1. Melilite Natural melilite from San Venanzo, Umbria, Italy Formula: (Ca 1.89 Sr 0.01 Na 0.08 K 0.02 )(Mg 0.92 Al 0.08 )(Si 1.98 Al 0.02 )O 7 Superspace group: P-42 1 m(αα0, α-α0) + translation part Cell parameters: a=7.860 (1), c=5.024 (1) Å q vectors : 0.2815(3)(a*+b*), 0.2815(3)(-a*+b*) l=1

3. Reflection -1 3 -2 -1 -1 (“h”)

4. Point detector measurement at various Ψ of reflection – 1 3 – 2 – 1 – 1 ( “ h ” ) -050-045-040-035 -030-025-020-015 -010-0050+040

5. Symmetry in SBN Formula: Sr 0.61 Ba 0.39 Nb 2 O 6 Superspace group: P4bm(αα½, α-α½) + translation part Cell parameters: a=12.4815 (9), c=3.9428 (6) Å q vectors : q 1 =(α,α,½), q 2 =(α,-α,½), α=0.3075(6)

6. Cell parameters: a=12.482, c=3.943 Å P4bm(αα½, α-α½) The rational part of q-vector must be removed by doubling c. Transformation matrix on indices (row): Resulting indices: (h, k, 2l+m+n, m, n) New super space group: X4bm(αα0, α-α0) Centring: (0,0,0,1/2,1/2,1/2)

7. Example: Five dimensional incommensurate structure of fresnoite Acta Cryst. (2006). B62, 1031–1037 The sample from Museo di Storia Naturale, Sezione di Mineralogia e Litologia, Universita di Firenze, Italy, catalogue number 44362/G originates from the sanbornite deposits of eastern Fresno County, California. Fresnoite, Ba 2 TiS i2 O 8, is structurally related to the melilite group minerals. Composition from electron microprobe: (Ba 1.962 Ca 0.038 ) = 2.000 Ti 1.000 Si 2.000 O 8.000 Unit cell 8.5133(3) 8.5166(3) 10.3913(5) 90.019(4) 89.972(3) 89.950(3) 753.42(6) q(1): 0.3023(3) 0.3023(3) 0 q(2): -0.3023(3) 0.3025(3) 0 PEAK TABLE 5865 main refl.; 6745 q1 ; 6668 q2 satellites; 1298 cross q1/q2 Instead of (a a ½) we use doubled c and X centering (0 0 ½ ½ ½ )

8. Reconstruction of reciprocal space, scaling factor 1000 Plane hk1 Plane hk0 Only first order satellites q 1, q 2 Main reflections and satellites q 1 +q 2, q 1 -q 2

9. Jana indexing tool We see both combined satellites and lambda/2 reflections

10. Verification in http://stokes.byu.edu/ssg.html by typing symmetry operators to finding toolhttp://stokes.byu.edu/ssg.html