Atomic form factors are widely used for the characterization of targets and specimens, from crystallography to biology. By using recent mathematical results, here we derive an analytical expression for the atomic form factor within the independent particle model constructed from nonrelativistic screened hydrogenic wave functions. The range of validity of this analytical expression is checked by comparing the analytically obtained form factors with the ones obtained within the Hartee-Fock method. As an example, we apply our analytical expression for the atomic form factor to evaluate the differential cross section for Rayleigh scattering off neutral atoms.

1.
K. K.
Seth
 et al.,
Phys. Rev. Lett.
110
,
022002
(
2013
).
2.
Z.
Ahmed
 et al.,
Phys. Rev. Lett.
108
,
102001
(
2012
).
3.
J. C.
Bernauer
 et al.,
Phys. Rev. Lett.
105
,
242001
(
2010
).
4.
A.
Courtoy
,
F.
Fratini
,
S.
Scopetta
, and
V.
Vento
,
Phys. Rev. D
78
,
034002
(
2008
).
5.
C. T.
Chantler
,
J. Phys. Chem. Ref. Data
29
,
597
(
2000
).
6.
A.
Tartari
,
A.
Taibi
,
C.
Bonifazzi
, and
C.
Baraldi
,
Phys. Med. Biol.
47
,
163
(
2002
).
7.
B. W.
Batterman
,
D. R.
Chipman
, and
J. J.
DeMarco
,
Phys. Rev.
122
,
68
(
1961
).
8.
A.
Alatas
 et al.,
Phys. Rev. B
77
,
064301
(
2008
).
9.
R.
Cesareo
 et al.,
Phys. Rep.
213
,
117
(
1992
).
10.
C.
Giacovazzo
 et al.,
Fundamentals of Crystallography
(
Oxford Science Publications
,
1992
), Chap. 2.
11.
D. B.
Williams
and
C. B.
Carter
,
Transmission Electron Microscopy: A Textbook for Material Science
(
Springer
,
2009
), Chap. 2 and 3.
12.
J. H.
Hubbell
 et al.,
J. Phys. Chem. Ref. Data
4
,
471
(
1975
).
13.
J. H.
Hubbell
and
I.
Øverbø
,
J. Phys. Chem. Ref. Data
8
,
69
(
1979
).
14.
A. J. C.
Wilson
and
V.
Geist
,
International Tables for Crystallography, Volume C: Mathematical, Physical and Chemical Tables
(
Kluwer Academic Publishers
,
Dordrecht
,
1992
), Chap. 6.1.
17.
D.
Belkic
and
H. S.
Taylor
,
Phys. Scr.
39
,
226
(
1989
).
19.
R. T.
Brown
,
Phys. Rev. A
1
,
1342
(
1970
).
20.
R. S.
Alassar
,
H. A.
Mavromatis
, and
S. A.
Sofianos
,
Acta Appl. Math.
100
,
263
(
2008
).
21.
D.
Schaupp
,
M.
Schumacher
,
F.
Smend
,
P.
Rullhusen
, and
J. H.
Hubbell
,
J. Phys. Chem. Ref. Data
12
,
467
(
1983
).
22.
R. D.
Cowan
,
The Theory of Atomic Structure and Spectra
(
University of California Press
,
Berkeley
,
1981
).
23.
W.
Muhammad
and
S. H.
Lee
,
PLoS One
8
,
e69608
(
2013
).
25.
F.
Smend
and
M.
Schumacher
,
Nucl. Phys. A
223
,
423
(
1974
).
26.
P. P.
Kane
,
L.
Kissel
,
R. H.
Pratt
, and
A. C.
Roy
,
Phys. Rep.
140
,
75
(
1986
).
27.
B. H.
Bransden
and
C. J.
Joachain
,
Physics of Atoms and Molecules
(
Longman
,
1983
).
28.
F.
Fratini
,
One- and Two-Photon Decays in Atoms and Ions
(
LAP
,
Saarbrücken, Germany
,
2011
).
29.
A.
Surzhykov
,
A.
Volotka
,
F.
Fratini
,
J. P.
Santos
,
P.
Indelicato
,
G.
Plunien
,
Th.
Stöhlker
, and
S.
Fritzsche
,
Phys. Rev. A
81
,
042510
(
2010
).
30.
E.
Clementi
and
D. L.
Raimondi
,
J. Chem. Phys.
38
,
2686
(
1963
).
31.
E.
Clementi
,
D. L.
Raimondi
, and
W. P.
Reinhardt
,
J. Chem. Phys.
47
,
1300
(
1967
).
32.
See http://www.webelements.com for information about the effective nuclear charges (Zeff).
33.
34.
A.
Costescu
,
K.
Karim
,
M.
Moldovan
,
S.
Spanulescu
, and
C.
Stoica
,
J. Phys. B: At., Mol. Opt. Phys.
44
,
045204
(
2011
).
35.
W. T.
Howell
,
Philos. Mag.
24
(
7
),
1082
(
1937
).
36.
L.
Carlitz
,
J. London Math. Soc.
36
,
399
(
1961
).
37.
M.
Abramowitz
and
I. A.
Stegun
,
Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables
,
National Bureau of Standards, Applied Mathematics Series
(
National Bureau of Standards, Superintendent of Documents, U.S. Government Printing
,
1964
).
38.
L.
Kissel
,
R. H.
Pratt
, and
S. C.
Roy
,
Phys. Rev. A
22
,
1970
(
1980
).
39.
L.
Kissel
,
Radiat. Phys. Chem.
59
,
185
(
2000
).
40.
S. C.
Roy
,
L.
Kissel
, and
R. H.
Pratt
,
Phys. Rev. A
27
,
285
(
1983
).
41.
L.
Safari
,
P.
Amaro
,
S.
Fritzsche
,
J. P.
Santos
, and
F.
Fratini
,
Phys. Rev. A
85
,
043406
(
2012
).
42.
L.
Safari
,
P.
Amaro
,
S.
Fritzsche
,
J. P.
Santos
,
S.
Tashenov
, and
F.
Fratini
,
Phys. Rev. A
86
,
043405
(
2012
).
43.
L.
Safari
,
P.
Amaro
,
J. P.
Santos
, and
F.
Fratini
,
Radiat. Phys. Chem.
106
,
271
(
2015
).
44.
V.
Florescu
,
M.
Marinescu
, and
R. H.
Pratt
,
Phys. Rev. A
42
,
3844
(
1990
).
46.
S. C.
Roy
,
L.
Kissel
, and
R. H.
Pratt
,
Radiat. Phys. Chem.
56
,
3
(
1999
).
47.
A. C.
Mandal
,
D.
Mitra
,
M.
Sarkar
, and
D.
Bhattacharya
,
Phys. Rev. A
66
,
042705
(
2002
).
48.
S.
Kumar
,
V.
Sharama
,
J. S.
Shahi
,
D.
Mehta
, and
N.
Singh
,
Eur. Phys. J. D
55
,
23
(
2009
).
49.
E.
Casnati
,
C.
Baraldi
, and
A.
Tartari
,
Phys. Rev. A
42
,
2627
(
1990
).
50.
P.
Latha
,
K. K.
Abdullah
,
M. P.
Unnikrishnan
,
K. M.
Varier
, and
B. R. S.
Babu
,
Phys. Scr.
85
,
035303
(
2012
).
51.
See http://www.ccdc.cam.ac.uk/products/csd/ for the Cambridge Structural Database (CDS).
52.
See http://www.fiz-karlsruhe.de/icsd_content.html for Inorganic Crystal Structure Database (ICSD).
53.
See http://www.rcsb.org/pdb for Protein Data Bank.
54.
F.
Fratini
,
L.
Safari
,
A. G.
Hayrapetyan
,
K.
Jänkälä
,
P.
Amaro
, and
J. P.
Santos
,
Europhys. Lett.
107
,
13002
(
2014
).
55.
E.
Caurier
,
G.
Martínez-Pinedo
,
F.
Nowacki
,
A.
Poves
, and
A. P.
Zuker
,
Rev. Mod. Phys.
77
,
427
(
2005
).
You do not currently have access to this content.