Doppler broadening is a general problem when determining the center frequency of a transition. Decreasing the velocity distribution along the propagation direction of the laser beam reduces the Doppler width. The narrower velocity distribution is the most substantial benefit of a collimated molecular beam, which propagates perpendicular to the laser axis. All molecules are contained inside a reservoir and satisfy the Maxwell-Boltzmann velocity distribution. The pressure inside the container is for example P0 ≈1 atm. A hole at the container allows the molecules to exit into a space of lower pressure Pb. The molecules escape collision-free if the hole diameter is smaller than the mean free path of the gas. Outside the container, the velocity distribution and the rotational and vibrational
7.1. Molecular Beam degrees of freedom stay the same as inside, which is characteristic for an effusive beam.
However, a hole diameter larger than the mean free path of the gas introduces collisions.
Escaping molecules frequently collide, which leads to adiabatic cooling of all degrees of freedom in the expansion region[167]. Sometimes it is also referred to as an isentropic expansion since it is adiabatic and reversible. The collisions contribute mainly to a cooling of the rovibrational states of the molecules, while the expansion is cooling the translational motion by narrowing the velocity distribution[168]. Clustering of the molecules at small velocity distributions limits the cooling. Thus a noble carrier gas counteracts the clustering.
Good choices are heavy noble gases like krypton or xenon. Their main advantage is the reduction of the velocity of the molecular beam. The translational temperature Ttr is mostly determined by the carrier velocity distribution ∆vtr. Thus, the temperature of the ideal gas is described by kBTtr = 12m∆vtr2 [169]. Typical translational temperatures of diatomic molecules are below 1 K. The rotational and vibrational temperatures are around 10 K and 100 K, respectively[45]. The velocity of the molecules is described by the Mach number M =u/a, with the mass flow velocity u and the local speed of sound a =p
γkBTtr/m[45]. The critical constant is the ratio of the heat capacities of constant pressure and constant volume γ =CP/CV ≈5/3, as an example of a monatomic gas. In terms of Mach numbers, the molecules start at M 1 and reach M ≈ 1 at the source exit (Figure 7.1a). This includes the assumption of a pressure ratio P0/Pb > 2.1[170]. In
(a) (b)
Figure 7.1:(a) Supersonic expansion of a molecular beam, with an isentropic core independent of the boundary condition ofPb. This zone of silence is surrounded by shock waves, depicted as black lines around the zone of silence. The ideal position of a skimmer is before the Mach disk inside the isentropic expansion (adapted from[170]).
this range, the pressure at the exit becomes independent of Pb and is roughly P0/2> Pb. Due to the pressure difference at the exit, the beam starts to expand. The flow velocity reaches values ofM 1. Thus, it is called a supersonic expansion. An exciting feature of particles moving at this velocity is the lack of information transfer. Information travels with the speed of sound, but the molecules move faster. Therefore, they are not affected by the boundary condition of Pb, and the Mach number continues increasing. The molecular beam even starts to overexpand, until it gets adjusted by shock waves. They are thin non-isentropic regions, in which all beam parameters experience a large gradient. The Mach number decrease in regions beyond the shock waves. The location of the Mach disk brakes the expansion in the forward direction to M < 1 and is estimated at[170]
xm = 0.67dp
P0/Pb. (7.1)
Chapter 7. Spectroscopy on a Molecular Beam
A background pressure Pb ≈10−4mbar and a hole diameter d= 1 mm lead, for example, to a distance ofxm ≈2 km. Thus, shock waves are negligible in the pulsed molecular beam in this thesis. The molecules passing the skimmer into the second vacuum chamber have a high Mach number, but a narrow velocity distribution in all dimensions. The maximum flow velocity is[169]
v0(T0) = s
2kBT0 m
γ
γ−1, (7.2)
with the stagnation temperature T0 = 293.15 K, correspondig to the start temperature of the molecules. The resulting velocity of the perfect isentropic expansion for an ideal gas of xenon is around v0 = 304 m/s. The actual value in an experiment will be slightly higher at 340 m/s. A detailed calculation includes the pressure and temperature dependence of γ[169]. As important as understanding the expansion of the molecular beam, is the creation of sample molecules. This thesis focuses only on the generation of an OH molecular beam.
The most prominent techniques for producing OH are photolysis[171], chemical reactions[172]
and electrical discharge[173]. However, the chemical reaction of H + NO2 −−→ OH + NO is limited to a continuous molecular beam. A modern electrical discharge source, which creates cold and intense OH beams[174] seems to be the most straightforward approach, though building such a device is a small project of its own.
7.1.1 Photodissociation of Nitric Acid
In this thesis, we used photodissociation of nitric acid to generate the OH. A mixture of gaseous nitric acid (HNO3) and the carrier gas xenon at 1 atm fill the pulsed valve.
Opening the valve lets the gas mixture flow first into an evacuated fused silica tube, which produces into the expansion chamber (Figure 7.1b). A perfectly timed laser pulse around 193 nm dissociates the nitric acid molecules inside the quartz tube
HNO3 −−→NO2+ OH. (7.3)
The OH radicals then undergo supersonic expansion, along with the carrier gas and NO2. In contrast to an ideal supersonic expansion, the quartz tube confines the molecules over a distance of a few millimeters. By the time, the first molecules reach the exit of the tube, the pressure difference to the vacuum chamber has already decreased. However, the cooling of the supersonic expansion is not endangered as long the relation P0/Pb >2.1 is satisfied. Effects like a potential tail of the supersonic expansion, due to the quartz tube are neglected. The most significant drawbacks of nitric acid dissociation are of technical nature. The vacuum pumps can handle only a limited amount of acid before they start to degrade, making a cold finger at liquid nitrogen temperature necessary. The cold finger takes frequent maintenance. It is emptied by bringing it to room temperature, purging
7.2. Doppler-Shift