There was a mistake in the letter that listed the Bijvoet pairs
for a monoclinic space group and that is confusing you. Let me
try.
The equivalent positions for a B setting monoclinic are
h,k,l; -h,k,-l.
The Friedel mates for the general position (h,k,l) are (-h,-k,-l).
This means that the equivalent positions also have Friedel mates at
h,-k,l.
The Bijvoet mates of h,k,l are therefore, according to the
definitions given in previous letters, -h,-k,-l; and h,-k,l.
There are more Bijvoet mates to a reflection then Fridel mates.
A centric reflection is a reflection that is BOTH a symmetry equivalent
reflection AND a Bijvoet mate to some other reflection. This is a
very small subset of all reflections.
Every reflection has one Friedel mate and has N Bijvoet mates,
where N is the number of equivalent positions. Only a small number
of reflections are centric (with the limiting case of only F000).
Dale Tronrud
Bernhard Rupp wrote:
Let's try this again, with definitions, and pls scream if I am wrong:
a) Any reflection pair hR = h forms a symmetry related pair.
R is any one of G point group operators of the SG.
This is a set of reflections (S). Their amplitudes
are invariably the same. They do not even show up
as individual pairs in the asymmetric unit of the reciprocal
space.
NB: their phases are restricted but not the same.
b) a set h=-h (set F) exist where reflections may or may not
carry anomalous signal. They form the centrosymmetrically related wedge
of the asymmetric unit of reciprocal space.
c) a centric reflection (set C) is defined as
hR=-h
and cannot carry anomalous signal. Example zone h0l in PG 2.
As Ian Tickle pointed out, the CCP4 wiki is wrong:
"Centric reflections in space group P2 and P21 are thus
those with 0,k,0." Not so; an example listing is attached at the end.
d) therefore, some e:F exist that carry AS (F.ne.C)
and some that do not carry AS (F.el.C).
I hope we can agree on those facts.
Now for the name calling:
(S) is simply the set of symmetry related reflections, defined as hR=h.
(F) is the set of Friedel pairs, defined as h=-h.
(C) are centric reflections, defined as hR=-h.
Thus, only if (F.ne.C), anomalous signal. I thought those
are Bijvoet pairs. They are, but it may not be the definition
of a Bijvoet pair.
Try 1:
Bijvoet pair is F(h) and any mate that is symmetry-related to F(-h),
e.g., F(hkl) and F(-h,k,-l) in monoclinic.
hkl is not related to -hk-l via h = -h. Only h0l is, and those are (e:C).
So, I cannot quote follow that, probably try 1 is not a good definition.
Try 2:
I've always thought that a Bijvoet pair is any pair for which an
anomalous difference could be observed.
Good start. I subscribe to that.
This includes Friedel pairs (h & h-bar)
Good. That's the definition of F.
but it also includes pairs of the form h & h', where h'
is symmetry-related to h-bar.
Ooops. That is the definition of a centric reflection.
Thus Friedel pairs are a subset of all possible Bijvoet pairs.
Cannot see that. I still maintain that Bijvoet pairs are
a subset of Friedel pairs (which does include Pat's definition).
I fail to see anything else but Friedel pairs in my list
of reflections - some of them carry AS (F.ne.C) and some
don't (F.el.C).
B = F.ne.C.
Seems to be a necessary and sufficient condition,
in agreement with Pat's definition (though not the explanation).
But - isn't that exactly what I said from the beginning?
"A Bijvoet pair is an acentric Friedel pair..."
Or - where are any other Bijvoet pairs hiding? Where did I miss them?
(NB: Absence of anisotropic AS assumed -let's not go there)
See reflection list P2 (hkl |F| fom phi 2theta stol2)
last 3 items: centric flag, epsilon, m(h)
0 0 1 993.54 1.00 179.99 65.61 0.0000581 1 1 2
0 0 -1 993.54 1.00 179.99 65.61 0.0000581 1 1 2
1 0 0 1412.58 1.00 0.14 38.22 0.0001711 1 1 2
-1 0 0 1412.58 1.00 0.14 38.22 0.0001711 1 1 2
0 0 2 3279.49 1.00 180.31 32.80 0.0002323 1 1 2
0 0 -2 3279.49 1.00 180.31 32.80 0.0002323 1 1 2
1 0 1 379.89 1.00 180.25 30.36 0.0002712 1 1 2
-1 0 -1 379.89 1.00 180.25 30.36 0.0002712 1 1 2
-1 0 2 1355.06 1.00 0.13 27.97 0.0003195 1 1 2
1 0 -2 1355.06 1.00 0.13 27.97 0.0003195 1 1 2
0 1 0 2432.85 1.00 21.09 24.35 0.0004216 0 2 1
0 -1 0 2434.14 1.00 339.65 24.35 0.0004216 0 2 1
0 1 1 621.36 1.00 101.67 22.83 0.0004797 0 1 2
0 -1 -1 623.27 1.00 258.49 22.83 0.0004797 0 1 2
1 0 2 319.68 1.00 359.98 22.65 0.0004874 1 1 2
-1 0 -2 319.68 1.00 359.98 22.65 0.0004874 1 1 2
0 0 3 426.17 1.00 180.99 21.87 0.0005227 1 1 2
0 0 -3 426.17 1.00 180.99 21.87 0.0005227 1 1 2
-1 0 3 1581.93 1.00 0.44 20.98 0.0005680 1 1 2
1 0 -3 1581.93 1.00 0.44 20.98 0.0005680 1 1 2
1 1 0 338.67 1.00 46.52 20.54 0.0005927 0 1 2
-1 -1 0 341.71 1.00 314.70 20.54 0.0005927 0 1 2
-1 1 1 1649.38 1.00 80.93 20.26 0.0006089 0 1 2
1 -1 -1 1652.55 1.00 279.90 20.26 0.0006089 0 1 2
0 1 2 343.14 1.00 66.86 19.55 0.0006540 0 1 2
0 -1 -2 345.84 1.00 293.42 19.55 0.0006540 0 1 2
-2 0 1 171.90 1.00 358.59 19.48 0.0006586 1 1 2
2 0 -1 171.90 1.00 358.59 19.48 0.0006586 1 1 2
2 0 0 1238.53 1.00 180.20 19.11 0.0006844 1 1 2
-2 0 0 1238.53 1.00 180.20 19.11 0.0006844 1 1 2
1 1 1 201.11 1.00 349.93 19.00 0.0006928 0 1 2
-1 -1 -1 188.82 1.00 12.34 19.00 0.0006928 0 1 2
-1 1 2 296.39 1.00 205.44 18.37 0.0007411 0 1 2
1 -1 -2 291.55 1.00 154.95 18.37 0.0007411 0 1 2
1 0 3 25.24 1.00 178.32 17.46 0.0008197 1 1 2
-1 0 -3 25.24 1.00 178.32 17.46 0.0008197 1 1 2
2 0 1 718.09 1.00 0.81 17.39 0.0008264 1 1 2
-2 0 -1 718.09 1.00 0.81 17.39 0.0008264 1 1 2
1 1 2 108.75 1.00 256.06 16.58 0.0009090 0 1 2
-1 -1 -2 109.68 1.00 109.85 16.58 0.0009090 0 1 2
0 0 4 16.54 1.00 354.30 16.40 0.0009293 1 1 2
0 0 -4 16.54 1.00 354.30 16.40 0.0009293 1 1 2
-1 0 4 375.95 1.00 179.77 16.37 0.0009326 1 1 2
1 0 -4 375.95 1.00 179.77 16.37 0.0009326 1 1 2
0 1 3 546.36 1.00 140.75 16.27 0.0009444 0 1 2
0 -1 -3 541.71 1.00 220.17 16.27 0.0009444 0 1 2
-2 0 3 56.64 1.00 357.74 16.18 0.0009554 1 1 2
2 0 -3 56.64 1.00 357.74 16.18 0.0009554 1 1 2
-1 1 3 256.58 1.00 106.38 15.89 0.0009896 0 1 2
1 -1 -3 246.70 1.00 251.95 15.89 0.0009896 0 1 2
2 0 2 197.51 1.00 359.40 15.18 0.0010846 1 1 2
-2 0 -2 197.51 1.00 359.40 15.18 0.0010846 1 1 2
-1 -1 -5 133.99 1.00 196.68 10.53 0.0022546 0 1 2
-2 0 6 163.47 1.00 1.91 10.49 0.0022718 1 1 2
2 0 -6 163.47 1.00 1.91 10.49 0.0022718 1 1 2
-2 2 1 458.26 1.00 271.66 10.32 0.0023451 0 1 2
2 -2 -1 455.12 1.00 88.27 10.32 0.0023451 0 1 2
-3 0 5 344.11 1.00 179.84 10.29 0.0023625 1 1 2
3 0 -5 344.11 1.00 179.84 10.29 0.0023625 1 1 2
2 2 0 610.75 1.00 80.56 10.27 0.0023710 0 1 2
-2 -2 0 608.51 1.00 279.87 10.27 0.0023710 0 1 2
2 1 4 402.06 1.00 236.16 10.27 0.0023711 0 1 2
-2 -1 -4 395.70 1.00 124.26 10.27 0.0023711 0 1 2
-3 1 4 148.85 1.00 341.90 10.23 0.0023873 0 1 2
3 -1 -4 139.22 1.00 17.04 10.23 0.0023873 0 1 2
-1 1 6 359.89 1.00 192.75 10.14 0.0024319 0 1 2
1 -1 -6 363.35 1.00 168.59 10.14 0.0024319 0 1 2
-2 2 2 362.68 1.00 251.52 10.13 0.0024355 0 1 2
2 -2 -2 356.29 1.00 108.72 10.13 0.0024355 0 1 2
3 0 3 341.40 1.00 359.75 10.12 0.0024404 1 1 2
-3 0 -3 341.40 1.00 359.75 10.12 0.0024404 1 1 2
3 1 2 572.31 1.00 91.40 10.11 0.0024457 0 1 2
-3 -1 -2 567.84 1.00 267.14 10.11 0.0024457 0 1 2
1 2 3 462.22 1.00 98.66 9.99 0.0025063 0 1 2
-1 -2 -3 455.44 1.00 263.08 9.99 0.0025063 0 1 2
0 1 6 172.20 1.00 322.68 9.97 0.0025126 0 1 2
0 -1 -6 173.26 1.00 38.38 9.97 0.0025126 0 1 2
2 2 1 355.05 1.00 173.38 9.97 0.0025130 0 1 2
-2 -2 -1 354.48 1.00 184.87 9.97 0.0025130 0 1 2
1 0 6 386.18 1.00 179.53 9.97 0.0025138 1 1 2
-1 0 -6 386.18 1.00 179.53 9.97 0.0025138 1 1 2
2 0 5 88.73 1.00 178.17 9.89 0.0025561 1 1 2
-2 0 -5 88.73 1.00 178.17 9.89 0.0025561 1 1 2
0 2 4 87.19 1.00 47.91 9.78 0.0026159 0 1 2
0 -2 -4 82.08 1.00 305.75 9.78 0.0026159 0 1 2
-1 2 4 209.29 1.00 234.28 9.77 0.0026191 0 1 2
1 -2 -4 205.72 1.00 128.29 9.77 0.0026191 0 1 2
-4 0 1 258.90 1.00 179.03 9.75 0.0026279 1 1 2
4 0 -1 258.90 1.00 179.03 9.75 0.0026279 1 1 2
-4 0 2 622.27 1.00 0.57 9.74 0.0026343 1 1 2
4 0 -2 622.27 1.00 0.57 9.74 0.0026343 1 1 2
-2 2 3 241.96 1.00 239.95 9.73 0.0026419 0 1 2
2 -2 -3 244.55 1.00 121.07 9.73 0.0026419 0 1 2
-2 1 6 226.30 1.00 179.63 9.63 0.0026935 0 1 2
2 -1 -6 221.30 1.00 179.01 9.63 0.0026935 0 1 2
-1 0 7 292.07 1.00 358.96 9.58 0.0027234 1 1 2
1 0 -7 292.07 1.00 358.96 9.58 0.0027234 1 1 2
4 0 0 29.23 1.00 348.55 9.56 0.0027377 1 1 2
-4 0 0 29.23 1.00 348.55 9.56 0.0027377 1 1 2
-4 0 3 20.80 1.00 180.51 9.52 0.0027569 1 1 2
4 0 -3 20.80 1.00 180.51 9.52 0.0027569 1 1 2
2 2 2 377.95 1.00 151.42 9.50 0.0027712 0 1 2
-2 -2 -2 374.79 1.00 207.12 9.50 0.0027712 0 1 2
-3 1 5 357.75 1.00 276.19 9.48 0.0027842 0 1 2
3 -1 -5 354.39 1.00 83.21 9.48 0.0027842 0 1 2
0 0 7 557.70 1.00 0.18 9.37 0.0028460 1 1 2
0 0 -7 557.70 1.00 0.18 9.37 0.0028460 1 1 2
3 1 3 260.52 1.00 260.70 9.35 0.0028620 0 1 2
-3 -1 -3 254.89 1.00 98.96 9.35 0.0028620 0 1 2
-3 0 6 35.11 1.00 180.75 9.32 0.0028756 1 1 2
3 0 -6 35.11 1.00 180.76 9.32 0.0028756 1 1 2
1 1 6 179.90 1.00 290.90 9.23 0.0029355 0 1 2
-1 -1 -6 188.84 1.00 67.96 9.23 0.0029355 0 1 2
-2 0 7 272.70 1.00 182.34 9.22 0.0029430 1 1 2
2 0 -7 272.70 1.00 182.34 9.22 0.0029430 1 1 2
1 2 4 199.34 1.00 23.09 9.20 0.0029548 0 1 2
-1 -2 -4 198.81 1.00 341.26 9.20 0.0029548 0 1 2
4 0 1 170.91 1.00 358.93 9.18 0.0029636 1 1 2
-4 0 -1 170.91 1.00 358.93 9.18 0.0029636 1 1 2
-2 2 4 324.59 1.00 281.88 9.18 0.0029646 0 1 2
2 -2 -4 333.72 1.00 78.81 9.18 0.0029646 0 1 2
3 0 4 620.31 1.00 179.59 9.17 0.0029728 1 1 2
-3 0 -4 620.31 1.00 179.59 9.17 0.0029728 1 1 2
2 1 5 515.94 1.00 316.87 9.16 0.0029778 0 1 2
-2 -1 -5 516.82 1.00 42.70 9.16 0.0029778 0 1 2
-4 1 1 137.14 1.00 39.42 9.05 0.0030496 0 1 2
4 -1 -1 135.68 1.00 321.72 9.05 0.0030496 0 1 2
-4 1 2 99.76 1.00 105.36 9.04 0.0030559 0 1 2
4 -1 -2 101.69 1.00 248.43 9.04 0.0030559 0 1 2
-1 2 5 38.13 1.00 78.86 8.98 0.0030999 0 1 2
1 -2 -5 43.37 1.00 279.05 8.98 0.0030999 0 1 2
