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].

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This research was supported by Poznan University of Technnlogy under Grants No. PB-24-215l97-BW and TB-24-217/97-DS.