Shift and broadening of resonance lines of antiprotonic helium atoms in solid helium
aa r X i v : . [ phy s i c s . a t o m - ph ] N ov Shift and broadening of resonance lines of antiprotonic heliumatoms in solid helium
Andrzej Adamczak ∗ Institute of Nuclear Physics, Polish Academy of Sciences,Radzikowskiego 152, PL-31342 Krak´ow, Poland
Dimitar Bakalov † Institute for Nuclear Research and Nuclear Energy,Bulgarian Academy of Sciences, Tsarigradsko chauss´ee 72, Sofia 1784, Bulgaria (Dated: October 14, 2018)
Abstract
We have estimated the shift and broadening of the resonance lines in the spectrum of antipro-tonic helium atoms ¯ p He + implanted in solid helium He. The application of the response functionfor crystalline helium has enabled determination of the contributions from the collective degrees offreedom to the shift and broadening. It occurs that the broadening due to the collective motionis negligible compared to the natural line width. The available pair-correlation functions for crys-talline He have been applied for estimating the resonance-line shift due to collisions of ¯ p He + atomwith the surrounding He atoms. The dependence of the line shift, which has been calculated inthe quasistatic limit, on the solid- He density is nonlinear.
PACS numbers: 36.10.-k, 32.70.Jz, 34.10.+x, 34.20.Cf ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The aim of this work is estimation of the shift and broadening of resonance lines of the¯ p He + atoms in solid He as functions of helium density. A similar study for liquid He,presented in Ref. [1], showed that the shift is a linear function of density in normal-fluid He and displays only small oscillations with temperature in superfluid He. The pair-distribution function, which describes the atom distribution around a given particle, is verysimilar for both the normal-fluid and the superfluid helium at various temperatures andsaturated-vapor pressure. Also, the density change for liquid He at such a pressure is lowerthan about 15%. Therefore, solid He gives the possibility of studying the resonance-lineshift and broadening at much higher densities and thus for shorter distances between theatoms.The antiprotonic helium atoms are created when antiprotons are decelerated in heliumtargets and then replace one electron in the helium atoms. About 3% of antiprotons arecaptured in metastable states ( n, ℓ ) with long lifetimes on the order of microseconds [2].This phenomenon enabled high-precision laser spectroscopy of ¯ p He + atoms. As a result,the antiproton-to-electron mass ratio [3] has been determined with the best accuracy up-to-date. Such high-precision measurements required the estimation of different systematiceffects. Among the most important effects are the shift and broadening of the spectrallines due to interaction with the helium atoms and varying with the density of the heliumtarget. In the case of helium gas, these effects have been calculated in the semiclassicalapproach with the use of a pairwise potential of interaction between the ¯ p He + atom and anordinary helium atom [4]. The calculation results agree well with the experimental data upto the gaseous-helium density ̺ = 127 g/l [2]. It has been found that the resonance lineshifts in these gas targets are proportional to the helium density within the experimentalaccuracy [5, 6]. The attempts for laser spectroscopy of antiprotonic helium atoms in liquidhelium, which were performed by the ASACUSA Collaboration at CERN [7], encouragedus to evaluate the corresponding shifts and broadening in fluid and superfluid He [1]. Inthe case of liquid helium, it was necessary to take into account the influence of collectivedynamics of helium atoms on the shifts and broadening of spectral lines, apart from thecollisional effects. Our calculations were performed for the target temperatures T = 1 . P up to 8 bars, which corresponds to the maximum pressure appliedin the gaseous-helium experiments. For such conditions, the maximum density of about146 g/l is reached at the λ point. Our calculations confirmed the linear dependence of theresonance-line shift in normal-fluid He. On the other hand, an appreciable oscillation (9%)of this shift as a function of temperature was found for the region of superfluid, where the He density is practically constant.At the atmospheric pressure, helium remains liquid even at absolute zero, due to theweak interaction between the helium atoms and the large zero-point motion of these atoms.However, the application of pressure P ≈
25 bar below a few kelvins leads to solidificationof He [8]. The solid helium at the melting line has a density significantly greater than thatof liquid helium. The solid- He density can be increased using even larger pressures.In this work, the numerical calculations are performed for the transition | i i = ( n, ℓ ) =(39 , → | f i = ( n ′ , ℓ ′ ) = (38 ,
34) between the initial | i i and final | f i states of the ¯ p He + atom (transition 1), which has been experimentally observed even at a relatively high densityof 127 g/l [3]. The resonance wavelength for this transition equals λ = 5972 .
570 ˚A [5] andthe corresponding resonance energy is E = 2 . R A ≈ . × s − [9] determines the natural line width Γ , according to therelation Γ ≈ ~ R A , (1)which gives Γ ≈ . × − eV. The corresponding frequency ν n = Γ /h equals 0.018 GHz.In Sec. II we estimate the changes of the line profile due to the collective motion inthe quantum crystal He, using the method presented in Ref. [10]. The line shift, which iscaused by collisions of ¯ p He + atom with neighboring He atoms, is calculated in Sec. III in thequasistatic approximation of Ref. [1] using the available pair-correlation functions g ( r ) forcrystalline He. It has been shown in Ref. [1] that in the quasistatic limit of slow collisionsthe semiclassical expression for the collisional shift of Ref. [4] takes the form of a mean valueof the interatomic potential, averaged over the spatial distribution of the perturbing heliumatoms around the emitting antiprotonic atoms. Under the assumption that the latter is closeto the distribution in pure helium, described by the pair-correlation function, this allows forusing experimental data about g ( r ). Unfortunately the semiclassical expression for the linebroadening [4] does not take any simple form in the quasistatic limit, so that the collisionalwidth remains to be evaluated by full scale semiclassical or quantum calculations, which isbeyond the scope of the present Brief Report. The results are briefly discussed in Sec. IV. II. LINE SHIFT AND BROADENING DUE TO THE COLLECTIVE DYNAMICSOF SOLID HELIUM
The shift and broadening of a resonance line can be evaluated using the method devel-oped by Singwi and Sj¨olander [10], which employs the Van Hove formalism of the responsefunction [11]. When a particle which absorbs or emits a photon is set at a fixed position,the absorption cross section σ a takes the following form σ a ( E ) = σ Γ / E − E ) + Γ / E is the photon energy, σ is the resonance maximum at the resonance energy E and Γ stands for the natural width of the resonance. In the case of a harmonic crystal,the resonance profile σ solid a ( E ) can be rigorously derived. For a monoatomic cubic Bravaislattice, the exact form of the profile is given as [10] σ solid a ( E ) = πσ Γ − W ) (cid:20) π Γ ( E − E ′ ) + Γ + ∞ X n =1 g n ( ω, T ) (2 W ) n n ! (cid:21) , (3)where ~ ω and ~ q denote the energy and momentum transfer to the crystal, respectively, and T is temperature. Although solid He has the hcp structure under specific conditions (seethe phase diagram, e.g., in Ref. [8]), apart from the cubic bcc and fcc structures, the aboveexpansion establishes a fair approximation also for this lattice. In the case of laser-stimulatedtransitions in the antiprotonic helium, the resonance energy in Eq. (3) equals E ′ = E +∆ E ,where ∆ E is the line shift due to the pairwise interaction. The exponent 2 W of the Debye-Waller factor exp( − W ) can be expressed as follows2 W = E r Z ∞ Z ( w ) w coth (cid:0) β T w (cid:1) d w, β T = 1 k B T , (4)3here Z ( w ) is the normalized density of vibrational states in the crystal, k B is Boltzmann’sconstant and E r denotes the recoil energy E r = ( ~ q ) M , (5)in which M is the mass of antiprotonic helium.The first term in the expansion (3) describes the recoil-less photon absorption or emissionin the rigid crystal lattice. The next terms of this expansion, which are proportional to q n ,describe the same process with simultaneous absorption or emission of one or more phonons.The functions g n from Eq. (3) are defined in Ref. [10]. In particular, the one-phonon term2 W g in the brackets of this equation takes the following form2 W g ( ω, T ) = E r Z ( ω ) ω [ n B ( ω, T ) + 1] , (6)where n B ( ω, T ) = [exp( β T ω ) − − (7)is the Bose population factor for phonons. The amplitudes of all the processes are deter-mined by the Debye-Waller factor. When 2 W ≪ ~ q = p , ~ ω = E − E ′ , (8)respectively. The absolute value of the photon momentum is denoted here by p . In the caseof transition 1, we have q = 2 π/λ = 0 . − (9)and the recoil energy equals E r = 0 . × − eV. Thus, the recoil energy is very smallcompared to the resonance energy E r /E ≈ . × − .The Debye-Waller factor can be estimated using the Debye model of isotropic crystal,which is a fair approximation also for quantum crystals such as solid helium, hydrogen ordeuterium. In this model, the density of vibrational states takes the form Z ( w ) = ( w /w if w ≤ w D w > w D , (10)in which the maximum energy of vibrations w D is determined by the Debye temperature θ D of the crystal: w D = k B θ D . The Debye temperature for solid He is greater than 25 K.For the pressures 26.7–129 bar and the corresponding temperatures 1 K–4 K, which areconsidered in this work, θ D ≈
25 K–38 K [12] and thus
T /θ D ≪
1. In the limit T → W = 32 E r w D , (11)which is a good approximation for T /θ D ≪
1. In the case of transition 1, one has 2 W ∼ − .Thus, the recoil-less term in the expansion (3) is dominant and the subsequent phonon4ontributions are negligible. This means that the resonance-line shift in solid helium issolely due to the collisional correction ∆ E . Let us note that in the case of a free ¯ p He + atomthe line shift is strictly equal to the recoil energy E r . On the other hand, in solid helium,the recoil effect disappears since the response of the He lattice to the resonance-photonabsorption or emission is practically the response of a rigid lattice.The phonon contribution to the line broadening is determined by the one-phonon term (6)with the width determined by w D ≈ W factor. Therefore, the collective degrees of freedom practically do not changethe resonance-line width in solid He. The width of the dominant recoil-less term is equalto the natural resonance width Γ . III. COLLISIONAL SHIFT OF RESONANCE LINES IN SOLID HELIUM
The collisional shift ∆ E of the resonance lines in crystalline He is estimated here in thequasistatic limit using the method that has been discussed in detail in Ref. [1]. The line shiftis expressed in terms of the pairwise potentials of ¯ p He + interaction with a single helium atomand the pair-correlation function g ( r ) of a condensed He target for a fixed temperature andpressure. The expression g ( r ) d r gives the probability of finding a He atom in the shell[ r, r + d r ] around a given atom. We use here the spherically symmetric pairwise potentials V i ( r ) and V f ( r ) of the ¯ p He + -He interaction in the initial and final states of antiprotonichelium [4, 13]. As a result, the collisional contribution ∆ E to the resonance-line shift isapproximated by the following expression:∆ E = Z r max d r g ( r )∆ V ( r ) , (12)where ∆ V ( r ) = V f ( r ) − V i ( r ) and r max is a cutoff. Let us note that the radius r in thefunction g ( r ) denotes the distance reckoned from a given He atom located in the origin. Inour case, we replace this atom by the implanted antiprotonic helium atom. However, thisis a reasonable approximation since the probability density calculated for the two-particlesystem ¯ p He + +He is very similar to g ( r ) at r . g ( r ) are scarce. We use here thetheoretical g ( r ) for the solid He near the melting curve at temperature T = 1 . P = 26 . ρ = 190 g/l) [14]. Also, we employ the theoretical g ( r ) for T = 2 . P = 57 bar ( ρ = 209 g/l) and T = 4 . P = 129 bar ( ρ = 234 g/l) [15]. The functions g ( r ) for solid and liquid He are compared in Fig. 1 for T = 1 K and T ≈ He is superfluid at 1 K and normal fluid at 4.27 K, the corresponding pair-correlationfunctions are very similar. Therefore, no significant change of behavior of the resonance-lineshift is expected in liquid helium. On the other hand, the functions g ( r ) for solid He atthe presented temperatures and densities differ significantly from each other and from thecorresponding functions for liquid helium. In particular, one can see that the probability offinding a neighboring He atom at 2 ˚A < r < . r gives a dominant contribution to the line shift, which is shown in Fig. 2.As a result, one can expect a significant change of the line-shift behavior in solid helium.The average number n ( r ) of He atoms, n ( r ) = 4 πN Z r d r ′ r ′ g ( r ′ ) , (13)5 solid( r =234 g/l, T=4.0 K)solid( r =190 g/l, T=1.0 K)liquid( r =124 g/l, T = 4.27 K)liquid( r =145 g/l, T = 1.0 K) r [ Å ] pa i r- c o rr e l a t i on f un c t i on g (r) FIG. 1: (Color online) The pair-correlation functions g ( r ) for solid [14, 15] and liquid (at thesaturated-vapor pressure) [16] He at several values of target density ̺ and temperature T . -1.5-1-0.500.511.52 2.5 3 3.5 4 4.5 5 D V [ meV ] g(190 g/l, 1.0 K)g(209 g/l, 2.5 K)g(234 g/l, 4.0 K) r [ Å ] FIG. 2: (Color online) The difference ∆ V ( r ) of the pairwise potentials V (39 , ( r ) and V (38 , ( r )together with the pair-correlation functions g ( r ) for solid He [14, 15] versus the distance r betweenthe antiprotonic helium and the He atom. The densities ̺ and and temperatures T of the threetargets are given in the plot. One can conclude from this figure that the contribution to theresonance energy shift, Eq. (12), practically comes from the interval 2.0 ˚A < r . V ( r ) and g ( r ) have significant values. For r ≤ . which are located within the sphere of radius r around the helium atom in the origin, is shownin Fig. 3 for the three pressures. Since n ( r ) < r . V ( r ) has the largest amplitude, the contribution to the resonance-line shift (12) from thisregion is dominant. Therefore, using the pairwise interaction potentials for determinationof the resonance shift in solid He is a reasonable approximation.The results of our calculations for solid He for the three different target densities aresummarized in Table I, where the reduced line shift ∆ E /̺ is given in the fifth column.A dependence of the calculated reduced line shift on the upper limit r max in the integral (12)is shown in Fig. 4. One can see that it is sufficient to perform integration in Eq. (12) up6 r = 190 g/l, T=1.0 K r = 209 g/l, T=2.5 K r = 234 g/l, T=4.0 K r [ Å ] ne i ghbo r nu m be r FIG. 3: (Color online) The number n ( r ) of He atoms within the sphere of radius r that surroundan atom placed at r = 0 in solid helium.TABLE I: The resonance-line shift ∆ E and the reduced line shift ∆ E /̺ for solid He.Temperature Pressure Density ∆ E ∆ E /̺ [K] [bar] [g/l] [GHz] [GHz l/g]1 . . − . − . . . − . − . . . − . − . to r max ≈ . E which are presented in Table I display a clearnonlinear dependence on the target density, which is in contrast to the behavior of analogousline shifts in He gas [5, 6] and liquid He above T = 2 .
17 K [1]. The absolute values ofthe reduced line shifts in solid helium are greater than the values of the corresponding lineshifts in liquid He (e.g., ∆ E /̺ = − .
427 GHz l/g for ̺ = 146 g/l at T = 2 .
27 K [1]).
IV. CONCLUSIONS
The experimental and theoretical investigation of the density dependence of theresonance-line shifts in gaseous helium and the projects of the ASACUSA Collaborationat CERN for the high-accuracy laser spectroscopy of ¯ p He + atoms in liquid He has moti-vated the present work. In particular, we have studied the influence of collective degreesof freedom in solid He on the broadening and shift of the resonance lines of antiprotonichelium located in this target.We have found that the resonant absorption or emission of a laser photon by the ¯ p He + atom implanted in solid He is a fully recoil-less process which takes place in the rigid latticeand thus is analogous the M¨ossbauer effect. This is due to a very small momentum transferto the lattice of about 0.001 ˚A − and extremely small amplitudes of phonon processes, whichare simultaneous with the resonant transition in the antiprotonic helium. Therefore, there is7 r = 190 g/l, T=1.0 K r = 209 g/l, T=2.5 K r = 234 g/l, T=4.0 Ktransition (39,35) fi (38,34) r max [ Å ] | D E (r m a x ) | / r [ G H z l / g ] FIG. 4: (Color online) The absolute value | ∆ E | /̺ of the reduced resonance-line shift in solid Heas a function of r max . exactly no contribution to the resonance-line shifts from the collective motion in solid helium.On the other hand, the resonance lines in the case of ¯ p He + atom in a dilute helium gas areshifted by the corresponding recoil energy, which is however very small. The broadeningof resonance-line shifts due to the collective motion in crystalline He is determined bythe one-phonon processes and equals about 2-3 meV, which is a typical maximum phononenergy in solid helium. However, the amplitude of such processes is smaller by many ordersof magnitude than the recoil-less process so that this broadening cannot be observed inexperiments. As a result, the total broadening and shift of the resonance lines in solidhelium are determined by the collisional effects. The magnitude of the reduced collisionalshift ∆ E /̺ for the resonance transition (39 , → (38 , − .
525 to − .
704 GHz l/g when the density of solid He varies from 190 to 234 g/l. Therefore, the density dependence of the total line shift isclearly nonlinear, which is in contrast to the behavior of the analogous line shifts in gaseousand normal liquid He.In order to improve the accuracy of the presented evaluation of the collisional contributionto the resonance-line shifts it is indispensable to calculate the potentials of ¯ p He + interactionwith at least two helium atoms. Also, this would enable a reliable estimation of the collisionalbroadening of the spectral lines. However, a calculation of the appropriate potentials is muchmore complicated than in the case of one neighboring He atom. Acknowledgments
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