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Diode voltage measurement needs to be done at certain currents.
Only the combination of voltage and current is meaningful.
You may draw a chart: V - I characteristic
That gives forward V I but the question relates to reverse breakdown voltage. (Avalanche or Zener knee)
Using a DVM wont work simply because it's high impedance will allow both ends of the diode to reach almost the same potential. You need to shunt the meter with a resistor so the diodes leakage current becomes insignificant compared to the breakdown current. I would suggest adding say 47K across the meter terminals, the voltage should then stay at almost zero until breakdown occurs.
Note that the SS34 has a rated maximum PIV of 40V so you need more than 40V to find the exact breakdown voltage.
According to datasheet, reverse blocking voltage has been measured with pulsed voltage source to avoid excessive heating. In so far the test setup in post #1 isn't well suited in several regards.
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