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Quantitative Analysis

Simultaneous refinement of X-ray diffraction patterns of multiple phases allows quantitative analysis using the following relationship, whose variables (aside from Z) are all refined in the fitting process:

Wp = (Sp (ZMV)p) / (S [S (ZMV)]) (1)

where

Wp = the mass fraction of phase p,
S = the Rietveld scale factor,
Z = the number of formula units per unit cell,
M = the mass of the formula unit, and
V = the unit cell volume.

Initial estimates of the phase fractions were used to estimate the linear absorption coefficient of the mixture. A second quantitative estimate using a Brindley absorption correction, based upon a mean particle size of 1 µm, established the final analyses. Phase abundance estimates of polymorphs of specific phases, such as -C2S and -C2S or cubic and orthorhombic C3A were summed for comparison with the optical microscopy data.

Figure 7. A straight normal probability plot of the errors indicates their distribution is well approximated by a Gaussian.

Table 2 Mass fraction (percent) by QXRD, RM 8486 optical data (OM), and round robin data (RR) following ASTM C 1356 [2]. QXRD analysis of nine vials (1-9) with two splits per vial (a, b), with each split analyzed in duplicate (1, 2).
  Alite Belite Ferrite Aluminate Periclase

1a1
1a2
1b1
1b2
2a1
2a2
2b1
2b2
3a1
3a2
3b1
3b2
4a1
4a2
4b1
4b2
5a1
5a2
5b1
5b2
6a1
6a2
6b1
6b2
7a1
7a2
7b1
7b2
8a1
8a2
8b1
8b2
9a1
9a2
9b1
9b2
OM1 a
OM1 b
OM1 c
OM1 d
RR1 a
RR1 b
RR2 a
RR2 b
RR3 a
RR3 b

56.4
56.8
55.1
56.1
56.8
56.5
58.5
58.5
56.8
57.0
58.6
57.4
56.3
56.2
56.7
56.9
55.8
55.7
57.2
57.1
56.9
56.8
57.3
57.0
55.9
56.5
56.5
56.9
56.7
57.0
56.9
56.3
55.8
56.9
56.1
57.2
59.0
56.3
60.3
58.1
56.8
56.2
60.0
60.2
61.8
60.6

21.5
21.8
21.1
20.3
20.9
21.7
20.8
21.0
21.5
21.2
20.4
20.6
21.1
21.5
20.9
20.9
20.6
20.8
21.1
21.0
21.5
21.4
20.7
20.5
21.4
20.7
21.5
21.1
21.2
21.1
20.6
21.1
21.4
20.8
21.0
20.8
23.0
25.3
20.6
23.7
23.6
24.1
23.6
24.6
23.9
25.0

13.8
14.0
16.1
15.7
14.6
14.4
13.9
13.5
14.2
14.5
13.7
14.5
15.0
14.9
14.8
14.7
15.7
15.8
14.1
14.1
14.1
14.1
14.2
15.0
14.9
14.8
14.3
14.2
14.4
14.3
15.0
14.8
15.1
14.6
15.3
14.8
13.0
13.1
14.2
14.1
13.5
14.3
9.6
8.1

3.9
3.3
3.6
3.7
3.5
3.4
3.1
3.2
3.3
3.4
3.3
3.5
3.4
3.3
3.5
3.5
3.5
3.5
3.4
3.4
3.4
3.4
3.6
3.5
3.5
3.4
3.6
3.5
3.5
3.4
3.5
3.5
3.5
3.5
3.4
1.1
1.1
1.0
1.0
3.2
2.5
1.3
2.3

4.3
4.0
4.2
4.3
4.2
4.0
3.7
3.8
4.2
3.9
3.9
4.1
4.2
4.1
4.1
4.1
4.4
4.2
4.2
4.3
4.0
4.3
4.2
3.9
4.3
4.6
4.3
4.1
4.2
4.0
4.1
4.2
4.3
4.2
4.1
3.8
3.0
3.7
3.7
3.7
1.9
2.0
4.7
3.8
2.5
2.8

 

Table 3. Mass fraction (percent) by QXRD, RM 8487 optical data (OM), and round robin data (RR) following ASTM C 1356 [2]. QXRD analysis of nine vials (1-9) with two splits per vial (a, b), with each split analyzed in duplicate (1, 2).

 

Alite

Belite

Ferrite

Aluminate

Arcanite

Free Lime*

1a1
1a2
1b1
1b2
2a1
2a2
2b1
2b2
3a1
3a2
3b1
3b2
4a1
4a2
4b1
4b2
5a1
5a2
5b1
5b2
6a1
6a2
6b1
6b2
7a1
7a2
7b1
7b2
8a1
8a2
8b1
8b2
9a1
9a2
9b1
9b2
OM1 a
OM1 b
OM1 c
OM1 d
RR1 a
RR1 b
RR2 a
RR2 b
RR3 a
RR3 b

74.7
74.2
74.3
74.4
74.4
74.2
74.6
74.2
74.4
74.4
74.2
73.5
74.5
73.8
73.8
73.6
73.8
73.8
74.2
74.4
74.1
74.0
74.2
74.5
74.2
74.7
74.7
74.9
73.9
74.3
74.1
73.7
73.8
74.0
74.0
74.1
73.19
71.64
73.25
75.46
72.24
72.51
70.42
71.92
76.93
76.11

11.2
11.2
11.5
11.7
11.6
11.8
11.5
11.6
11.4
11.4
11.4
11.4
11.1
11.4
11.4
11.6
11.4
11.5
11.9
11.8
11.6
11.4
11.6
11.4
11.2
10.8
11.3
11.0
11.3
11.2
11.2
11.7
11.5
11.6
11.9
11.5
7.29
9.59
7.14
6.98
9.10
9.79
9.00
8.50
8.31
8.29

4.3
4.7
4.3
4.3
4.3
4.3
4.4
4.3
4.4
4.4
4.6
4.5
4.5
4.8
4.8
4.7
4.8
4.7
4.2
4.3
4.4
4.8
4.5
4.4
4.5
4.5
4.2
4.3
4.9
4.7
4.6
4.7
4.7
4.5
4.3
4.6
4.00
3.72
2.78
2.56
1.46
0.88
0.92
1.20
1.00
1.00

8.3
8.5
8.3
8.1
8.3
8.3
8.0
8.3
8.3
8.3
8.2
9.1
8.4
8.3
8.3
8.4
8.5
8.5
8.1
8.0
8.3
8.3
8.3
8.3
8.6
8.5
8.4
8.4
8.5
8.4
8.6
8.3
8.4
8.3
8.3
8.3
12.43
12.29
12.82
10.82
13.23
13.32
15.63
14.50
13.40
10.40

0.6
0.6
0.7
0.6
0.6
0.6
0.6
0.7
0.6
0.5
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.5
0.5
0.6
0.6
0.6
0.5
0.5
0.6
0.6
0.6
0.6
0.6
0.7
0.6
0.82
0.62
0.98
1.49
1.69
1.48
0.00
0.00
0.00
0.00

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2.25
1.91
3.01
2.63
2.24
1.99
4.03
3.88
1.86
1.57

* XRD data did not measure free lime as it was altered during sample preparation and storage.

 

Table 4. Mass fraction (percent) by QXRD, RM 8488 optical data (OM), and round robin data (RR) following ASTM C 1356 [2]. QXRD analysis of nine vials (1-9) with two splits per vial (a, b), with each split analyzed in duplicate (1, 2).

 

Alite

Belite

Ferrite

Aluminate

1a1
1a2
1b1
1b2
2a1
2a2
2b1
2b2
3a1
3a2
3b1
3b2
4a1
4a2
4b1
4b2
5a1
5a2
5b1
5b2
6a1
6a2
6b1
6b2
7a1
7a2
7b1
7b2
8a1
8a2
8b1
8b2
9a1
9a2
9b1
9b2
OM1 a
OM1 b
OM1 c
OM1 d
RR1 a
RR1 b
RR2 a
RR2 b
RR3 a
RR3 b

67.2
68.2
67.5
66.9
66.7
67.3
67.0
67.1
65.8
66.7
67.3
67.5
68.0
68.0
67.3
67.3
66.6
67.4
67.4
67.4
66.1
66.5
66.9
67.4
68.2
68.3
67.5
68.0
67.3
68.0
68.7
69.1
69.0
68.9
68.1
69.2
65.14
64.98
64.20
65.54
63.40
64.06
65.52
67.72
67.35
67.82

14.8
13.9
14.1
14.7
14.9
14.0
15.0
15.5
15.2
14.8
14.0
14.1
13.7
14.1
14.8
14.9
15.6
14.5
14.8
14.8
15.4
14.7
15.2
15.0
14.6
14.5
15.1
14.7
14.4
13.9
15.2
15.3
16.0
15.4
15.8
14.5
18.30
19.31
18.48
17.94
19.82
19.37
19.92
17.21
18.09
18.16

14.3
14.3
14.8
14.7
14.8
14.9
14.3
13.7
15.1
14.6
14.8
14.6
14.6
14.5
14.2
14.2
14.1
14.3
14.2
14.0
14.7
14.8
14.3
14.0
13.5
13.6
13.9
13.9
14.5
14.3
12.7
12.2
11.7
12.3
12.6
12.7
13.46
11.56
13.20
10.25
12.77
12.03
7.80
7.54
9.80
8.20

3.7
3.7
3.6
3.7
3.6
3.7
3.8
3.7
3.8
3.9
3.8
3.8
3.7
3.5
3.6
3.6
3.7
3.9
3.7
3.7
3.8
4.0
3.7
3.6
3.7
3.6
3.5
3.5
3.9
3.7
3.4
3.4
3.4
3.4
3.5
3.6
3.08
4.15
3.87
6.24
4.02
4.53
6.76
7.54
4.80
8.20

 

Table 5. RM Clinker QXRD Summary: 95 % Confidence Limits for the Mean (Mass Percent).
Limits LowerMeanUpper
RM 8486  
Alite 56.50 56.75 57.00
Belite 20.91 21.04 21.17
Aluminate 3.41 3.46 3.50
Ferrite 14.40 14.61 14.81
Periclase 4.07 4.13 4.20
Limits Lower Mean Upper
RM 8487  
Alite 73.51 74.18 74.85
Belite 10.95 11.44 11.92
Aluminate 7.98 8.35 8.73
Ferrite 4.11 4.51 4.91
Periclase 0.77 0.91 1.05
Limits LowerMeanUpper
RM 8488 
Alite 65.91 67.55 69.18
Belite 13.64 14.77 15.90
Aluminate 3.36 3.66 3.96
Ferrite 12.33 14.02 15.71

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