REDUCTION OF UNIT CELLS OF Me7C3-TYPE CARBIDES BY NIGGLI METHOD
Dominik Senczyk, Katarzyna Lis
Poznan University of Technology, pl. M. Sklodowskiej-Curie
5, PL-60-965 Poznan, Poland
senczyk@sol.put.poznan.pl
Keywords: Carbides, polytypes, crystal structure, Niggli method, unit cell, reduction of unit cell, lattice constants, Bravais lattice
The structure of carbides formed by some transition metals (i. e. chromium, manganese, iron) has been described as hexagonal, orthorhombic or monoclinic, but considerably different values of the lattice constants have been given at the same time. Thus arises the question: what is the real structure of this carbide? The first aim of this work is to answer this question. The second purpose is to consider the possibility of polytypism for this carbide.
In order to determine the proper crystallographic system and type of Bravais lattice the Niggli cell reduction method was used [1]. Initial crystallographic data for Cr7C3, Mn7C3 and Fe7C3 are given in Tables l, 2, 3 and 4.
Table 1. Lattice constants of the Cr7C3 unit cell before and after reduction
Unit Cell |
Reduced Unit Cell |
Ref. | ||||||
Crystal system | a [nm] | b [nm] | c [nm] | Crystal system | a [nm] | b [nm] | c [nm] | |
H | 1.398 | 0.4523 | OR | 1.398 | 1.2107 | 0.4523 | [2] | |
H | 1.4008 | 0.4532 | OR | 1.4008 | 1.2131 | 0.4532 | [3,4] | |
OR | 0.7010 | 0.2142 | 0.452 | OR | 0.7010 | 1.2142 | 0.452 | [6] |
H | 1.402 | 0.452 | OR | 1.402 | 1.2141 | 0.452 | [6] | |
H | 0.7010 | 0.452 | OR | 0.607 | 0.701 | 0.452 | [6] |
Table 2. The crystal structure of (Cr, Fe)7C3 carbides [7]
Carbide | Unit cell | Reduced unit cell | |||||
Crystal system |
a [nm] | c [nm] | Crystal system |
a [nm] | b [nm] | c [nm] | |
Cr7C3 | H | 1.40266 | 0.45316 | OR | 1.21473 | 1.40266 | 0.45316 |
Cr6FeC3 | H | 1.40064 | 0.45107 | OR | 1.21298 | 1.40064 | 0.45107 |
Cr5Fe2C3 | H | 1.39649 | 0.45105 | OR | 1.20939 | 1.39649 | 0.45105 |
Cr4Fe3C3 | H | 1.39187 | 0.45062 | OR | 1.20539 | 1.39187 | 0.45062 |
Cr3Fe4C3 | H | 1.38665 | 0.45053 | OR | 1.20087 | 1.38665 | 0.45053 |
Cr2.05Fe4.95C3 | H | 1.38336 | 0.45052 | OR | 1.19802 | 1.38336 | 0.45052 |
Table 3. The crystal structure of Mn7C3 carbide
Unit Cell |
Reduced Unit Cell |
Ref. | ||||||
Crystal system | a [nm] | b [nm] | c [nm] | Crystal system | a [nm] | b [nm] | c [nm] | |
H | 1.3898 | 0.4539 | OR | 1.3898 | 1.2036 | 0.4539 | [8,9] | |
H | 0.695 | 0.454 | OR | 0.695 | 0.6018 | 0.454 | [10] | |
OR | 0.4546 | 0.6959 | 1.1979 | OR | 0.4546 | 0.6959 | 1.1979 | [11] |
OR | 0.6959 | 1.1976 | 0.4546 | OR | 0.6959 | 1.1976 | 0.4546 | [10] |
H | 1.390 | 0.455 | OR | 1.390 | 1.2037 | 0.455 | [12] | |
H |
1.3870 | 0.4541 | OR | 1.3870 | 1.2011 | 0.4541 | [13] |
Table 4. The crystal structure of Fe7C3 carbide
Unit Cell |
Reduced Unit Cell |
Ref. | ||||||
Crystal system | a [nm] | b [nm] | c [nm] | Crystal system | a [nm] | b [nm] | c [nm] | |
H | 0.688 | 0.688 | 0.454 | OR | 0.688 | 0.5959 | 0.454 | [14] |
OR | 0.6879 | 1.1942 | 0.454 | OR | 0.6879 | 1.1942 | 0.4542 | [15,16] |
Because of the fact that the greatest lattice constants this carbide are "a" and "b" (see Tables 1-4) the reduction was started using the relation:
a1 = a + (1/2)b or b1 = b + (1/2) a
where: |a|, |b| - lattice constants before reduction, |a1|, |b1| - lattice constant after reduction, because then a1b1 = 0. The results from the cell reduction procedure results are given in Tables 1-4. From the data Niggli matrix [1] was formed:
s11 | s22 | s33 |
s23 | s31 | s12 |
where s11 =a1a1, s22 = b1b1, s33 = c1c1, s23 = b1c1, s31 = c1a1, s12 = a1b1.
For all data s11 <>s22 <>s33 as well as s23 =s31 =s12 = 0 and in this case the matrix describes a primitive orthorhombic unit cell [1].
This research was supported by Poznan University of Technnlogy under Grants No. PB-24-215l97-BW and TB-24-217/97-DS.