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Detectability of a resonance curve of Lorentz or dispersion type
Nuclear Instruments and Methods 179 (1981) 405-406 © North-Holland Publishing Company
LETTERS TO THE EDITOR DETECTABILITY OF A RESONANCE CURVE OF LORENTZ OR DISPERSION TYPE M. FUJ1OKA and S. SHIBUYA Cyclotron-Radioisotope Center and Department o f Physics, Tohoku University, Sendal Japan
Received 8 May 1980
The detectability of a resonance curve of Lorentz or dispersion type superposed on a large constant background is estimated on the basis of a least-squares method. It is found that the dispersion curve should be easier to detect than the Lorentz curve.
In our laboratory practice we sometimes encounter the detection of a small resonance on a large background. The resonance may have a Lorentz or a dispersion shape. As an example, we take the case of a stroboscopic observation of the quadrupole interaction using a pulsed beam as recently performed by Raghavan and Raghavan [1 ]. They put the detectors at 4>= 0 ° and 90 °, and observed Lorentz-type beat resonances. In this example one can equally detect dispersion-type resonance of the same amplitude [see eq. (1)], putting the detectors at 45 ° and 135 ° with respect to the beam direction [2,3]. In this Letter we estimate the detectability or the detection efficiency of the resonance amplitude for the two types of curves on the basis of a least-squares algorithm. We normalize the resonance curves in the following way: 1 fL(x) = l + x 2
x and fD(x) - 1 +X 2 .
assuming that the expected amplitude, a~, is the same for the two curves:
We assume that the points of measurement, xi, are equidistant with an interval o f h and that - X ~ < xi <~X . We approximate FtF by integrals [5], e.g.,
(FtFh,
=~
F q F- , I ~ 1 {,X
fxlfxl
dx
-x
1 x
= h- f x (
dx
1 (tan_~X+
1 +x~) 2
h
X
)
]-+X 2 '
for the Lorentz shape. Thus we obtain the error matrices as
(1)
EL . . . . . . .
h
2X( tan-IX + 1 ~ )
It is noted that the total excursion is the same and is equal to unity for the Lorentz IlL(X)] and the dispersion brD(X)] curves. The measured valuesyi are to be fitted by
×
- 4(tan-IX)2
2X
-2 tan-iX
-2 tan-iX
tan-~X+ 1 X
];
Yi ~ aLf(xi) + a~ ,
where the error attached to Y i is assumed to be constant (a large background), e.g., oCvi)= 1. Following the usual procedure of least-squares fitting [4] we obtain the error matrix of ax as
~D =
h 2X (tan -1X
1 X x 2)
(3)
× = (FtF) -1
with/7/1
=
f(Xi)
and Fi2 : 1 .
0
For the detectability, or the degree of accuracy, A, of the resonance we take the inverse variance of a~,
X t a n - I X - 1-+X~-
It is noted that for the dispersion curve al and a2 are 405
406
M. Fu]ioka, S. Shibuya / Detectability o f a resonance curve
independent (off-diagonals are zero) and that the accuracy of a2 is 2X/h which is just the total counts measured; we assumed o(yi) = 1. Putting eq. (3) into eq. (2), we arrive at the expressions of the detectability of the amplitude as AL = h~-~L ;
EL=2- L
+---21+Xa
--
;
2X AD = -h--ED ;
1 ~tan-tX
1 )
RD=yy
(4)
I+-X 2
where we defined the detection efficiency, E = - A / (2X/h); this quantity is a sort of indicator of the fraction o f total statistics (total counts) which is utilized in determining a parameter, in this case, the amplitude o f a resonance. Fig. 1 compares the efficiencies o f Lorentz and
.
.
.
.
] ....
I
X=1.83
dispersion curves as a function of X which is half the width of measurement. It is somewhat suprising that ED is always larger than EL, showing that the dispersion curve is more favourably detected than the Lorentz curve, provided that the expected amplitude is the same. One might consider that the Lorentz curve could be more easily identified because the maximum deviation from the background is two times that of the dispersion curve; see eq. (1). The present results of eq. (4) and fig. 1 can be understood, however, seeing that the excursion of the Lorentz curve is one-sided, making the discrimination between resonance and background more difficult; the ratio of EL to E o becomes very small for small X. For large values of X (X 2 10, i.e., the width of measurement >~10 times the half-width of the Lorentz curve) ED approaches E L but both efficiencies become small. The present discussion shows that the dispersion curve should be easier to detect than the Lorentz curve, when superimposed on a large background, especially, when the width of measurement is limited because of a long time required for the measurement. It is noted that a more refined treatment including the fitting of resonance position and width does not change the essential feature of the present discussion, although the detection efficiency of the position of the resonance of the Lorentz type is somewhat larger than that of the dispersion type. The authors are grateful to Dr. N. Kawamura and Prof. T. Ishimatsu for discussions.
10-' o
EL
w
References
10-2 S
' .
0.5
.
.
.
.
.
.
.
1
5
10
20
x
Fig. 1. Detection efficiency of the amplitude of a resonance of Lorentz type (EL) and of dispersion type (ED) superposed on a large background, as a function of half the width of the region of measurement (X); X = 1 corresponds to a width of measurement equal to the fwhm of the Lorentz shape. The value of X is indicated for which the efficiency becomes maximum. See eq. (4).
[1] P. Raghavan and R.S. Raghavan, Hyp. Interactions 3 (1977) 371. [2] R.M. Steffen and K. Aider, The electromagnetic interaction in nuclear spectroscopy (ed. W.D. Hamilton; NorthHolland Publ. Co., Amsterdam, 1975) p. 505. [3] R. Coussement, private communication (Feb., 1980). [4] A.H. Wapstra, G.J. Nijgh and R. van Lieshout, Nuclear spectroscopy tables (North-Holland Publ. Co., Amsterdam, 1959) ch. 1. [5] M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions (Dover Publ., N. Y., 1970) p. 12.
LETTERS TO THE EDITOR DETECTABILITY OF A RESONANCE CURVE OF LORENTZ OR DISPERSION TYPE M. FUJ1OKA and S. SHIBUYA Cyclotron-Radioisotope Center and Department o f Physics, Tohoku University, Sendal Japan
Received 8 May 1980
The detectability of a resonance curve of Lorentz or dispersion type superposed on a large constant background is estimated on the basis of a least-squares method. It is found that the dispersion curve should be easier to detect than the Lorentz curve.
In our laboratory practice we sometimes encounter the detection of a small resonance on a large background. The resonance may have a Lorentz or a dispersion shape. As an example, we take the case of a stroboscopic observation of the quadrupole interaction using a pulsed beam as recently performed by Raghavan and Raghavan [1 ]. They put the detectors at 4>= 0 ° and 90 °, and observed Lorentz-type beat resonances. In this example one can equally detect dispersion-type resonance of the same amplitude [see eq. (1)], putting the detectors at 45 ° and 135 ° with respect to the beam direction [2,3]. In this Letter we estimate the detectability or the detection efficiency of the resonance amplitude for the two types of curves on the basis of a least-squares algorithm. We normalize the resonance curves in the following way: 1 fL(x) = l + x 2
x and fD(x) - 1 +X 2 .
assuming that the expected amplitude, a~, is the same for the two curves:
We assume that the points of measurement, xi, are equidistant with an interval o f h and that - X ~ < xi <~X . We approximate FtF by integrals [5], e.g.,
(FtFh,
=~
F q F- , I ~ 1 {,X
fxlfxl
dx
-x
1 x
= h- f x (
dx
1 (tan_~X+
1 +x~) 2
h
X
)
]-+X 2 '
for the Lorentz shape. Thus we obtain the error matrices as
(1)
EL . . . . . . .
h
2X( tan-IX + 1 ~ )
It is noted that the total excursion is the same and is equal to unity for the Lorentz IlL(X)] and the dispersion brD(X)] curves. The measured valuesyi are to be fitted by
×
- 4(tan-IX)2
2X
-2 tan-iX
-2 tan-iX
tan-~X+ 1 X
];
Yi ~ aLf(xi) + a~ ,
where the error attached to Y i is assumed to be constant (a large background), e.g., oCvi)= 1. Following the usual procedure of least-squares fitting [4] we obtain the error matrix of ax as
~D =
h 2X (tan -1X
1 X x 2)
(3)
× = (FtF) -1
with/7/1
=
f(Xi)
and Fi2 : 1 .
0
For the detectability, or the degree of accuracy, A, of the resonance we take the inverse variance of a~,
X t a n - I X - 1-+X~-
It is noted that for the dispersion curve al and a2 are 405
406
M. Fu]ioka, S. Shibuya / Detectability o f a resonance curve
independent (off-diagonals are zero) and that the accuracy of a2 is 2X/h which is just the total counts measured; we assumed o(yi) = 1. Putting eq. (3) into eq. (2), we arrive at the expressions of the detectability of the amplitude as AL = h~-~L ;
EL=2- L
+---21+Xa
--
;
2X AD = -h--ED ;
1 ~tan-tX
1 )
RD=yy
(4)
I+-X 2
where we defined the detection efficiency, E = - A / (2X/h); this quantity is a sort of indicator of the fraction o f total statistics (total counts) which is utilized in determining a parameter, in this case, the amplitude o f a resonance. Fig. 1 compares the efficiencies o f Lorentz and
.
.
.
.
] ....
I
X=1.83
dispersion curves as a function of X which is half the width of measurement. It is somewhat suprising that ED is always larger than EL, showing that the dispersion curve is more favourably detected than the Lorentz curve, provided that the expected amplitude is the same. One might consider that the Lorentz curve could be more easily identified because the maximum deviation from the background is two times that of the dispersion curve; see eq. (1). The present results of eq. (4) and fig. 1 can be understood, however, seeing that the excursion of the Lorentz curve is one-sided, making the discrimination between resonance and background more difficult; the ratio of EL to E o becomes very small for small X. For large values of X (X 2 10, i.e., the width of measurement >~10 times the half-width of the Lorentz curve) ED approaches E L but both efficiencies become small. The present discussion shows that the dispersion curve should be easier to detect than the Lorentz curve, when superimposed on a large background, especially, when the width of measurement is limited because of a long time required for the measurement. It is noted that a more refined treatment including the fitting of resonance position and width does not change the essential feature of the present discussion, although the detection efficiency of the position of the resonance of the Lorentz type is somewhat larger than that of the dispersion type. The authors are grateful to Dr. N. Kawamura and Prof. T. Ishimatsu for discussions.
10-' o
EL
w
References
10-2 S
' .
0.5
.
.
.
.
.
.
.
1
5
10
20
x
Fig. 1. Detection efficiency of the amplitude of a resonance of Lorentz type (EL) and of dispersion type (ED) superposed on a large background, as a function of half the width of the region of measurement (X); X = 1 corresponds to a width of measurement equal to the fwhm of the Lorentz shape. The value of X is indicated for which the efficiency becomes maximum. See eq. (4).
[1] P. Raghavan and R.S. Raghavan, Hyp. Interactions 3 (1977) 371. [2] R.M. Steffen and K. Aider, The electromagnetic interaction in nuclear spectroscopy (ed. W.D. Hamilton; NorthHolland Publ. Co., Amsterdam, 1975) p. 505. [3] R. Coussement, private communication (Feb., 1980). [4] A.H. Wapstra, G.J. Nijgh and R. van Lieshout, Nuclear spectroscopy tables (North-Holland Publ. Co., Amsterdam, 1959) ch. 1. [5] M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions (Dover Publ., N. Y., 1970) p. 12.