A Possible Solution to the Matrix Problem in X-Ray Fluorescence Spectroscopy E. J. FELTEN, ISIDOR FANKUCHEN, and JOSEPH STEIGMW Departments of Chemistry and Physics, Polytechnic Institute of Brooklyn, Brooklyn, N. Y .
,The impregnation of filter paper with solutions containing binary unknowns of the transition metals is followed by measurement of the intensities of fluorescent radiations from the two components for increasing numbers of filter papers. The ratios of these intensities a t the various thicknesses are calculated and extrapolated to zero thickness. The extrapolated or limiting ratio is found to b e directly proportional to the ratio of the molar concentrations of the elements involved, thus reducing the absorption and enhancement effects commonly encountered. Although the absolute values of the intensities vary from filter paper to
filter paper for the same binary solution, the extrapolated ratio is constant If an unknown consists to within 1%. of only two components, the limiting ratio will yield the absolute percentage of each element. For the more general case, two methods of analysis are examined. A dilution method, in which a known quantity of one of the constituents is added to a definite quantity of unknown, shows errors as large as 70%. An internal standard method works well, with errors of 5% or less. In addition, a rapid procedure involving intensity ratios at a single thickness appears to have some utility, but shows errors up to 7%.
T
sample under fixed operating conditions. For the transition metals this thickness has been reported to be approximately 0.004 cm. (8). As the sample thickness is decreased below that corrcsponding to the niaximum fluorescent intrnsity, the matrix absorption effects brronw smaller ( f 2 ) . However, decreasing the thickness of very thin (40-micron) solid samplcs in a known manner is difficult. A modification of the method of Pfeiffer antl Zemany (fI), who evaporated a small portion of an aqueous solution on a hardened filter papcr, \vas uscd. Absorbent filter kapers were iinprcgnated with solutions of the samplcs to be studied, antl the intrnsities of the fluorescent radiations \vcre measured for increasing n u n i b m of papcrs. For 0.1 to 0.3J1 solutions, the maximum fluorescent intrnsity \vas not achievrti for w e n six thickncsses of iniprcbgnated paper. Thus it is possihlt, to cwntrol sample thickness bdow that corresponding to the maximum fluorescent intensity. Thc: ahsolutc, intrmity of fluorcsccncc will approach z w o as the thickness of the spwinicn rl(ywascs, but for a binary sJ.stem the ratio of the intensities of two fluorescing c4cmcmts might bc expected to approach a finite quantity.
principal analytical problem in x-ray fluorescence spectroscopy (3, 10) is the interpretation of the intensities of the characteristic secondary radiations for the different fluorescing elements in terms of the concentrations of these elements in the unknown. It would be desirable if the intensities of these radiations w r e dircetly proportional to th(A quantities of the elements involved. In gcwmil, howvt,r, the relationship is complicated by absorption and enhancement effects which occur within that part of the sample which is transparent to the primary x-ray beam ( 5 ) . These effects do not permit the estahlishment of a simple relationship between the intensity of fluorescent radiation and the atomic percentage of the fluorescing component in a mixture. Jlethods of relating fluorescent radiation intcmity and composition have included an empirical approach with chemically nnal!.zed standards ( 2 ) , addition of an intcmal standard ( g ) , and dilution of the specimen with n fixed amount of a n inrrt salt mixture, (.ither in the dry (6) or fused ( 4 ) state. The method for minimizing matrix cffwts depends upon the measurement of intcmities of fluorcscent radiation in progressively thinntbr samples. As the thickness of a sample increases, the intensity of fluorescent radiation increases, ultimatd>-approaching a liniiting value which reprrsrnts the maximum fluorescent intensity for the particular HE
EXPERIMENTAL
T h x-ray rquipmrnt uscd in this work was a Nordco (Sorth American
Philips Co., Inc.) diffraction unit with fluoresccncc aecwsorios. Thv unit was equipprd with both currchnt and voltage stabilizers. Rock salt and lithium fluoride crystals wcw usod as diffrartion gratings. Thc d-spacings w r i b 2.8 19 and 2.014 A., rcqwctiveiy. -4 krypton-filltd Geigvr tulw send as tho dvtector for tlic fluorrscvnt ratliation. Thc: primary x-ray sourw was a Jlachlctt OEG tub(, with a tungstvn target, opcratcd at 40 ki.. antl 20 ma. Unless othrrwisc nototl, th(: limiting ratios w r c obtaincd from nwasurcments with thc rovk salt crystal. The following compounds, of rchagcbnt grade, w r v uscd without purificatjon: cupric sulfate, CuS04‘ 5 H 2 0 ;ainnioriium ferrosulfatc: hexahydratr , (KH,)tF(:(SO&. 6 H 2 0 ; niek(,l sulfatc: monohydrate, NiSO,. H 2 0 ; manganous sulfate, MnS04.4 H 2 0 ; sodium chroniatc nioiiohydratcx, S a z C r 0 4 .H,O; cohaltous sulfatr, CoS04.H 2 0 . Srvoral litcw of 0.1.11 solution of each compound w’rc prepartd as stock solutions, arid varying proportions of the diffwcnt solutions w r e mixed to give iniprcgnating solutions of known ratios of componmts. Where ncccssary, mow concmtratcd solutions up to 0.3.U w r c usvd. The specimens for counting w c w pr+ parcd by immersing strips of IVhatman No. 1 filter paper in the solutions, airdrying hy suspension (using an ortlinary clothespin), and thcn prrssing thr strips flat. The strips w r e initially 102 X 55 mm. After drying and flattrning, a scetion (used for suspnsion) 102 x 10 mm. was cut away and thv remaining paper, 102 X 45 mm., was cut into three strips. cacnh 34 X 4’o mm. The latter fittrd thc spwimen holdvr which was provided with thv instrumrnt. The bottom of each shwt might t w expected to show a larger quantity of each component than the rest of t h r sheet, because of thv drops which formed by drainagc antl whirh subs(:quently dried. IIowwer, the portion of the specimen which was exposed to thc beam did not include the bottom of thv sheet. S o such rffect is appartmt from the experimcmtal results. The intrnsities of tho characteristic fluorescrnt radiations were dctrrmined by counting at the peak maxima, mvasuring thc: time requirrd for 25,000 counts to accumulatcb, and rc~alculating, after correction for background to counts per second. Cnlrss otherwise notcd, each samplr was counted VOL. 31, NO. 1 1 , NOVEMBER 1959
1771
Table 1.
Reproducibility of Instrument and Positioning
Stationary Sample Co(Ku) Fe(Ka) Intensity, counts/sec. Std. dev.
Table 11.
R
234.1 2.03
194,7 1.27
CoK,, and
Intensity of FeK,,
Table
I CoK,
I FeK,
Repositionedkmple CO(K u ) Fe( K a )
as a Function of Specimen Thickness
234.3 1.92
Iv.
Least Squares Treatment of Intensity Ratio Data
Moles Co/Mole Fe Thickness
.. 166 1 276 8 3;. ratio is calculated from the measurements for a single thickness of paper and is repeated for two and three thicknesses. A plot of the ratio as a function of thickness is then made on semilog paper. This plot yields a straight line, which is extrapolated to zero thickness. The extrapolated ratio a t zero thickness, R., may then be used for subsequent calculations. Figure I shows such ratios for up to six thicknesses derived from nine solutions containing cobalt and iron in varying proportions. Although there , is distinct curvature in some of the solutions somewhere in the entire range of thicknesses, there is a linear relationship among the first three thicknesses of all samples. A more reliable procedure for determining the value of R, lies in a least squares calculation, assuming thc. following linear relationship: log R = a
+ bT
(1)
where R is the intensity ratio determined experimentally a t thickness 2'. b is the slope of the line, and a represents log R,. Table IV shows thc results of such calculations for three solutions containing cobalt and iron in different proportions, together with the experimentally determined valuer a t the different thicknesses. The assumption of a straight-line relationship in Equation 1 is justifled. All limiting ratios in this paper werc calculated by least squares, unless otherwise noted. Absorption and enhancement effects are presumed to be minimized with sufficiently thin samples. If this is correct, the limiting intensity ratios, R., of a series of, for example, ironcobalt solutions should be related to one another by the h o l e fraction ratio of the two components. That is, if one solution is equimolar in iron and cobalt and a second has twice as much
Figure 1 . Variation of intensity ratio with thickness for a number of cobalt-iron mixtures
6.0
5040-
U 1c w 30-
LL >
c
--
-
-
--
-
--
-
Towards the end thc sodium chloridc c u s t a l split, and n lithiuni fluoriticb c v s t a l uas used in its p1:1c~. T:iblv VI11 shows the limiting intcmity ratios of iron-cobalt samplrs with the lithium fluoride crystd. The limiting intcnsity r:itios ohtaincd with this cqstnl arc' t l i f f ( s i , i m t from those of Table V. A nrw sodium chloride crystal and :i new lithium fluorid(% crystal gave intmsit!. r:,tios for an iron-cobalt sample which w r r i n much closcr agrermvnt. l h v origind sodium vhloridc c v s t a l w s iintlvr possible internal strain :ml the :tt)solutc~v:~lucs of the r:itios obtr.incd witti it :ire in doubt. Hon.cwr, a nunil)c.r of s p t ~ i mcns which were pi,rio(lic:i,lly rc-exxmincd shoivcd 110 signifiwnt clinngcxs in ratios over the pcriod of timi, devotrd to exploring thc mrthod tlwcrihrtl hcrc, :!nd therefore thc an:iIytic:d rcwlts obtaincd with this cr?.stnI can hc taken : ' s valid. Quantitative Analysis with Limiting Intensity Ratios. If thc unknown is compos(d of only two c~ompon(~nts, thrn tlic w i g h t pcr rent of (m:h of tho t)y mc:isc~l(~mcnts can I)c detc~rininc~tl uring the intmsitios of thv two, calc,ulnting t h r limiting ratio, R,, for this unknown, and determining its molar ratio. f, from the previously rstablished R , x f constant for the particular system. An rquntion n.hich is useful in this connrction is, for thc cohaltiron systcm,
e-4
--
-3
4
2
3 20-
i t t
1.0
IL 0.5
0.33
I
0
3
2
I
4
5
THICKNESSES OF ?APER MEASURED
cuhalt a s iron, thrn the limiting ratio of the second should be tnicc as much as that of the first, as s h o w in t,he f'ourth column in Table V. Nine solutions containing cobalt and iron in molar ratios extending from 5 : 1 to 1 : 5 were measured for from one to three thicknesses. The intensity ratio a t each thickness was calculated, :ind the limiting intensity ratio, R,, {vas cstimatcd by least squares. Each ratio was thrn multiplied by a normalizing factor which reprcseiitcd the ratio of moles of iron to molcs of cobalt in the solution. The product of the factor and the limiting ratio is eswntially indcpendcnt of thc composition of the solution. The average value of this product for the nine solutions is :tssociated with a relative error of about 2.5%. The matrix effects become more :kpparent at greater thicknesses. Thus, as' the proportion of iron to cobalt increases, there is an upward trend in the value of the product of ratio and normalizing factor ( R x f), particularly noticeable in the molar ratios which are greater than unity. This trend, however, is small in the limiting ra,tios. 'The effect of dilution was investigated on solutions of iron and cobalt in which the total metal ion concentration was 0.05M. Measurements on papers imprcgnated with these solutions were made with the x-ray tube a t 40 kv. :ind 30 ma. and the sodium chloride crystal. The results are shown in Table VI. These results compare favorably with those of Table V, although the rrror is somewhat larger.
The iron-cobalt system exhibits only absorption effects. A more srvcre test of the limiting ratio method is found in the iron-copper system, mhich shons not only absorption but enhancement effccts as well. The coppcr rndintion is sufficiently energetic to excite iron fluorescence. Table VI1 shows the limiting ratios calculated from mepsurcments made on iron-copper samples prepared from 0.2 and 0.3A1 solutions. The rock-salt crystal was used, and the primary x-ray tube was operated at 40 kv. and 10 ma. These results show that the limiting intensity ratio method is applicable to the iron-copper system. Similar results with comparable rclative errors were obtained for binary mixtures of iron with chromium, manganese, and nickel. The molar ratios ranged from 4 t o 0.2561, and the total metal concrntration in each solution was 0.1M. Similar results were also obtained with the copper-zinc system.
Table V.
Moles Co Mole Fe 5 4 3 2
in which Aco and AFe arc the atomic I COK~ of the clcmmts, R,
rFeKp
weights
is thc ctxpcrimentally detcrmincd limiting ratio of the unknown, and R , X f is the previously established constant characteristic of the s ~ s t c m , as in Tables V and VI. It is already evident from the constancy of the product R, X f in cither table, that comparable errors
I CoK,
Relationship between Sample Thickness and R - -I FeK,
H,
J
R, X f
RI
It, X f
K,
6: 294
0 20 0 25 0 33 0 50 100 2 00 3 00 400 500
1 259 1 175 1 253 1.238 1 246 1 280 1 2?0 1 320 1.320 1.276 0.0301
6.333 5.1C5 3.7c5 2.503 1.277 0.654 0.443 0.344 0.275
I . 26;
6 371
1.276 1.265 1.252 1 277 1 ,308 1 329 1.376 1.375 1.303 0.0474
5 128 3 832 2 539 1 309 0 669 0 457 0 357 0 289
5 100
1
3 759 2 475 1 246
0.5 0.33 0.25 0.20
0 640 0 430 0.330 0 264
Av. Std. dev.
Kz
X
f
1.274 1.282 1.277 1.270 1.309 1.338 1.371 1.428 1.445 1,338 0.0681
R3
6 409 5 142 3 870 2 555 1 342 0 683 0 471 0 371 0 303
VOL. 31, NO. 1 1, NOVEMBER 1959
h 3
X .f
282 286 290 278 I 342 1 366 1 413 1 484 1 515 1 362 0.0906 1 1 1 1
1773
will be obtained if the data arc recalculated and expressed in tcrms of per cent composition. A procedure to be preferred is the following: To the solutions used for tha determination of the R, X f product for some system, a known amount of one of the components is added, new papers are impregnated, and new limiting intensity ratios are determined. The percentage compositions of these solutions can then be calculated, using Equation 2. Table IX shows the results of such a procedure applied to
Table VI.
Limiting Intensity ti^^ of 0.05M Iron-Cobalt Solutions Moles Fe I CoKa f R, R,, x j M X o ’ I FeKO ~
0.33 0.50 1.0 2.0 3.0
1,180 1.232 1.277 1.280 1,245 1.243 0.041
3.540 2.463 1.277 0.640 0.415
Av. Std.dev.
Table VII. Limiting Intensity Ratios of Iron-Copper Samples
CU/ Mole Fe, .f
RoICuKa I FeKO x f(0.2.M
Solutions)
0.25 0.33 0.50 1.00 2.00 3.00 400
R, IFeKO - x I CuKa f (0.y Solutions)
1.104 1.123 1.122 1.115 1.024 1.092 1.040
1.120 1.130 1.077 1.081 1.108 1.125 1.168 -_ 1.116 0.031
Av. 1.089 Std. dev. 0 . 0 4 0 Table VIII.
Limiting Intensity Ratios of Iron-Cobalt Specimens with Lithium Fluoride Crystal
Moles __ Fe Mole Cu ’
.~
R, Z C O K ~ I FeKO 3.384 2.685 1.727 0.854 0.443 0.303 0.223
0.25 0.33 0.50 1.00 2.00 3.00 4.00
.4v. Std. dev.
Chromium % Fe, 76 Fe, % present found error 74.11 60.70 62.29 47.22 31.82 23.99 19.26
1774
74.22 69.88 61.60 47.00 31.15 23.66 19.43
+0.15 +0.26 -1.11 -0.47 -2.11 -1.38 +0.88
Ro X f 0,846 0.895 0.864 0.854 0.886 0.909 0.892 0.878 0.024
thc determination of iron in a series of nents to a definite qultntity of unknown, binary solutions. The lithium fluoride and redetermining the limiting intensity crystal was used in these measurements. ratio of the two elements. It is a The average error for the 35 detervariant of the well known isotopic minations in Table IX is less than 1%, dilution method. and the iron concentrations range from If the two components show a con18 to 80%. The limiting ratio method stant R. X f product over a range of is useful for quantitative analysis in concentrations, then: unknowns consisting of two components. The unknown, if metallic, would a’R, a = require chemical attack to put it into (3) ( R ’ O - Ro) solution. Usually this involves mineral acids. I n practice, the bulk of the where a represents the Of grams acid should be removed and the reof one of the ConlPonents in a sample of mainder neutralized before analysis. T~ determine the effectsof acid anions the particular unknown, a’ represents the number Of a,dded grams Of comon the limiting ratios, a solution which was 0.litf in both copper and iron was ponent, R o represents the limiting intensity ratio in the original unknown, prepared, and to portions of this soluadded equal volumes of 0 . 2 ~ and R’, the limiting intensity ratio tion after the additionOf a’ grams. solutions of ammonium chloride (representing perchlorate as well), amThis method, however, is susceptible monium nitrate, and ammonium brato large errors because the differrnce of mide. The limiting ratios obtained the ratios may be small. Thus, results from these solutions with the sodium obtained from 35 mixtures of iron with various binary partners showed errors chloride crystal are given in Table X. There is a and probably negliin the iron concentration which \vc‘r(’ gible effectof added ammonium %its greater than 5%, ranging, in fact, u p to 70%. This method is not recotnon the limiting ratio, particularly in view of the data of Table IX. I n mended the js to practice, the quantity of residual acid some extent in a.dvance, so that the R’o) Can bc adjusted could be reduced further, and the difference (Ro R, x f products could be determined in until it is approximately equal to the corresponding ammonium salt sohI n the internal standard method, tions. I n systems where the two components known quantities Of One Or Inore elements not found in the unknown are up onlv Dart of the composition of the the ratio is added to the latter and the appropriate obvious~y insufficient for quantitative Limiting intensity ratios are detcrmin,:d. analysis. T,,.~ suggestthemFrom these, the elemental concentraselves: a dilution method and an tions of the desired constituents can be internal standard method. The calculated. The principles involved in tion technique consistsin determining this type of analysis have been thorOughlY discussed by Adkr and Axelrod the limiting ratio of the two components in the particular unknown, adding a (1). The iron-coppfr system investigated because it had been studied knoffn quantity of one of the as a binary system, and because it exhibits both absorption and enhancrment effects. A single internal standard-nickelTable X. Effects of Foreign Ions on was used in place of two because the R, in Iron-Copper Solutions early results obtained with it looked Fe KO promising. In this connection, one problem which is only partially anRo C X Salt Added swered (Table X) was that of a change None 1.115 in the limiting intensity ratio of a pair 1.085 NH,Cl of elements caused by the introduction XH,NOa 1.055 NHIBr 1.077 of a third element. There is practically no change in R , of an equimolar iron-
-
Table IX. Determination of Iron in Binary Solutions Manganese Cobalt Nickel % Fe, % Fe, % % Fe, % Fe, ‘36 70 Fe, % Fe, present found error present found error present found
73.05 68.54 61.00 45.86 30.65 23.02 18.43
ANALYTICAL CHEMISTRY
73.24 68.48 61.19 4639 30.64 23.44 18.16
+0.26 -0.09 +0.31 +1.16 -0.03 +1.82 -1.47
81.00 76.31 68.54 53.20 38.13 30.66 26.21
80.98 76.13 68.63 53.67 38.47 30.20 26.80
-0.03 -0.24 $0.13 +0.88 +0.89 -1.50 +2.25
71.73 67.10 59.45 44.23 29.24 .21.87 17.46
73.00 67.24 60.17 4422 20.32 21.70 17.49
Copper
~~~
error
5% Fe,
present
5;; Fe,
+1.77 $0.21 $1.21 -0.02 $0.27 -0.78 $0.17
70.81 i4.02 66.80 51.32 36.34 29.07 24.78
80.31 74.77 67.13 50.58 36.70 28 04 24.77
70
found
56
error $0 63 -0.20 $0.36 -1.46 +O.99 -0.45 -0.04
copper solution to which is added increasing quantities of a nickel-salt solution. For the experiments reported in this section, 0.2M solutions of iron, copper, and nickel salts were prepared separately. Varying proportions of iron and copper solutions were first prepared, and to each of these were added different quantities of the standard nickel solution. Solution
Iron, Parts Copper, Parts 4 1 :3 1 2 1 1 1
A B
c
n E
1
F
3 4
Sickel, Part 1
Solution .4, Parts
1 1.5
yo Fe, Present
C D E
A
H
Papers were impregnated with the test solutions and the fluorescent intensities were measured with the sodium chloride crystal. Initially, the R, values for the nickel to iron and nickel to copper intensities were obtainrd for equimolar solutions of each pair, yielding values of 1.285 and 1.397, rrspcctively. These values were usrd to determine the concentrations of iron and copper by means of the following aquations:
ANi
+
.4Fe
7"Cu, Present
-0 26 +0 17
13.1i 16 52
13 42 15 93
+l.89
-0 49 -1 33
16 40 20 55 21 7 3 2 i 1T 32.16 40 10
16 37 21 61
-0.18 +5 1ti -4.69 f1.51 1.83 -0.X +l.Ol
/G
-0 92 +1 72
42 33 52 61
43 14
77
13 06 18 06 10 71 13 31
+ 4 18 + 3 79
47 32 58 7 2
47 O!)
57 71
-0.40 -1.72
+0 28 -3 41
50 28 Gt' 33
49 82 ti2 :39
-0,!)l $0.10
13 40 17 40 10 68 13 78
15 01 43
-3 $0 -3 -1
58 43 39
20 27 32 39
71 58
i5 99
52 ti1
-3.5i
+
0 00
Iron and Copper Concentrations Obtained Using Nickel as an Internal Standard and Ratios a t One Thickness
C
-2 12 - 1 67
13 17
14 IO
1G 52
1 G 8T
43 22 54 15 38 16 47 74
42 58 5'2 80
8G 49 -3 06 -0 ':I
16 40 20 55
21 03
21 i 3 27 17
21 5G 28 I!)
28 25 35 22
27 61 :35 19
-2
27 -0 09
-12 l(i
40 10
32 31 40 00
+0 4ti -0 25
18.59 28. 10
17.65 21.90
-5.06 -5.20
42 :1:% 5'2.61
41.01 53 O!)
+:I. 97 +0 !)I
18 40 17.40
1:1 88 18.13
+3.58 $4.20
58. 72
57 08
- 2 79
10 68 13.78
10 58 13.30
-0.94 -3.48
50 28 6:! :. decreased, and the necessit?. for graphical extrapolation or least squares calculations would be climinated. The suggestion \vas tvstcd with the data used for Table XI. The intensity ratios at a single thickness w r e used in place of the limiting ratios. Thus, R I S i K a was 1.306, and R1 ' I FeK= I NiKp I-CFKi was 1.376. The rrsults arc rcPorted in 'l'able
VOL. 31, NO. 1 1 , NOVEMBER 1959
e.
1775
even a t a single thickness. However, this in the method may be useful in other systems, and it does offer a rapid method of obtaining results ranging from semiquantitative to quantitative for a wide range of element concentrxtions. LITERATURE CITED
( 1 ) Adler, I., Axelrod, J. M., Spectrochim.
Acla7,91(1955).
(2) Beattie, H. J., Brissie, R. M., ANAL. CHEM.26, 980 (1054).
( ( 3 ) Birks, L.S., Brooks, E. J., Friedman, H., Zbid., 25, 692 (1953). (4) Claisse, F., Norelco Reptr. 3, 3 (1957). (5) Cullity, B, D,, "Elements of x : R ~ ~
Diffraction," Addison-Wesley Publishing Co., Reading, Mass., 1956. (6) G'Jnn, E. L.1 A N A L . CHEM. 29, 184
( 1957). ( 7 ) .Kaufman, H. S., private communica-
tion.
(8) Kohl P. K., Caugherty, B., J . Appl. Phys. 23, 427 (1952). (9) Kokotailo, G . T., Damon, G. F., ANAL.CHEM.2 5 , 1185 (1953). (10) Parrish, W., Philips Tech. Rev. 17, 269 (1956).
(11) Pfeiffer, H. G., Zemany, P. D., . v ~ J t w e 174,397 (1954). (12) Rhodin, T. N., ANAL.CHEM.27, 1857 (1955).
RECEIVED for review October 22, 1958. Accepted August 7, 1959. Division of Analytical Chemistry, 132nd Meeting, ACS, New York, N . Y., September 1957 Taken from a thesis submitted by E . J. Felten to the Polytechnic Institute of Brooklyn in partial fulfillment of the requirements for the degree of doctor of philosophy in chemistry in June 1958.
Precision in X-Ray Emission Spectrography Background Present PAUL
D. ZEMANY,
HEINZ G. PFEIFFER, and HERMAN A. LIEBHAFSKY
General Electric Co., Schenectady,
N. Y,
Considerations stemming from the probability theory and from the theory of errors determine the best precision attainable in x-ray emission spectrography under realizable operating conditions, even when the background is significant. This statement is supported by 90 res,ilts for a spot containing less than 2 X lo-' gram of zinc and by 216 results for another containing 4 X lop5 gram of strontium.
V% for N e . According to the rule for the error of a difference, the counting error for N T - N g is: ~-
SC =
d.TT
f
.vg
(2)
The complexity of sc increases with the number of quantities counted to establish it. One objective of this investigation is to see whether s = sc
(3)
the units being identical for both.
P
( 2 ) has shown that x-ray emission spectrography under ideal operating conditions and with background nrgligible is a random process, simibr in this respect to radioactive decay. Undw such conditions, the individual counts N , . . N , lie upon the unique Gaussian distribulion of mean and standard deviation sc = V% The present investigation deals with the practically more important case in which the background N e is not negligiblc and mav be comparable with the total count N T mqde a t the goniometer position corresponding to the peak of the chararteristic line being used 2,s the analytical line. Samples for the experiments were standard spots (3) of zinc and strontium on filter paper. For these samples, the amount of elemcnt E present may be assumed proportion21 to N T - N e . If n detcrminations of E give the results el?-", the standard deviation is. RPVIOUS WORK
.
N
The counting error, due to statistical for N T and fluctuations only, is
4%
1776
ANALYTICAL CHEMISTRY
Another objective is to see \\hether
N r - N e is distributed according to the
unique Gaussian for which the standprd deviation is sc. Achievement of either objective indicatcs that precision in x-ray emission spectrography as actunlly practiced can be calculatcd satisfactorily from counting data. Establishment of distribution is the niorr conclusive and time-consuming test. EXPERIMENTS W I T H ZINC, 1954
Instrumental Details. The tungsten-target tube was operated a t 50 kv. and 50 ma. A crystal of lithium fluoride was used. The detector was an argon-filled Geiger counter. The analytical line was zinc K , a t 28 = 41.77', with a n auxiliary setting for background determination a t 39.80'. Formation and Counting of Zinc Spot. About 0.02 ml. of aqueous zinc nitrate (zinc content near 10 y per ml.) was evaporated upon a 1.25-cm. square of Whatman No. 1 filter paper supported in the sample holder of the x-ray siectrograph upon a strip of Mylar, 0.001 cm. thick and 0.6 cm. wide. Ninety values of NT a t 28 = 41.77' and 90 values of N e a t 28 = 39.80' were obtained for this spot, each N r alternating
with an ATB. Counts obtained o v w longer periods on other samplw showed that no significant error due to resetting the goniometcr was present. The true zinc content of the spot and the background adjustmrnt factor wercb determined by extended counting to a precision much greater than that of an!. individual difference N T - N e .
Determination of Ne. A filter paper was treated as in the formation of the zinc spot, but with the zinc, omitted. The following data wcrc, obtained . times for 16,384 counts, 1366.99 seconds a t 39.80" and 1159.72 seconds a t 41.77". For the individual determinations, N e was estimated h!. multiplying a count made a t 39.80' on the zinc spot by the br.ckground correction factor 1366.99/1159.72, or 1.179. Counting Rate for Zinc. Spots containing large amounts of zinc (-1 containing 4.0 y each, and 4 more containing 8 y each) were prepared and counted as described above. These were averaged to give the true counting rate for zinc. The result is: Rzn = 16.0 counts per second per
7
(4J
Zinc Content of Spot. The fol1on.ing data give the zinc content of thc spot. Time for .VT = 16,384 counts at 4 1 . i i 0 , 970.33 seconds. From preceding data, N E for this intervd is 13,829.
Zinr content of spot: t)Iz,, = (16,384 - 13,829)/ (970.33 X 16.0) = 0.165
7
(51
Individual Experiments. A counting interval, At, of approximately 40 seconds was selected for the individual experiments. On the zinc snot, counts a t 39.80" were alternated with counts a t
