As discussed above, helium droplets were produced in free jet expansions as early as 1961. It was not long before the potential of helium droplets to provide information on nite quantum uids was realized. Unfortunately, since helium is optically transparent below ∼24 eV, few experimental methods were available which could directly probe the
system. In recent years, however, methods have been developed to controllably dope the droplets with almost any atom or molecule [7073], providing indirect probes of the quantum uid. In fact, a helium droplet will pick-up virtually anything it comes into contact with as it ies through the apparatus. Indeed, while liquid helium is a rather poor solvent, it is certainly better than vacuum. As a result, if a source can be designed to introduce sucient vapor pressure into the beam path, then the droplets can be doped with almost any material imaginable.
The droplets are doped in the following experiments by passing the beam through one or more pick-up cells, containing the vapor(s) of interest. In our apparatus, there are two ways to introduce dopants to the droplet beam. The rst method consists of leaking gas into a static, dierentially pumped3, gas pick-up cell, using a very-ne metering needle
valve. The second method consists of a load lock design which allows various pick-up sources to be introduced either upstream or downstream the static gas pick-up cell. The beauty of this design is that the vacuum does not have to be broken to introduce the source, and multiple sources can be utilized on a day to day basis. The dierential pumping in the static cell spatially connes the vapor to a small region, such that the location of other pick-up sources can be varied to control the pick-up order of two or more 3The static cell was pumped with a small 100 l/s turbomolecular pump, resulting in a baseline (gas o) pressure of 1 x10−7 Torr.
distinct species. The load lockable pick-up sources used for the experiments presented here consists mainly of metal oven sources designed to introduce about 1011 atoms/cm3
metal vapor into the beam path. Specic descriptions of these metal oven sources will be given in later chapters as they arise.
When a droplet picks-up an atom or molecule, the dopants kinetic energy, internal energy, and solvation energy is quickly cooled. The energy is transfered from dopant to droplet by creating excitations in the uid. These excitations will eventually exchange energy with atoms on the surface, leading to quantum evaporation. Each helium atom takes away∼5cm−1 (7.2 K) of energy4 [25]. However, the rate of evaporation eventually
becomes negligible, leading to a microcanonical [75] system cooled to 0.37 K [16, 17]. If the vapor pressure is high enough, the droplet may pick-up another atom or molecule. The second atom or molecule is cooled as well through helium evaporation. The two picked-up species rapidly nd each other5 in the droplet, attracted to each other due to
their long-range intermolecular interactions. As the two molecular sub-units condense into a van-der-Waals complex (or perhaps a chemically bound system), the condensation energy is similarly drained away by the evaporation process. Forming a complex such as the HCN dimer [59] results in the evaporation of approximately 600 helium atoms. In principle, the vapor pressure can be increased further such that extremely large [77] clusters are formed in the droplets. In practice, though, the size of the cluster is limited by the largest accessible cooling capacity, i.e. the largest possible droplet size.
As discussed above, the column density of the vapor determines the number of atoms or molecules picked-up by the droplet. In fact, dopant pick-up is a statistical process, and sequential pick-up of dopants leads to a distribution of cluster sizes formed in the droplets. Each capture event (pick-up of dopant) is completely independent of all others, resulting 4In the context of the liquid drop model [74], this eect is droplet size dependent, with E(N) =
−7.21 + 17.71N1/3−5.95N−2/3(Kelvin).
5For HF-Arn [76], the ight time between the two pick up zones (<50 µs) was long enough for the Ar atoms to form a cluster before encountering the HF molecule downstream.
in Poisson pick-up statistics. It is reasonable to suspect that the probability of picking up a dopant will be related to the geometric cross section of the droplet,σ =πR2 = 15.5N2/3
(R= 2.22√3
N fromρ= 0.0218Å−3andV = 43πR3), and the column density of the vapor.
Indeed, it is now well accepted [72, 78, 79] that the probability that an N atom droplet picks up k molecules at a given gas density, η, is given by the Poisson distribution,
P(k, N) = (βN
2/3)k
k! exp(−βN
2/3) (2.8)
whereβ = 15.5ηL is a dimensionless parameter, andL is the length of the pick-up zone. The maximum of the Poisson distribution occurs when βN2/3 =k, and, as a result, the
pick-up of k = 2 molecules will optimize at twice the gas density required to optimize
the monomer.
We can now dene the occupation probability, Pocc, as the probability that a droplet
of size N has been produced and has picked up exactly k atoms or molecules:
Pocc(k, N) =PN(N)∗P(k, N) (2.9)
with PN(N) given in equation 2.1. Since Pocc is a probability, by denition,
∞ X k=0 Z ∞ 0 Pocc(k, N)dN = 1 (2.10)
With this equation, it is simple to determine the fraction of droplets in the distribution that contain k molecules when the experimental conditions are such that Pocc(k, N) is
a maximum for a given k. For k = 1, at most ∼ 35% of all droplets contain only one
molecule, and this percent drops askincreases. We will revisit the denition ofPocc(k, N)
in Chapter 4. It is interesting to note that the pressure (p=ηkT) required to maximize Pocc(k, N)fork = 1, N¯ = 3000, andL= 2 cm is on the order of 10−4 Torr. This value is
104 times lower than the pressure typically required to coexpand a species with a carrier
gas in a supersonic expansion. This point cannot be overemphasized; doping beams of liquid helium droplets makes possible the spectroscopic study of sensitive materials that would otherwise thermally decompose at the temperatures required for gas-phase coexpansions.