In a previous work, Alexandrowicz et al. [3] reported on quasi-elastic broadening measurements from
3 He-spin-echo experiments on CO/Cu(100). They
observe a quasi-elastic broadening that varies from 0 to about 1 leV as a function of
the momentum transferred to CO molecules along either the 100
h
i or the 100
h
i
direction.
The observed quasi-elastic energy broadening can in principle be related to the
width Γ of the dynamical structure factor (DSF) S(q, E), which is the temporal
Fourier transform of the ISF. The ISF is the spatial Fourier transform of the pair
correlation function proposed by van Hove [4], who also derived a general
expression for the DSF in terms of the eigenvalues and eigenfunctions pertaining to
the stationary vibrational states of the adsorbates. This expression was never evaluated, to our knowledge, from within a fully quantum mechanical treatment of the
adsorbate’s dynamics. In the present work, we perform such an evaluation using
eigenvalues and eigenfunctions derived from global potential energy surfaces (PES)
for the H/Pd(111) [5] and CO/Cu(100) [6] systems. The problem is treated in full
dimensionality for the hydrogen/palladium system with static metal atoms. The
carbon monoxide quantum dynamics will be treated in one dimension, only, in
the present work. More extended, full dimensional quantum wave packet studies of
the dynamics are in preparation for both systems and will be published elsewhere.
A key parameter in the present work is the intrinsic energy broadening C i related
to the lifetime s i / 1=C i of vibrational eigenstates. This finite lifetime can be
related to the coupling of the vibrational motion of the adsorbates with either the
motion of the substrate atoms (phonons), or to the motion of electrons beyond the
Born-Oppenheimer approximation (electron-hole pair formation), or to both.
Depopulation of vibrational eigenstates of adsorbates on metal substrates via formation of electron-hole pairs is expected to proceed on the picosecond time scale
[7, 8], or even faster [9], which would be much faster than the relaxation due to the
coupling to phonons [10]. Note that the picosecond time scale is about the time
scale that can be reached with the
3 He spin-echo technique. However, a lifetime
s ¼ 1 ps corresponds to an energy broadening of 1.3 meV, which is about two to
three orders of magnitude larger than the broadening typically observed in the
aforementioned
3 He spin-echo experiments. We shall see that it is possible to
accommodate the different time domains by setting
C ¼ C i þ C d
ð1Þ
where Γ is the overall width (full width at half maximum, FWHM) of the DSF; C i is
the aforementioned intrinsic broadening and C d can be interpreted as the portion of
the broadening that is caused by diffusion.
178
T. Firmino et al.
3 He-spin-echo experiments on CO/Cu(100). They
observe a quasi-elastic broadening that varies from 0 to about 1 leV as a function of
the momentum transferred to CO molecules along either the 100
h
i or the 100
h
i
direction.
The observed quasi-elastic energy broadening can in principle be related to the
width Γ of the dynamical structure factor (DSF) S(q, E), which is the temporal
Fourier transform of the ISF. The ISF is the spatial Fourier transform of the pair
correlation function proposed by van Hove [4], who also derived a general
expression for the DSF in terms of the eigenvalues and eigenfunctions pertaining to
the stationary vibrational states of the adsorbates. This expression was never evaluated, to our knowledge, from within a fully quantum mechanical treatment of the
adsorbate’s dynamics. In the present work, we perform such an evaluation using
eigenvalues and eigenfunctions derived from global potential energy surfaces (PES)
for the H/Pd(111) [5] and CO/Cu(100) [6] systems. The problem is treated in full
dimensionality for the hydrogen/palladium system with static metal atoms. The
carbon monoxide quantum dynamics will be treated in one dimension, only, in
the present work. More extended, full dimensional quantum wave packet studies of
the dynamics are in preparation for both systems and will be published elsewhere.
A key parameter in the present work is the intrinsic energy broadening C i related
to the lifetime s i / 1=C i of vibrational eigenstates. This finite lifetime can be
related to the coupling of the vibrational motion of the adsorbates with either the
motion of the substrate atoms (phonons), or to the motion of electrons beyond the
Born-Oppenheimer approximation (electron-hole pair formation), or to both.
Depopulation of vibrational eigenstates of adsorbates on metal substrates via formation of electron-hole pairs is expected to proceed on the picosecond time scale
[7, 8], or even faster [9], which would be much faster than the relaxation due to the
coupling to phonons [10]. Note that the picosecond time scale is about the time
scale that can be reached with the
3 He spin-echo technique. However, a lifetime
s ¼ 1 ps corresponds to an energy broadening of 1.3 meV, which is about two to
three orders of magnitude larger than the broadening typically observed in the
aforementioned
3 He spin-echo experiments. We shall see that it is possible to
accommodate the different time domains by setting
C ¼ C i þ C d
ð1Þ
where Γ is the overall width (full width at half maximum, FWHM) of the DSF; C i is
the aforementioned intrinsic broadening and C d can be interpreted as the portion of
the broadening that is caused by diffusion.
178
T. Firmino et al.