990)
		8.0994		32.0192	3.9533	32	h	Floating zone	Leisegang (2010)
		8.1072		8.1072	1.0000	8	Er₂RhSi₃ (190/194)	800°C, 7 days	Mo et al. (2015 )	192586
		4.0459		3.9977	0.9881	1	AlB₂	750°C, 5 days	Szytuła et al. (1999)
		8.1000		32.0000	3.9506	32	Ho₂PdSi₃	Floating zone	Tang et al. (2011)
		4.0460		3.9977	0.9881	4	Ce₂CoSi₃/U₂RuSi₃	750°C, 5 days	Zajdel et al. (2015)
	Rh	8.0860		7.8040	0.9651	8	Er₂RhSi₃	800°C, 4 days	Bazela et al. (2003)	97374
		8.0860		7.8040	0.9651	8	Er₂RhSi₃ ( $[P\overline{6}2c]$ )	800°C, 4 days	Bazela et al. (2003)	97373
		8.0720		7.7710	0.9627	8	Er₂RhSi₃	800°C, 4 days	Chevalier et al. (1984)	639636
	Si	3.8070	6.5939	4.1060	1.0785	2	Er₃□Si₅	800°C, 1 day	Auffret et al. (1991)
		4.0290	3.9170	13.2770	3.2954	4	Nd□_xSi_2−x	800°C, 1 day	Auffret et al. (1991)
		3.8087	6.5969	4.1030	1.0773	2	Er₃□Si₅	1100°C, 8 days	Eremenko et al. (1995 )
		4.0230	3.9140	13.2820	3.3015	4	Nd□_xSi_2−x	1100°C, 8 days	Eremenko et al. (1995)
		3.8160		4.1070	1.0763	1	AlB₂	–	Gladyshevskii (1963)	20249
		3.8160		4.1070	1.0763	1	AlB₂	700°C, 3 days	Iandelli et al. (1979)	639729
		3.8100	6.5991	4.1035	1.0770	2	Er₃□Si₅	1000°C, 10 days	Ji et al. (2004)
		6.5991		4.1100	0.6228	3	Yb₃□Si₅	–	Knapp & Picraux (1985)
		3.8160	6.6095	4.1070	1.0763	2	Er₃□Si₅	–	Koleshko et al. (1986)	639748
		4.0300	3.9700	13.3100	3.3027	4	o	–	Mayer et al. (1962)
		3.8000		4.1000	1.0789	1	AlB₂	450°C, 0.5 days	Mayer et al. (1967)	56250
		3.9610		13.6450	3.4448	4	ThSi₂	–	Nesper et al. (1979)	639750
		4.0150	3.9060	13.2200	3.2927	4	GdSi₂	–	Pierre et al. (1988)	639731
		3.9900	3.9400	13.3000	3.3333	4	GdSi₂	–	Sekizawa & Yasukouchi (1966)	639743
		4.0280	3.9120	13.2870	3.2987	4	GdSi₂	–	Weitzer et al. (1991)
		4.0100	3.9120	13.2550	3.3055	4	GdSi₂	–	Weitzer et al. (1991)
La	Al	4.3030		14.2100	3.3023	4	ThSi₂	1000°C, 4 days	Raman & Steinfink (1967)
	Co	4.1880		4.3660	1.0425	1	AlB₂	–	Bodak & Gladyshevskii (1985)
		8.1850		4.3500	0.5315	4	Ce₂CoSi₃/U₂RuSi₃	750°C, 7 days	Majumdar et al. (1999a)
	Cu	4.0840		4.3950	1.0762	1	AlB₂	–	Hwang et al. (1996)
		4.1440		4.2860	1.0343	1	AlB₂	–	Raman (1967)	103037
		4.0710		4.3830	1.0766	1	AlB₂	–	Raman (1967)
		4.0840		4.3950	1.0762	1	AlB₂	–	Tien et al. (1997 )
	Fe	4.0800		4.3500	1.0662	1	AlB₂	–	Bodak & Gladyshevskii (1985)
		4.0690		4.1010	1.0079	1	AlB₂	–	Raman (1967)
		4.0970		4.3310	1.0571	1	AlB₂	–	Raman (1967)
	Ni	4.0930		4.3540	1.0638	1	AlB₂	–	Bodak & Gladyshevskii (1985)
		4.0770		4.3670	1.0711	1	AlB₂	–	Gladyshevskii & Bodak (1965)	20305
		4.0450		4.3810	1.0831	1	AlB₂	700°C, 2 days	Mayer & Felner (1972)	641574
		4.0770		4.3000	1.0547	1	AlB₂	–	Raman (1967)
		4.0570		4.3880	1.0816	1	AlB₂	–	Raman (1967)
		4.0711		4.3737	1.0743	1	AlB₂	800°C, 7 days	Rojas et al. (2010)
		4.0689		4.3753	1.0753	1	AlB₂	–	Szlawska & Kaczorowski (2012)
	Pt	8.2900		4.4170	0.5328	4	Ce₂CoSi₃/U₂RuSi₃	750°C, 14 days	Majumdar et al. (2001)
	Rh	8.2330		8.5940	1.0438	8	Er₂RhSi₃	800°C, 4 days	Chevalier et al. (1984)	641751
		8.2800		8.6500	1.0447	8	Er₂RhSi₃ (190/194)	800°C, 5 days	Sengupta et al. (2003)
	Si	4.3700		13.5600	3.1030	4	ThSi₂	–	Bertaut & Blum (1950 )	174010
		4.3100		13.2800	3.0812	4	ThSi₂	–	Binder (1960)
		4.2612		13.7118	3.2178	4	ThSi₂	–	Brauer & Haag (1950)	641982
		4.2810		13.7500	3.2119	4	ThSi₂	–	Brauer & Haag (1952)	25663
		4.3300		13.8300	3.1940	4	ThSi₂-defect	800°C, 1 day	Houssay et al. (1989)	641955
		4.3100		13.8000	3.2019	4	ThSi₂	–	Lawrence et al. (1984)	641973
		4.1900	4.2700	13.9400	3.3270	4	GdSi₂	450°C, 0.5 days	Mayer et al. (1967)	641958
		4.2900		13.8700	3.2331	4	ThSi₂	–	Mayer & Eshdat (1968)	641961
		4.3260		13.8400	3.1993	4	ThSi₂	–	Nakano & Yamanaka (1994)	78028
		4.3100		13.8000	3.2019	4	t	–	Perri et al. (1959b)
		4.3000		13.8400	3.2186	4	ThSi₂	–	Pierre et al. (1988)
		4.3050		13.8400	3.2149	4	ThSi₂	1000°C, 4 days	Raman & Steinfink (1967)
Lu	Pd	4.0267		3.9218	0.9739	1	AlB₂	Floating zone	Cao et al. (2013 , 2014 )	250596, 250597
	Si	3.7450		4.0500	1.0814	1	AlB₂	–	Gladyshevskii (1963)	20253
		3.7470		4.0460	1.0798	1	AlB₂	700°C, 3 days	Iandelli et al. (1979)	642610
		6.4952		4.0500	0.6235	3	Yb₃□Si₅	–	Knapp & Picraux (1985)
		3.7450	6.4865	4.0500	1.0814	2	Er₃□Si₅	–	Koleshko et al. (1986)	642613
		3.7400		4.0400	1.0802	1	AlB₂	–	Mayer et al. (1962)	642611
		3.7500		4.0500	1.0800	1	AlB₂	450°C, 0.5 days	Mayer et al. (1967)	642607
Nd	Ag	4.1750		14.3100	3.4275	4	ThSi₂	–	Mayer & Felner (1973b)	605613
	Cu	8.0760		8.4400	1.0451	8	Er₂RhSi₃ (190/194)	800°C, 7 days	Yubuta et al. (2009)
	Ni	4.0420		4.1630	1.0299	1	AlB₂	–	Gladyshevskii & Bodak (1965)	20307
		4.0130		4.2020	1.0471	1	AlB₂	700°C, 2 days	Mayer & Felner (1972)	76594
		4.0200		4.2070	1.0465	1	AlB₂	–	Mayer & Felner (1973b)	645635
	Pd	8.1970		8.4020	1.0250	8	h	750°C, 5 days	Kotsanidis et al. (1990)
		4.1050		4.2040	1.0241	1	AlB₂	750°C, 10 days	Li et al. (2003)
		4.1033		4.2039	1.0245	1	AlB₂	750°C, 5 days	Szytuła et al. (1999)
	Pt	4.0927		4.2582	1.0404	1	AlB₂	900°C, 23 days	Li et al. (2001)
		8.2170		4.2820	0.5211	4	Ce₂CoSi₃/U₂RuSi₃	750°C, 14 days	Majumdar et al. (2001)
	Rh	8.1860		8.2720	1.0105	8	Er₂RhSi₃	800°C, 4 days	Chevalier et al. (1984)	645781
		8.1710		8.2760	1.0129	8	Er₂RhSi₃ ( $[P\overline{6}2c]$ )	800°C, 54 days	Szytuła et al. (1993)	57432
	Si	4.1800	4.1500	13.5600	3.2440	4	GdSi₂	–	Binder (1960)
		4.1016		13.4223	3.2725	4	ThSi₂	–	Brauer & Haag (1950)	645987
		4.1110		13.5600	3.2985	4	ThSi₂	–	Brauer & Haag (1952)	25666
		4.1600	4.2000	13.6000	3.2692	4	Nd□_xSi_2−x	800°C, 1 day	Houssay et al. (1989)	645941
		4.1800	4.1500	13.5600	3.2440	4	GdSi₂	–	Lawrence et al. (1984)
		4.1700	4.1300	13.6500	3.2734	4	GdSi₂	450°C, 0.5 days	Mayer et al. (1967)	645948
		4.1800	4.1600	13.6300	3.2608	4	GdSi₂	–	Mayer & Eshdat (1968)
		4.1800	4.1500	13.5600	3.2440	4	GdSi₂	–	Mayer & Felner (1973b)
		4.1650		13.6420	3.2754	4	GdSi₂	–	Nesper et al. (1979)	645989
		4.1110		13.5600	3.2985	4	ThSi₂	–	Perri et al. (1959b)	645972
		4.1740	4.1540	13.6100	3.2607	4	GdSi₂	–	Pierre et al. (1988)	645963
		3.9480	6.8381	4.2690	1.0813	2	Er₃□Si₅	–	Pierre et al. (1990)
		4.1350	4.1010	13.7400	3.3229	4	Nd□_xSi_2−x	–	Pierre et al. (1990)
		4.1470	4.1250	13.6700	3.2964	4	Nd□_xSi_2−x	–	Pierre et al. (1990)
		4.1620		13.5800	3.2629	4	ThSi₂	1000°C, 4 days	Raman & Steinfink (1967)	645949
		4.1620		13.5800	3.2629	4	ThSi₂	–	Raman (1968 )	645985
		4.1850	4.1600	13.6100	3.2521	4	GdSi₂	1050°C, 10 days	Schobinger-Papamantellos et al. (1991 )
Np	Si	3.9680		13.7150	3.4564	4	ThSi₂	–	Yaar et al. (1992 )	657647
		3.9700		13.7000	3.4509	4	ThSi₂	–	Zachariasen (1949)	31644
Pr	Cu	4.0520		4.2550	1.0501	1	AlB₂	–	Tien et al. (1997)
		4.0420		4.2050	1.0403	1	AlB₂	900°C, 20 days	Wang et al. (2014 )
	Ni	4.0450		4.2260	1.0447	1	AlB₂	–	Gladyshevskii & Bodak (1965)	20306
		4.0210		4.0250	1.0010	1	AlB₂	–	Mayer & Felner (1973b)	646272
	Pd	8.2210		8.4660	1.0298	8	h	750°C, 5 days	Kotsanidis et al. (1990)
		4.0250		4.2070	1.0452	1	AlB₂	Floating zone	Xu et al. (2010)
	Pt	8.2300		4.3000	0.5225	4	Ce₂CoSi₃/U₂RuSi₃	750°C, 14 days	Majumdar et al. (2001)
	Si	4.2000		13.7600	3.2762	4	ThSi₂	–	Binder (1960)	649371
		4.2100		13.7300	3.2613	4	ThSi₂-defect	800°C, 5 days	Boutarek et al. (1994 )	658012
		4.1315		13.4922	3.2657	4	ThSi₂	–	Brauer & Haag (1950)
		4.1480		13.6700	3.2956	4	ThSi₂	–	Brauer & Haag (1952)	25665
		4.2100		13.7300	3.2613	4	ThSi₂-defect	800°C, 1 day	Houssay et al. (1989)	649364
		4.2900		13.7600	3.2075	4	ThSi₂	–	Lawrence et al. (1984)
		4.1700	4.1200	13.8200	3.3141	4	GdSi₂	450°C, 0.5 days	Mayer et al. (1967)	649365
		4.1600		13.7600	3.3077	4	ThSi₂	–	Mayer & Eshdat (1968)
		4.2900		13.7600	3.2075	4	ThSi₂	–	Mayer & Felner (1973b)
		4.2000		13.7600	3.2762	4	ThSi₂	–	Perri et al. (1959b)	649376
		4.2000		13.7600	3.2762	4	ThSi₂	–	Perri et al. (1959a)
		4.1840		13.7300	3.2815	4	ThSi₂	–	Pierre et al. (1988)
		4.2000		13.7300	3.2690	4	ThSi₂-defect	–	Pierre et al. (1990)
Pu	Si	3.9670		13.7200	3.4585	4	ThSi₂	–	Coffinberry & Ellinger (1955 )	649973
		3.8750	6.7117	4.1020	1.0586	2	Er₃□Si₅	840°C, 42 days	Land et al. (1965 )
		3.9680		13.7100	3.4551	4	ThSi₂	–	Land et al. (1965)	649969
		3.8840		4.0820	1.0510	1	AlB₂	–	Runnals & Boucher (1955 )	44867
		3.9800		13.5800	3.4121	4	ThSi₂	–	Zachariasen (1949)	31645
Sc	Si	3.6600	6.3393	3.8700	1.0574	2	Er₃□Si₅	–	Gladyshevskii & Émes-Misenko (1963 )
		3.6600	6.3393	3.8700	1.0574	2	Er₃□Si₅	–	Koleshko et al. (1986)	651822
		3.6620	6.3428	3.8790	1.0593	2	Er₃□Si₅	–	Kotroczo & McColm (1994 )
		3.6620	6.3428	3.8790	1.0593	2	Er₃□Si₅	–	Kotroczo & McColm (1994)	657975
		3.6600		3.8700	1.0574	1	AlB₂	–	Nörenberg et al. (2006 )
Sm	Ni	4.0020		4.1600	1.0395	1	AlB₂	–	Gladyshevskii & Bodak (1965)	20308
	Si	4.1050	4.0350	13.4600	3.2789	4	GdSi₂	–	Binder (1960)
		4.0417		13.3126	3.2938	4	ThSi₂	–	Brauer & Haag (1950)
		4.0490		13.3600	3.2996	4	ThSi₂	–	Brauer & Haag (1952)	25667
		4.1100	4.0600	13.4900	3.2822	4	GdSi₂	–	Mayer & Eshdat (1968)	652268
		4.1050	4.0350	13.4600	3.2789	4	o	–	Perri et al. (1959b)	652273
		4.0800		13.5100	3.3113	4	ThSi₂	–	Perri et al. (1959b)	652274
		4.1040	4.0350	13.4600	3.2797	4	GdSi₂	–	Perri et al. (1959a)
Sr	Au	8.3407	9.2664	14.4465	1.7320	8	Ca₂AgSi₃	650°C, 7 days	Zeiringer et al. (2015 )
	Ni	4.0690		4.6630	1.1460	1	AlB₂	–	Bodak & Gladyshevskii (1968)	20301
	Si	4.4380		13.8300	3.1163	4	ThSi₂	–	Evers et al. (1977a)	1455
	Si	4.4380		13.8300	3.1163	4	ThSi₂	–	Evers et al. (1978b)
		4.4290		13.8420	3.1253	4	ThSi₂	–	Palenzona & Pani (2004 )	99238
		4.4390		13.8380	3.1174	4	ThSi₂	–	Palenzona & Pani (2004)
Tb	Pd	4.0480		4.0370	0.9973	1	h	Floating zone	Frontzek (2009)
		8.1210		8.1000	0.9974	8	h	750°C, 5 days	Kotsanidis et al. (1990)
		4.0650		4.0520	0.9968	1	AlB₂	750°C, 10 days	Li et al. (2003)
		4.0643		4.0502	0.9965	1	AlB₂	750°C, 5 days	Szytuła et al. (1999)
	Pt	8.1223		8.2368	1.0141	8	Er₂RhSi₃ ( $[P\overline{6}2c]$ )	900°C, 23 days	Li et al. (2002a)
	Rh	8.1100		7.8600	0.9692	8	Er₂RhSi₃	800°C, 4 days	Chevalier et al. (1984)	650328
		8.1400		7.8120	0.9597	8	Er₂RhSi₃ ( $[P\overline{6}2c]$ )	800°C, 54 days	Szytuła et al. (1993)	57483
	Si	3.8460	6.6615	4.1430	1.0772	2	Er₃□Si₅	800°C, 1 day	Auffret et al. (1991)
		4.0570	3.9650	13.3770	3.2973	4	Nd□_xSi_2−x	800°C, 1 day	Auffret et al. (1991)
		3.8470		4.1460	1.0777	1	AlB₂	–	Gladyshevskii (1963)	20247
		3.8470		4.1460	1.0777	1	AlB₂	700°C, 3 days	Iandelli et al. (1979)	652359
		6.6684		4.1500	0.6223	3	Yb₃□Si₅	–	Knapp & Picraux (1985)
		3.8470	6.6632	4.1460	1.0777	2	Er₃□Si₅	–	Koleshko et al. (1986)	652375
		4.0450	3.9600	13.3800	3.3078	4	o	–	Mayer et al. (1962)
		3.8400		4.1400	1.0781	1	AlB₂	450°C, 0.5 days	Mayer et al. (1967)	652354
		3.9600	4.0500	13.3800	3.3788	4	GdSi₂	–	Mayer & Eshdat (1968)	652355
		3.9902		13.6920	3.4314	4	ThSi₂	–	Nesper et al. (1979)	652377
		4.0500	3.9650	13.3600	3.2988	4	GdSi₂	–	Pierre et al. (1988)	652360
		4.0400	3.9600	13.3900	3.3144	4	GdSi₂	–	Sekizawa & Yasukouchi (1966)	652370
Th	Au	4.1972		14.3030	3.4077	4	ThSi₂	800°C, 7 days	Albering et al. (1994)	658096
	Co	4.0520		4.1510	1.0244	1	AlB₂	800°C, 7 days	Albering et al. (1994)	658085
		4.0430		4.1890	1.0361	1	AlB₂	950°C, 8 days	Zhong et al. (1985)	53078
	Cu	4.0230		4.1910	1.0418	1	AlB₂	800°C, 7 days	Albering et al. (1994)	108410
	Fe	4.0993		14.1850	3.4603	4	ThSi₂	800°C, 7 days	Albering et al. (1994)	658089
	Ir	4.1366		14.3640	3.4724	4	ThSi₂	800°C, 7 days	Albering et al. (1994)	658094
		4.1200		14.3100	3.4733	4	ThSi₂	–	Lejay et al. (1983), Chevalier et al. (1986)
	Mn	4.1069		14.1130	3.4364	4	ThSi₂	800°C, 7 days	Albering et al. (1994)	658088
	Ni	4.0322		4.1891	1.0389	1	AlB₂	800°C, 7 days	Albering et al. (1994)	54299
	Os	4.1384		14.3784	3.4744	4	ThSi₂	800°C, 7 days	Albering et al. (1994)	658093
	Pd	4.1570		14.2820	3.4357	4	ThSi₂	800°C, 7 days	Albering et al. (1994)	658092
	Pt	4.1592		14.2850	3.4346	4	ThSi₂	800°C, 7 days	Albering et al. (1994)	658095
	Rh	4.1241		14.3870	3.4885	4	ThSi₂	800°C, 7 days	Albering et al. (1994)	658091
		4.1100		14.3200	3.4842	4	ThSi₂	–	Lejay et al. (1983), Chevalier et al. (1986)
	Ru	4.1242		14.4470	3.5030	4	ThSi₂	800°C, 7 days	Albering et al. (1994)	658090
	Si	4.1180		14.2210	3.4534	4	ThSi₂	–	Benesovsky et al. (1966)
		4.1340		14.3750	3.4773	4	ThSi₂	–	Brauer & Mittius (1942)	77320
		4.1260		14.3460	3.4770	4	ThSi₂	–	Brauer & Mittius (1942)	660234
		4.1360		4.1260	0.9976	1	AlB₂	–	Brown & Norreys (1959 )
		3.9850	6.9022	4.2280	1.0610	2	Er₃□Si₅	–	Brown & Norreys (1959)
		4.1360		4.1260	0.9976	1	AlB₂	–	Brown (1961 )	15449
		4.1350		14.3750	3.4764	4	ThSi₂	–	Brown (1961)	652390
		3.9850		4.2200	1.0590	1	AlB₂	–	Jacobson et al. (1956 )	26569
		4.1270		14.1940	3.4393	4	ThSi₂	950°C, 8 days	Zhong et al. (1985)
Tm	Pd	4.0570		3.9700	0.9786	1	h	Floating zone	Frontzek (2009)
		8.0710		7.8500	0.9726	8	h	750°C, 5 days	Kotsanidis et al. (1990)
	Si	3.7730		4.0700	1.0787	1	AlB₂	–	Gladyshevskii (1963)	20251
		3.7680		4.0700	1.0801	1	AlB₂	700°C, 3 days	Iandelli et al. (1979)	52468
		6.5298		4.0700	0.6233	3	Yb₃□Si₅	–	Knapp & Picraux (1985)
		3.7730	6.5350	4.0700	1.0787	2	Er₃□Si₅	–	Koleshko et al. (1986)	604540
		3.7600		4.0700	1.0824	1	AlB₂	–	Mayer et al. (1962)	652455
		3.7700		4.0700	1.0796	1	AlB₂	450°C, 0.5 days	Mayer et al. (1967)	652451
U	Au	4.1450		3.9890	0.9624	1	AlB₂	800°C, 60 days	Chevalier et al. (1996)
		4.1450		3.9890	0.9624	1	AlB₂	800°C, 8 days	Pöttgen & Kaczorowski (1993)	106295
	Co	3.9870		3.8830	0.9739	1	AlB₂	800°C, 60 days	Chevalier et al. (1996)
		3.9880		3.8830	0.9737	1	AlB₂	800°C, 10 days	Kaczorowski & Noël (1993)	106494
		3.9880		3.8830	0.9737	1	AlB₂	800°C, 8 days	Pöttgen & Kaczorowski (1993)
		3.9765		3.8980	0.9803	1	AlB₂	–	Szlawska et al. (2011)
	Cu	3.9710		13.9260	3.5069	4	ThSi₂	800°C, 10 days	Kaczorowski & Noël (1993)	603112
		4.0090		3.9570	0.9870	1	AlB₂	600°C, 49 days	Pechev et al. (2000 )	92357
		3.9710		13.9260	3.5069	4	ThSi₂	800°C, 8 days	Pöttgen & Kaczorowski (1993)	602804
	Fe	4.0030		3.8570	0.9635	1	AlB₂	800°C, 60 days	Chevalier et al. (1996)
		4.0040		3.8640	0.9650	1	AlB₂	800°C, 10 days	Kaczorowski & Noël (1993)	603109
		4.0100		3.8400	0.9576	1	AlB₂	800°C, 7 days	Lourdes Pinto (1966 )	53551
		4.0040		3.8640	0.9650	1	AlB₂	800°C, 8 days	Pöttgen & Kaczorowski (1993)
		8.0030		3.8540	0.4816	4	Ce₂CoSi₃/U₂RuSi₃	800°C, 10 days	Yamamura et al. (2006 )
	Ir	4.0650		3.9140	0.9629	1	AlB₂	800°C, 60 days	Chevalier et al. (1996)
		4.0720		3.8950	0.9565	1	AlB₂	800°C, 8 days	Pöttgen & Kaczorowski (1993)	57398
		4.0830		3.9320	0.9630	1	AlB₂-like	800°C, 7 days	Yubuta et al. (2006 )
		4.0900		3.8540	0.9423	1	AlB₂-like	800°C, 7 days	Yubuta et al. (2006)
	Mn	8.0450		3.8082	0.4734	4	Ce₂CoSi₃/U₂RuSi₃	800°C, 60 days	Chevalier et al. (1996)
	Ni	3.9790		3.9460	0.9917	1	AlB₂	800°C, 60 days	Chevalier et al. (1996)
		3.9790		3.9490	0.9925	1	AlB₂	800°C, 10 days	Kaczorowski & Noël (1993)	54300
		3.9790		3.9490	0.9925	1	AlB₂	800°C, 8 days	Pöttgen & Kaczorowski (1993)
		3.9720		3.9461	0.9935	1	AlB₂	–	Schröder et al. (1995 )
	Os	8.1600		3.8440	0.4711	4	Ce₂CoSi₃/U₂RuSi₃	800°C, 60 days	Chevalier et al. (1996)
		4.0666		3.8517	0.9472	1	AlB₂	800°C, 8 days	Pöttgen & Kaczorowski (1993)	57453, 54310
		8.1600		3.8440	0.4711	4	Ce₂CoSi₃/U₂RuSi₃	800°C, 60 days	Pöttgen et al. (1994)
	Pd	4.0800	7.0670	3.9390	0.9654	2	U₂RhSi₃	800°C, 60 days	Chevalier et al. (1996)	57172
		4.0830		3.9320	0.9630	1	AlB₂	800°C, 3 days	Li et al. (1998b)
		4.0850		3.9350	0.9633	1	AlB₂	800°C, 8 days	Pöttgen & Kaczorowski (1993)	57467
	Pt	4.0730		3.9650	0.9735	1	AlB₂	800°C, 60 days	Chevalier et al. (1996)
		4.0840		3.9730	0.9728	1	AlB₂	800°C, 10 days	Kaczorowski & Noël (1993)
		4.0810		3.9700	0.9728	1	AlB₂	800°C, 10 days	Li et al. (1997)
		4.0670		3.9640	0.9747	1	AlB₂	800°C, 8 days	Pöttgen & Kaczorowski (1993)	602802
		4.0840		3.9730	0.9728	1	AlB₂	850°C, 5 days	Sato et al. (1991 )	54345
		4.0840		3.9730	0.9728	1	AlB₂	–	Sato et al. (1992 )
		4.0730		3.9600	0.9723	1	AlB₂	800°C, 10 days	Yamamura et al. (2006)
	Rh	4.0620	7.0360	3.9290	0.9673	2	U₂RhSi₃	800°C, 60 days	Chevalier et al. (1996)	57171
		4.0740		3.8810	0.9526	1	AlB₂	800°C, 3 days	Li et al. (1999)
		4.0760		3.8830	0.9526	1	AlB₂	800°C, 8 days	Pöttgen & Kaczorowski (1993)	57485
		8.1011		3.9477	0.4873	4	Ce₂CoSi₃	–	Szlawska et al. (2016)
	Ru	8.1480		3.8550	0.4731	4	Ce₂CoSi₃/U₂RuSi₃	800°C, 60 days	Chevalier et al. (1996)
		4.0750		3.8380	0.9418	1	AlB₂	800°C, 8 days	Pöttgen & Kaczorowski (1993)	108727
		8.1450		3.8496	0.4726	4	Ce₂CoSi₃/U₂RuSi₃	800°C, 60 days	Pöttgen et al. (1994)	78530
		8.1480		3.8550	0.4731	4	Ce₂CoSi₃/U₂RuSi₃	800°C, 60 days	Pöttgen et al. (1994)
	Si	3.8600		4.0700	1.0544	1	AlB₂	–	Benesovsky et al. (1966)
		3.9500		13.6800	3.4633	4	ThSi₂	–	Benesovsky et al. (1966)
		3.8520		4.0280	1.0457	1	AlB₂	–	Brown & Norreys (1959)	652472, 52469
		3.8430	6.6563	4.0690	1.0588	2	Er₃□Si₅	–	Brown & Norreys (1959)
		3.8520		4.0280	1.0457	1	AlB₂	650°C	Brown & Norreys (1961)
		3.8430	6.6563	4.0690	1.0588	2	Er₃□Si₅	650°C	Brown & Norreys (1961)
		3.8390		4.0720	1.0607	1	AlB₂	–	Dwight (1982 )	106053
		3.8390		4.7200	1.2295	1	AlB₂	–	Dwight (1982)	652476
		3.9220		14.1540	3.6089	4	ThSi₂	–	Sasa & Uda (1976 )	203
		3.8600		4.0700	1.0544	1	AlB₂	–	Zachariasen (1949)	31646
		3.9800		13.7400	3.4523	4	ThSi₂	–	Zachariasen (1949)	31643
Y	Pd	8.1380		8.0410	0.9881	8	h	750°C, 5 days	Kotsanidis et al. (1990)
		8.0910		8.0920	1.0001	8	Er₂RhSi₃ (190/194)	750°C, 7 days	Mallik & Sampathkumaran (1996)
	Pt	8.0990		8.1940	1.0117	8	Er₂RhSi₃ (190/194)	750°C, 14 days	Majumdar et al. (2001)
	Rh	8.0860		7.8290	0.9682	8	Er₂RhSi₃	800°C, 4 days	Chevalier et al. (1984)	650353
		8.1300		7.8800	0.9692	8	Er₂RhSi₃ (190/194)	800°C, 5 days	Sengupta et al. (2003)
	Si	3.8400	6.6511	4.1400	1.0781	2	Er₃□Si₅	–	Baptist et al. (1988 )
		6.6511		4.1400	0.6225	3	Yb₃□Si₅	–	Baptist et al. (1990 )
		4.0400	3.9500	13.3300	3.2995	4	GdSi₂	–	Binder (1960)
		3.8420	6.6545	4.1400	1.0776	2	Er₃□Si₅	–	Gladyshevskii & Émes-Misenko (1963)
		3.8415	6.6537	4.1425	1.0784	2	Er₃□Si₅	1000°C, 10 days	Ji et al. (2004)
		6.6511		4.1400	0.6225	3	Yb₃□Si₅	–	Knapp & Picraux (1985)
		3.8420	6.6545	4.1400	1.0776	2	Er₃□Si₅	–	Koleshko et al. (1986)	652588
		3.8383		4.1310	1.0763	1	AlB₂	–	Kotur & Mokra (1994 )	658906
		4.0500	3.9500	13.2200	3.2642	4	GdSi₂	–	Lazorenko et al. (1974 )	652570
		3.8500		4.1400	1.0753	1	AlB₂	–	Mayer et al. (1962)	652584
		4.0500	3.9500	13.4000	3.3086	4	o	–	Mayer et al. (1962)
		3.8300		4.1400	1.0809	1	AlB₂	450°C, 0.5 days	Mayer et al. (1967)	652566
		3.8430		4.1430	1.0781	1	AlB₂	800°C, 2 days	Mayer & Felner (1972)	52478
		4.0400	3.9500	13.2300	3.2748	4	GdSi₂	–	Perri et al. (1959b)	652582
		4.0400		13.4200	3.3218	4	ThSi₂	–	Perri et al. (1959b)	150662
		4.0400	3.9500	13.3300	3.2995	4	GdSi₂	–	Perri et al. (1959a)
Yb	Au	8.2003	14.1870	16.8690	2.0571	16	Ba₄Li₂Si₆	800°C, 5 days	Sarkar et al. (2013)	250525
	Si	3.7710		4.0980	1.0867	1	AlB₂	–	Gladyshevskii (1963)	20252
		3.7840		4.0980	1.0830	1	AlB₂	700°C, 3 days	Iandelli et al. (1979)	52480
		6.5120		4.0900	0.6281	3	Yb₃□Si₅	700°C, 3 days	Iandelli et al. (1979)
		6.5472		4.1000	0.6262	3	Yb₃□Si₅	–	Knapp & Picraux (1985)
		3.7710	6.5316	4.0980	1.0867	2	Er₃□Si₅	–	Koleshko et al. (1986)	652601
		3.7700		4.1000	1.0875	1	AlB₂	–	Mayer et al. (1962)	652598
		3.7610		4.0920	1.0880	1	AlB₂	–	Nesper et al. (1979)	652603
		3.9868		13.5410	3.3965	4	ThSi₂	850°C, 3 days	Peter & Kanatzidis (2012)
		6.5120		4.0900	0.6281	3	Yb₃□Si₅	700°C, 21 days	Pöttgen et al. (1998 )

We used calculations based on density functional theory (DFT) to predict the stability of not yet reported RSi₂ and R₂TSi₃ compounds. The formation energy ΔE^tot is the difference of the total energy E^tot of the compound and E^tot of its elements, normalized to six atoms (R₂Si₄ or R₂TSi₃). Appendix B presents the space groups of the unary R crystals. The more negative the formation energy, the more thermodynamically favorable is the formation of that compound. We considered a formation energy of up to −25 meV per atom as potentially stable at room temperature. However, this assumption does not take into account potential energy barriers which might kinetically hinder the formation of the ground state. The projector-augmented wave (PAW) method (Kresse & Joubert, 1999 ) in spin-polarized Perdew–Burke–Ernzerhof parametrization (Perdew et al., 1996 ) was employed as implemented in the VASP code (Kresse & Furthmüller, 1996 ). Total energies have been converged better than 10⁻⁷ eV with a maximum kinetic energy of 320 eV for the planewave basis set and Γ-centered k-point meshes with spacings less than 0.02 × 2π Å⁻¹. All structures have been fully relaxed, with respect to atomic positions as well as cell geometry within the space group, to forces less than 10⁻³ V Å⁻¹. A Hubbard U correlation correction was not used because the Si framework with s- and p-orbitals governs the stability of the structure and because it would complicate the comparability of the formation energies within the R₂TSi₃ series.

3. Results and discussion

In this article, we treat the R₂TSi₃ compounds as a distinct phase with a fixed composition and not as a solid solution. As ternary phase diagrams are scarce for these compounds, we checked all available data, in particular the thermodynamic assessment of Bodak & Gladyshevskii (1985 ), for compositional degrees of freedom in the corresponding phase diagram region and possibly prevailing solid solutions. Nevertheless, the vast majority of compounds were reported to form superstructures which, in general, allow only slight variations in stoichiometry. We discuss those structures as distinct phases due to the changes in symmetry at these particular compositions in the phase diagrams. Many ternary phase diagrams are often determined at elevated temperatures, which is beyond the scope of this work. The phase diagrams given by Bodak & Gladyshevskii (1985) are not at room temperature.

3.1. Structural relationships

The many structure types within compounds RSi₂ and R₂TSi₃ compounds are related to each other according to their space groups and occupied Wyckoff positions. Starting from the highest symmetric structure, different perturbations induce symmetry reductions. Bärnighausen diagrams are the perfect tool to visualize these group–subgroup relationships in a simple and descriptive way. Fig. 1 presents the full Bärnighausen diagram for the RSi₂ and R₂TSi₃ compounds analyzed in this work. This diagram is partially based on a diagram by Hoffmann & Pöttgen (2001), but is greatly extended.

Figure 1
Bärnighausen diagram for RSi₂ and R₂TSi₃ compounds. The header of each box comprises the Hermann–Mauguin symbol of the space group, the range of ordering n and the structure type, whereas the body contains the occupied Wyckoff sites sorted by element. The arrows display the type of transformation between the structures: t is translationengleich, k is klassengleich and i is isomorphic. Fig. 2

comprises the respective structure plots. The fourth branch of AlB₂-like compounds comprises the superstructures caused by interplays with vacancies (R₃□Si₅). A potential tetragonal superstructure is presented in the right-hand part of the diagram.

The presented Bärnighausen diagram would allow for further group–subgroup transitions; thus the authors cannot exclude the existence of further structure types within the RSi₂ and R₂TSi₃ compounds and thus also additional branches in the diagram. However, the space groups we present here already have a high number of free parameters. The extension of the diagram by further symmetry reduction accompanied with further degrees of freedom without losing the rough lattice and symmetry is challenging.

Our diagram provides information about the type of transition (klassengleiche with perpetuation of lattice symmetry, translationengleiche with perpetuation of translational symmetry and isomorphous with perpetuation of both), the change of the lattice (direction and distance), the characteristics of the structure (space group, structure type and Wyckoff positions) as well as the absolute occurrence of the structure types in the literature. Additionally, Fig. 2 visualizes the atom arrangements of the different structures and presents their relationships in a hierarchical structure similar to the Bärnighausen diagram. In contrast, it focuses on the structural models and only shows these branches that include new structure types compared to Hoffmann & Pöttgen (2001). Appendix A includes tables with Wyckoff positions of all structure types taken into account within this article (Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17).

Table 2
Wyckoff positions of the hexagonal aristotypic structure type AlB₂ with space group P6/mmm (No. 191) and lattice parameters a_h ≈ 3.00, c_h ≈ 3.24 Å

Element	Wyckoff symbol	x	y	z
R	1a	0	0	0
Si/T	2d	$[1\over 3]$	$[1\over 3]$	½

Table 3
Wyckoff positions of the hexagonal structure type Ce₂CoSi₃ with space group P6/mmm (No. 191) and lattice parameters a ≈ 2a_h, c ≈ c_h

Element	Wyckoff symbol	x	y	z
R	1a	0	0	0
R	3f	½	0	0
T	2d	$[1\over 3]$	$[2\over 3]$	½
Si	6m	x_Si ≈ $[1\over 6]$	2x_Si ≈ $[2\over 6]$	½

Table 4
Wyckoff positions of the hexagonal structure type U₂RuSi₃ with space group P6/mmm (No. 191) and lattice parameters a ≈ 2a_h, c ≈ c_h

The Si site is only half occupied.

Element	Wyckoff symbol	x	y	z
R	1a	0	0	0
R	3f	½	0	0
T	2d	$[1\over 3]$	$[2\over 3]$	½
Si	12o	x_Si ≈ $[1\over 6]$	2x_Si ≈ $[2\over 6]$	z_Si ≈ ½

Table 5
Wyckoff positions of the hexagonal structure type Er₂RhSi₃ with space group P6₃/mmc (No. 194) and lattice parameters a ≈ 2a_h, c ≈ 2c_h

Element	Wyckoff symbol	x	y	z
R	2b	0	0	¼
R	6h	x_R ≈ ½	2x_R ≈ 0	¼
T	4f	$[1\over 3]$	$[2\over 3]$	z_T ≈ 0
Si	12k	x_Si ≈ $[1\over 6]$	2x_Si ≈ $[1\over 3]$	z_Si ≈ 0

Table 6
Wyckoff positions of the hexagonal structure type Er₂RhSi₃ with space group $[P{\overline 6}2c]$ (No. 190) and lattice parameters a ≈ 2a_h, c ≈ 2c_h

Element	Wyckoff symbol	x	y	z
R	2b	0	0	¼
R	6h	x_R ≈ ½	y_R ≈ ½	¼
T	4f	$[1\over 3]$	$[2\over 3]$	z_T ≈ 0
Si	12h	x_Si ≈ $[1\over 6]$	y_Si ≈ $[1\over 3]$	z_Si ≈ 0

Table 7
Wyckoff positions of the orthorhombic structure type Ho₂PdSi₃ with space group I112/b (No. 15) and lattice parameters a ≈ 2a_h, c ≈ 8c_h

Element	Wyckoff symbol	x	y	z
R	4e	0	¼	z_R,1 ≈ 0
R	4e	0	¼	z_R,2 ≈ $[{1\over {8}}]$
R	4e	0	¼	z_R,3 ≈ $[{2\over {8}}]$
R	4e	0	¼	z_R,4 ≈ $[{3\over {8}}]$
R	4e	0	¼	z_R,5 ≈ $[{4\over {8}}]$
R	4e	0	¼	z_R,6 ≈ $[{5\over {8}}]$
R	4e	0	¼	z_R,7 ≈ $[{6\over {8}}]$
R	4e	0	¼	z_R,8 ≈ $[{7\over {8}}]$
T	8f	x_T,1 ≈ $[1\over 6]$	y_T,1 ≈ $[{1\over {12}}]$	z_T,1 ≈ $[{7\over {16}}]$
T	8f	x_T,2 ≈ $[1\over 6]$	y_T,2 ≈ $[{1\over {12}}]$	z_T,2 ≈ $[{{13}\over {16}}]$
Si	8f	x_Si,1 ≈ $[1\over 6]$	y_Si,1 ≈ $[{1\over {12}}]$	z_Si,1 ≈ $[{{1}\over {16}}]$
Si	8f	x_Si,2 ≈ $[1\over 6]$	y_Si,2 ≈ $[{1\over {12}}]$	z_Si,2 ≈ $[{{3}\over {16}}]$
Si	8f	x_Si,3 ≈ $[1\over 6]$	y_Si,3 ≈ $[{1\over {12}}]$	z_Si,3 ≈ $[{{5}\over {16}}]$
Si	8f	x_Si,4 ≈ $[1\over 6]$	y_Si,4 ≈ $[{1\over {12}}]$	z_Si,4 ≈ $[{{9}\over {16}}]$
Si	8f	x_Si,5 ≈ $[1\over 6]$	y_Si,5 ≈ $[{1\over {12}}]$	z_Si,5 ≈ $[{{11}\over {16}}]$
Si	8f	x_Si,6 ≈ $[1\over 6]$	y_Si,6 ≈ $[{1\over {12}}]$	z_Si,6 ≈ $[{{15}\over {16}}]$

Table 8
Wyckoff positions of the orthorhombic structure type Er₃□Si₅ with space group Pmmm (No. 47) and lattice parameters a ≈ a_h, b ≈ $[\sqrt{3}a_{{\rm h}}]$ , c ≈ c_h

Element	Wyckoff symbol	x	y	z
R	1a	0	0	0
R	1f	½	½	0
Si/T	2p	½	y_Si/T,1 ≈ ¼	½
Si/T	2n	0	y_Si/T,2 ≈ ¼	½

Table 9
Wyckoff positions of the orthorhombic structure type U₂RhSi₃ with space group Pmmm (No. 47) and lattice parameters a ≈ a_h, b ≈ $[\sqrt{3}a_{{\rm h}}]$ , c ≈ c_h

Element	Wyckoff symbol	x	y	z
R	1a	0	0	0
R	1f	½	½	0
Si/T	2n	0	y_T ≈ $[1\over 3]$	½
Si	2p	½	y_Si ≈ $[5\over 6]$	½

Table 10
Wyckoff positions of the orthorhombic structure type Ca₂AgSi₃ with space group Fmmm (No. 69) and lattice parameters a ≈ 2a_h, b ≈ 2c_h, c ≈ $[2\sqrt{3}a_{{\rm h}}]$

Element	Wyckoff symbol	x	y	z
R	8i	0	0	z_R ≈ ¼
R	8f	¼	¼	¼
T	8h	0	y_T ≈ $[2\over 3]$	0
Si	8h	0	y_Si,1 ≈ $[1\over 6]$	0
Si	16o	x_Si,2 ≈ ¼	y_Si,2 ≈ $[{1\over {12}}]$	0

Table 11
Wyckoff positions of the orthorhombic structure type Ho₃□Si₅ with space group P2mm (No. 25) and lattice parameters a ≈ 3a_h, $[b\approx\sqrt{3}a_{{\rm h}}]$ , c ≈ 2c_h

Element	Wyckoff symbol	x	y	z
R	1a	0	y_R,1 ≈ 0	0
R	1b	$[3\over 6]$	y_R,2 ≈ ½	0
R	2g	x_R,1 ≈ $[2\over 6]$	y_R,3 ≈ 0	0
R	2g	x_R,2 ≈ $[1\over 6]$	y_R,4 ≈ ½	0
□	1c	0	y_□,1 ≈ $[ {{2}\over {6}}]$	½
□	1d	$[3\over 6]$	y_□,1 ≈ $[{{5}\over {6}}]$	½
Si	1c	0	y_Si,1 ≈ $[{{4}\over{6}}]$	½
Si	1d	$[3\over 6]$	y_Si,2 ≈ $[{{5}\over{6}}]$	½
Si	2h	x_Si,1 ≈ $[{{2}\over{6}}]$	y_Si,3 ≈ $[{{4}\over{6}}]$	½
Si	2h	x_Si,2 ≈ $[{{1}\over{6}}]$	y_Si,4 ≈ $[{{5}\over{6}}]$	½
Si	2h	x_Si,3 ≈ $[{{1}\over{6}}]$	y_Si,5 ≈ $[{{1}\over{6}}]$	½
Si	2h	x_Si,4 ≈ $[{{2}\over{6}}]$	y_Si,6 ≈ $[{{2}\over{6}}]$	½

Table 12
Wyckoff positions of the orthorhombic structure type Ba₄Li₂Si₆ with space group Fddd (No. 70) and lattice parameters a ≈ 2a_h, $[b\approx 2\sqrt{3}a_{{\rm h}}]$ , c ≈ 4c_h

Element	Wyckoff symbol	x	y	z
R	16g	$[{1\over 8}]$	$[{1\over 8}]$	z_R,1 ≈ $[{2\over 8}]$
R	16g	$[{1\over 8}]$	$[{1\over 8}]$	z_R,2 ≈ $[{6\over 8}]$
T	16f	$[{1\over 8}]$	y_T ≈ $[{7\over {24}}]$	$[{1\over 8}]$
Si	16f	$[{1\over 8}]$	y_Si,1 ≈ $[{11\over {24}}]$	$[{1\over 8}]$
Si	32h	x_Si,2 ≈ $[{1\over 8}]$	y_Si,2 ≈ $[{7\over {24}}]$	z_Si,2 ≈ $[{1\over 8}]$

Table 13
Wyckoff positions of the Si vacancy cell of structure type Yb₃□Si₅ with space group $[P\overline{6}2m]$ (No. 189) and lattice parameters $[a\approx\sqrt{3}a_{{\rm h}}]$ , c ≈ c_h

Element	Wyckoff symbol	x	y	z
R	3f	x_R ≈ $[2\over 3]$	0	0
□	1b	0	0	½
Si	3g	x_Si ≈ $[1\over 3]$	0	½
Si	2d	$[1\over 3]$	$[2\over 3]$	½

Table 14
Wyckoff positions of the Si vacancy cell of structure type Tb₃□Si₅ with space group $[P\overline{6}2c]$ (No. 190) and lattice parameters $[a\approx\sqrt{3}a_{{\rm h}}]$ , c ≈ 2c_h

Element	Wyckoff symbol	x	y	z
R	6g	x_R ≈ $[1\over 3]$	0	0
□	2c	0	0	¼
Si	6h	x_Si ≈ $[1\over 3]$	y_Si ≈ $[1\over 3]$	¼
Si	2d	$[2\over 3]$	$[1\over 3]$	¼
Si	2b	0	0	¼

Table 15
Wyckoff positions of the tetragonal structure type ThSi₂ with space group I4₁/amd (No. 141) with lattice parameters a_t ≈ a_h, c_t ≈ 13.4–14.4 Å

Element	Wyckoff symbol	x	y	z
R	4a	0	0	$[1\over 8]$
Si/T	8e	0	0	z_Si/T ≈ $[{7\over {24}}]$

Table 16
Wyckoff positions of the orthorhombic structure type GdSi₂ with space group Imma (No. 74) and lattice parameters a ≈ a_t, c ≈ c_t

Element	Wyckoff symbol	x	y	z
R	4e	0	¼	z_R ≈ $[1\over 8]$
Si/T	4e	0	¼	z_Si/T,1 ≈ $[{7\over {24}}]$
Si/T	4e	0	¼	z_Si/T,2 ≈ $[{{11}\over {24}}]$

Table 17
Wyckoff positions of the proposed orthorhombic superstructure of the tetragonal branch with space group C222₁ (No. 20) and lattice parameters $[a\approx\sqrt{2}a_{{\rm t}}]$ , b ≈ c_t, $[c\approx\sqrt{2}a_{{\rm t}}]$

Element	Wyckoff symbol	x	y	z
R	4a	x_R ≈ ¼	0	0
R	4b	0	y_R ≈ ¼	¼
T	4b	x_T ≈ 0	y_T ≈ $[{4\over {12}}]$	z_T ≈ ¼
Si	4b	x_Si,1 ≈ 0	y_Si,1 ≈ $[{2 \over {12}}]$	z_Si,1 ≈ ¼
Si	8c	x_Si,2 ≈ ¼	y_Si,2 ≈ $[{1 \over {12}}]$	z_Si,2 ≈ 0

Figure 2
Models of the different observed structure types within RSi₂ and R₂TSi₃ compounds (unit cell outlined in black). The AlB₂-like structures are depicted such that the view onto the two-dimensional R network is almost identical. The common structure pattern of the ordered AlB₂-like structures (gray frame at right top) is highlighted with a light-gray frame and red Si/T bonds. The structure types Ce₂CoSi₃ and U₂RuSi₃ are almost identical. In contrast to the U₂RuSi₃ type, the Si atoms of the Ce₂CoSi₃ type are on the highly symmetric z = ½ position. This is highlighted by the blurred Si location along the c direction. The tetragonal structures (gray frame at center top) compose a 3D Si/T subnetwork with incomplete hexagons at the faces (highlighted in orange). The structures are connected according to their symmetry relations (dashed lines, if the transition is not minimal; labels comprise the lattice transformation).

3.1.1. Compounds deduced from the AlB₂ structure type

First, we will present the relationships of RSi₂ and R₂TSi₃ compounds derived from the AlB₂ structure. The lattice parameters are in the range of a_h ≈ 3.8–4.2 Å and c_h ≈ 3.9–4.5 Å, which is much higher than for the parent structure AlB₂ itself (a_AlB₂ = 3.00 Å, c_AlB₂ = 3.24 Å).

Hoffmann & Pöttgen (2001) gave an overview of the hexagonal and orthorhombic transitions of AlB₂-related compounds. Only three of Hoffmann's Bärnighausen branches are applicable for the stoichiometries addressed here (RSi₂ and R₂TSi₃). We identify further structure types not discussed by Hoffmann & Pöttgen (2001), analyze the relationships of all structure types in the following paragraphs and show the new structure types in the Bärnighausen diagram (Fig. 2). Our Bärnighausen diagram (Fig. 2) thus exhibits four main branches which result from interactions with a T element or an Si vacancy □.

The first branch of the Bärnighausen diagram describes the symmetrical relationships between the hexagonal derivatives of the AlB₂ type. Fig. 2 shows that Ce₂CoSi₃ (Gordon et al., 1997 ) has the same structural motif as the aristotype. The difference is the ordering of the T atoms resulting in isolated [Si₆] rings, see top right of Fig. 2. Only a certain part of this pattern is visible in the unit cell of Ce₂CoSi₃ and in other structure types of the RSi₂ and R₂TSi₃ compounds, indicated by red bonds. Besides [Si₆] rings, [T₂Si₄] hexagons also occur, with the T atoms opposing each other in the ring. This ordering change indicates the doubling of the unit-cell parameter a in the Ce₂CoSi₃ type and an isomorphous symmetry reduction. If the Si atoms are shifted along the c direction, the layers are no longer perfectly planar, but puckered. This arrangement can be described with the same space group as Ce₂CoSi₃, but with half-occupied Wyckoff site 12o, instead of fully occupied 6m, known as the structure type U₂RuSi₃ (Pöttgen et al., 1994 ). Fig. 2 shows both structure types within one subfigure with the different Si positions indicated by a series of atoms.

Compared to their ideal crystallographic positions, the Er₂RhSi₃ (P6₃/mmc) type (Gladyshevskii et al., 1992 ) exhibits shifts of the T atoms along the c direction accompanied by distortions of the R atoms centering the [T₂Si₄] rings. This puckering results in a doubling of the c parameter and thus a further klassengleiche reduction of the symmetry of the Ce₂CoSi₃ or U₂RuSi₃ type. The reported noncentrosymmetric structure for Er₂RhSi₃ ( $[P\overline{6}2c]$ ) (Chevalier et al., 1984 ) assumes additional distortions of the [Si₆] rings and their centering R atoms by decoupled x and y coordinates resulting in a translationengleiche symmetry reduction of centrosymmetric Er₂RhSi₃ (P6₃/mmc).

The second branch only includes the Ho₂PdSi₃ structure type (Tang et al., 2011 ) with monoclinic space group I112/b (Nentwich et al., 2016 ). This structure contains eight Si/T layers with stacking sequence ABCDBADC. Each layer exhibits the same Si/T occupation pattern as the Ce₂CoSi₃ type. The [T₂Si₄] rings of adjacent layers are shifted and rotated by multiples of 60° around the c axis with respect to each other. The 12-fold coordinated R elements are located on two different Wyckoff positions, either coordinated by two [T₂Si₄] rings or by one [T₂Si₄] ring and one [Si₆] ring. The Ho₂PdSi₃ type contains 32 subcells and is thus one of the largest structures within the AlB₂ Bärnighausen diagram. The atoms are assumed to be on the ideal crystallographic position, without any distortions, although the space group would allow this. The transition from AlB₂ type to Ho₂PdSi₃ involves several symmetry reduction steps, detailed in Fig. 1.

The third branch comprises the orthorhombic derivatives of the AlB₂ type. The starting point for further reductions is an orthohexagonal setting with space group Cmmm and Wyckoff sequence 2a, 4k. This setting is still a missing link (Hoffmann & Pöttgen, 2001), meaning that no report about a compound with this structure has been found. This space group has independent lattice parameters a and b – in contrast to all previous structure types – causing a translationengleiche symmetry reduction and making it an important starting point for five further structure types.

One of them is Ba₄Li₂Si₆ (von Schnering et al., 1996 ), which has perfectly ordered Si/T layers with the same occupational pattern as the Ce₂CoSi₃ type. As in the Ho₂PdSi₃ structure type, the Si/T atoms are perfectly ordered and form an ABCD stacking sequence, which is consistent with the two differently coordinated R sites as mentioned before. Accompanied with the anisotropic available space of the R site surrounded by one [T₂Si₄] and one [Si₆] ring, its z component is not on the ideal crystallographic position resulting in a puckering of the R and Si/T layers. Identical R elements are connected along the former hexagonal a direction. These structural changes are accompanied with three consecutive klassengleiche symmetry reductions doubling the a and b parameters and quadrupling the c parameter.

A second structure type is U₂RhSi₃ (Pöttgen & Kaczorowski, 1993 ) with space group Pmmm (No. 47). Its Si/T atoms are partially ordered and only shifted along the b direction. These shifts induce a break in translational symmetry and a klassengleiche reduction. The Ho₂PdSi₃, Ba₄Li₂Si₆ and Ca₂AgSi₃ structure types (Gordon et al., 1997) have perfectly ordered Si/T layers and the same local arrangements around the R atoms. The R elements of the same Wyckoff site are connected along the orthorhombic a direction. These structural changes indicate the doubling of lattice parameters and a klassengleiche transition from structure type U₂RhSi₃. Hoffmann & Pöttgen (2001) have already reported a second structure type with the same space group as U₂RhSi₃, but with a different Wyckoff sequence, namely Er₃□Si₅. This type represents the disordered nonstoichiometric disilicides. In addition to the disordered ones, we also found reports about ordered versions. The otherwise very detailed review by Hoffmann & Pöttgen (2001) did not discuss these variants, which form due to vacancy ordering. According to the real stoichiometry of RSi_1.67, one Si atom is regularly missing in the Si hexagons (Roge et al., 1995 ). This arrangement can be realized by a hexagonal and a orthohexagonal setting (Auffret et al., 1990 ). The hexagonal setting will be discussed in the fourth branch. The orthohexagonal arrangement requires a triplication of the a parameter. We will refer to this setting as Ho₃□Si₅ type. We prepared a list of its atomic parameters in space group P1 (No. 1) and inserted it to the software FINDSYM (Stokes & Hatch, 2005 ), which determined the highest possible space group as Pmm2 (No. 25). We changed the setting to P2mm (No. 25) for a better comparability to its supergroup Pmmm (No. 47). Thus, the triplication causes a translationengleiche and a klassengleiche symmetry reduction, which is accompanied with potential shifts of all atoms within the a,b plane.

The fourth branch comprises the ordered R₃□Si₅ structures, which are not related to the disordered Er₃□Si₅ type within the Bärnighausen diagram.

d'Avitaya et al. (1989 ) described a $[\sqrt{3}\times\sqrt{3}]$ low-energy electron diffraction (LEED) pattern of Er₃□Si₅ thin films. Iandelli et al. (1979 ) determined the space group of this arrangement for Yb₃□Si₅ as $[P\overline{6}2m]$ (No. 189), only allowing the x parameter of R and Si to deviate from its ideal crystallographic position. To consider the underlying symmetries of this arrangement, the cell needs to be enlarged and rotated with respect to the AlB₂ unit cell using an isomorphous symmetry reduction. The location of the vacancy on an independent Wyckoff site is accompanied by a further translationengleiche symmetry reduction and an origin shift from space group P6/mmm to $[P\overline{6}2m]$ .

Another model proposed by Stauffer et al. (1992 ) is based on the aforementioned arrangement, but every second Si/T layer is rotated by 120° around c. We determined the space group of this vacancy ordering as $[P\overline{6}2c]$ , assuming that only the occupational pattern of the Si lattice would adapt, without changing the atomic positions. This results in a doubling of the c parameter, accompanied by a klassengleiche transition. The first reports concerning this arrangements used the compound Er₃□Si₅. However, this type name is already used for the disordered nonstoichiometric disilicides. Thus, we will refer to this structure type as Tb₃□Si₅ in accordance with the report by Luo et al. (1997 ).

We did not consider cells based on the Ho₃□Si₅ type with doubled c parameter, as it is only reported for the $[\sqrt{3}\times\sqrt{3}]$ type cells.

Further remarks. Gordon et al. (1997) reported a further superstructure for Ce₂PdSi₃ with doubled lattice parameter a and quadrupled c, but did not focus on the specific space group. Therefore, we could not implement this report for the construction of the Bärnighausen diagram. During the literature research we additionally found structures of the EuGe₂-type with space group $[P\overline{3}1m]$ (No. 164). This structure type is very similar to the AlB₂ type, but with a puckered Si sublattice, inducing a translationengleiche transition. Reports about this structure type refer to binary alkaline earth disilicides at non-ambient conditions (Evers et al., 1977b ; Bordet et al., 2000 ; Brutti et al., 2006 ) or with mixed R sites (Eisenmann et al., 1970 ; Evers et al., 1979 ) as well as theoretical considerations about the puckering only (Gemming & Seifert, 2003 ; Gemming et al., 2006 ; Enyashin & Gemming, 2007 ; Flores-Livas et al., 2011 ). As these reports do not meet the requirements of experiments at ambient conditions, we did not consider this group of compounds within this work.

All aforementioned structure types will be termed AlB₂-like in the following sections. By studying the atomic coordinates of the addressed space groups, we observed that the R elements form a rigid frame for the structure, as they are mostly the heaviest and largest elements in the structure and, thus, the most immobile. This also means that the Si/T atoms are more mobile and thus puckering of these layers is rather common.

3.1.2. Compounds deduced from ThSi₂ structure type

Compounds of the ThSi₂ type (Brauer & Mittius, 1942) crystallized in space group I4₁/amd (No. 141), see gray box of Fig. 2 (with tetragonal lattice parameters a_t ≈ a_h, c_t ≈ 13.4–14.4 Å). The Si/T atoms form a complex 3D network, in contrast to the 2D honeycombs in AlB₂. So far, the only reported variation of the ThSi₂ type is the GdSi₂ structure (Perri et al., 1959b; Binder, 1960 ) with independent lattice parameters a and b. This degree of freedom causes a translationengleiche symmetry reduction to space group Imma (No. 74).

If the ThSi₂ or GdSi₂ type structures exhibit Si vacancies, these do not order regularly and only cause partially occupied Wyckoff positions. The proportion of vacancies is generally 10% (RSi_1.8), thus almost one Si ion per tetragonal or orthorhombic unit cell is vacant. The resulting structures remain in the original space group and are called ThSi₂-defect and Nd□_xSi_2−x, respectively.

In contrast to the distortive modulation of ThSi₂, we did not find evidence for a tetragonal superstructure induced by ordering. This absence may be partially due to the small number of reports concerning tetragonal R₂TSi₃ compounds [18 structure reports in ten articles (Gordon et al., 1997; Albering et al., 1994 ; Kaczorowski & Noël, 1993 ; Lejay et al., 1983; Chevalier et al., 1986; Li et al., 2008 ; Mayer & Felner, 1973b ; Pöttgen & Kaczorowski, 1993; Raman & Steinfink, 1967 ; Raman, 1967 )]. In order to shed light on a potential ordering, we constructed a tetragonal superstructure based on geometrical, chemical and electronic considerations. First, every Si atom has exactly one T element in its coordination. Second, every T element is coordinated by exactly three Si atoms. Third, every zigzag chain fulfills the 1:3 ratio of T:Si (zigzag chains explained in Section 3.2). And fourth, short-range periodicity is mandatory; thus, no doubling of the unit cell along the c direction is expected. By choosing an arbitrary atom within the tetragonal Si/T network as the first T element, only two positions unfold positioning the next T element. Two atomic arrangements resulted following the aforementioned conditions. We transferred these patterns onto the simple space group P1 (No. 1) and imported them into the tool FINDSYM (Stokes & Hatch, 2005) to determine the space group. Both variants proved to be identical and to exhibit the space group C222₁ (No. 20). We will refer to this new structure type with eight instead of four formula units as POTS (proposed ordered, tetragonal structure). The gray box in Fig. 2 visualizes the Si/T-ordering. As this structure has not been reported so far for R₂TSi₃ compounds, we decided to perform DFT calculations to estimate its stability, see Section 3.3.

These three structure types introduced in this section (§3.1.2) will be addressed as ThSi₂-like in the following.

3.2. Structure description

The hexagonal and the tetragonal subgroups of RSi₂ and R₂TSi₃ compounds do not seem to be symmetrically related at first glance. The AlB₂-like compounds exhibit graphite-like 2D networks of planar Si/T hexagons, whereas the Si/T atoms of ThSi₂-like compounds form 3D networks. Still, the structures show similarities due to the trigonal coordination of the Si atoms. Fig. 3 illustrates the Si/T atoms in trigonal prisms, the 12-fold coordinated R atoms (connectors in black) and the Si/T zigzag chains (bonds in red/orange) in both structures.

Figure 3
Differences in the arrangement of Si/T (blue) zigzag chains in hexagonal (left) and tetragonal (right) RSi₂ and R₂TSi₃ compounds. The consecutively added zigzag chains (red bonds) in hexagonal compounds always lie within the same plane, whereas in tetragonal compounds these layers are rotated by 90° along the bonds shown in orange. The 12-fold coordination of the R elements is highlighted for one atom ,as an example, with bonds shown in black.

Not only are the hexagonal honeycombs similar to graphite but also the tetragonal 3D network. The typical net exists simultaneously in planes perpendicular to the tetragonal a_t and b_t directions which are interconnected by bonds along the c_t direction. More precisely, two consecutive Si/T zigzag chains are rotated by 90° along the c_t direction, thereby spanning the (100)_t and (010)_t faces of the unit cell and causing incomplete hexagons (see the orange bonds in the ThSi₂ structure type in Fig. 2). This additional symmetry degree of freedom causes a slight deformation of the trigonal Si/T arrangement in the tetragonal network. The Si—T bonds along the c_t direction (in orange, interchain) elongate in comparison to the intrachain bonds (in red), see Fig. 3. Further, the angle within the zigzag chains increases, whereas the other two angles decrease (between bonds shown in red and orange). Therefore, the chains with stronger bonds are slightly flattened compared to the ideal structure with perfect trigonal coordination. These structural differences between hexagonal and tetragonal structure types cause different crystal symmetries that permit a common origin in the Bärnighausen diagram for the RSi₂ and R₂TSi₃ compounds.

3.3. Elemental combinations and stability analysis of missing links with DFT calculations

During the literature search, we collected numerous structure reports of various RSi₂ and R₂TSi₃ compounds. Fig. 4 gives an overview of the reported compounds according to their appearance within the R–T grid. In this R–T diagram, we marked the number of reports with different colors, see Fig. 4. This diagram does not include the elements of the Zn group as those compounds were only analyzed at elevated temperatures (Demchenko et al., 2002 ; Malik et al., 2013 ; Nasir et al., 2010 ; Romaka et al., 2012 ; Salamakha et al., 1998 ), which are out of the scope of this article. Additionally, we did not find any reports which include R₂CrSi₃ compounds. We assume that certain electron configurations are necessary for the formation of R₂TSi₃ compounds. Furthermore, some elements rarely appear within the R₂Si and R₂TSi₃ compounds, such as Sm and Yb, which are highly volatile (Cao, 2014, private communication), Tc, which has a very low radio-active half-life and is very scarce (Holleman & Wiberg, 2007 ), or Pm, which is radioactive (Cao, 2014, private communication; Frontzek, 2014, private communication). The interest in using La and Lu was lower as most of the research aimed for the magnetic properties that do not exist for these two elements (Frontzek, 2014, private communication). The cost of the elements seems to play a subordinate role, e.g. the more expensive Rh (89 000 USD per kg) compounds were analyzed more frequently than the ones containing Ir (36 000 USD per kg) (Haynes, 2012 ).

Figure 4
Overview of the literature reports of RSi₂ and R₂TSi₃ crystals. The number of reports is visualized with numbers and colors (few to very frequent: red – yellow – green – blue – purple). Additionally, to predict the stability for selected unreported structures, this study performed DFT calculations for the highlighted compounds (black circles).

These distributions are emphasized in Figs. 5, 6 and 7, which show systematic approaches in the literature. Fig. 5 gives an overview of RSi₂ series with the corresponding authors and R elements. This summary shows the high interest in the lanthanide compounds compared to R elements of the alkaline earth metals and the actinides. Fig. 6 shows a similar illustration of T series within the R₂TSi₃ compounds. Sorted by T element and author, the corresponding R elements are highlighted. Within the 3d elements the largest variety was analyzed, mostly in combination with La and Ce. In contrast, the heavy lanthanides were more favored when 4d elements were used, which have been intensively studied. Finally, Fig. 7 shows the R series, sorted by R element and author, with highlighted T elements. Again, the focus on the 3d elements as well as La and Ce is clear. The most complete investigations were carried out for U and Th, which emphasizes their importance for reactor technology.

Figure 5
Overview of the RSi₂ compounds that were analyzed systematically by the same first author. Some of the results were published in more than one article: Brauer (Brauer & Mittius, 1942

; Brauer & Haag, 1950

; Brauer & Haag, 1952

), Evers (Evers et al., 1977a

, 1978a

, 1983

; Evers, 1979

, 1980

), Mayer:1 (Mayer et al., 1962

, 1967

; Mayer & Eshdat, 1968

), Perri (Perri et al., 1959a

), Pierre (Pierre et al., 1988

, 1990

Figure 6
Overview of the R₂TSi₃ compounds that were analyzed systematically by the same first author according to their T element (see color code). Some of the results were published in more than one article: Mayer:2 (Mayer & Tassa, 1969

; Mayer & Felner, 1972

, 1973a

), Szlawska (Szlawska et al., 2007

, 2009

, 2011

, 2016

; Szlawska & Kaczorowski, 2011

, 2012

), Li (Li et al., 1997

, 1998,a

, 1999

, 2001

, 2002a

, 2003

, 2008

, 2013

), Frontzek (Frontzek et al., 2004

, 2006

; Frontzek, 2009

), Mallik (Mallik & Sampathkumaran, 1996

; Mallik et al., 1998a

), Xu (Xu et al., 2010

, 2011a

), Li (Li et al., 1998b

, 2001

, 2002a

, 2003

, 2013

Figure 7
Overview of the R₂TSi₃ compounds that were analyzed systematically by the same first author according to their R element (see color code). Some of the results were published in more than one article: Mayer:2 (Mayer & Tassa, 1969

; Mayer & Felner, 1972

, 1973a

), Szlawska (Szlawska et al., 2007

, 2009

, 2011

, 2016

; Szlawska & Kaczorowski, 2011

, 2012

), Li (Li et al., 1997

, 1998a

, 1999

, 2001

, 2002a

, 2003

, 2008

, 2013

), Frontzek (Frontzek et al., 2004

, 2006

; Frontzek, 2009

), Pottgen (Pöttgen & Kaczorowski, 1993

; Pöttgen et al., 1994

), Majumdar (Majumdar et al., 1998

, 1999a

, 2000

, 2001

By studying the R–T diagram of Fig. 4 one main question arises: What are the stability relationships of those R₂TSi₃ compounds that are missing? To clarify this question, we sorted the compounds according their R element and discuss the Co, Rh and Pt series in the following sections.

We assumed ordered structures as DFT cannot evaluate mixed positions, except in the framework of virtual crystal approximations (VCA) using potential mixing. We adapted the structure type of the adjacent compounds within the R–T grid or used the highly symmetric Ce₂CoSi₃ structure type with space group P6/mmm (No. 191) as the basis for the unknown compounds. Table 17 summarizes the formation energies and lattice parameters all considered compounds. We will compare the formation energy of an unreported compound with those of similar reported compounds to evaluate its relative stability.

The DFT results of all models indicate metallic structures, although the DFT band gap problem may suppress the appearance of small band gaps. Thus, all structures have an intrinsic buffer of electronic states at the Fermi level to account for stability considerations of the T coordination within the ionic Si/T subnetwork according to molecular orbital theory, see Nentwich et al. (2020).

The first compound of interest is Nd₂CoSi₃. The series of Nd compounds is fairly complete, compare Fig. 4, for example, with reported Nd₂RhSi₃ (Chevalier et al., 1983 , 1984; Szytuła et al., 1993 ; Mitsufuji et al., 1996 ; Gribanov et al., 2010 ; Zajdel et al., 2015 ), which is the 4d analog compound to Nd₂CoSi₃. Additionally, we found comments on this compound in two publications, but without any information concerning property, structure and phase purity (Chevalier et al., 1984; Szytuła et al., 1993). The formation energies and existing structure types of La₂CoSi₃ and Ce₂CoSi₃ serve as references. Furthermore, the likewise hypothetical compound Pr₂CoSi₃ was also calculated. The blue markers in Fig. 8 show the respective formation energies ranging from −4.61 eV to −4.37 eV. The lowest energy results for R = Ce and the highest for R = Nd. As the formation energy of Pr₂CoSi₃ lies in between the reported compounds, we expect it to be stable. The energy difference between Nd₂CoSi₃ and La₂CoSi₃ (the reported compound with highest energy) is 25 meV per atom. This corresponds to the tolerance limit; thus, we conclude that Nd₂CoSi₃ could also be stable. This conclusion is supported by the reports of Mayer & Tassa (1969 ) and Felner & Schieber (1973 ) on Pr₂Co_0.8Si_3.2 and Nd₂Co_0.8Si_3.2. They also synthesized samples with higher T content, which lead to `the disappearance of the AlB₂ type phase, and the X-ray patterns obtained could not be interpreted' (Mayer & Tassa, 1969). Nevertheless, we think that the synthesis of Pr₂CoSi₃ and Nd₂CoSi₃ and the interpretation of the corresponding X-ray patterns would be successful nowadays due to improved hardware and measurement techniques. Additionally, an enhanced thermal treatment would certainly improve the crystal quality regarding the Si/T ordering. Thus, we advise reinvestigating the R₂TSi₃ compounds discussed by Mayer & Tassa (1969), with R = La, Ce, Pr, Nd, Sm, Gd and T = Fe, Co, Ni.

Figure 8
Formation energies of some R₂TSi₃ compounds in different structure types.

Another interesting compound is Eu₂RhSi₃. The Rh series is well represented in the R–T diagram and its 3d analog Eu₂CoSi exists. However, the R element Eu supposedly only forms a compound with Co, but not with Rh (Mayer & Tassa, 1969; Mayer & Felner, 1973a ). We also modeled R₂RhSi₃ compounds with R elements Gd, Tb, Dy and Ho again and used the formation energies of existing structures as references. For the Rh series, the formation energies range from −6.68 eV to −4.34 eV, with the not yet reported Eu₂RhSi₃ having the highest formation energy. Both tested symmetries –the higher symmetric Ce₂CoSi₃ and the lower symmetric Er₂RhSi₃ – gave almost the same results, for formation energies (−4.34 eV) and interatomic distances [d_a(R,R) ≈ 4.13 Å, d_c(R,R) ≈ 4.27 Å]. The formation energy of Eu₂RhSi₃ differs from the second highest formation energy of Ho₂RhSi₃ by 160 meV per atom which exceeds the limit of 25 meV per atom, see green markers in Fig. 8. Therefore, the Eu₂RhSi₃ compound in Ce₂CoSi₃ or Er₂RhSi₃ structure type is significantly less stable.

The third compound of interest is Eu₂PtSi₃. In the R₂PtSi₃ series only a few element combinations have not yet been experimentally confirmed. Nevertheless, we identified missing compounds for R between Nd and Gd. Due to the radioactivity and low abundance of Pm and the volatility of Sm, we chose the Eu compound for further investigation. In analogy to the Rh series, we additionally chose R = Gd, Tb, Dy as references for formation energy and structure. In addition we modeled the not-yet-reported compound Ho₂PtSi₃. We decided to calculate the compounds in the reported Er₂RhSi₃ ( $[P\overline{6}2c]$ ) symmetry and additionally in the higher symmetric type Ce₂CoSi₃ as well as in the lowest possible symmetry P1 (No. 1) to evaluate the influence of the degrees of freedom onto the formation energies. The energies for the R₂PtSi₃ compounds range from −6.18 eV to −5.11 eV, see orange markers in Fig. 8. Except for Eu, the energies of different compounds and also different structure types are very similar. As expected, the energies of the lower symmetric Er₂RhSi₃ structure types are always lower than those of the highly symmetric type Ce₂CoSi₃, due to the additional degrees of freedom in atomic positions. The spread is between 0 meV for Gd and 28 meV for Ho per atom and about additional 1 meV going down to P1 (No. 1). The energies of the low-symmetric versions of the R₂PtSi₃ compounds are even lower than that of existing Gd₂PtSi₃. The formation energy of the (still) hypothetical Ho₂PtSi₃ in Ce₂CoSi₃ type structure is 33 meV per atom higher than that of Gd₂PtSi₃, thus this high-symmetry type is certainly not stable. However, the lower symmetry types will very probably be stable. The formation energy of Eu₂PtSi₃ is 14 meV per atom higher than for Gd₂PtSi₃; therefore, the compound is in the two considered symmetries most probably accessible as the thermodynamically stable phase. On the one hand, these data show that in some cases (Eu₂RhSi₃, Eu₂PtSi₃ and Gd₂PtSi₃) the formation energy hardly changes for different structure types. On the other hand, the formation energy of different structure types may change so strongly that our relative limit of 25 meV per atom is by far exceeded and only the lower symmetric variations may be stable. This is the case for Tb₂PtSi₃, Dy₂PtSi₃ and Ho₂PtSi₃.

After analyzing those three R series, we discovered further characteristics in the R–T diagram worth studying for different reasons. Compound La₂PdSi₃ attracted our attention because Chaika et al. (2001 ) and Behr et al. (2008 ) have already successfully synthesized this compound, but did not determine the lattice parameters or structural information during their investigations. We performed DFT calculations for La₂PdSi₃ using the Ce₂CoSi₃ structure type as well. The formation energy is lower than for the chemically similar compound La₂CoSi₃ which was reported in the ordered structure type Ce₂CoSi₃. Thus, we conclude that the Ce₂CoSi₃ type may be a stable configuration for La₂PdSi₃, next to the disordered AlB₂ type. The relaxed parameters a = 8.34 Å and c = 4.38 Å are very close to the lengths expected from the adjacent compounds La₂RhSi₃ and Ce₂PdSi₃ (a ≈ 8.25 Å, c ≈ 4.3 Å). We recommend checking La₂PdSi₃ for indicators of an ordered Si/T site, e.g. satellite reflections.

Furthermore, we wondered which structure would arise for stoichiometric BaSi₂. Most reported space groups of BaSi₂ are orthorhombic (Imai & Watanabe, 2010 ; Evers, 1980 ; Janzon et al., 1970 ; Kitano et al., 2001 ; Migas et al., 2007 ; Schäfer et al., 1963 ; Evers et al., 1977b, 1978a ) and do not fit into our Bärnighausen diagram and are, therefore, not listed in Table 1 nor depicted in Figs. 4 and 9. The only exception is a hexagonal phase determined by Gladyshevskii (1959 ). In fact, the original sample had Li impurities and exhibits the structure type Ba₄Li₂Si₆, discovered by von Schnering et al. (1996). This finding explains the discrepancy with the tetragonal phases of the related alkaline earth compounds CaSi₂ and SrSi₂, e.g. Evers et al. (1977a ,b ). We tested both an hexagonal and a tetragonal variant for BaSi₂ to evaluate which symmetry is more stable. Additionally, we modeled SrSi₂ in both the hypothetical AlB₂ and the already reported ThSi₂ structure type to compare the formation energies. As expected, the formation energy of tetragonal SrSi₂ is lower than the one of hexagonal SrSi₂. The energies for both BaSi₂ models are almost identical (−2.06 eV) and, thus, expected to be equally stable. Nevertheless, these data alone are not sufficient to convey the stability of BaSi₂ to SrSi₂ as the elements Ba and Sr are too different. Furthermore, given the degrees of freedom, the tetragonal model of BaSi₂ relaxed into an orthorhombic lattice with differences in lattice parameters a and b in the order of 0.4%. It should be noted that the a parameters of hexagonal and tetragonal symmetry differ for both BaSi₂ and SrSi₂ compounds (see Table 18), although they are alike for dimorphic compounds of the family, e.g. GdSi₂.

Table 18
Formation energies (eV) and lattice parameters (Å) calculated with DFT

Formation energies are given for R₂Si₄ and R₂TSi₃ compounds, respectively (same amount of atoms within calculated range). Compounds marked with * have already been reported in the literature.

		Reported			Calculated
Compound	Structure type	a	b	c	a	b	c	ΔE^tot
Co series
La₂CoSi₃	Ce₂CoSi₃*	8.185	a	4.350	8.14	a	4.34	−4.52
Ce₂CoSi₃	Ce₂CoSi₃*	8.110	a	4.220	8.01	a	4.08	−4.61
Pr₂CoSi₃	Ce₂CoSi₃	–	–	–	8.03	a	4.11	−4.58
La₂CoSi₃	Ce₂CoSi₃*	8.185	a	4.350	8.14	a	4.34	−4.52
Ce₂CoSi₃	Ce₂CoSi₃*	8.110	a	4.220	8.01	a	4.08	−4.61
Pr₂CoSi₃	Ce₂CoSi₃	–	–	–	8.03	a	4.11	−4.58
Nd₂CoSi₃	Ce₂CoSi₃	–	–	–	8.04	a	4.15	−4.37

Rh series
Eu₂RhSi₃	Ce₂CoSi₃	–	–	–	8.26	a	4.27	−4.35
	Er₂RhSi₃	–	–	–	8.26	a	8.55	−4.34
Gd₂RhSi₃	Er₂RhSi₃*	8.112	a	7.976	8.21	a	8.02	−6.68
Tb₂RhSi₃	Er₂RhSi₃*	8.110	a	7.860	8.18	a	7.90	−5.51
Dy₂RhSi₃	Er₂RhSi₃*	8.097	a	7.823	8.18	a	7.90	−5.45
Ho₂RhSi₃	Er₂RhSi₃*	8.086	a	7.804	8.18	a	7.89	−5.31

Pt series
Eu₂PtSi₃	Ce₂CoSi₃	–	–	–	8.27	a	4.34	−5.11
	Er₂RhSi₃	–	–	–	8.27	a	8.67	−5.11
Gd₂PtSi₃	Ce₂CoSi₃	–	–	–	8.17	a	4.14	−5.97
	Er₂RhSi₃*	8.139	a	8.303	8.17	8.17	8.28	−5.97
Tb₂PtSi₃	Ce₂CoSi₃	–	–	–	8.15	a	4.08	−5.92
	Er₂RhSi₃ ( $[P\overline{6}2c]$ )*	8.122	a	8.237	8.16	a	8.18	−6.17
	P1	–	–	–	8.16	a	8.17	−6.18
Dy₂PtSi₃	Ce₂CoSi₃	–	–	–	8.16	a	4.07	−5.84
	Er₂RhSi₃ ( $[P\overline{6}2c]$ )*	–	–	–	8.22	8.23	8.33	−6.14
	P1	8.100	a	8.200	8.16	a	8.14	−6.14
Ho₂PtSi₃	Ce₂CoSi₃	–	–	–	8.16	a	4.07	−5.77
	Er₂RhSi₃	–	–	–	8.16	a	8.13	−6.04
	Er₂RhSi₃ ( $[P\overline{6}2c]$ )	–	–	–	8.16	8.16	8.10	−6.04
	P1	–	–	–	8.16	a	8.11	−6.05

La₂PdSi₃
La₂PdSi₃	Ce₂CoSi₃*	–	–	–	8.34	a	4.38	−5.54

SrSi₂ versus BaSi₂
SrSi₂	ThSi₂*	4.438	a	13.830	4.46	4.46	13.82	−2.21
	AlB₂	–	–	–	4.14	a	4.64	−1.90
BaSi₂	ThSi₂	–	–	–	4.67	4.67	14.16	−2.06
	AlB₂	–	–	–	4.17	a	5.06	−2.06

Sr₂AgSi₃ versus Ba₂AgSi₃
Sr₂AgSi₃	Ba₄Li₂Si₆	–	–	–	8.48	14.69	18.56	−2.83
	Ca₂AgSi₃	–	–	–	8.48	9.28	14.67	−2.74
Ba₂AgSi₃	Ba₄Li₂Si₆*	8.613	14.927	19.639	8.63	14.97	19.84	−2.74
	Ca₂AgSi₃	–	–	–	9.11	10.19	15.58	−2.36

Potential tetragonal structure with ordered Si/T sites
NdSi₂	ThSi₂*	3.968	a	13.715	4.12	a	14.05	−3.97
	AlB₂	–	–	–	4.08	a	4.13	−4.20
Nd₂AgSi₃	ThSi₂*	4.175	a	14.310	–	–	–	–
	POTS	–	–	–	5.96	5.93	14.54	−3.72
	Ce₂CoSi₃	–	–	–	8.35	a	4.28	−3.69
Nd₂PdSi₃	AlB₂*	4.103	a	4.204	–	–	–	–
	Ce₂CoSi₃	–	–	–	8.26	a	4.24	−5.17
Nd₂CuSi₃	Ce₂CoSi₃	–	–	–	8.06	a	4.26	−4.23
	Er₂RhSi₃ ( $[P\overline{6}2c]$ )*	8.076	a	8.440	8.07	a	8.46	−4.54
	P1	–	–	–	8.06	a	8.44	−4.14
Nd₂NiSi₃	Ce₂CoSi₃*	4.020	a	4.190	7.98	a	4.14	−6.32

Figure 9
R–T diagram of the RSi₂ and R₂TSi₃ compounds. The color of the markers symbolizes the range of ordering n, see Section 3.4

. If the structure is disordered (AlB₂, ThSi₂, GdSi₂), then n = 0 and the symbol is gray. If the structure is ordered, the range of ordering accords to the number of stacks along c in the unit cell. Up to three markers on one grid position are possible, representing different publications.

Subsequently, we use the chemical similarity of Ba and Sr to evaluate which orthorhombic structure type is more favorable for compound Sr₂AgSi₃, as it is the only alkaline earth compound that has not yet been synthesized. Both, the Ba₄Li₂Si₆ type of (Ba,Eu)₂AgSi₃ and the Ca₂AgSi₃ type are reasonable. We excluded other structure types as other chemically similar compounds only crystallize in those two structures. Here, chemically similar means a noble metal T and R preferring the +II oxidation state (e.g. alkaline earth metals, Eu and Yb). For T = Ag, Sr₂AgSi₃ is the only alkaline earth compound that has not yet been synthesized.

As a reference, we used Ba₂AgSi₃, also in both structure types. For Ba₂AgSi₃, the respective formation energies exhibited a clear preference for the reported Ca₂AgSi₃ type structure. However, the formation energies for both Sr₂AgSi₃ models are almost identical with a value of −2.83 eV, therefore we conclude that both structure types are equally stable. The formation energy of Sr₂AgSi₃ is slightly lower than that of Ba₂AgSi₃, which supports a stable structure.

Finally, we consider the potential tetragonal R₂TSi₃ superstructure as determined in Section 3.1. We did not find reports on this ordered tetragonal structure and expect that it is energetically unfavored. Only a few articles on suitable compounds exist, mainly containing Th compounds (Albering et al., 1994; Lejay et al., 1983; Chevalier et al., 1986; Li et al., 2008; Raman, 1967; Kaczorowski & Noël, 1993; Pöttgen & Kaczorowski, 1993) as well as U₂CuSi₃ (Albering et al., 1994; Lejay et al., 1983; Chevalier et al., 1986), La₂AlSi₃ (Raman & Steinfink, 1967), Ce₂AuSi₃ (Gordon et al., 1997), Er₂CuSi₃ and Nd₂AgSi₃. We chose Nd₂AgSi₃ for better comparability, as several compounds with either Nd or Ag have already been examined in the previous discussions. To compare our hypothetical tetragonal superstructure with an existing structure, we chose the hexagonal Ce₂CoSi₃ type, since the most obvious tetragonal ThSi₂ type exhibits mixed positions. We further took the disilicide NdSi₂ into account in both ThSi₂ and AlB₂ type structures.

Please note that the lattice parameters of the POTS type (calculated) are related to those of the ThSi₂ type (experimental) by rotation and elongation by a factor of $[\approx\sqrt{2}]$ . Thus, the interatomic distances of both tetragonal structure types of Nd₂AgSi₃ are approximately the same a_ThSi2 = 4.12 Å ≈ 4.21 Å = a_POTS/ $[\sqrt{2}]$ . For Nd₂CuSi₃, we compared three different symmetries, the high symmetry Ce₂CoSi₃, experimentally confirmed Er₂RhSi₃ $[(P\overline{6}2c)]$ and low symmetry P1 (No. 1). The lattice parameters of all three models are a = 8.06 Å and c ≈ 4.24 Å, which is in good agreement with the experimental values [Er₂RhSi₃ $[(P\overline{6}2c)]$ -type].

The formation energies of Nd₂AgSi₃ stoichiometry are −3.69 eV for the Ce₂CoSi₃ type and −3.72 eV for the tetragonal superstructure. With an absolute formation energy which is lower by 0.30 eV per atom, the tetragonal type is clearly favored. In general, the superstructural order for tetragonal symmetries may be suppressed for further reasons. On the one hand, the 3D Si/T network itself may present kinetic barriers. On the other hand, the entropy of mixing may hinder structural ordering more severe for the degeneracies of the 3D Si/T network than for the planar stacking of hexagonal symmetries.

3.4. Structure distribution

Fig. 9 gives an overview of the scatter of structure types within the RSi₂ and R₂TSi₃ compounds. This figure adapts the R–T grid of Fig. 4 with symbols announcing symmetry and range of order. To quantify the ordering within the different structure types, we defined the range of order as zero if the Si/T atoms do not order and otherwise as the number of Si/T layers along c in the unit cell. The range of order is highlighted by the color of the marker. The symmetry is marked by shape: hexagon for hexagonal AlB₂-like, open star for orthorhombic AlB₂-like, diamond for tetragonal ThSi₂, elongated diamond for orthorhombic GdSi₂. For technical reasons, this diagram shows at most three reports of the same compound (left, right, bottom). Our algorithm chooses the datasets with the highest as well as the lowest a parameter and an additional dataset with a different structure type, to depict the most significant variations. Fig. 9 visualizes the range of order in dependence on the atomic number of the R and T cations; it depicts the following trends:

First, most of the compounds in the grid exhibit an hexagonal AlB₂-like lattice. The other lattice types are mainly determined by the included R and T element. For example, the orthorhombic GdSi₂ structure type arises exclusively for lanthanide disilicides. The tetragonal lattice is dominant for R = Th compounds as well as for the disilicides with light rare earth elements. Additional compounds with tetragonal lattice are Ce₂AuSi₃, Nd₂AgSi₃ and Er₂CuSi₃, all possessing a noble metal T element. Thus, the Fermi level of the T element affects the structural stability, see Nentwich et al. (2020).

Furthermore, the completely ordered orthorhombic structure types Ca₂AgSi₃ and Ba₄Li₂Si₆ are only reported for R₂TSi₃ compounds with the monovalent ions T = Ag, Au and the divalent ions R = Ca, Ba, Eu, Yb (Cardoso Gil et al., 1999 ; Sarkar et al., 2013 ). The partially ordered structure type U₂RhSi₃ additionally arises for U₂PdSi₃ (Chevalier et al., 1996 ). Here, we do not consider the compound Ba₂LiSi₃ itself, since Li does not accord with our limitations to the T elements. Thus, the ordered orthorhombic AlB₂-like structure types are more probable if the T element is a monovalent atom and if the R element prefers the +II oxidation state – as for the alkaline earth metals.

Second, tetragonal LaSi₂ does not follow the hexagonal symmetry of the disilicides with third group elements Sc and Y. This phenomenon illustrates the affiliation of Sc and Y to the heavy and of La to the light rare earth elements (RÖMPP Online, 2011 ).

Third, with increasing atomic number of R within the lanthanide disilicides, three structure types succeed each other. The tetragonal ThSi₂ type is the dominant one for light rare earth elements (Ce–Eu), followed by the orthorhombic GdSi₂ type in the intermediate range and the hexagonal AlB₂ type for the heavy rare earth elements (according to the classification by Sitzmann; RÖMPP Online, 2011). This development is present in all samples independent of their thermal treatment, see Nentwich et al. (2020). This meets an observation of Mayer et al. (1967): upon heating the samples to 1600°C, they discovered two phase transformations, one from AlB₂ type to GdSi₂ type and another one from GdSi₂ type to ThSi₂ type. These transformations are reversible. A decreasing atomic number within the lanthanide group is accompanied with a significantly increasing radius and therefore with a higher space requirement. Increased thermal lattice vibrations at higher temperatures also cause higher space requirements. Thus, annealing has the same effect as decreasing the atomic number of R.

4. Conclusions

We present an extensive literature study of the RSi₂ and R₂TSi₃ compounds crystallizing in AlB₂- and ThSi₂-like structures complemented by DFT calculations. The local similarities between these structures, e.g. threefold planar coordination of the Si/T atoms, twelvefold coordination of the R elements, are highlighted and discussed. Additionally, we systematized the structure data and arranged them in a Bärnighausen diagram showing the relationships between structure types. We were able to determine the space groups of the ordered nonstoichiometric disilicides as piezoelectric $[P\overline{6}2m]$ (No. 189), $[P\overline{6}2c]$ (No. 190) and P2mm (No. 25).

According to Bodak & Gladyshevskii (1985), compounds La₂FeSi₃, La₂CoSi₃, La₂NiSi₃, Ce₂CuSi₃ and Ce₂NiSi₃ form a solid solution of structure type AlB₂ (disordered Si/T sites). Nevertheless, as evident from the discussion, we conclude that superstructures are expected to be the thermodynamic equilibrium structures, although they may be hard to synthesize, as they require obtaining the exact chemical composition on the one hand and for a careful thermal treatment on the other hand.

Comparison of the symmetry distribution within the R–T grid showed a special characteristic of the structure types Ca₂AgSi₃ and Ba₄Li₂Si₆. These structure types only arise if R has the formal +II oxidation state and T is either Au or Ag. Additionally, these structures are reported to have ionic character, whereas all other compounds are reported to be metallic. The given R–T diagram also shows a transition from tetragonal ThSi₂ to orthorhombic GdSi₂ to hexagonal AlB₂ type within the lanthanide disilicides with increasing atomic number of R. The structure types behave similarly with increasing temperature when respective crystals are heated.

Figs. 5 to 7 emphasize the number of systematic investigations of the RSi₂ and R₂TSi₃ compounds. On the one hand, these systematic investigations reduce systematic errors. On the other hand, the author's expectations may also have an impact on the evaluation (such as the structure type).

Concluding the DFT analysis, hypothetical compounds Ho₂PtSi₃, Pr₂CoSi₃, Eu₂PtSi₃ and Nd₂CoSi₃ are suggested to be stable, whereas Eu₂RhSi₃ will be unstable. Due to the positive results for Pr₂CoSi₃ and Nd₂CoSi₃, we recommend reinvestigating the R₂TSi₃ compounds reported by Mayer & Tassa (1969), with R = La, Ce, Pr, Nd, Sm, Gd and T = Fe, Co, Ni (originally with R₂T_0.8Si_3.2 stoichiometry). To complete the crystal structure information of La₂PdSi₃, we predict the lattice parameters a = 8.34 Å and c = 4.38 Å in a Ce₂CoSi₃ type structure. With respect to the question whether Sr₂AgSi₃ prefers the Ca₂AgSi₃ or the Ba₄Li₂Si₆ structure type, both models result in almost identical formation energies of −2.83 eV and are equally stable from a theoretical point of view. Likewise, BaSi₂ may exhibit hexagonal as well as tetragonal symmetry, as the formation energy of both models is −1.03 eV. In comparison, the potential tetragonal superstructure is less favorable than a highly symmetric hexagonal structure. The results of this work do not exclude the existence of structures that are equally or more stable than the ones presented here. The solid solutions with disorder at the Si/T position may always present potential candidates for the ground state of a specific R₂TSi₃ compound.

At this point, the question of particular driving forces for a certain type of symmetry and the multiplicity of the superstructure symmetry types and structure types remains. This question will be addressed in the second part of this work (Nentwich et al., 2020) focusing on the electronic structure.

APPENDIX A

Wyckoff positions of the different superstructures

Wyckoff positions of the different superstructures are presented here in Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17.

APPENDIX B

Fundamentals of the DFT calculations

To calculate the formation energies with DFT, it is necessary to know the energy of the components that make up the compound. Table 19 contains a list of the underlying single-element compounds used to calculate the formation energies in Table 18.

Table 19
Space groups of the unary R crystals used for standardization of the formation energies

Atomic number	Element	Space group	ICSD code
14	Si	$[Fd\overline{3}m]$ (No. 227)	51688
27	Co	P6₃/mmc (No. 194)	184251
28	Ni	$[Fm\overline{3}m]$ (No. 225)	646089
38	Sr	$[Fm\overline{3}m]$ (No. 225)	652875
45	Rh	$[Fm\overline{3}m]$ (No. 225)	171677
46	Pd	$[Fm\overline{3}m]$ (No. 225)	76148
47	Ag	$[Fm\overline{3}m]$ (No. 225)	181730
56	Ba	$[Im\overline{3}m]$ (No. 229)	108091
57	La	P6₃/mmc (No. 194)	641382
58	Ce	$[Fm\overline{3}m]$ (No. 225)	620620
59	Pr	$[Fm\overline{3}m]$ (No. 225)	649185
60	Nd	P6₃/mmc (No. 194)	164281
63	Eu	$[Im\overline{3}m]$ (No. 229)	604033
64	Gd	P6₃/mmc (No. 194)	184250
65	Tb	$[R\overline{3}mH]$ (No. 166)	652944
66	Dy	P6₃/mmc (No. 194)	95172
67	Ho	$[R\overline{3}mH]$ (No. 166)	639322
78	Pt	$[Fm\overline{3}m]$ (No. 225)	649490

Footnotes

‡These authors contributed equally to this work

Funding information

Funding for this research was provided by: European regional development fund (grant No. 100109976); Federal Ministry of Education and Research (grant No. 03EK3029A; grant No. 03SF0542A); Helmholtz Excellence Network (grant No. ExNet 0026); Deutsche Forschungsgemeinschaft (grant No. 324641898).

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