esea ch pape s
J. Appl. C ys . (2017). 50, 131–144 h ps://doi.o g/10.1107/S1600576716018914 131
Recei ed 27 July 2016
Accep ed 26 No embe 2016
Edi ed by K. Chapman, A gonne Na ional
Labo a o y, USA
Keywo ds: X- ay di ac ion; s ess
measu emen ; a ea de ec o s; po able X- ay
de ices.
P ecision and accu acy o s ess measu emen wi h
a po able X- ay machine using an a ea de ec o
Seung-Yub Lee,
a
Jinjing Ling,
a
Shenghe Wang
a
and Joaquin Rami ez-Rico
b
*
a
Applied Physics and Applied Ma hema ics, Columbia Uni e si y, New Yo k, NY 10027, USA, and
b
Fisica de la Ma e ia
Condensada – ICMS, Uni e sidad de Se illa – CSIC, 41012 Se illa, Spain. *Co espondence e-mail: [email p o ec ed]
The use o po able X- ay s ess analyze s, which u ilize an a ea de ec o along
wi h he newly adop ed ‘cos’ o ull- ing i ing me hod, has ecen ly a ac ed
inc easing in e es . In labo a o y condi ions, hese measu emen s a e as ,
con enien and p ecise because hey employ a single-exposu e echnique ha
does no equi e sample o a ion. In addi ion, he e ec s o g ain size and
o ien a ion can be e alua ed om he Debye ing eco ded on he a ea de ec o
p io o da a analysis. The accu acy o he measu ed s ess, howe e , has been
ques ioned because in mos cases jus a single e lec ion is analyzed and he
sample- o-de ec o dis ances a e ela i ely sho . This a icle p esen s a
comp ehensi e analysis o he unce ain y associa ed wi h a s a e-o - he-a
comme cial po able X- ay de ice. Annealed e i e e e ence powde s we e
used o quan i y he ins umen p ecision, and he accu acy o he s ess
measu emen was es ed by in si u ensile loading on 1018 ca bon s eel and 6061
aluminium alloy ba samples. The esul s show ha he p ecision and accu acy
a e sensi i e o he ins umen (o sample) il angle (
0
)aswellas o he
selec ed hkl e lec ion o he sample. The ins umen , sample and da a analysis
me hods all a ec he o e all unce ain y, and each con ibu ion is desc ibed o
his speci ic po able X- ay sys em. Finally, on he basis o he conclusions
eached, desi able measu emen /analysis p o ocols o accu a e s ess assess-
men s a e also p esen ed.
1. In oduc ion
X- ay di ac ion echniques ha e been used o de e mine he
nea -su ace s ess s a e o c ys alline ma e ials o nea ly a
cen u y, and signi ican ad ances ha e been made in bo h
ha dwa e echnologies and analysis me hods (Noyan &
Cohen, 1987). The o me includes high- esolu ion de ec o s,
mode n op ical de ices and associa ed elec onics. The la e
co esponds o he ad anced g ain in e ac ion models
(K one , 1958; Hashin & Sh ikman, 1962; Do
¨lle, 1979), as
well as o he in eg a ion o ini e elemen simula ions (Chen
& Ko ace ic, 2003; Clausen, Lee e al., 2003). High p ecision
and accu acy o he measu ed s ess mus be balanced wi h
o he impo an c i e ia, such as measu emen speed and
po abili y o indus ial applica ions. Fo me ly, small de ices
wi h po able ilms p o ided less eliable esul s compa ed o a
labo a o y di ac ome e , bu ad ances in he eliabili y o
po able equipmen ha e opened he doo o in si u and in-line
measu emen s, which a e o g ea indus ial in e es .
The idea o minia u iza ion o he X- ay sys em i sel is no
new a all. Ea ly epo s o po able de ices om he 1960s
desc ibed esidual s ess measu emen in ai c a pa s using
back- e lec ion pho og aphic ilm echniques (Bols ad, 1967;
Homicz, 1967). La e gene a ions o po able de ices (James
& Cohen, 1978a; Ruud e al., 1984; A aki, 1989; B auss e al.,
1996; Monin e al., 2003; Fa ell, 2010) u ilized posi ion
ISSN 1600-5767
#2017 In e na ional Union o C ys allog aphy
sensi i e de ec o s (PSDs) o minimize o comple ely emo e
de ec o o a ion because non-digi ized ilm eading is labo -
ious (Monin e al., 2000) and any mo ion o sample o de ec o
can in oduce signi ican e o s due o he sho sample- o-
de ec o dis ance. Fo example, he double exposu e ech-
nique is en imes mo e sensi i e o he displacemen e o
han a single exposu e (Ruud e al., 1984). Two-dimensional
image pla es, also known as s o age phospho s, we e de el-
oped in he 1980s, i s o diagnos ic adiog aphy (Sonoda e
al., 1983) and la e o X- ay di ac ion (Miyaha a e al., 1986;
Amemiya & Miyaha a, 1988; Yoshioka & Ohya, 1992). This
lase -s imula ed luminescence echnique eplaced he old ilms
and e en one-dimensional PSDs in po able X- ay sys ems.
O he ypes o a ea de ec o s, such as wo-dimensional PSDs
(mul i-wi e PSDs) and cha ge coupled de ices (CCDs), also
compe ed wi h he image pla e. De ailed desc ip ions o he
ad an ages and disad an ages o each de ec o (Ea hough e
al., 1999) and a pe o mance compa ison be ween image
pla es and CCDs (Kiss e al., 2002) can be ound elsewhe e.
Ano he c i ical componen in he success o new po able
de ices u ilizing a ea de ec o s is he de elopmen o s ess
de e mina ion me hods ha ake ull ad an age o wha has
been e med ‘ wo-dimensional’ di ac ion: cos(Tai a e al.,
1978; Tai a & Tanaka, 1979), XRD
2
(He, 2011) and ull- ing
i ing me hods (Kamp e e al., 2000),
1
all o which make use o
he con ou o he Debye ing. Rega dless o de ec o ype, he
sin
2
me hod was ubiqui ous un il he 1980s; since he 1990s,
wo-dimensional me hods ha e become mo e a ailable along
wi h high- esolu ion a ea de ec o s.
Cu en ly, he e a e many compe ing e o s de eloping
high- esolu ion po able X- ay de ices
2
o indus ial appli-
ca ions. Ini ially hese we e a ge ed o he s eel indus y
(Ganesh e al., 2013; Fa ell, 2010), bu
applica ions a e expanding alongside he
sh inking scale o unc ional ma e ials.
Despi e he ob ious bene i s o a
po able de ice, he e ha e been no
objec i e e alua ions o he unce ain y
budge o ad anced, comme cial,
po able X- ay uni s om hi d pa ies
wi hou con lic o in e es . Fu he mo e,
he e has been no ag eemen among
expe s abou he mos e icien , p ecise
and accu a e s ess de e mina ion
me hods. Fo hese easons, we chose
one o he s a e-o - he-a po able X- ay
de ices equipped wi h an image pla e
and ca ied ou a se ies o sys ema ic
es s on bo h p ecision and accu acy o
he s ess measu emen .
A schema ic o he po able de ice
used in his s udy is displayed in Fig. 1(a),
and a ypical Fe 211 powde pa e n is shown in Fig. 1(b). We
p e iously epo ed ha he s ain p ecision o his po able
machine is abou 9 mic os ain (m") wi h e i e e e ence
powde s (Ling & Lee, 2015), and ha in essence he sin
2
and
cosme hods a e heo e ically iden ical (Rami ez-Rico e al.,
2016) and yield s a is ically indis inguishable expe imen al
esul s (Rami ez-Rico e al., 2016; Ling & Lee, 2015). The
ma hema ical equi alency be ween sin
2
and coswas also
shown e y ecen ly (Miyazaki & Sasaki, 2016). S ess
measu emen wi h an image pla e and he cosme hod is now
known o be as (90 s o s eel), con enien (no cen e o
o a ion alignmen ) and p ecise (2 MPa epea abili y in
e i ic s eel), in a well con olled en i onmen . Howe e ,
measu ed s ess alues o en di e om he known applied
ones, and only a na ow ange o measu emen condi ions
ha e been es ed. Thus i is necessa y o in es iga e
measu emen accu acy in mo e elaxed en i onmen s ha
esemble ield o p oduc ion line condi ions.
In his a icle, we epo comp ehensi e unce ain y
analyses o a po able X- ay de ice, including he measu e-
men pa ame e dependency o bo h s ess p ecision (x4.1)
and accu acy (x4.2), as well as he e ec o he choice o a
pa icula e lec ion (x4.3). Unce ain ies om each e o
sou ce a e discussed in x5.
2. S ess de e mina ion me hods
Di ac ion-based s ess de e mina ion is an indi ec con e -
sion p ocess om a s ain, ", o a s ess, , enso ia a
cons i u i e law such as
ij ¼Cijkl"kl;ð1Þ
whe e C
ijkl
a e he componen s o he elas ic s i ness enso . I
a ma e ial is elas ically iso opic, equa ion (1) can be simpli-
ied o
esea ch pape s
132 Seung-Yub Lee e al. P ecision and accu acy o s ess measu emen J. Appl. C ys . (2017). 50, 131–144
Figu e 1
(a) A schema ic diag am o he po able X- ay de ice used in his s udy. The de ec o (shaded in
blue) has a diame e o 60 mm wi h a 6 mm hole a he cen e o beam pene a ion. (b) A ypical
di ac ion pa e n o Fe 211 iewed om he op. Da a used o he sin
2
and cosme hods a e
ma ked wi h hollow ci cles and dashed lines, espec i ely.
1
XRD
2
and he ull- ing i ing me hod a e basically he same, bu only pa o
he ing is used in he XRD
2
sys em.
2
m-X360 (Puls ec), Sma Si e RS (Rigaku), mXRD (P o o), Xs ess 3000
(S ess eck), MAX (TEC), Xsolo (Inel) e c. a e cu en ly a ailable po able X-
ay de ices o s ess measu emen s.
"ij ¼1þ
Eij ij

Ekk;ð2Þ
wi h wo independen pa ame e s, Young’s modulus Eand
Poisson’s a io . He e ij ¼0i i6¼ jo ij ¼1i i¼j.The
in e plana spacing d
hkl
is used as a buil -in s ain gauge and
calcula ed om he di ac ion peak posi ion in he labo a o y
coo dina e sys em (
0
,’
0
); co esponding s ains a e ans-
o med in o he sample coo dina es ( ,’) be o e s ess
con e sion. S a ing om he basic o mula o equa ion (2),
he nex wo sub-sec ions b ie ly desc ibe he equa ions used
in he sin
2
and cosme hods.
2.1. The sin
2
wme hod
The s ain measu ed along he sca e ing ec o a angles
( ,’) om he zand xaxes can be w i en in e ms o he
s ain componen "ij in he sample coo dina e sys em as
" ’ ¼d ’ d0
d0
¼"11 cos2’þ"12 sin 2’þ"22 sin2’"33

sin2 þ"33 ð3Þ
by assuming a bi-axial plane s ess condi ion wi hou s eep
s ess g adien s in he nea -su ace egion. He e d
’
is he
in e plana spacing a and ’,andd
0
is he uns essed
in e plana spacing. I he ma e ial’s elas ic beha io is
iso opic and homogeneous, equa ion (3) can be exp essed in
e ms o s ess ia equa ion (2), as ollows:
d ’ d0
d0
¼1þ
E’sin2 
E11 þ22
ðÞ;ð4Þ
whe e ’¼11 cos2’þ12 sin 2’þ22 sin2’.
I he in-plane s ess is no dependen on angle ’, he
equa ion can be u he simpli ied as
11 ’¼0ðÞ¼
E
1þ
@"
@sin2

;ð5Þ
which is he well known sin
2
equa ion. The bi-axial in-plane
s ess can hen be de e mined om he slope o a d e sus
sin
2
plo . I he d e sus sin
2
plo is linea , only wo
measu emen s a = 0 and , he so-called ‘ wo- il ’ me hod,
a e equi ed, as exp essed in equa ion (6) wi h ’=0:
11 ’¼0ðÞ¼
E
1þ
1
sin2
co 
2202

¼K2; ð6Þ
whe e he s ess cons an
K¼E
1þ
1
sin2
co 
2:ð6aÞ
is he B agg angle, which is abou (
0
+
)/2, whe e 
0
and 
a e he di ac ing angles a =0and , espec i ely. This
wo- il sin
2
me hod using an a ea de ec o is e y use ul
because he displacemen e o due o sample o a ion is
elimina ed.
2.2. The cosame hod
The wo-dimensional cosme hod was i s p oposed in he
la e 1970s (Tai a e al., 1978; Tai a & Tanaka, 1979) using
pho og aphic ilm. I was applied o he image pla e in he
ea ly 1990s (Yoshioka & Ohya, 1992), and has been u he
de eloped o ad anced use since he la e 1990s (Sasaki e al.,
1997; Sasaki & Kobayashi, 2009; Sasaki e al., 2014). Suppose
ha a di ac ion ing o ms on he image pla e h ough which
he inciden beam passes wi h an angle o
0
and ’
0
om he z
and xaxes, as shown in Fig. 1. The s ain "p ojec ed along a
di ec ion wi h angle can be exp essed as he ollowing, unde
he same assump ions as in he sin
2
me hod:
"¼11
1
E
n2
1n2
2þn2
3

þ22
1
E
n2
2n2
1þn2
3

þ12
21þðÞ
E

n1n2
ðÞ;ð7Þ
whe e n1,n2,n3a e he di ec ional cosines o he no mal o he
di ac ing plane wi h espec o he axes in he sample coo -
dina es and depend on ,, 0and ’0. Subs i u ing ,þ
and  o in equa ion (7), we can hen ob ain he
ollowing ela ions:
a1’0
ðÞ¼
1
2ð""þÞþð""Þ

;ð8Þ
a2’0
ðÞ
¼1
2ð""þÞð""Þ

:ð9Þ
Exp essing equa ions (8) and (9) in e ms o s ess a e e-
exp essing he di ec ional cosines leads o he inal ela ion-
ships o his me hod:
11 ’0¼0ðÞ¼
E
ð1þÞ
1
sin 2
1
sin 2 0
@a1
@cos 

;ð10Þ
12 ’0¼0ðÞ¼
E
2ð1þÞ
1
sin 2
1
sin 0
@a2
@sin 

;ð11Þ
whe e is de ined in Fig. 1. Fo a gi en
0
and , we can ge
he in-plane no mal s ess om he slope o he a1 e sus cos 
plo when ’0¼0 om equa ion (10).
The e a e o he wo-dimensional me hods such as XRD
2
o
ull- ing i ing. Howe e , only cosis chosen in his s udy as a
ep esen a i e o wo-dimensional me hods because he
undamen al equa ion o he XRD
2
me hod is basically iden-
ical o equa ion (7), and i measu es only pa o he Debye
ing (Miyazaki & Sasaki, 2016), while he cosand ull- ing
me hods use all o he in o ma ion. We ha e also shown
expe imen ally ha , when using he same da ase , he e was
no di e ence be ween he cosand ull- ing i ing me hods
(Rami ez-Rico e al., 2016).
In summa y, equa ions (5) and (10) a e he basic equa ions
o he sin
2
and cosanalysis in iso opic, homogeneous and
bi-axial s ess condi ions. Fo aniso opic, he e ogeneous o
mul i-axial s ess cases, hese equa ions ha e o be modi ied as
epo ed p e iously by Noyan & Cohen (1987) and Do
¨lle
(1979) o he sin
2
me hod and by Sasaki and co-wo ke s
(Sasaki e al., 2014; Sasaki & Kobayashi, 2009) o he cos
me hod.
esea ch pape s
J. Appl. C ys . (2017). 50, 131–144 Seung-Yub Lee e al. P ecision and accu acy o s ess measu emen 133
3. Expe imen and da a analysis
3.1. Samples
Th ee samples we e es ed: (1) Fe
e e ence powde s (E: 224 GPa, :
0.28) p o ided by he manu ac u e
o he po able X- ay de ice, (2) solid
ba s o 1018 mild ca bon s eel
(E: 203 GPa, : 0.28) and (3) solid
ba s o 6061 aluminium alloy
(E: 69 GPa, : 0.33) pu chased om
he McMas e –Ca supply company.
The Fe e e ence powde s we e
annealed o emo e any p e-exis ing
de ec s and esidual s ess. Wi h
a e age g ain size > 1 mm, size o
s ain b oadening e ec s we e no
expec ed in he di ac ion pa e ns o
hese powde s. Measu emen s o he
e e ence powde s we e he e o e
used o cha ac e ize he in insic ins umen p ecision o he
po able X- ay de ice. The solid ba samples o Fe and Al,
bo h wi h dimensions 0.25 0.25 2400 (100 = 25.4 mm), we e
loaded in ension o accu acy es ing. Since he Fe is he main
es ing sample, we measu ed he mic os uc u e o he ca bon
s eel ba by op ical mic oscopy (ZEISS Axio Scope) and
elec on backsca e di ac ion (EBSD) (JEOL 5600
equipped wi h HKL No dlys and Channel 5 so wa e; Ox o d
Ins umen s, Abingdon, Ox o dshi e, UK), as shown in Fig. 2.
These measu emen s clea ly display elonga ed g ains along
he e ical (Z) di ec ion. X- ay di ac ion om 20 o 110in
2wi h a Cu sou ce (D500 di ac ome e ) p obing bo h he
op c oss sec ion (XY plane) and he side wall (XZ plane)
e ealed ha some (110) ex u e was de eloped in he Z
di ec ion, simila o wi e ex usion, while g ains pa allel o he
side wall (XZ plane) a e dis ibu ed qui e iso opically. The
110 peak in ensi y a io [I
110
/(I
110
+I
200
+I
211
+I
220
)] is 0.74,
0.66 and 0.63 o he op c oss sec ion, side wall and JCPDS
e e ence, espec i ely. The EBSD analysis om he op c oss
sec ion also shows a la ge popula ion o (110) g ains han
seen in he side walls.
3.2. In si u di ac ion measu emen s in ensile loading
The 1018 mild ca bon s eel and 6061 Al alloy ba samples
we e placed in an Ins on 5984 Uni e sal Tes ing Machine o
ensile loading up o 300 and 200 MPa, espec i ely. Expe i-
men s we e ca ied ou unde load con ol o keep he elas ic
s ess cons an du ing di ac ion measu emen s, and samples
we e held by pneuma ic g ips as shown in Fig. 3. A minia u e
po able X- ay appa a us was moun ed nex o he sample
(sample- o-de ec o dis ance 30–60 mm) wi h a ious il
angles (
0
in he ange 5–48). The load was inc eased in 25 o
50 MPa s eps. A each load we pe o med di ac ion
measu emen s o abou 90 s o Fe and 120 s o Al. The
speci ica ions o he po able X- ay uni a e he ollowing:
ube ol age (30 kV), cu en (1 mA), sou ce (C Kwi h 
il e ), X- ay ene gy (=2.29A
˚,E= 5.4 keV), collima o
(1 mm diame e ), beam size (2 mm diame e ), senso uni
weigh (5 kg), powe supply weigh (6 kg), ene gy consump-
ion (80 W in ope a ion, 30 W in s andby mode). The speci ic
model and company in o ma ion abou his po able de ice is
no disclosed he e o objec i e e alua ion.
3.3. Measu emen pa ame e s
Ins umen al pa ame e s which in luence he p ecision and
accu acy o he s ess measu emen include beam size and
di e gence, use o a K il e , sample- o-de ec o dis ance,
choice o il angle
0
, de ec o esolu ion, peak i ing
esea ch pape s
134 Seung-Yub Lee e al. P ecision and accu acy o s ess measu emen J. Appl. C ys . (2017). 50, 131–144
Figu e 2
Op ical (a), (b) and EBSD (c), (d) images o he 1018 ca bon s eel sample om he op c oss sec ion
(a), (c) and side su ace (b), (d), which show elonga ed g ains in he e ical di ec ion. The measu ed
a e age g ain size is abou 15–20 mm.
Figu e 3
(a) Expe imen al se up o he in si u ensile loading es wi h a 2400 long
ec angula ba . (b) Schema ic illus a ion wi h di ac ion geome y. The
po able de ice was il ed o p obe a ious di ac ion ec o s. No e ha
2=22, =
0
.
unc ion, scanning ime e c. Among hese pa ame e s, he mos
impo an a e sample il angle (
0
) and sample- o-de ec o
dis ance (abb e ia ed as ‘SD’ he ea e ).
Fig. 4 shows he one-dimensional di ac ion p o iles in e-
g a ed om he Debye ings o Fe e e ence powde s. I he
displacemen e o is au oco ec ed by SD adjus men so ha
he peak cen e s a e shi ed o he posi ion expec ed om
B agg’s law, he e is no no iceable di e ence due o sample il
angle (
0
), as shown in Fig. 4(a). Howe e , he 2 ange and
in ensi y a e di ec ly in luenced by he SD. The use o la ge
SDs esul s in a na owe 2 ange and hus in highe eso-
lu ion, bu a he cos o sac i icing in ensi y. Meanwhile, as he
SD becomes smalle , he peak in ensi y inc eases wi h
inc easing 2co e age. Fo example, o a gi en de ec o
con igu a ion (29.7 mm ou e adius wi h 3 mm inne adius,
50 mm pi ch, 534 adial pixels), a 60 mm SD co e s 23in 2
esul ing in 0.04 esolu ion, whe eas co e age doubles when
he SD is educed o 20 mm a he cos o inc easing he
esolu ion o 0.08. In gene al, he e is a ade-o be ween
in ensi y and esolu ion in he chosen sample- o-de ec o
dis ance. This e ec is somewha mi iga ed because, as seen in
Fig. 4(b), he peak- o-backg ound a io emains abou he
same. Thus i is be e o keep he SD la ge as long as he
e lec ion peak o in e es alls wi hin he ange co e ed by he
image pla e.
3.4. Ins umen esolu ion and beam cen e calib a ion
Be o e add essing ques ions o p ecision and accu acy
unde ield measu emen condi ions, he in insic ins umen
esea ch pape s
J. Appl. C ys . (2017). 50, 131–144 Seung-Yub Lee e al. P ecision and accu acy o s ess measu emen 135
Figu e 4
Fe 211 powde di ac ion pa e ns ob ained a = 0, wi h a ious (a)
0
a SD = 35 mm and (b)SDa
0
=35
. The in ensi y and 2 ange a e s ongly
dependen on SD.
Figu e 5
(a) Peak cen e posi ion e sus angle (blue solid line) and calib a ion cu e ( ed dashed line) esul ing om he 25 mm beam cen e o se . (b)
Calib a ed peak cen e plo ob ained by adding he wo cu es om (a). This co ec ed p o ile is used o he s ess calcula ions exp essed in
equa ion (10).

esolu ion, i.e. he bes a ainable p ecision in labo a o y
condi ions, and aw da a calib a ion p ocedu e mus be
es ablished.
The spa ial esolu ion o each de ec o pixel is de e mined
by he pi ch o he lase scan, which can be adjus ed om 20 o
100 mm; a de aul alue o 50 mm was used h oughou all he
measu emen s in his s udy. Fo a gi en de ec o size (59.4 mm
diame e wi h 6 mm diame e hole in he cen e o beam
pene a ion and collima o ins alla ion), 50 mm adial pi ch
and 0.72angula s ep in a spi al eading gene a es 534 da a
poin s in 2and 500 da a alues a each 2in e ms o he 
angle.
We pe o med p elimina y es s wi h annealed Fe e e ence
powde s a one o he ecommended se ings: 35
0
,38mm
SD. A o al o 80 measu emen s we e epea ed o e he cou se
o 48 h wi h mul iple cycles, u ning he appa a us on/o , bu
wi hou mo ing any pa o he sample o machine o he bes
ue ins umen p ecision. In o de o quan i y e e y sou ce o
unce ain y, all da a analyses we e done ully manually in he
ollowing s eps: (1) Raw in ensi y e sus 2da a we e
expo ed o 500 angles. (2) Peak cen e posi ions in 2we e
ound ia pseudo-Voig peak i ing o 40 000 da ase s
(80 scans 500 angles). (3) Residual s esses we e calcula ed
ia he sin
2
and cosme hods o i e selec ed da ase s, bo h
wi h and wi hou beam cen e co ec ion using equa ions (5)
and (10), and hen compa ed wi h coss ess alues ob ained
om he ins umen ’s own analysis so wa e.
Fig. 5(a) shows he a e aged peak cen e om 80
measu emen s e sus azimu hal angle, , as a solid blue line.
E o ba s (one s anda d de ia ion, abou 0.005) a e
displayed only in a ew places, bu hey we e ai ly consis en
o all angles. The blue cu e wi h 0.025ampli ude in 2
indica es ha he beam cen e is o se by 25 mm om he
exac cen e poin , which is hal o he pixel esolu ion. Owing
o he di icul y o he X- ay beam alignmen and he o se
being less han he pi ch esolu ion, his de ice has he in e nal
calib a ion cu e shown as a ed dashed line in Fig. 5(a).
Adding hose wo cu es gi es he co ec ed inal peak posi-
ions shown in Fig. 5(b). Since hese calib a ion cu es change
acco ding o he SD, he measu emen e o in sample heigh
(SD) in oduces e o in he s ess measu emen . The ypical
way o calcula e he SD is o sp ead a mix u e o e e ence Fe
powde s in acuum g ease on he sample and le he machine
calcula e he SD om he Fe 211 peak. A e wa ds, he g ease
is wiped o , exposing he sample su ace o measu emen .
Since he wa eleng h and Fe 211 B agg angle a e known, he
accu a e SD can be calcula ed and use s can employ his alue
o hei ac ual samples. The p ecision in SD measu emen is
less han 10 mm, so he s ess e o associa ed wi h SD
unce ain y is negligible.
Du ing he 50 s pe iod o da a eading and analysis a e he
40 s beam exposu e, he po able machine goes h ough i ing
and calib a ion p ocesses, and hen p o ides an in-plane
esidual s ess o 0 2 MPa, which is iden ical o ha
p o ided by ou manual cosanalysis. Howe e , i he cali-
b a ion s ep is missed, i.e. he blue line in Fig. 5(a) is used, he
s ess om he cosme hod is 35 2 MPa. In he sin
2
me hod,
3
he measu ed s esses a e 22 MPa o non-
calib a ed da a (Fig. 5a, blue) and 0 2 MPa o he calib a ed
da a (Fig. 5b), which p o es ha he sin
2
me hod is no
sensi i e o he calib a ion p ocess because i p obes
-dependen s ain a a ixed , while he cosme hod uses
-dependen s ain a a ixed (Table 1). Howe e , he sin
2
me hod equi es samples o be a he exac cen e o o a ion
o an accu a e measu emen , which is challenging and ime
consuming o such a small po able X- ay de ice. One can use
he ‘ wo- il ’ sin
2
me hod o a oid he sample o a ion issue
by using only wo poin s a = 0 and 180 om he Debye
ing, bu i s accu acy is also sensi i e o he calib a ion p ocess,
jus as in he cosme hod, when non-calib a ed da a a e used.
Unce ain y a ises a each analysis s ep and possible e o
sou ces a e summa ized in Table 2. The o e all unce ain y is a
combina ion o he con ibu ions ela ed o (1) di ac ion
peak i ing e o (Type 1), (2) he epea abili y o he peak
cen e calcula ion (Type 2), and (3) linea i ing o equa ion
(10) o (5) o s ess de e mina ion in he cos(Type 3) o
sin
2
(Type 4) me hods, espec i ely. As shown in Table 1,
wi hou sample he e ogenei y o displacemen e o s, he
o e all s ess e o can be as good as 2 MPa o e i ic s eel.
Types 5–7 in Table 2 a e lis ed as bounds o maximum e o s.
No e ha hese unce ain ies a e measu ed a a ixed
0
and
SD, and he p ecision was ob ained a he bes s a ic labo a-
o y condi ions, bu hose pa ame e s a e no o en achie able
in ield measu emen s. The e o e, he e ec s o
0
and SD on
he p ecision and accu acy need o be measu ed o p ac ical
applica ions.
esea ch pape s
136 Seung-Yub Lee e al. P ecision and accu acy o s ess measu emen J. Appl. C ys . (2017). 50, 131–144
Table 1
Measu ed s ess om he sin
2
o cosme hod om di e en da a ypes.
The e o (2 MPa) is he s anda d de ia ion o he i e measu emen s. Scans
we e done a 35
0
and 38 mm SD.
Da a ype sin
2
cos
Non-calib a ed da a (Fig. 5a, blue solid line) 22 MPa 35 2MPa
Calib a ed da a (Fig. 5b, black solid line) 0 2MPa 02MPa
Table 2
Lis o unce ain y ypes and alues.
Measu emen p ecision can be ep esen ed by he Type 3 e o , which is abou
2 MPa in he case o Fe. All s ess alues calcula ed in x3.4 a e based on he
e e ence powde modulus, E= 224 GPa.
Type Unce ain y
2
()
d/d
(m")

(MPa)
1 Di ac ion peak i ing e o (2, Fig. 4) 0.003 5 1
2 A e age s anda d de ia ion om all angles
(Fig. 5b)
0.005 9 2
3 cos i ing e o [equa ion (10)] NA NA 2
4 sin
2
i ing e o [equa ion (5)] NA NA 4
5 Maximum de ia ion among angles (Fig. 5b) 0.015 28 6
6 De ec o spa ial esolu ion (50 mm) 0.05 90 20
7 FWHM o he peak (Fig. 4) 2.3 4000 NA
3
Fo he sin
2
me hod, six
0
coun e clockwise o a ions (0, 5, 10, 15, 20, 25)
we e made in labo a o y coo dina es, which co esponds o he six angles
(11.8, 16.8, 21.8, 26.8, 31.8, 36.8) in sample coo dina es a = 180.
4. Resul s
4.1. P ecision es wi h e i e e e ence powde s
The dependence o he p ecision on he choice o
measu emen pa ame e s (
0
and SD) was in es iga ed using
annealed e i e e e ence powde s. The sample il (
0
) angle
a ied om 5 o 60 o each sample- o-de ec o dis ance
(SD), which i sel anged om 20 o 60 mm. Fo he s a is ical
analysis, his whole p ocess was epea ed i e imes ( o a o al
o 540 measu emen s) a di e en sample posi ions. Since
equi alency be ween cosand sin
2
was shown in x3.4 and by
o he au ho s (Ling & Lee, 2015; Rami ez-Rico e al., 2016;
Miyazaki & Sasaki, 2016) unde a homogeneous bi-axial s ess
s a e, we used s ess alues calcula ed wi h he ins umen
so wa e by he cosme hod. Re e ence powde s we e
expec ed o exhibi no s ess i espec i e o he chosen alue
o he a o emen ioned pa ame e s.
The a e aged s ess alues e sus il angles (
0
) a e plo ed
in Fig. 6, showing ha he e is a p ope ange o sample il
angle o p ecise and accu a e measu emen s: 20 
0
50.
Ou side ha ange, measu emen s become imp ecise and
inaccu a e. A lowe il angles,
0
15, he measu ed
s esses s ongly depend on SD, while mono onic dec eases
a e obse ed a
0
55. This is consis en wi h ou p e ious
simula ion esul s (Rami ez-Rico e al., 2016) in ha o ien a-
ion e o , 
0
, is mo e signi ican a lowe
0
, and ha beam
de ocusing becomes impo an a highe
0
. The inse o Fig. 6
shows ha o his pa icula appa a us an SD o 35 mm
esul s in good accu acy ega dless o il angle. The same da a
as in Fig. 6 a e plo ed in Fig. 7, as a unc ion o SD o gi en
0
angles. The calcula ed s ess ends o inc ease wi h
inc easing SD, and i s e ec becomes la ge a lowe il angles.
Fig. 8 displays one s anda d de ia ion om i e measu e-
men s a each measu emen condi ion (
0
and SD) as a
ep esen a i e pa ame e quan i ying p ecision. A in e -
media e anges (20 
0
50,20SD 60 mm), he mean
s ess is 3 MPa; he p ecision is abou 4 MPa o i e di e en
spo s and 3 MPa o i e con inuous scans a one spo . Fig. 8
also illus a es ha he sample- o-de ec o dis ance is no a
c i ical pa ame e unless
0
is oo low o high. Mo e e o s a
e y low il angles a e ine i able because below
0
= 11.8
4
we lose he independen in o ma ion olume ha he Debye
ing cap u es o coss ess calcula ion. The wide beam
sp ead a high
0
angle also causes e o s in bo h p ecision
and accu acy. In gene al, hese undesi able measu emen
condi ions should be a oided, al hough in ield condi ions his
migh no be possible. Fo such cases, i is necessa y o know
he unce ain y budge s caused by measu emen pa ame e s
as shown in Figs. 6, 7 and 8.
4.2. Accu acy es wi h solid samples unde in si u ensile
loading
Ob aining he ue s ess s a e is a challenging ask o bo h
di ac ion-based echniques and des uc i e mechanical
esea ch pape s
J. Appl. C ys . (2017). 50, 131–144 Seung-Yub Lee e al. P ecision and accu acy o s ess measu emen 137
Figu e 7
Measu ed s ess alues e sus SD. The SD e ec is mino a ound
in e media e il angles (20 
0
50), bu becomes signi ican a lowe
0
.
Figu e 6
Measu ed s ess alues e sus
0
angles. Measu ed s esses de ia e om
ze o conside ably a
0
15and
0
55.
Figu e 8
Con ou plo o s anda d de ia ion (p ecision) ou o i e measu emen s
a each
0
and SD. The p ecision e o is less sensi i e o SD a ound
in e media e il angles (20 
0
50), while
0
shows a wide a ia ion
in all SD anges.
4
=
0
,= 11.8 o Fe 211, so he in o ma ion becomes edundan a
0
11.8.
me hods; hus hey a e o en used oge he as complemen a y
ools. The bes way o e alua e measu emen accu acy is o
compa e measu emen s wi h known applied s esses. Two
ec angula solid ba s (1018 s eel and 6061 aluminium) we e
loaded unde ensile s ess using he appa a us and me ho-
dology desc ibed in xx3.1 and 3.2; a ensile s ess was applied
a se e al il angles (
0
= 12, 25, 35 and 45)in50o 25MPa
s eps. Fig. 9 shows aw wo-dimensional di ac ion p o iles
and he Debye ing dis o ion om each sample a a ious
0
.
The ca bon s eel shows no p e e ed o ien a ion, a leas in he
211 e lec ions wi hin he di ac ing olume, and he peak
cen e is e y s able h oughou all angles. Howe e , he
esea ch pape s
138 Seung-Yub Lee e al. P ecision and accu acy o s ess measu emen J. Appl. C ys . (2017). 50, 131–144
Figu e 9
No malized wo-dimensional di ac ion pa e ns o (a) mild ca bon s eel 1018 and (b) aluminium 6061 alloy a ze o applied s ess. The Debye ing mo es
ou side as
0
inc eases owing o he co esponding SD inc ease. The ca bon s eel shows a homogeneous g ain dis ibu ion, while he aluminium alloy is
ex u ed. The le column displays how much he Debye ing is dis o ed by he peak cen e posi ions (pola scale om 2000 o 2000 m"). The sample-
o-de ec o dis ances a e 34, 34, 50 mm o s eel and 41, 41, 52 mm o aluminium.
Figu e 10
Measu ed e sus applied s ess om (a) he 1018 ca bon s eel 211 peak and (b) he 6061 aluminium alloy 222 peak. Fo s eel, bo h sin
2
and cosshow
iden ical esul s o a gi en
0
, while aluminium does no show eliable esul s. Ini ial esidual s esses we e o se o ze o. The s eel da a a
0
=12
a e
plo ed ins ead o a
0
=25
, because he da a a 25a e nea ly he same as hose a 35.
aluminium 222 in ensi y is weak and inhomogeneous. The
Debye ing is also dis o ed as peak cen e s luc ua e a ound
he ing. The e o e, s eel samples a e expec ed o show mo e
eliable esul s han aluminium. Also no e ha he SD was
a he la ge (abou 50 mm) o
0
=45
, because he minimum
sample- o-de ec o dis ance inc eases wi h inc easing il angle
owing o he ac ha he sample is e y long and in e e es
wi h he ins umen a high il angle (see Fig. 3).
Measu ed s esses om bo h he cosand he wo- il sin
2
me hods a e plo ed agains applied s ess in Fig. 10. As a
measu e o accu acy, he slope ( a io o measu ed o e applied
s ess) was calcula ed o each se o il angle condi ions and
no ed in Fig. 10. The s eel sample shows ha bo h analysis
me hods yield simila esul s and he e is a clea
0
depen-
dency in e ms o accu acy. Fo example, measu emen was
mos accu a e and p ecise a
0
=45
. As o he aluminium
sample,
0
dependency can be in e ed, bu cosand sin
2
gene a e qui e di e en esul s, nei he o which is accu a e o
p ecise o he es ed angles.
Knowing ha he accu acy depends on he il angle and
ha he s eel sample is much mo e eliable han he alumi-
nium sample, a se ies o new in si u ensile loading expe i-
men s (0, 100, 200, 300 MPa) we e pe o med wi h a 2400 long
s eel ba sample o a
0
-dependen accu acy es . A o al o
en
0
angles we e es ed om 5 o 48
0
wi h 5in e als as
0
=0
does no gi e a coss ess and 48is he maximum il
angle o cap u e he 211 peak o a gi en sample geome y.
This se o measu emen s was epea ed i e imes (4 s eps 
10
0
5 = 200 measu emen s). Since he wo- il sin
2
esul s
shown in Fig. 10(a) a e e y simila o he cos esul s as
expec ed, cos alues epo ed by he appa a us so wa e
we e used h oughou .
The a ios o measu ed o e applied s ess a e plo ed in
Fig. 11 o all i e measu emen s and he a e age s esses a e
lis ed in Table 3 along wi h s anda d de ia ions. As in he
p e ious p ecision analysis, he measu emen s a e qui e
epea able a 20 
0
48, bu he accu acy changes om 70
o 100% o he ac ual s ess alues. In o he wo ds, he
condi ions o he bes p ecision (
0
=35
, wi h bo h Fe
powde s and he solid ba ) and accu acy (
0
=19o 45
) a e
di e en . This esul does no mean ha he machine cali-
b a ion is pe ec a 45o o se by 30% a 35because he e
may be o he ac o s a ec ing he accu acy, which a e
discussed in he nex sec ion. The measu ed/applied s ess
a ios a each alue o
0
a e summa ized in Table 4.
esea ch pape s
J. Appl. C ys . (2017). 50, 131–144 Seung-Yub Lee e al. P ecision and accu acy o s ess measu emen 139
Table 3
Applied and measu ed coss esses (MPa) wi h ca bon s eel a each il angle (
0
).
A e age alues and e o ba s we e ob ained om i e epea ed measu emen s. No e ha a e age alues (accu acy) can be di e en i he in e nal calib a ion
p o ile changes, while e o ba s (p ecision) a e mo e sensi i e o measu emen condi ions. Fo he accu acy e alua ion, he measu ed/applied s ess a io, as shown
in Table 4, is he be e way because he ue esidual s ess s a e a ze o load is unknown. The bes p ecision is achie ed a
0
=35
.
Measu ed s ess (MPa) a each il angle (
0
)
Applied s ess (MPa) 5101520253035404548
0 67 (9) 30 (11) 27 (12) 25 (9) 20 (10) 1 (13) 19 (8) 50 (15) 98 (26) 138 (22)
100 236 (23) 173 (14) 143 (8) 123 (12) 98 (8) 70 (13) 54 (7) 39 (15) 5 (21) 38 (23)
200 403 (37) 323 (16) 258 (6) 218 (12) 170 (10) 138 (13) 127 (7) 125 (15) 102 (21) 65 (23)
300 558 (68) 463 (28) 372 (7) 311 (9) 254 (30) 205 (9) 198 (6) 208 (17) 198 (22) 166 (23)
Figu e 11
0
angle e ec on he a io o measu ed s ess (cos) o e applied s ess
o he 1018 ca bon s eel sample. A a io o 1 indica es accu a e s ess
measu emen . Each a io (slope) was de e mined om he ou ensile
loading s eps (0, 100, 200, 300 MPa) and is summa ized in Table 4.
Figu e 12
SD e ec on he a io o measu ed s ess (cos) e sus applied s ess o
he 1018 ca bon s eel sample. A a io o 1 indica es accu a e s ess
measu emen . The accu acy o s ess measu emen s is no sensi i e o he
SD excep o he low
0
angle egions whe e he measu ed s ess
inc eases wi h inc easing SD.