Microsecond

The measurements being considered here go beyond the measurement of fast reaction kinetics. When a chemical reaction occurs in a beaker, one usually measures a statistical rate of reaction that involves diffusion of reagents in a solvent. These kinetic measurements, which involve an ensemble of molecules, are much slower and do not reveal the fundamental steps of the reaction such as bond formation. The introduction of ultrafast pulses allowed, for the first time, the measurement of the motion of the atoms in real time, as if the motion was frozen by ultrafast flash photography. Capturing this motion helps to reveal the internal forces, described by a potential energy surface, that act on the atoms in the molecule during the chemical reaction. This concept is illustrated in Figure 1, where we see the reagents and the products for the reaction. Typically, the starting and ending point of a chemical reaction are very well known and the compounds are very well characterized. However, how the reagents become the products is not always known. Chemists typically deduce a reaction mechanism based on a large body of experiments where different parameters including the structure of the reagents are modified to evaluate their effect on the outcome of the reaction. Unfortunately in most cases, reaction mechanisms, even when consistent with all available experimental evidence, may not accurately reflect how a particular chemical reaction takes place.

Figure 1. A sketch of a chemical reaction, AB+CD→A+BCD. The reactants and products are well known, but it is not known how they change from reactants to products. This mechanism may involve a transition state which is (a) a linear complex with vibrational motion, (b) a nonlinear complex, (c) a bent complex with rotational motion, or many other possibilities and combinations. The elucidation of the mechanism and transition state dynamics are the goals of femtosecond time-resolved studies.

The ideal method to make the determination involves direct observation. As mentioned earlier, direct observation requires femtosecond laser pulses, just like a fast camera shutter is required to take pictures of fast-moving objects. The concept of a femtosecond pump-probe measurement is illustrated in Figure 2. The chemical reaction is initiated at time t=0 fs by the pump laser. The pump laser provides the energy required for initiating the chemical transformation. The probe laser, delayed in time, probes the formation of the product. Figure 2 illustrates a number of pump-probe measurements each obtained at a different time delay between the pump and probe laser pulses. When a series of pump-probe measurements are gathered as a function of time delay, one obtains a transient that contains a record for the evolution of the reagents to product as a function of time.

Figure 2. Energy schematic of the reaction of AB+CD→A+BCD assuming vibrational motion in a linear transition state. The pump pulse provides enough energy to excite the reactants to the transition state at t=0. This transition state is probed by the second pulse. This pulse arrives at different time delays, from no delay (t=0), to increasing delays (t₁, t₂,…), to infinite delay (t_∞). The signal varies depending on the time delay between the two pulses and these changes in signal intensity correspond to vibrational and rotational information about the transition state. To obtain a transient as shown at the bottom, the time delay between the pump and probe pulses is scanned. The dashed line shows the transient that would be obtained when the transition state is monitored; the solid line shows the transient that would be obtained when the product is monitored.

Almost any portion of an electronic design can be realized in hardware (using logic gates and registers, etc.) or software (as instructions to be executed on a microprocessor). One of the main partitioning criteria is how fast you wish the various functions to perform their tasks:

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Picosecond and nanosecond logic: This has to run insanely fast, which mandates that it be implemented in hardware (in the FPGA fabric).

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Microsecond logic: This is reasonably fast and can be implemented either in hardware or software (this type of logic is where you spend the bulk of your time deciding which way to go).

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Millisecond logic: This is the logic used to implement interfaces such as reading switch positions and flashing light-emitting diodes (LEDs). It's a pain slowing the hardware down to implement this sort of function (using huge counters to generate delays, for example). Thus, it's often better to implement these tasks as microprocessor code (because processors give you lousy speed—compared to dedicated hardware—but fantastic complexity).

Fig. 10 shows the experimental setup where a nanosecond pulsed laser is used as pump beam, which heats the sample and a probe laser almost collinear to the pump beam. This is the responsible for the detection of the thermal response of the sample by means of a photomultiplier.

Fig. 10. Pulsed photothermal mirror technique setup.

The photoconductor, as shown in Fig. 7, depends upon the creation of holes or electrons in a uniform bulk semiconductor material, and the responsivity, temporal response, and wavelength cutoff are unique to the individual semiconductor. An intrinsic photoconductor utilizes “across-the-gap” photoionization or hole-electron pair creation. An extrinsic photoconductor depends upon the ionization of impurities in the material and in this case only one carrier, either hole or electron, is active. The same is true for a quantum-well photoconductor, in which electrons or holes can be photoexcited from a small potential well in the narrower band-gap regions of the semiconductor. The quantum efficiency for the structure in the figure is determined by the absorption coefficient, α, and may be written as η = (1 − R)[1 − e^−αd), where R is the reflection coefficient at the top surface. Carriers produced by the radiation, P, flow in the electric field and contribute to this current flow for a time, τ_r, the recombination time. The value of the current is

FIGURE 7. Photoconductor. Intrinsic operation shown. In extrinsic mode, mobile electrons are photoexcited from fixed positive donor impurities, or holes from negative acceptors.

where τ_t is the carrier transit time through the device and G_p is the photoconductive gain. In the first term of the equation, the fraction in parentheses is the current produced by a carrier of charge, q, moving at a velocity, v, between electrodes spaced at a distance, L. The second term is the average number of carriers, the rate of production times the lifetime. The transit time from one end of the structure to the other is τ_t = L/v = L/μE = L²/μV, with μ the carrier mobility. The photoconductive gain will be greater than unity if the lifetime exceeds the transit time. In this case, an unrecombined excess carrier leaving the material is immediately replaced by a new carrier at the opposite end, maintaining charge neutrality. A 1-mm-long detector might have a transit time as short as a microsecond, so that for a 1-ms recombination time, the gain could be as high as 1000. The current noise, produced by the signal-induced random (g)eneration of carriers as well as the random (r)ecombination, is called g-r noise and is given by

not too different from the noise current of the photomultiplier above. The circuit of Fig. (2b) is applicable to the photoconductor, but the dark current associated with residual thermally excited carriers can produce enough noise so that the detector is still not signal or photon noise limited, even with gains of 1000 or greater. The dark current noise is given by $\overline{{\text{i}}_{\text{n}}^2}=4\text{q}{\text{G}}_{\text{p}}{\text{i}}_{\text{D}}$. In addition, the dark resistance shunts the external load, R, in Fig. (2b), reducing the signal voltage.

The unit for time is s (second). The commonly used smaller units are

1ms (millisecond) = 10⁻³ s

1 μs (microsecond) = 10⁻⁶ s

1ns (nanosecond) = 10⁻⁹ s

1ps (picosecond) = 10⁻¹² s

Solution

1

To convert from units to multiples or submultiples of units it is necessary to divide by the multiple or submultiple. To find the number of milliseconds in 1 second we simply divide by the submultiple 10⁻³. Thus 1 second = 1/10⁻³ = 10³ milliseconds. In 10 seconds there are therefore 10 × 10³ = 10⁴ ms.

2

To find the number of microseconds in 10 seconds we divide by the submultiple 10⁻⁶. Thus in 10 s there are 10/10⁻⁶ = 10⁷ µs.

The following built-in time functions are available:

micros(): This function returns (in microseconds) the number of microseconds since a program started running.

millis(): This function returns (in milliseconds) the number of milliseconds since a program started running.

delay(ms): This function pauses the program for the amount of milliseconds specified in the argument. In the following example, the program is paused for 1 s:

delay(1000);

delayMicroseconds(μs): This function pauses the program for the amount of microseconds specified in the argument.