l frequency of the spring/molecular mass system is kept well below the vibrational resonance frequencies of the molecule itself. The following discussion reports the results of a simple lumped-element analysis of the effects of internal vibration on the massometer completed for the author by J. Soreff, which confirms the previously described performance estimates for the massometer.
Suppose that it is desired to weigh some mass M to a precision DM by attaching it to a weak measuring spring (spring constant kweak) and measuring the shift in resonant frequency. The shift in frequency due to coupling to an internal mode of the sample must be less than (1/2) (DM / M) (1 / 2 p) (kweak/M)1/2. If the sample has some lowest internal mode frequency nint, then approximating the sample as two M/2 masses separated by xstrong connected by a strong spring (kstrong) gives nint = (1/p) (kstrong/M)1/2. If this sample is attached to a weak measuring spring of length xweak, which is in turn attached to a rigid support, then the equations of motion are:
Setting xweak = est and xstrong = R est where t = time, s = time constant, and R = -(1 + (2 kweak / M s2)), and since the angular frequencies of the two springs are wstrong = 2 p nint = 2 (kstrong/M)1/2 and wweak = (kweak/M)1/2, then the lower frequency solution for wweak << wstrong may be approximated by s = i wweak (1 - wweak2 / 2 wstrong2) and so the (undesired) fractional change in frequency due to the flexibility of the object to be weighed is
For a molecule sample size of Lsample, nint ~ vsound / Lsample ~ 500 GHz for vsound ~ 1500 m/sec and Lsample ~ 3 nm; M = r Lsample3 = 2.7 x 10-23 kg for r = 1000 kg/m3. To detect the mass difference of a single proton in a sample molecule, sensor Q = M / mp = 16,000; from Eqn. 4.28, the sensor may operate at a factor of Q1/2 below nint, or nspring ~ nint / Q1/2 ~ 4 GHz. A period of 0.25 nanosec is slightly too short to allow a phase shift of Dtmin ~ 1 nanosec, since the period is only one-quarter of this. However, a phase shift of p radians should accumulate in tmeas = Q / 2 nspring ~ 2 microsec and should be detectable by looking at magnitudes of forces. Interestingly, an SPM microprobe of circular cross-section and exponential profile with a 0.1 nm2 tip has been proposed that could also reach ~proton mass detection sensitivity at room temperature and 1 atm pressure with an eigenmode frequency of 10 GHz;1195 an electrically-driven carbon nanotube-based resonant-beam balance demonstrated ~10-17 kg (~1010 proton or ~0.01 micron3 mass) sensitivity in 1999.3023
Existing surface acoustic wave (SAW) devices458 are chemical sensors that can measure areal mass densities as small as 80 picogram/cm2, or 0.01 N2 molecule per nm2 of surface, though current devices are limited to ~micron2 areas. A proposed quartz microresonator microbalance could detect a 10-6 molecular monolayer,459 the equivalent of 5 x 10-9 molecule per nm2. Detection of ~400 million protons using an MRFM vibrating rod ~60 nm thick was demonstrated by Rugar3256 in 1998.
Last updated on 17 February 2003