- Email: [email protected]
Measurement of the K-line intensity ratios in muonic hydrogen between 0.25 and 150 torr gas pressures
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
9 August 1984
M E A S U R E M E N T O F T H E K-LINE I N T E N S I T Y R A T I O S IN M U O N I C H Y D R O G E N B E T W E E N 0.25 A N D 150 ton" G A S P R E S S U R E S *
H. A N D E R H U B , H.P. yon ARB, J. B O C K L I N , F. D I T T U S , R. F E R R E I R A M A R Q U E S 1, H. H O F E R , F. K O T T M A N N , D. T A Q Q U and J. U N T E R N ~ , H R E R Institut ff4r Hochenergiephysik, ETH-H~nggerberg, CH-8093 Zurich, Switzerland Received 22 February 1984
Precise measurements of the K X-ray energy spectrum of muonic hydrogen ( # - p) were made at H 2 gas pressures between 0.25 and 150 torr ( T = 300 K). The measured intensity ratios l(Ka)/l(Ktot) and I(K/j)/I(Kv~...) are compared with the results of new cascade calculations. The pressure dependence of the relative (/~P)2s population was investigated. Its value is (4.24+0.14)% at 10 torr and (2.21 5:0.22)% at 0.25 torr.
Introduction. The formation of muonic hydrogen in the metastable 2S state is a prerequisite to perform a precise measurement of the 2 S - 2 P energy difference in the # p atom by laser spectroscopy [1]. Previous measurements have shown that the (#P)2s lifetime at 150 and 600 torr is too short to perform a resonance experiment [2,3]. The collisional quenching rates are predicted to become small at pressures below 1 torr [4]. This low pressure region is experimentally accessible with the " m u o n bottle" apparatus and the #-stop detector described in ref. [5]. Three stages can be distinguished in the "history" of a (#P)2s atom: the slowing down of the free muon until atomic capture in a highly excited level [5]; the subsequent cascade with a fraction ~2s of all # p ' s ending in the metastable 2S state; and the disintegration of the ( # P ) 2 s a t o m [6]. In this letter we report measurements investigating the second stage, in particular the muonic K-line intensity ratios I(K,,)/I(Ktot) and I(K#)/I(Kr) .1 at H 2 pressures ranging from 150 tort down to * This work was supported by SIN. 1 Present address: Departamento de Fisica, Universidade de Coimbra, P-3000 Coimbra, Portugal. ,1 K~: 2 P ~ I S , Ka: 3 P ~ I S , and K r designating the sum of all transitions nP ~ 1S with n > 3.
0.25 torr. The fraction ~2s call be directly deduced from the K-line intensity ratios. A comparison of the measured ratios (covering a pressure range spanning three orders of magnitude) with the predictions of a cascade calculation gives information on the various interaction processes of the excited # p atom. At pressures below 10 torr, the intensity ratios become sensitive to cascade processes occurring at high n states. Before this experiment, there was practically no information about these early cascade processes. A more detailed description of this experiment is given in ref. [7].
Experimental set-up. Negative muons from pions decaying in flight were trapped in a magnetic bottle [2,5]. The measurements were made with the 40 M e V / c pion b e a m available in the erE3 area at the Swiss Institute for Nuclear Research (SIN). Typically, 150 # - / s were stopped at a H2-pressure of 1 torr and a rate of 2 x 105 ~ r - / s entering the apparatus. The muonic K X-rays were detected by two xenon gas scintillation proportional counters (GSPC); they are improved versions of those described in ref. [8]. Each GSPC has a total sensitive area of 346 cm 2, an overall energy resolution of 20% ( F W H M ) for 2 keV X-rays, and a time resolution of 80 ns ( F W H M ) at 2keV. The timing char65
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
acteristics of the GSPC are described in ref. [9]. An improved window support and the use of only one foil (5.2 ttm Mylar + 73 nm A1) for the window (cathode) increased the effic!ency by about a factor two relative to the earlier version. The amount of xenon scintillation light detected by the PM array has a radial dependence which is corrected by a procedure described in ref. [8]. Since the energy resolution depends also on the radial position of an event, the whole sensitive area was divided into five concentric rings of comparable areas, and their spectra were analyzed separately. A typical X-ray energy spectrum is shown in fig. 1 for the innermost ring; the relative energy resolution is 16% ( F W H M ) at 1.9 keV.
Analysis. The function used to fit the ttp K X-ray spectra is a sum of two gaussians with energies E , = 1.895 keV and E# = 2.246 keV, and a third gaussian representing all lines of K r. The energy E, and resolution R r of K r were left as free parameters in the fit. The three gaussians were superimposed on a constant background. The physically interesting free parameters were the intensity ratios I ~ t - I ( K , ~ ) / I ( K t o t ) and I#~I(K~)/I(K~). Replacing the constant background by a parabolic one did not change the results appreciably. The effect of the small peak asymmetry (determined by measurements with a 55Fe source with Cr filter) on I~t was about - 0 . 3 % . The 4
o
3
/:
f-
2 f-
........... .....~; .... ::.:;::. ........~x~ 0
0
I
2 Ex
3
(keY)
Fig. 1. Energy spectrum o f / t p K-line X-rays at H 2 gas pressure of 6 tort. The three fitted gaussian components K , , K/3 and K r and a flat background are shown. The chi-square is 40.4 for 35 degrees of freedom.
66
energy dependence of the efficiency was calculated from the distribution of incident angles obtained in a Monte Carlo simulation, and the known absorption coefficients in the window and the Xe gas ( = 6.5 m m absorption layer).
Results. The resulting values for I~t, I#r and Ef are given in table 1. The errors of I=t include a systematic uncertainty caused by a small dependence of I,t on the radial position in the GSPC. Fig. 2 shows the data together with the results of previous measurements and cascade calculations. We define the 2S population cES as the ratio between the number of ttp atoms reaching the 2S state and those ending in the 1S or 2S state. At subatmospheric pressures, both the 1S and 2S states are populated by pure radiative transitions from the same (n,l) states (n>~ 3, 1 = 1) with k n o w n b r a n c h i n g ratios f , = W [ ( n , l ) --~ 2S]/W[(n,l) ~ 1S] [10], where 14I is the radiative transition probability. This formula gives the values f3 = 0.134 and f> 3 -- 0.144. Thus cES can be deduced from the measured intensity ratios: :2s = l#tA + I~,f>3,
(1)
where Ipt and Irt, the relative intensities of K# and K r, are calculated from I~t and I/~r. Eq. (1) describes the case where the 2S state is coUisionally quenched and decays radiatively within a very short time. The K , transitions produced by this process are included in the measured intensity I(K~). Conversely, if the quenching rate is negligibly small, (/.tP)Es atoms are not detected [11,12], and eq. (1) must be replaced by the following relation: E 2 S = (1--CP2S)(IBtf3~-irt/>3).
(J
9 August 1984
(2)
The time distribution of the X-rays shows that there is no long-lived 2S component ,2 for pressures above 1 tort [6], hence the corresponding values of C~s are omitted in table 1. At lower pressures, the real 2S population lies between E2s and C~s.
:~2 Non-radiative 2S quenching (external Auger effect) can be neglected [4].
Volume 143B, number 1, 2, 3
9 August 1984
PHYSICS LETTERS
Table 1 Final results. pH 2 (torr) ")
N u m b e r of events ( X 103)
E, (keY)
I ,t (%)
Iflr
150 10 6 4 1
27 106 398 197 69
2.427 + 0.019 2.433 + 0.009 2.440+0.006 2.431+0.008 2.435 + 0.011
65 + 46 + 52 + 42 + 49 +
23
2.454 + 0.009
55.3 _+_+3.3 69.9 + 1.0 71.5+0.7 73.9+0.9 78.9 + 1.0 (77.9 + 2.2) b) 84.3 + 1.6 (84.5 + 2.4) b)
0.25
¢2S (%)
E2S (%)
18 6 4 6 8
6.26 + 0.46 4.24 + 0.14 4.02+0.09 3.68+0.13 2.97 + 0,14
2.89 + 0.14
56 _+ 7
2.21 + 0.22
2.16 _+0.22
(%)
") 760 torr = 1 atm = 101.3 kPa. T = 300 K b) Results of previous measurements taken with an earlier version of the X-ray detectors.
,
,
'
,,
' " ,
.
,
.
,"
,',1
,
,
•
,,
.,..
,
,
.
T . ,
,,.
,
,
'
r,
',,i
'
• KJKtot
•
90
(%)
" "'~ "-
~'~....
•
Ref. 2
,
i,
' " l
'
~ present work | J
0 K,,IK r :>~
c
.
//
]
,
'
'
""
.
"
"
70
/ \
.,"
r
~ ,
/"
50
,, 0.1
l . . . . 1
10
10 2
10 3
10 4
10 5
10 6
p[Torrl
Fig. 2. The pressure dependence of the ratios I,~t = l ( K , ) / l ( K t o t ) and Ipr = I ( K I D / I ( K ~ ) . (The data point from ref. [3] is the averaged PWC-value; the Si(Li)-value of 0.46 + 0.06 is not shown.) The curves a - e are calculated as described in the text with the following parameter values: (a) kst,r k ~ 0.5, fR = 1, faucet = 2; (b) k s t , k = 1, fR = 1, faucet = 2; (C) kst,r k = 0.5, fR = 0, faucet = 2; (d) kstar k = 0.5, fR = 1, fAuger = 4; (e) kstar k = 0.5, fR = 1, fAuger = 1. The parameters kind = 2, fchem = 3 are c o m m o n to a-e. The curves f and g correspond to two limits for the intensity of the Stark effect in the calculation of Borie and Leon [15]. Curve h interpolates the points calculated by Markushin [13].
Cascade calculations. T h e i n t e n s i t y
lated
by
Markushin
ratios calcu-
[13] f o r t h e h i g h e r
range are in agreement fig. 2). T h i s c a l c u l a t i o n
pressure
with the data (curve h of commenced with the level
n = 10 a n d w a s r e s t r i c t e d
to pressures
above
10 - 2
atm. This calculation, in common with other calcul a t i o n s o f t h e # p - c a s c a d e [14,15], d o e s n o t t r e a t i n detail the cascade processes at high n levels (n >
14),
which
play
an
d e n s i t i e s ,3. A n e w
important cascade
role
model
at
very
low
was therefore
developed to predict the K-intensity ratios and cascade times observed at low pressures. Wherever possible, known mation
theoretical
and deexcitation
results on the/~p
for-
were used.
$3 At 1 torr, the cascade is almost purely radiative for n _<10. 67
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
The # p atom was assumed to be initially populated by n-states from 14 to 25, with a probability proportional t o n - 3 , and with a statistical distribution of the (m, l)-substates [16-18]. A fixed kinetic energy of the # p atom was adopted (1 eV). The following cascade effects were taken into account: non-radiative inelastic collision processes between the /tp atom and H 2 molecules, Stark mixing, external Auger effect, and radiative transitions. The strength of these effects could be varied in the cascade calculation using a set of parameters described below. The intensity ratios were calculated assuming high radiative 2S quenching rates at all pressures [cf. eq. (1)]. There are two inelastic processes: The chemical reaction /~p~ + H 2 --~ #p~, + H + H [14] and the Coulomb
deexcitation
/xpn + H 2 ~ / . t p n ,
+ H E
[14,19]. For Coulomb deexcitation, A n = 1 is dominant, whereas for the chemical reaction small An-values are forbidden by energy conservation. We assumed that the cross sections of both processes have the same n-dependence as the cross section onoulfor the atomic process/~p~ + H --*/tp~, + H, which has been calculated in ref. [19]. The total strength of the inelastic processes K i n d = (Ochem ÷ aCoul )//OcHoul and the relative intensity fchem = ach~m/OCou~ were introduced in the cascade calculation as free parameters. The calculated ratio I~t is changed by only 1% when kinel is varied by a factor 2. To reproduce the observed short cascade time at 0.25 torr (%as < 200 n s [6]), kin d is required to be >t 2 for a fixed ratio fch~m = 3. The present measurements do not allow a determination of f~hem" Smaller values of fchem would require even higher values of kin~t. Hence, the short cascade time indicates that at high n levels Och¢m is much larger than assumed in ref. [14]. The cross sections for Stark mixing were deduced from ref. [20]. To correct for the size of the mesic atom, a fraction fR of the/~p radius was added to the radius of that cross section. The resulting area, multiplied by a factor kstark, w a s used in the cascade calculation as the average cross section for complete statistical redistribution of the different /-states. It is the parameter ksta~k that primarily determines I~t. The measurements at pressures below 10 torr are reproduced well with kstar k = 0.5 and f a = 1 (fig. 2, curve a for Iat 68
9 August 1984
and a' for Ior). The data at 150 and 600 torr are in better agreement with curve b ( k s t a r k = 1). fR affects only the results at the lowest pressures, where the/~p system stays for a long time in high n levels (cf. curve c, f R = 0). The cross sections for the Auger effect were calculated according to ref. [14], but were restricted to a maximum v a l u e fAuger~a02 t o avoid the very large values obtained in the Born approximation for some n levels. Varying fAuger slightly modifies I~t at higher pressures (curves d and e). To sum up, the intensity ratios and cascade times could be reproduced in a cascade calculation with suitable extrapolations of the commonly accepted cross sections for Stark- and Auger-effect, whereas very high probabilities for chemical dissociation at high n levels were needed. We thank our technicians R. Schaeren and G. Wemmers, as well as the technical staff of SIN for their help. L. Felawka is acknowledged for correcting the manuscript.
References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
H. Anderhub et al., SIN proposal R - 7 8 - 1 5 (1979). H. Anderhub et al., Phys. Lett. 71B (1977) 443. P.O. Egan et al., Phys. Rev. A23 (1981) 1152. G. Kodosky and M. Loen, Nuovo Cimento 1B (1971) 41; G. Carboni and G. Fiorentini, Nuovo Cimento 39B (1977) 281; R.O. Mueller et al., Phys. Rev. A l l (1975) 1175; J.S. Cohen and J.N. Bardsley, Phys. Rev. A23 (1981) 46. H. Anderhub et al., Phys. Lett. 101B (1981) 151. H. Anderhub et ai., Search for the metastable 2S-state in muonic hydrogen at low gas pressures, to be published. R. Ferreira Marques, Dissertation ETH Nr. 7111 (Ziarich, 1982). J. B~Rzklin et al., Nucl. lnstrum. Methods 176 (1980) 105. H.P. von Arb et al., Nucl. Instrum. Methods 207 (1983) 429. A. Bertin et al., Lett. Nuovo Cimento 18 (1977) 277. A. Placci et al., Phys. Lett. 32B (1970) 413. B. Budick et al., Phys. Lett. 34B (1971) 539. V.E. Markushin, Sov. Phys. JETP 53 (1981) 16. M. Leon and H.A. Bethe, Phys. Rev. 127 (1962) 636. E. Borie and M. Leon, Phys. Rev. A21 (1980) 1460. M. Leon, in: Exotic atoms '79, eds. K. Crowe et al. (Plenum, New York, 1980) p. 141. J.S. Cohen et al., Phys. Rev. A24 (1981) 33. J.S. Cohen, Phys. Rev. A27 (1983) 167. L. Braeci and G. Fiorentini, Nuovo Cimento 43A (1978) 9. J.L. Vermeulen, Nucl. Phys. B12 (1969) 506.
PHYSICS LETTERS
9 August 1984
M E A S U R E M E N T O F T H E K-LINE I N T E N S I T Y R A T I O S IN M U O N I C H Y D R O G E N B E T W E E N 0.25 A N D 150 ton" G A S P R E S S U R E S *
H. A N D E R H U B , H.P. yon ARB, J. B O C K L I N , F. D I T T U S , R. F E R R E I R A M A R Q U E S 1, H. H O F E R , F. K O T T M A N N , D. T A Q Q U and J. U N T E R N ~ , H R E R Institut ff4r Hochenergiephysik, ETH-H~nggerberg, CH-8093 Zurich, Switzerland Received 22 February 1984
Precise measurements of the K X-ray energy spectrum of muonic hydrogen ( # - p) were made at H 2 gas pressures between 0.25 and 150 torr ( T = 300 K). The measured intensity ratios l(Ka)/l(Ktot) and I(K/j)/I(Kv~...) are compared with the results of new cascade calculations. The pressure dependence of the relative (/~P)2s population was investigated. Its value is (4.24+0.14)% at 10 torr and (2.21 5:0.22)% at 0.25 torr.
Introduction. The formation of muonic hydrogen in the metastable 2S state is a prerequisite to perform a precise measurement of the 2 S - 2 P energy difference in the # p atom by laser spectroscopy [1]. Previous measurements have shown that the (#P)2s lifetime at 150 and 600 torr is too short to perform a resonance experiment [2,3]. The collisional quenching rates are predicted to become small at pressures below 1 torr [4]. This low pressure region is experimentally accessible with the " m u o n bottle" apparatus and the #-stop detector described in ref. [5]. Three stages can be distinguished in the "history" of a (#P)2s atom: the slowing down of the free muon until atomic capture in a highly excited level [5]; the subsequent cascade with a fraction ~2s of all # p ' s ending in the metastable 2S state; and the disintegration of the ( # P ) 2 s a t o m [6]. In this letter we report measurements investigating the second stage, in particular the muonic K-line intensity ratios I(K,,)/I(Ktot) and I(K#)/I(Kr) .1 at H 2 pressures ranging from 150 tort down to * This work was supported by SIN. 1 Present address: Departamento de Fisica, Universidade de Coimbra, P-3000 Coimbra, Portugal. ,1 K~: 2 P ~ I S , Ka: 3 P ~ I S , and K r designating the sum of all transitions nP ~ 1S with n > 3.
0.25 torr. The fraction ~2s call be directly deduced from the K-line intensity ratios. A comparison of the measured ratios (covering a pressure range spanning three orders of magnitude) with the predictions of a cascade calculation gives information on the various interaction processes of the excited # p atom. At pressures below 10 torr, the intensity ratios become sensitive to cascade processes occurring at high n states. Before this experiment, there was practically no information about these early cascade processes. A more detailed description of this experiment is given in ref. [7].
Experimental set-up. Negative muons from pions decaying in flight were trapped in a magnetic bottle [2,5]. The measurements were made with the 40 M e V / c pion b e a m available in the erE3 area at the Swiss Institute for Nuclear Research (SIN). Typically, 150 # - / s were stopped at a H2-pressure of 1 torr and a rate of 2 x 105 ~ r - / s entering the apparatus. The muonic K X-rays were detected by two xenon gas scintillation proportional counters (GSPC); they are improved versions of those described in ref. [8]. Each GSPC has a total sensitive area of 346 cm 2, an overall energy resolution of 20% ( F W H M ) for 2 keV X-rays, and a time resolution of 80 ns ( F W H M ) at 2keV. The timing char65
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
acteristics of the GSPC are described in ref. [9]. An improved window support and the use of only one foil (5.2 ttm Mylar + 73 nm A1) for the window (cathode) increased the effic!ency by about a factor two relative to the earlier version. The amount of xenon scintillation light detected by the PM array has a radial dependence which is corrected by a procedure described in ref. [8]. Since the energy resolution depends also on the radial position of an event, the whole sensitive area was divided into five concentric rings of comparable areas, and their spectra were analyzed separately. A typical X-ray energy spectrum is shown in fig. 1 for the innermost ring; the relative energy resolution is 16% ( F W H M ) at 1.9 keV.
Analysis. The function used to fit the ttp K X-ray spectra is a sum of two gaussians with energies E , = 1.895 keV and E# = 2.246 keV, and a third gaussian representing all lines of K r. The energy E, and resolution R r of K r were left as free parameters in the fit. The three gaussians were superimposed on a constant background. The physically interesting free parameters were the intensity ratios I ~ t - I ( K , ~ ) / I ( K t o t ) and I#~I(K~)/I(K~). Replacing the constant background by a parabolic one did not change the results appreciably. The effect of the small peak asymmetry (determined by measurements with a 55Fe source with Cr filter) on I~t was about - 0 . 3 % . The 4
o
3
/:
f-
2 f-
........... .....~; .... ::.:;::. ........~x~ 0
0
I
2 Ex
3
(keY)
Fig. 1. Energy spectrum o f / t p K-line X-rays at H 2 gas pressure of 6 tort. The three fitted gaussian components K , , K/3 and K r and a flat background are shown. The chi-square is 40.4 for 35 degrees of freedom.
66
energy dependence of the efficiency was calculated from the distribution of incident angles obtained in a Monte Carlo simulation, and the known absorption coefficients in the window and the Xe gas ( = 6.5 m m absorption layer).
Results. The resulting values for I~t, I#r and Ef are given in table 1. The errors of I=t include a systematic uncertainty caused by a small dependence of I,t on the radial position in the GSPC. Fig. 2 shows the data together with the results of previous measurements and cascade calculations. We define the 2S population cES as the ratio between the number of ttp atoms reaching the 2S state and those ending in the 1S or 2S state. At subatmospheric pressures, both the 1S and 2S states are populated by pure radiative transitions from the same (n,l) states (n>~ 3, 1 = 1) with k n o w n b r a n c h i n g ratios f , = W [ ( n , l ) --~ 2S]/W[(n,l) ~ 1S] [10], where 14I is the radiative transition probability. This formula gives the values f3 = 0.134 and f> 3 -- 0.144. Thus cES can be deduced from the measured intensity ratios: :2s = l#tA + I~,f>3,
(1)
where Ipt and Irt, the relative intensities of K# and K r, are calculated from I~t and I/~r. Eq. (1) describes the case where the 2S state is coUisionally quenched and decays radiatively within a very short time. The K , transitions produced by this process are included in the measured intensity I(K~). Conversely, if the quenching rate is negligibly small, (/.tP)Es atoms are not detected [11,12], and eq. (1) must be replaced by the following relation: E 2 S = (1--CP2S)(IBtf3~-irt/>3).
(J
9 August 1984
(2)
The time distribution of the X-rays shows that there is no long-lived 2S component ,2 for pressures above 1 tort [6], hence the corresponding values of C~s are omitted in table 1. At lower pressures, the real 2S population lies between E2s and C~s.
:~2 Non-radiative 2S quenching (external Auger effect) can be neglected [4].
Volume 143B, number 1, 2, 3
9 August 1984
PHYSICS LETTERS
Table 1 Final results. pH 2 (torr) ")
N u m b e r of events ( X 103)
E, (keY)
I ,t (%)
Iflr
150 10 6 4 1
27 106 398 197 69
2.427 + 0.019 2.433 + 0.009 2.440+0.006 2.431+0.008 2.435 + 0.011
65 + 46 + 52 + 42 + 49 +
23
2.454 + 0.009
55.3 _+_+3.3 69.9 + 1.0 71.5+0.7 73.9+0.9 78.9 + 1.0 (77.9 + 2.2) b) 84.3 + 1.6 (84.5 + 2.4) b)
0.25
¢2S (%)
E2S (%)
18 6 4 6 8
6.26 + 0.46 4.24 + 0.14 4.02+0.09 3.68+0.13 2.97 + 0,14
2.89 + 0.14
56 _+ 7
2.21 + 0.22
2.16 _+0.22
(%)
") 760 torr = 1 atm = 101.3 kPa. T = 300 K b) Results of previous measurements taken with an earlier version of the X-ray detectors.
,
,
'
,,
' " ,
.
,
.
,"
,',1
,
,
•
,,
.,..
,
,
.
T . ,
,,.
,
,
'
r,
',,i
'
• KJKtot
•
90
(%)
" "'~ "-
~'~....
•
Ref. 2
,
i,
' " l
'
~ present work | J
0 K,,IK r :>~
c
.
//
]
,
'
'
""
.
"
"
70
/ \
.,"
r
~ ,
/"
50
,, 0.1
l . . . . 1
10
10 2
10 3
10 4
10 5
10 6
p[Torrl
Fig. 2. The pressure dependence of the ratios I,~t = l ( K , ) / l ( K t o t ) and Ipr = I ( K I D / I ( K ~ ) . (The data point from ref. [3] is the averaged PWC-value; the Si(Li)-value of 0.46 + 0.06 is not shown.) The curves a - e are calculated as described in the text with the following parameter values: (a) kst,r k ~ 0.5, fR = 1, faucet = 2; (b) k s t , k = 1, fR = 1, faucet = 2; (C) kst,r k = 0.5, fR = 0, faucet = 2; (d) kstar k = 0.5, fR = 1, fAuger = 4; (e) kstar k = 0.5, fR = 1, fAuger = 1. The parameters kind = 2, fchem = 3 are c o m m o n to a-e. The curves f and g correspond to two limits for the intensity of the Stark effect in the calculation of Borie and Leon [15]. Curve h interpolates the points calculated by Markushin [13].
Cascade calculations. T h e i n t e n s i t y
lated
by
Markushin
ratios calcu-
[13] f o r t h e h i g h e r
range are in agreement fig. 2). T h i s c a l c u l a t i o n
pressure
with the data (curve h of commenced with the level
n = 10 a n d w a s r e s t r i c t e d
to pressures
above
10 - 2
atm. This calculation, in common with other calcul a t i o n s o f t h e # p - c a s c a d e [14,15], d o e s n o t t r e a t i n detail the cascade processes at high n levels (n >
14),
which
play
an
d e n s i t i e s ,3. A n e w
important cascade
role
model
at
very
low
was therefore
developed to predict the K-intensity ratios and cascade times observed at low pressures. Wherever possible, known mation
theoretical
and deexcitation
results on the/~p
for-
were used.
$3 At 1 torr, the cascade is almost purely radiative for n _<10. 67
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
The # p atom was assumed to be initially populated by n-states from 14 to 25, with a probability proportional t o n - 3 , and with a statistical distribution of the (m, l)-substates [16-18]. A fixed kinetic energy of the # p atom was adopted (1 eV). The following cascade effects were taken into account: non-radiative inelastic collision processes between the /tp atom and H 2 molecules, Stark mixing, external Auger effect, and radiative transitions. The strength of these effects could be varied in the cascade calculation using a set of parameters described below. The intensity ratios were calculated assuming high radiative 2S quenching rates at all pressures [cf. eq. (1)]. There are two inelastic processes: The chemical reaction /~p~ + H 2 --~ #p~, + H + H [14] and the Coulomb
deexcitation
/xpn + H 2 ~ / . t p n ,
+ H E
[14,19]. For Coulomb deexcitation, A n = 1 is dominant, whereas for the chemical reaction small An-values are forbidden by energy conservation. We assumed that the cross sections of both processes have the same n-dependence as the cross section onoulfor the atomic process/~p~ + H --*/tp~, + H, which has been calculated in ref. [19]. The total strength of the inelastic processes K i n d = (Ochem ÷ aCoul )//OcHoul and the relative intensity fchem = ach~m/OCou~ were introduced in the cascade calculation as free parameters. The calculated ratio I~t is changed by only 1% when kinel is varied by a factor 2. To reproduce the observed short cascade time at 0.25 torr (%as < 200 n s [6]), kin d is required to be >t 2 for a fixed ratio fch~m = 3. The present measurements do not allow a determination of f~hem" Smaller values of fchem would require even higher values of kin~t. Hence, the short cascade time indicates that at high n levels Och¢m is much larger than assumed in ref. [14]. The cross sections for Stark mixing were deduced from ref. [20]. To correct for the size of the mesic atom, a fraction fR of the/~p radius was added to the radius of that cross section. The resulting area, multiplied by a factor kstark, w a s used in the cascade calculation as the average cross section for complete statistical redistribution of the different /-states. It is the parameter ksta~k that primarily determines I~t. The measurements at pressures below 10 torr are reproduced well with kstar k = 0.5 and f a = 1 (fig. 2, curve a for Iat 68
9 August 1984
and a' for Ior). The data at 150 and 600 torr are in better agreement with curve b ( k s t a r k = 1). fR affects only the results at the lowest pressures, where the/~p system stays for a long time in high n levels (cf. curve c, f R = 0). The cross sections for the Auger effect were calculated according to ref. [14], but were restricted to a maximum v a l u e fAuger~a02 t o avoid the very large values obtained in the Born approximation for some n levels. Varying fAuger slightly modifies I~t at higher pressures (curves d and e). To sum up, the intensity ratios and cascade times could be reproduced in a cascade calculation with suitable extrapolations of the commonly accepted cross sections for Stark- and Auger-effect, whereas very high probabilities for chemical dissociation at high n levels were needed. We thank our technicians R. Schaeren and G. Wemmers, as well as the technical staff of SIN for their help. L. Felawka is acknowledged for correcting the manuscript.
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