These are the principal quadrupole moments of the molecule. The sum of the above quadrupole moments per definition gives
Θxx+ Θyy+ Θzz = 0, (F.33) so that only two of them are independent. Because the charge distribution of the hydrogen molecule is symmetric around, say, the z-axis, there is:
Θ = Θxx = Θyy=− 1
2Θzz . (F.34)
It can be shown that the spatially averaged electric field caused by the quadrupole moment of the hydrogen molecule is approximately
EAvg' 3Θ
r4 (F.35)
where Θ = 0.6×10−26 esu cm2 [159].
At short distances (r < a0) the hydrogen molecule can be treated as two independent
hydrogen atoms. The spherically symmetric Coulomb potential generated by the hydro- gen atom in the region r < a0 is expanded in r. The first non–vanishing contribution
gives [160]:
|E(r)|=r−2 1 + 2r+ 2r2e−2r e a2 0
(F.36) where r is expressed in atomic units. Both the molecular quadrupole potential and the atomic potential are calculated with Eq. (F.29) and shown for comparison in Fig. F.1. The potential at large distances is dominated by the quadrupole term. At short distances (r . 2a0) the atomic transition is so far shifted from its unperturbed value that the
exact knowledge of the potential is unnecessary for the calculation of the pressure shift. As will be shown below only collisions with impact parameter b & 2 plays a role in the determination of the collisional line shift.
F.3.4 Numerical results
Equations (F.22 – F.24) combined with Eqs. (F.27 – F.29), and (F.35) give the pressure shift and broadening. The phase shift η for a complete collision has to be computed for every impact parameter b. The functionsbsin (η) andb[1−cos (η)] are plotted in Fig. F.2 versus the impact parameter b. The integrals of these two curves are proportional to the pressure shift and broadening (see Eqs. (F.23) and (F.22)). This illustrates which impact parameter region is relevant for the shift and the broadening. The line shift, proportional
F.3 153
Impact parameter b [atomic units]
b sin(Phase) [ atomic units ] ˙ -1.5 -1 -0.5 0 0.5 1 1.5 2 0.5 1 1.5 2 2.5 3 3.5 4
Impact parameter b [atomic units]
b cos(Phase) [ atomic units ] ˙ 0 0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 2.5 3 3.5 4 Shift Broadening
Figure F.2: Plotted are functions whose integral is proportional to the pressure shift (left) and
broadening (right). It shows that only soft collisions (b&2) contribute to the shift and predomi-
nantly hard collisions (b.2) contribute to the broadening.
to the integral of the function in Fig. F.2 (left), turns out to be determined only by collisions with impact parameter greater than∼2a0. Therefore a precise knowledge of the
interatomic potential for distances smaller than∼2a0 is not required for the computation
of the line shift. At larger distances the long–range quadrupole potential turns out to be correct and it is the relevant potential causing the pressure shift. Only soft collisions contribute to the line shift, whereas the broadening is given by harder collisions with impact parameters b . 2a0. At these short distances the multi–pole expansion leading
to the quadrupole potential is not valid. Hence the pressure broadening value computed using the quadrupole potential is only qualitatively correct.
A pressure shift and broadening of
∆νshift= 1.3 MHz×p[hPa]
∆νbroadening= 2.4 MHz×p[hPa]
(F.37) are expected. This is three orders of magnitude smaller than the accuracy in the de- termination of the 2S−2P centroid position we are aiming at (1.5 GHz), and therefore completely negligible.
Appendix G
Population and lifetime of the 2S
state
The feasibility of our experiment relies crucially on a sizable fraction of muonic hydrogen atoms in the 2S state with sufficiently long lifetime. This Appendix is devoted to the description of the muonic hydrogen formation in highly excited states and subsequent deexcitation mechanisms (cascade). Focus is given to the fraction of formed µp which reach the 2S state, to the distinction between “long–lived” and “short–lived” 2S states and to the lifetime of the “long–lived” 2S state which corresponds to the initial state relevant for the laser experiment.
G.1
Muonic hydrogen formation and cascade processes
A negatively charged muon introduced into H2 gas will slow down to a kinetic energy of
about 15 eV and is then captured by a hydrogen molecule. The hydrogen molecule breaks up and muonic hydrogen is formed [161]. Deceleration of the muon is caused by inelastic processes,i.e.,by ionization and excitation of hydrogen molecules.
Based on measurements withµ− [162] and ¯p [163] the stopping power ofµ− in H2 gas
forµ−energies between 1 and 5 keV ranges fromS= 2.4×10−15eV/cm2toS= 3.7×10−15 eV/cm2, respectively. Knowing the stopping power of µ−, it is possible to calculate its
energy loss via dE/dx = −nS where n is the atomic density. The resulting range and slowing down time until the µ− is captured is respectively 20 cm and 300 ns for a 1 keV
µ−, and 60 cm and 500 ns for a 5 keVµ−in 0.6 hPa H
2gas at room temperature, assuming
a capture energy of 15 eV [16].
As a rule of thumb, muonic hydrogen is formed in an orbit with similar energy and dis- tance from the nucleus as the displaced electron, since then the overlap between electronic and muonic wave function is maximal. The corresponding principal quantum number of the initial state is then ni ∼
p
mµpr /mepr ∼ 14 where mµpr and mepr are respectively the reduced masses of µp and H atoms.
After the formation of the µp atom in a highly excited state a number of different processes take place until the metastable 2S or the 1S ground state are reached: radiative transition, Auger emission, Coulomb deexcitation, Stark mixing and elastic collisions. Figure G.1 and Table G.1 give a summary of the processes included in the present cascade model of T. S. Jensen and V. Markushin [12, 164–169].
156 Population and lifetime of the 2S state
Table G.1: Processes involved in the deexcitation ofµp after its formation at highly excited states.
Stark mixing: (µp)nli+ H2 →(µp)nlf + H2
External Auger effect: (µp)i+ H2 →(µp)f + H+2 +e
Coulomb deexcitation: (µp)ni+ H2 →(µp)nf + H2 (nf < ni) Elastic collision: (µp)nl+ H2 →(µp)nl+ H2
Radiative transition: (µp)nili →(µp)nflf +γ Weak decay: µ−→e−νµν¯e
Nuclear capture: µ−+p→n+ν
µ
1. Radiative transitions: The radiative rates of muonic hydrogen are related to those of atomic hydrogen by Γradnili→nflf(µp) = mrµp/mepr Γradnili→nflf(H) (as follows from Eq. (E.16)). Only electric dipole transitions are considered in the cascade model:
(µp)nili →(µp)nflf +γ (G.1) with lf = li±1. This is the only cascade process which does not depend on den- sity and kinetic energy. The radiative rates strongly increase with decreasing n
due to the radiative rate dependence on the energy difference (Γrad ∼∆Enn3 0) and
the wave–function overlap [154]. Hence below a specific density–dependent nvalue which increases with lower hydrogen density, the radiative transitions dominate the cascade. For the same reason (Γrad ∼∆Enn3 0), the population of the circular states
|n, l = n−1i is strongly enhanced by ∆n 1 radiative dipole transitions since ∆l=±1 . From such states the radiative decay can proceed exclusively via ∆n= 1 dipole transitions. Therefore the circular transitions are the most important source feeding the 1S ground state. The 2S state is not fed via radiative transitions from these circular states since the 2P state “always” decays radiatively to the ground state (except at very high densities, i.e.,close to that of liquid hydrogen).
2. Stark mixing: Since muonic hydrogen is neutral and rather small in the atomic scale, it approaches closely the nuclei of neighboring atoms, experiencing their Cou– lomb field. Hence the corresponding cross section is given by the size of the hydrogen atom. The electric field experienced during a collision mixes the pure parity states |nlmi with states of different angular momentum (linear Stark effect):
(µp)nli+ H2 →(µp)nlf + H2 (G.2)
The rates of this process increase with increasing kinetic energy and principal quan- tum numbern. Radiative transitions preferably populate the circular states whereas Stark mixing is reestablishing a statistical distribution of the orbital angular momen- tum l. Since the radiative transition rates are pressure–independent and the Stark mixing rate depends linearly on the pressure, an increase of the pressure will lead to an increase of the fraction ofµp atoms reaching the 2S state (see Fig. G.2).
3. Coulomb deexcitation: This process is important in the upper part of the cas- cade n > 10 (at 0.6 hPa) and is the only process which accelerates the µp atoms considerably,
G.1 157 1 -2530 2 -632 3 -281 4 -158 5 -101 6 -70 7 -51 8 -39 10 -25 12 -1714 -12 EB (eV) n n=14
Quantum number n
Rate 10
12s
-1 Radiative Stark Coulomb Auger 10-7 10-6 10-5 10-4 10-3 10-2 4 6 8 10 12 14 16 18 20Figure G.1: (Left): Schematic view of the atomic cascade in µp. The µp atom is formed in a
highly excited state with principal quantum number n ∼ 14. At ∼ 1 hPa pressure the most
important deexcitation mechanism in the upper part of the cascade is Coulomb deexcitation with
large jumps ∆n= 1−4. Below n∼ 10the radiative transitions are the dominant deexcitation
mechanism. (Right): The l-average rates at 1 eV kinetic energy for 1 hPa pressure: Coulomb
deexcitation (empty red squares), Stark mixing (full blue squares), Auger transitions (magenta stars) and radiative deexcitation (black full circles). The two extreme cases of radiative rates are
shown: np−→1sand the circularn(n−1) −→ (n−1)(n−2). These cross sections have been
computed by T. Jensen.
wherenf < ni. The transition energy is shared between the colliding particles, i.e., the released energy ∆Enn0 is partially converted into kinetic energy of the µp atom
producing exotic atoms with energies E1 eV [10].
4. External Auger deexcitation: This is a deexcitation which occurs via ionization of the H2 molecule in a collision:
(µp)i+ H2 →(µp)f + H+2 +e . (G.4)
In contrast to the Coulomb deexcitation, nearly all the transition energy is carried away by the released electron, so that the recoil energy of the µp atom is rather small. The Auger rate has its maximum at a critical level nc= 7 above which only ∆n >1 transitions are energetically possible (to be compared with the H2 ionization
potential of 15.4 eV). Above nc therefore the cross section decreases. Below nc, ∆n = 1 transitions are possible but the probability of electron emission decreases rapidly withnsince the size of the neutral µp atoms decreases much below the size of the electron wave–function.
158 Population and lifetime of the 2S state
5. Elastic scattering: Elastic scattering describes the collision ofµp atoms with the H2 molecule:
(µp)nl+ H2 →(µp)nl+ H2 (G.5)
in which the quantum state is not changed. Together with Stark mixing (inelastic scattering), elastic scattering is decelerating the µp atoms, counteracting the accel- eration caused by the Coulomb deexcitation. However elastic scattering does not dominate the evolution of the energy distribution, and when the µp atoms reach the ground state or the 2S metastable state they are far from being thermalized. However this process is responsible for the thermalization of the 2S state which is of fundamental importance for the laser experiment (cf. §G.3).
The cascade model developed by Jensen and Markushin accounts also for the time evo- lution of the kinetic energy of the µp atoms during the cascade. The present cascade model reproduces well the measured x-ray yields [14] and kinetic energy distributions [10] whereas it gives too long values for the cascade time [16]. The measured cascade time at 0.6 hPa is (30±7) ns [16].